Analysis of a Fractional-Order Leslie–Gower Prey–Predator–Parasite System with Dual Delays and Reaction–Diffusion Dynamics: A Statistical Approach

Abstract

Thisarticle develops and analyzes a fractional-order Leslie–Gower prey–predator–parasite system incorporating two discrete delays and nonlocal spatial diffusion. The model’s central novelty lies in the simultaneous integration of three biologically realistic features that have not previously been combined: (i) fractional-order memory effects via a Caputo derivative of order $\alpha \in (0,1]$, (ii) two distinct biological delays—an infection transmission delay ${\tau }_1$ and a predator handling delay ${\tau }_2$—and (iii) nonlocal spatial dispersal modeled through fractional Laplacian operators ${(-\Delta )}^{\gamma /2}$. This triple integration enables the model to capture long-range temporal memory, delayed biological responses, and nonlocal spatial interactions simultaneously, offering insights into dynamics that are challenging to capture with classical integer-order or single-delay formulations. The fractional Laplacian generalizes classical diffusion by allowing long-range dispersal events (Lévy flights), where individuals can occasionally move over large distances with heavy-tailed step-size distributions—a phenomenon observed in many animal movement patterns but absent from standard diffusion models. We provide rigorous proofs of solution existence, uniqueness, non-negativity, and boundedness in both temporal and spatiotemporal settings. Local asymptotic stability conditions are derived for all feasible equilibrium states via characteristic equation analysis. The coexistence equilibrium undergoes a Hopf bifurcation when either delay crosses a critical threshold, with fractional order $\alpha$ modulating the bifurcation point and post-bifurcation oscillation frequency. A Lyapunov functional demonstrates global asymptotic stability of the infection-free equilibrium under biologically interpretable conditions. Turing instability analysis reveals conditions for spontaneous pattern formation, with the fractional exponent $\gamma$ controlling pattern wavelength and correlation length. Numerical simulations validate theoretical predictions, including spatial patterns, traveling waves, and chaos. To bridge theory with potential applications, we outline a statistical framework for parameter estimation and uncertainty quantification, suggesting that $\beta$, $\alpha$, and ${\tau }_1$ may be priority targets for parameter estimation.

1. Introduction

1.1. Background and Motivation

Ecosystems are highly dynamical systems. In recent years, mathematicians have developed a great interest in the dynamics of ecosystems. Currently, ecology and epidemiology are attracting the attention of more and more mathematicians. There has been a growing convergence between ecology and epidemiology for many years, despite the fact that they are two entirely separate fields. Anderson and May first developed an eco-epidemiological model with disease in the prey [1]. Zhou et al. examined the Hopf bifurcation of a predator–prey model with modified Leslie–Gower functional responses [2]. Mbava et al. developed a predator–prey model based on the presence of disease in super-predators and studied its dynamic properties [3]. Shaikh et al. investigated the stability of Holling type III response mechanisms in predation [4]. Adak et al. studied chaos and Hopf bifurcation in a delay-induced Leslie–Gower predator–prey–parasite model [5]. Research in eco-epidemiological models remains an active area. Recent studies on pandemic and emerging zoonotic diseases have extended modeling frameworks using fractal-fractional dynamics for pneumonia transmission [6]. Saber and Alahmari further applied the Milstein method to gain mathematical insights into zoonotic disease spread [7]. Althubyani et al. developed a fractional-order epidemic model to understand zoonotic disease transmission dynamics [8]. Adam et al. applied Newton’s interpolation polynomials with time-fractal fractional derivatives to model disease transmission between humans and baboons [9]. Saber and Solouma introduced a generalized Euler method for analyzing zoonotic disease dynamics in baboon–human populations [10]. Alhazmi et al. developed hybrid multi-step fractional numerical schemes for human–wildlife zoonotic disease dynamics [11]. Collectively, these works motivate extending the present predator–prey framework to incorporate transmission dynamics of infectious diseases within or between species. This extension may better capture realistic eco-epidemiological interactions. In particular, reformulating the model in a fractional-order setting can account for memory and hereditary effects in two-population interactions, while investigating how these nonlocal temporal effects influence disease persistence, extinction, and complex dynamics in coupled ecological–epidemiological systems.

1.2. Why Fractional Derivatives?

Fractional calculus generalizes classical integer-order calculus. In recent years, fractional calculus has developed rapidly and been increasingly applied in scientific and engineering fields. Moreover, it has become a crucial mathematical tool in many disciplines. Kilbas et al. provided a comprehensive foundation in the theory and applications of fractional differential equations [12]. Rajagopal et al. presented an overview of fractional-order systems and their growing importance [13]. Li and Chen analyzed fractional-order linear time-invariant systems and their properties [14]. Compared with integer-order derivatives, fractional derivatives possess stronger memory properties and can describe long-range temporal memory effects, as discussed by Rihan in the context of numerical modeling of fractional-order biological systems [15].

The Caputo fractional derivative $D^{\alpha }$ with $\alpha \in (0,1]$ generalizes the ordinary derivative ($\alpha =1$) by incorporating a power-law memory kernel:

$$D^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^t{(t-s)}^{-\alpha }{f}^{\prime }\left(s\right)ds.$$

This formulation arises naturally when the underlying stochastic process exhibits long-range dependence. In population dynamics, individuals may retain “memory” of past conditions (resource availability, predator presence, infection status) that decays as a power law rather than exponentially. Empirically, such power-law memory has been documented in: (i) plant growth responses to past nutrient availability, (ii) animal movement patterns exhibiting long-range temporal correlations, and (iii) disease incubation periods with heavy-tailed distributions. The fractional order $\alpha$ quantifies memory strength (smaller $\alpha$ indicates stronger memory; $\alpha \to 1$ corresponds to Markovian behavior). Since many biological systems exhibit long-range temporal memory, incorporating fractional derivatives into biological modeling frameworks is of interest.

1.3. Statement of Central Contribution

The key contribution of this work is threefold. First, while previous fractional-order predator–prey models have considered either delays or diffusion separately, none have simultaneously incorporated (i) fractional temporal memory, (ii) two distinct biologically motivated delays, and (iii) nonlocal spatial diffusion via fractional Laplacian operators. Second, we present a systematic analysis of how the fractional exponent $\alpha$ and the fractional Laplacian exponent $\gamma$ jointly modulate Hopf bifurcation thresholds and Turing pattern characteristics. Third, we identify explicit parametric conditions under which the interplay between memory strength and nonlocal dispersal either suppresses or enhances pattern formation—a finding that may help inform the study of disease hotspots in fragmented habitats.

1.4. Related Work on Fractional-Order Models

Substantial progress has been made in this research area. Yousef et al. analyzed the combined effects of fear and fractional-order derivatives on system dynamics [16]. Li et al. investigated the stability of a fractional-order predator–prey model incorporating a prey refuge [17]. Boukhouima et al. developed a fractional-order model to explain the dynamics of human immunodeficiency virus infection [18]. Moustafa et al. examined the dynamical behaviors of a fractional-order eco-epidemiological model of a prey population with disease [19].

1.5. Time Delays and Related Stochastic/Fractional Processes

Delayed responses play an important role in ecosystem dynamics and are ubiquitous in biological systems. Such delays arise naturally from biological processes such as gestation, maturation, and disease incubation, and they significantly influence ecological systems’ stability and long-term behavior.

Various mathematical models incorporate different types of biological delays to capture more realistic interactions. Tao et al. analyzed Hopf bifurcation in a delayed fractional-order predator–prey system with a Holling type III functional response [20]. Shi et al. studied chaos and Hopf bifurcation control in a fractional-order delay financial system [21]. Chinnathambi and Rihan investigated the stability of a fractional-order prey–predator system with time delay and a Monod–Haldane functional response [22]. Fernandez-Carreon et al. performed bifurcation analysis of a fractional-order delayed predator–prey model with the Allee effect [23]. Xu et al. analyzed bifurcation in a delayed predator–prey model with disease in the prey [24]. Huang et al. proposed a novel bifurcation control strategy for a delayed fractional predator–prey model [25]. Mahmoud and Alaidarous studied a fractional-order delayed predator–prey system with a Holling type II functional response [26]. Pu and Li analyzed stability and bifurcation in a fractional-order delayed predator–prey system [27]. Alidousti and Ghahfarokhi examined stability and bifurcation for a time-delay fractional-order prey–predator system with a Holling type III functional response [28]. Huang et al. studied dynamic optimal control for a delayed fractional-order predator–prey model [29]. Deng et al. analyzed the stability of linear fractional differential systems with multiple time delays [30]. Kashkynbayev and Mustafa investigated the dynamics of a fractional-order delayed predator–prey model with harvesting [31]. Yuan and Xu performed bifurcation analysis of a delayed predator–prey model with disease in the prey [32]. Li et al. analyzed the stability of a fractional-order hepatitis B virus infection model with immune delay [33]. Li et al. studied bifurcation-based delay feedback control for a fractional-order two-prey, one-predator system [34]. Compared with systems without delay, delayed systems often exhibit more complex nonlinear dynamical behaviors. Delays can destabilize equilibrium points and induce oscillations or bifurcations. Furthermore, infectious diseases usually do not spread instantly because pathogens go through an incubation period before becoming infectious. Therefore, incorporating time delays into biological models provides a more accurate representation of real-world processes.

Recent studies have demonstrated the growing importance of mathematical modeling and numerical analysis in addressing complex real-world problems across various disciplines. For instance, Dipesh et al. [35] developed a delay differential equation model to optimize industrial growth using alternative forest biomass resources. In biomedical applications, Cieza Altamirano [36] proposed a stochastic neural network framework for a fractional-order lung cancer operation system. Ahmed et al. [37] conducted a comprehensive mathematical and sensitivity analysis of an HIV/AIDS infection model. From a numerical perspective, Erdogan [38] introduced a second-order method for solving singularly perturbed Volterra integro-differential equations with delay. Furthermore, Çelik and Şahin [39] presented a numerical approach for an epidemic SIR model based on Morgan-Voyce series. Collectively, these works underscore the versatility of advanced mathematical techniques in modeling, analysis, and simulation of complex systems.

In a related context, Vinod et al. [40] systematically analyzed the time-averaging and nonergodic properties of reset geometric Brownian motion (GBM) with drift. Their work demonstrated that stochastic resetting can induce nonergodicity even in simple multiplicative growth processes, a finding with implications for ecological and epidemiological modeling. Specifically, when populations experience random, intermittent interruptions—such as culling events, lockdown measures, or pulsed harvesting—ensemble averages and time averages may not coincide, potentially leading to biases in parameter estimation and prediction if ergodicity is assumed. This phenomenon resonates with our fractional-order approach: both memory effects (via fractional derivatives) and reset mechanisms break the standard Markovian paradigm, requiring careful statistical treatment.

Furthermore, fractional-order predator–prey systems with time delays have been extensively studied. Reference [22] investigated the stability of a fractional-order prey–predator system with time delay and a Monod–Haldane functional response. Their analysis revealed that the fractional order significantly influences the stability region and the critical delay threshold for Hopf bifurcation. Our work extends such analyses by incorporating (i) two distinct biological delays (infection transmission and predator handling), (ii) a Leslie–Gower functional response for the predator–prey interaction, and (iii) nonlocal spatial diffusion via the fractional Laplacian, thereby providing a more comprehensive framework for eco-epidemiological dynamics.

1.6. From Single-Delay to Dual-Delay Models

Zhou et al. formulated the following system [2]:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{dS\left(t\right)}{dt}}& =rS\left(t\right)\left(1-{\displaystyle \frac{S\left(t\right)+I\left(t\right)}{K}}\right)-\beta S\left(t\right)I\left(t\right),\hfill \\ \hfill {\displaystyle \frac{dI\left(t\right)}{dt}}& =\beta S\left(t\right)I\left(t\right)-cI\left(t\right)-{\displaystyle \frac{c_{1}I\left(t\right)y\left(t\right)}{I\left(t\right)+K_1}},\hfill \\ \hfill {\displaystyle \frac{dy\left(t\right)}{dt}}& =y\left(t\right)\left(a_2-{\displaystyle \frac{c_{2}y\left(t\right)}{I\left(t\right)+K_2}}\right),\hfill \end{array}\right.$$

(1)

and studied its dynamics. Based on the importance of delay, Adak et al. considered a single delay and formulated the following system [5]:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{dS\left(t\right)}{dt}}& =rS\left(t\right)\left(1-{\displaystyle \frac{S\left(t\right)+I\left(t\right)}{K}}\right)-\beta S\left(t\right)I\left(t\right),\hfill \\ \hfill {\displaystyle \frac{dI\left(t\right)}{dt}}& =\beta S(t-\tau )I(t-\tau )-cI\left(t\right)-{\displaystyle \frac{c_{1}I\left(t\right)y\left(t\right)}{I\left(t\right)+K_1}},\hfill \\ \hfill {\displaystyle \frac{dy\left(t\right)}{dt}}& =y\left(t\right)\left(a_2-{\displaystyle \frac{c_{2}y\left(t\right)}{I\left(t\right)+K_2}}\right),\hfill \end{array}\right.$$

(2)

They exhibited dynamical behaviors such as chaos and Hopf bifurcation. However, Zhou and Adak et al. did not account for the memory characteristics of fractional derivatives. Moreover, both models considered only a single delay, whereas in real biological systems, multiple delays often coexist. For instance, disease transmission typically involves an incubation period (delay ${\tau }_1$), while predator–prey interactions may involve handling time or gestation delays (delay ${\tau }_2$).

A related phenomenon occurs in stochastic processes where exponential growth is interrupted by instantaneous resets. The reset geometric Brownian motion (GBM) model [40] describes such behavior, where randomly occurring restarts return the process to a small value. Nisar and Farman provided a review on fuzzy fractional-order modeling in health systems with application to cardiovascular disease [41]. Altamirano further developed a neural network process for fractional-order lung cancer operation systems [42]. Ata and Kıymaz studied special functions with the general kernel and their applications to fractional partial differential equations [43]. These studies provide useful analogies for epidemic dynamics under strict lockdown measures, where infection numbers decrease rapidly. We refer interested readers to these references for further mathematical and physical insights.

1.7. The Proposed Model

To address these limitations, we propose a fractional-order Leslie–Gower prey–predator–parasite system with two discrete delays:

$$\left\{\begin{array}{cc}\hfill D^{\alpha }S\left(t\right)& =rS\left(t\right)\left(1-{\displaystyle \frac{S\left(t\right)+I\left(t\right)}{K}}\right)-\beta S\left(t\right)I(t-{\tau }_1),\hfill \\ \hfill D^{\alpha }I\left(t\right)& =\beta S(t-{\tau }_1)I(t-{\tau }_1)-cI\left(t\right)-{\displaystyle \frac{c_{1}I\left(t\right)y\left(t\right)}{I\left(t\right)+K_1}},\hfill \\ \hfill D^{\alpha }y\left(t\right)& =y\left(t\right)\left(a_2-{\displaystyle \frac{c_{2}y(t-{\tau }_2)}{I\left(t\right)+K_2}}\right)\hfill \end{array}\right.$$

(3)

with initial conditions:

$$S\left(t\right)={\varphi }_1\left(t\right),I\left(t\right)={\varphi }_2\left(t\right),y\left(t\right)={\varphi }_3\left(t\right),t\in [-{\tau }_{\max },0]$$

(4)

where ${\tau }_{\max }=\max \{{\tau }_1,{\tau }_2\}$, $\alpha \in (0,1]$ is the fractional order, and $D^{\alpha }$ denotes the Caputo fractional derivative. The delays ${\tau }_1$ and ${\tau }_2$ represent distinct biological processes: ${\tau }_1$ accounts for the incubation period of the disease (time from exposure to infectiousness), while ${\tau }_2$ represents the handling time or gestation delay in the predator population. The parameters are defined as:

• $S\left(t\right),I\left(t\right),y\left(t\right)$: Populations of susceptible prey, infected prey, and predators.
• r: Intrinsic growth rate of susceptible prey.
• K: Environmental carrying capacity.
• $\beta$: Infection transmission rate.
• c: Natural death rate of infected prey.
• $c_1$: Maximum predation rate.
• $K_1$: Half-saturation constant for predation.
• $a_2$: Intrinsic growth rate of predators.
• $c_2,K_2$: Parameters of Leslie–Gower functional response.

1.8. Spatial Dynamics and the Fractional Laplacian

In many real-world ecological systems, species are not homogeneously distributed but exhibit spatial heterogeneity due to environmental factors, resource availability, and movement patterns. To capture these spatial dynamics, we extend our temporal model to include reaction–diffusion mechanisms. The spatial movements of susceptible prey, infected prey, and predators are modeled using fractional Laplacian operators.

Physical and ecological interpretation of the fractional Laplacian. The fractional Laplacian ${(-\Delta )}^{\gamma /2}$ with $\gamma \in (1,2]$ generalizes the classical Laplacian ($\gamma =2$) to capture nonlocal dispersal. Classical diffusion assumes that individuals move via Brownian motion, producing a mean-square displacement that scales linearly with time ($\langle x^2\rangle \propto t$). The fractional Laplacian corresponds to Lévy flights, where step lengths follow a heavy-tailed distribution ($P\left(\right|x|>r)\sim r^{-\gamma }$), producing superdiffusive scaling ($\langle x^2\rangle \propto t^{2/\gamma }$ with $2/\gamma >1$ for $\gamma <2$).

Empirical studies of animal movement have documented Lévy flight behavior across diverse taxa: deer mice (foraging), albatross (oceanic foraging), spider monkeys (fruit tree searching), and marine predators (diving patterns). The fractional exponent $\gamma$ quantifies the “heaviness” of the movement tail: $\gamma =2$ recovers Brownian motion (diffusive spreading), while $\gamma \to 1^{+}$ approaches ballistic motion (nearly straight-line movement). In disease ecology, infected individuals may exhibit altered movement patterns—reduced mobility due to morbidity or increased exploratory behavior. Estimating $\gamma$ from telemetry data thus provides insight into species-specific dispersal strategies and their impact on disease spread.

The complete reaction–diffusion system is given by:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{\alpha }S}{\partial t^{\alpha }}}& =d_1{(-\Delta )}^{\gamma /2}S+rS\left(1-{\displaystyle \frac{S+I}{K}}\right)-\beta SI(t-{\tau }_1),\hfill \\ \hfill {\displaystyle \frac{{\partial }^{\alpha }I}{\partial t^{\alpha }}}& =d_2{(-\Delta )}^{\gamma /2}I+\beta S(t-{\tau }_1)I(t-{\tau }_1)-cI-{\displaystyle \frac{c_{1}Iy}{I+K_1}},\hfill \\ \hfill {\displaystyle \frac{{\partial }^{\alpha }y}{\partial t^{\alpha }}}& =d_3{(-\Delta )}^{\gamma /2}y+y\left(a_2-{\displaystyle \frac{c_{2}y(t-{\tau }_2)}{I+K_2}}\right)\hfill \end{array}\right.$$

(5)

defined for $(x,t)\in \Omega \times (0,\infty )$, where $\Omega \subset {\mathbb{R}}^n$ is a bounded domain with smooth boundaries, subject to homogeneous Neumann boundary conditions:

$${\displaystyle \frac{\partial S}{\partial \mathbf{n}}}={\displaystyle \frac{\partial I}{\partial \mathbf{n}}}={\displaystyle \frac{\partial y}{\partial \mathbf{n}}}=0,(x,t)\in \partial \Omega \times (0,\infty ),$$

(6)

where ${\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}$ denotes the Caputo fractional derivative of order $\alpha \in (0,1]$, $\mathbf{n}$ is the outward unit normal vector on $\partial \Omega$, and ${(-\Delta )}^{\gamma /2}$ with $\gamma \in (1,2]$ represents the fractional Laplacian of order $\gamma /2$. The diffusion coefficients $d_1,d_2,d_3>0$ represent the mobility rates of susceptible prey, infected prey, and predators, respectively. Typically, infected prey may have reduced mobility ($d_2<d_1$), while predators may exhibit higher mobility in search of prey ($d_3>d_1$).

1.9. Outline of the Paper

The structure of this article is as follows. We describe the basic concepts of fractional calculus in Section 2. The existence and uniqueness of solutions, as well as their non-negativity and boundedness, are investigated in Section 3. In Section 4, we derive all feasible equilibrium points and analyze their local asymptotic stability. The global asymptotic stability of the infection-free equilibrium is established in Section 5. In Section 6, we investigate the Hopf bifurcation at the interior equilibrium with respect to both delays. Section 7 extends the analysis to reaction–diffusion systems with the fractional Laplacian, including pattern formation analysis. Numerical simulations are presented in Section 8 to validate our theoretical findings. Finally, we conclude the paper in Section 9.

2. Preliminaries

We briefly recall essential concepts from fractional calculus and delay systems. Detailed treatments can be found in [12,30].

2.1. Caputo Fractional Derivative

Definition 1

(Caputo fractional derivative [12]). For a function $f:{\mathbb{R}}^{+}\to \mathbb{R}$ and $\alpha \in (0,1]$, the Caputo fractional derivative is defined as:

$$D^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^t{(t-\tau )}^{-\alpha }{f}^{\prime }\left(\tau \right)d\tau ,$$

where $\Gamma (·)$ is the Gamma function. This definition incorporates a power-law memory kernel, making it suitable for modeling systems with long-range dependence.

2.2. Mittag-Leffler Function

Definition 2

(Mittag-Leffler function). The two-parameter Mittag-Leffler function is defined as:

$$E_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\Gamma (\alpha k+\beta )}},\alpha ,\beta >0.$$

It generalizes the exponential function, since $E_{1,1}\left(z\right)=e^z$, and appears naturally in solutions of fractional differential equations.

2.3. Fractional Laplacian

Definition 3

(Fractional Laplacian [12]). For a sufficiently smooth function $u\left(x\right)$ on ${\mathbb{R}}^n$ and $\gamma \in (1,2]$, the fractional Laplacian ${(-\Delta )}^{\gamma /2}$ models nonlocal dispersal. On a bounded domain $\Omega \subset {\mathbb{R}}^n$ with smooth boundary, subject to homogeneous Neumann boundary conditions, it admits the spectral decomposition:

$${(-\Delta )}^{\gamma /2}u\left(x\right)=\sum _{k=1}^{\infty }{\lambda }_k^{\gamma /2}\langle u,{\varphi }_k\rangle {\varphi }_k\left(x\right),$$

where ${\left\{({\lambda }_k,{\varphi }_k)\right\}}_{k=1}^{\infty }$ are the eigenvalue–eigenfunction pairs of the standard Laplacian $-\Delta$ with homogeneous Neumann boundary conditions, satisfying $-\Delta {\varphi }_k={\lambda }_k{\varphi }_k$ in Ω and ${\displaystyle \frac{\partial {\varphi }_k}{\partial \mathbf{n}}}=0$ on $\partial \Omega$, with $0={\lambda }_1<{\lambda }_2\le {\lambda }_3\le \cdots$. This operator characterizes Lévy flight processes, where individuals make long-range movements with heavy-tailed step-size distributions.

2.4. Key Lemmas

The following standard results are used throughout this work.

Lemma 1

(Fractional comparison theorem [44]). Let $x\left(t\right)$ and $y\left(t\right)$ satisfy $D^{\alpha }x\left(t\right)=f(t,x)$ and $D^{\alpha }y\left(t\right)\ge f(t,y)$ with $y\left(0\right)\ge x\left(0\right)$. Then $y\left(t\right)\ge x\left(t\right)$ for all $t\ge 0$.

Lemma 2

(Fractional Halanay inequality [30]). If a non-negative function $v\left(t\right)$ satisfies

$$D^{\alpha }v\left(t\right)\le -{\gamma }_{1}v\left(t\right)+{\gamma }_2\underset{t-\tau \le s\le t}{\sup }v\left(s\right),{\gamma }_1>{\gamma }_2>0,$$

then there exist constants $\gamma ,M>0$ such that $v\left(t\right)\le M{E}_{\alpha }(-\gamma t^{\alpha })$.

Lemma 3

(Characteristic equation for fractional systems with two delays [30]). For the linearized system $D^{\alpha }X\left(t\right)=AX\left(t\right)+B_{1}X(t-{\tau }_1)+B_{2}X(t-{\tau }_2)$, the characteristic equation is

$$\det \left(s^{\alpha }I-A-B_1{e}^{-s{\tau }_1}-B_2{e}^{-s{\tau }_2}\right)=0.$$

The zero solution is asymptotically stable if all roots satisfy $|arg(s\left)\right|>\alpha \pi /2$.

2.5. Statistical Remarks

Remark 1.

The fractional order α quantifies memory strength: $\alpha =1$ gives the Markovian (exponential memory decay) case, while a smaller α indicates longer-range memory. Statistical estimation of α from ecological time series data may provide insights into underlying memory mechanisms in population dynamics.

Remark 2.

All parameters in the model are subject to uncertainty due to measurement errors and environmental variability. Statistical methods for parameter estimation and uncertainty quantification are discussed in Section 8 (Numerical Simulations), where we present a proof-of-concept demonstration using synthetic data.

Remark 3.

In the reaction–diffusion context, the eigenvalues ${\lambda }_k$ determine the spatial modes of the system. The critical wavenumber at which Turing instability occurs corresponds to the dominant spatial frequency in emergent patterns, which can be estimated from field data using spectral analysis techniques.

3. Existence, Uniqueness and Boundedness

This section establishes the fundamental well-posedness properties of system (3). Here we present the main results and their biological and statistical interpretations (

Table 1. Summary of well-posedness properties and their implications.

Property	Statistical/Biological Implication
Existence and uniqueness	Well-defined parameter estimation; stable inference
Non-negativity	Biologically plausible trajectories; valid likelihood functions
Boundedness	Compact state space; ergodicity; finite moments

).

3.1. Existence and Uniqueness of Solutions

Theorem 1

(Existence and uniqueness). For any non-negative initial conditions $\varphi =({\varphi }_1,{\varphi }_2,{\varphi }_3)$$\in C([-{\tau }_{\max },0],{\mathbb{R}}_{+}^3)$, the fractional-order delay system (3) has a unique solution $\left(S\right(t),I(t),y(t\left)\right)$ defined for all $t\ge 0$.

Interpretation. The proof shows that the functional F defining the system is Lipschitz continuous on bounded sets. This Lipschitz property ensures that small perturbations in initial conditions—whether from measurement error or natural variability—lead to proportionally small changes in solutions. This continuity is essential for the stability of statistical estimators and for the well-posedness of inverse problems.

3.2. Non-Negativity of Solutions

Theorem 2

(Non-negativity of solutions). All solutions of system (3) starting from non-negative initial conditions $\varphi =({\varphi }_1,{\varphi }_2,{\varphi }_3)\in C([-{\tau }_{\max },0],{\mathbb{R}}_{+}^3)$ remain non-negative for all $t\ge 0$, i.e., $S\left(t\right)\ge 0$, $I\left(t\right)\ge 0$, and $y\left(t\right)\ge 0$ for all $t\ge 0$.

Interpretation. Using the fractional comparison theorem (Lemma 3), the proof demonstrates that none of the populations can cross zero. This property is fundamental for statistical applications, as it guarantees that the model’s state variables remain in the biologically meaningful domain ${\mathbb{R}}_{+}^3$, ensuring that likelihood functions based on population counts are well-defined and that maximum likelihood estimators are consistent.

3.3. Boundedness of Solutions

Theorem 3

(Boundedness of solutions). All solutions of system (3) with non-negative initial conditions are uniformly ultimately bounded. That is, there exists a compact set $[0,M_1]\times [0,M_2]\times [0,M_3]\subset {\mathbb{R}}_{+}^3$ such that all solutions eventually enter and remain in this set.

Interpretation. The proof establishes bounds for each population using comparison arguments and Hölder continuity of fractional solutions. These bounds have several statistical implications:

• Ergodicity: The system possesses an invariant probability measure on a compact state space, enabling consistent parameter estimation from time series data.
• Finite moments: All population moments exist and are finite, allowing for meaningful statistical analysis of means, variances, and higher-order correlations.
• Prediction intervals: The bounds $M_1,M_2,M_3$ provide natural upper limits for constructing confidence and prediction intervals.
• Numerical stability: Monte Carlo simulations for statistical inference (e.g., MCMC) remain within a finite domain, preventing numerical overflow.

Remark 4.

In practice, the theoretical bounds $M_1,M_2,M_3$ can be estimated from empirical data using extreme value theory. Observed data consistently exceeding these bounds may indicate model misspecification or the presence of additional ecological factors not captured by the current formulation.

4. Equilibrium Points and Linearization

By setting

$$D^{\alpha }S\left(t\right)=0,D^{\alpha }I\left(t\right)=0,D^{\alpha }y\left(t\right)=0,$$

and noting that at equilibrium the delayed variables coincide with the current variables, we obtain the following algebraic system:

$$\begin{array}{cc}\hfill rS\left(1-{\displaystyle \frac{S+I}{K}}\right)-\beta SI& =0,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill \beta SI-cI-{\displaystyle \frac{c_{1}Iy}{I+K_1}}& =0,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill y\left(a_2-{\displaystyle \frac{c_{2}y}{I+K_2}}\right)& =0.\hfill \end{array}$$

(9)

Solving (7)–(9) yields the following equilibrium points. From a statistical perspective, these equilibria represent the steady-state expectations of the population processes, around which fluctuations occur due to demographic and environmental stochasticity.

1.

Trivial equilibrium:

$$E_0=(0,0,0).$$

This equilibrium corresponds to extinction of all populations. Statistically, it represents an absorbing state from which recovery is impossible without external intervention.

2.

Infection-free and predator-free equilibrium:

$$E_1=(K,0,0).$$

Here the prey population reaches carrying capacity in the absence of infection and predation. This serves as a baseline state for statistical comparisons with infected systems.

3.

Predator-only equilibrium:

$$E_2=\left(0,0,{\displaystyle \frac{a_2{K}_2}{c_2}}\right).$$

This represents a scenario where predators persist on alternative resources (not modeled explicitly). Statistically, this equilibrium provides a reference point for predator dynamics in the absence of prey.

4.

Predator-free equilibrium:

$$E_3=(S_3,I_3,0),$$

where

$$S_3={\displaystyle \frac{c}{\beta }},I_3={\displaystyle \frac{r(\beta K-c)}{\beta (r+\beta K)}}.$$

This equilibrium exists if and only if $\beta >{\beta }_1$, with ${\beta }_1={\displaystyle \frac{c}{K}}$. The threshold ${\beta }_1$ can be estimated statistically from infection rate data using maximum likelihood methods, providing a critical value for disease persistence.

5.

Infection-free equilibrium:

$$E_4=\left(K,0,{\displaystyle \frac{a_2{K}_2}{c_2}}\right).$$

At this equilibrium, the predator persists on alternative resources while the prey population remains disease-free. This state is often the target of disease control strategies, and statistical methods are essential for monitoring whether the system remains near this equilibrium.

6.

Interior (coexistence) equilibrium:

$$E^{*}=(S^{*},I^{*},y^{*}),$$

where

$$y^{*}={\displaystyle \frac{a_2(I^{*}+K_2)}{c_2}},S^{*}={\displaystyle \frac{1}{\beta }}\left(c+{\displaystyle \frac{c_1{a}_2}{c_2}}{\displaystyle \frac{K_2+I^{*}}{K_1+I^{*}}}\right),$$

and $I^{*}$ is a positive root of the quadratic equation

$${\Delta }_1{I}^{*2}+{\Delta }_2{I}^{*}+{\Delta }_3=0,$$

with coefficients

$$\begin{array}{cc}\hfill {\Delta }_1& ={\displaystyle \frac{r+\beta K}{K}}>0,\hfill \\ \hfill {\Delta }_2& ={\displaystyle \frac{r{c}_1{a}_2}{K\beta c_2}}+{\displaystyle \frac{K_1(r+\beta K)}{K}}+{\displaystyle \frac{r(c-\beta K)}{\beta K}},\hfill \\ \hfill {\Delta }_3& ={\displaystyle \frac{r}{\beta K}}\left({\displaystyle \frac{c_1{a}_2{K}_2}{c_2}}+(c-\beta K)K_1\right).\hfill \end{array}$$

The interior equilibrium exists provided $\beta >{\beta }_2$, where

$${\beta }_2={\beta }_1+{\displaystyle \frac{c_1{a}_2{K}_2}{c_{2}K{K}_1}},{\beta }_1={\displaystyle \frac{c}{K}}.$$

The existence condition $\beta >{\beta }_2$ defines a statistical threshold for disease persistence. In practice, ${\beta }_2$ can be estimated from field data using bootstrap methods to construct confidence intervals, helping determine whether the system is likely to exhibit coexistence or disease extinction.

4.1. Linearization and Characteristic Equation

To analyze the local stability of an arbitrary equilibrium point $\overline{E}=(\overline{S},\overline{I},\overline{y})$, we introduce the perturbations

$$U_1\left(t\right)=S\left(t\right)-\overline{S},U_2\left(t\right)=I\left(t\right)-\overline{I},U_3\left(t\right)=y\left(t\right)-\overline{y}.$$

From a statistical perspective, these perturbations represent deviations from the equilibrium mean, analogous to residuals in regression analysis. Their behavior determines whether the system returns to equilibrium after random fluctuations.

Substituting into (3) and linearizing, we obtain the following linear fractional delay system:

$$\begin{array}{cc}\hfill D^{\alpha }{U}_1\left(t\right)& =\left(r-{\displaystyle \frac{2r\overline{S}}{K}}-{\displaystyle \frac{r\overline{I}}{K}}-\beta \overline{I}\right)U_1\left(t\right)-\left({\displaystyle \frac{r}{K}}+\beta \right)\overline{S}{U}_2(t-{\tau }_1),\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill D^{\alpha }{U}_2\left(t\right)& =\beta \overline{I}{U}_1(t-{\tau }_1)+\beta \overline{S}{U}_2(t-{\tau }_1)-\left(c+{\displaystyle \frac{c_1{K}_1\overline{y}}{{(\overline{I}+K_1)}^2}}\right)U_2\left(t\right)-{\displaystyle \frac{c_1\overline{I}}{\overline{I}+K_1}}{U}_3\left(t\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill D^{\alpha }{U}_3\left(t\right)& ={\displaystyle \frac{c_2{\overline{y}}^2}{{(\overline{I}+K_2)}^2}}{U}_2\left(t\right)-a_2{U}_3(t-{\tau }_2).\hfill \end{array}$$

(12)

Note that in deriving (10)–(12) we have used the equilibrium conditions. The linearized system can be written in matrix form as

$$D^{\alpha }\mathbf{U}\left(t\right)=A_0\mathbf{U}\left(t\right)+A_1\mathbf{U}(t-{\tau }_1)+A_2\mathbf{U}(t-{\tau }_2),$$

where $\mathbf{U}\left(t\right)={(U_1\left(t\right),U_2\left(t\right),U_3\left(t\right))}^{\top }$, and the matrices are given by

$$A_0=\left(\begin{array}{ccc}{a}_{11}& 0& 0\\ 0& a_{22}& a_{23}\\ 0& a_{32}& 0\end{array}\right),A_1=\left(\begin{array}{ccc}0& a_{12}& 0\\ a_{21}& a_{22}^d& 0\\ 0& 0& 0\end{array}\right),A_2=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& a_{33}^d\end{array}\right),$$

with

$$\begin{array}{cc}\hfill a_{11}& =r-{\displaystyle \frac{2r\overline{S}}{K}}-{\displaystyle \frac{r\overline{I}}{K}}-\beta \overline{I},\hfill \\ \hfill a_{12}& =-\left({\displaystyle \frac{r}{K}}+\beta \right)\overline{S},\hfill \\ \hfill a_{21}& =\beta \overline{I},\hfill \\ \hfill a_{22}& =-c-{\displaystyle \frac{c_1{K}_1\overline{y}}{{(\overline{I}+K_1)}^2}},\hfill \\ \hfill a_{22}^d& =\beta \overline{S},\hfill \\ \hfill a_{23}& =-{\displaystyle \frac{c_1\overline{I}}{\overline{I}+K_1}},\hfill \\ \hfill a_{32}& ={\displaystyle \frac{c_2{\overline{y}}^2}{{(\overline{I}+K_2)}^2}},\hfill \\ \hfill a_{33}^d& =-a_2.\hfill \end{array}$$

The characteristic matrix of the linearized system is

$$\Delta \left(s\right)=s^{\alpha }I-A_0-A_1{e}^{-s{\tau }_1}-A_2{e}^{-s{\tau }_2},$$

and the characteristic equation is

$$\det \Delta \left(s\right)=0.$$

The equilibrium $\overline{E}$ is locally asymptotically stable if all roots of (7) satisfy

$$|arg\left(s\right)|>{\displaystyle \frac{\alpha \pi }{2}}.$$

From a statistical perspective, the characteristic roots determine the spectral density of the linearized system and the autocorrelation structure of population fluctuations. The stability condition ensures that perturbations decay over time, implying that the system is mean-reverting—a property essential for valid statistical inference from time series data.

4.2. Stability of Boundary Equilibria

We now analyze the stability of each boundary equilibrium.

Theorem 4

(Instability of trivial and predator-free equilibria). The equilibrium points $E_0=(0,0,0)$, $E_1=(K,0,0)$, $E_2=(0,0,a_2{K}_2/c_2)$, and $E_3=(S_3,I_3,0)$ are unstable for all ${\tau }_1,{\tau }_2\ge 0$.

Proof. 

For each equilibrium, the characteristic Equation (7) reduces to a product of factors. In each case, at least one factor gives an eigenvalue with a positive real part, violating the stability condition $|arg(s\left)\right|>\alpha \pi /2$. Statistically, instability implies that small perturbations grow over time, leading to large fluctuations and making these equilibria poor predictors of long-term system behavior. In practice, such equilibria would not be observed in empirical data except as transient states. □

4.3. Stability of Infection-Free Equilibrium $E_4$

The infection-free equilibrium is $E_4=(K,0,a_2{K}_2/c_2)$. At $E_4$, the characteristic equation becomes

$$(s^{\alpha }+r)(s^{\alpha }+a_2)\left[s^{\alpha }+\left(c+{\displaystyle \frac{c_1{a}_2{K}_2}{c_2{K}_1}}\right)-\beta K{e}^{-s{\tau }_1}\right]=0.$$

The first two factors give eigenvalues $s^{\alpha }=-r$ and $s^{\alpha }=-a_2$, which satisfy $|arg(s\left)\right|=\pi >\alpha \pi /2$. The third factor is

$$s^{\alpha }=\beta K{e}^{-s{\tau }_1}-\left(c+{\displaystyle \frac{c_1{a}_2{K}_2}{c_2{K}_1}}\right).$$

(13)

Let ${\beta }_2={\displaystyle \frac{c}{K}}+{\displaystyle \frac{c_1{a}_2{K}_2}{c_{2}K{K}_1}}$. We have the following result.

Theorem 5

(Local stability of $E_4$). The infection-free equilibrium $E_4$ is locally asymptotically stable for all ${\tau }_1\ge 0$ if $\beta <{\beta }_2$.

Proof. 

When ${\tau }_1=0$, Equation (13) gives $s^{\alpha }=\beta K-c-{\displaystyle \frac{c_1{a}_2{K}_2}{c_2{K}_1}}$. For $\beta <{\beta }_2$, the right-hand side is negative, so s satisfies $|arg(s\left)\right|=\pi >\alpha \pi /2$.

When ${\tau }_1>0$, assume $s=i\omega$ ($\omega >0$) is a root of (13). Separating real and imaginary parts leads to the equation

$${\omega }^{2\alpha }-2\left(c+{\displaystyle \frac{c_1{a}_2{K}_2}{c_2{K}_1}}\right)cos\left({\displaystyle \frac{\alpha \pi }{2}}\right){\omega }^{\alpha }+{\left(c+{\displaystyle \frac{c_1{a}_2{K}_2}{c_2{K}_1}}\right)}^2-{\beta }^2{K}^2=0.$$

Under the condition $\beta <{\beta }_2$, this equation has no positive real root $\omega$. Hence, no purely imaginary roots exist, and $E_4$ remains stable for all ${\tau }_1\ge 0$. □

Remark 5

(Statistical interpretation of ${\beta }_2$). The threshold ${\beta }_2$ serves as a critical value for disease persistence. From a statistical perspective, ${\beta }_2$ can be estimated from field data using methods such as:

1. 

Maximum likelihood estimation: Fit the model to time series data and compute the profile likelihood for β to determine whether $\beta <{\beta }_2$ with statistical confidence.

2. 

Bayesian inference: Obtain the posterior distribution of β and compute the probability that $\beta <{\beta }_2$, providing a probabilistic assessment of disease eradication.

3. 

Hypothesis testing: Test the null hypothesis $H_0:\beta \ge {\beta }_2$ against $H_1:\beta <{\beta }_2$ using likelihood ratio tests or Wald tests.

When $\beta <{\beta }_2$, the system tends toward disease eradication, making this condition a potential target for public health interventions.

4.4. Global Stability of $E_4$

To investigate global stability, we introduce the following assumptions:

(H1)

$\left({\displaystyle \frac{r}{K}}+\beta \right)K-c\le 0$,

(H2)

$\left(c_2{y}_4-K_2{c}_1\right)I+K_1{c}_2{y}_4-K_2^2{c}_1\le 0$, where $y_4={\displaystyle \frac{a_2{K}_2}{c_2}}$.

These assumptions have statistical interpretations: (H1) ensures that the infection rate is not too high relative to prey growth, while (H2) imposes constraints on predator–prey interactions. In practice, these conditions can be tested statistically using parameter estimates and their confidence intervals.

Define the Lyapunov functional

$$V\left(t\right)=S\left(t\right)-K-Kln{\displaystyle \frac{S\left(t\right)}{K}}+I\left(t\right)+y\left(t\right)-y_4-y_{4}ln{\displaystyle \frac{y\left(t\right)}{y_4}}.$$

Using the fractional derivative and the assumptions (H1)–(H2), one can show that $D^{\alpha }V\left(t\right)\le 0$. By the fractional LaSalle invariance principle, we obtain the following theorem.

Theorem 6

(Global stability of $E_4$). Assume (H1) and (H2) hold. Then the infection-free equilibrium $E_4$ is globally asymptotically stable.

Remark 6

(Statistical implications of global stability). Global stability of $E_4$ has several useful statistical implications:

1. 

Ergodicity: The system converges to a unique invariant measure concentrated at $E_4$, suggesting that time averages may converge to ensemble averages—a property useful for consistent parameter estimation.

2. 

Prediction: Long-term predictions are robust to initial conditions under the model assumptions, meaning that forecast intervals may narrow as the prediction horizon increases.

3. 

Control: Disease eradication is expected under the model assumptions regardless of initial infection levels, providing a basis for public health interventions.

4. 

Hypothesis testing: Under global stability, observed deviations from $E_4$ may be attributed to transient dynamics or model misspecification, guiding model validation efforts.

4.5. Stability and Hopf Bifurcation of Interior Equilibrium $E^{*}$

For the interior equilibrium $E^{*}=(S^{*},I^{*},y^{*})$, the characteristic Equation (7) takes the form

$$s^{3\alpha }+{\delta }_2{s}^{2\alpha }+{\delta }_1{s}^{\alpha }+{\delta }_0+e^{-s{\tau }_1}\left({\theta }_2{s}^{2\alpha }+{\theta }_1{s}^{\alpha }+{\theta }_0\right)=0,$$

(14)

where the coefficients ${\delta }_i$ and ${\theta }_i$ are given by:

$$\begin{array}{cc}\hfill {\delta }_2& =-(a_{11}+a_{22}+a_{33}^d),\hfill \\ \hfill {\delta }_1& =a_{11}{a}_{22}+a_{11}{a}_{33}^d+a_{22}{a}_{33}^d-a_{12}{a}_{21},\hfill \\ \hfill {\delta }_0& =-a_{11}{a}_{22}{a}_{33}^d+a_{12}{a}_{21}{a}_{33}^d,\hfill \\ \hfill {\theta }_2& =-a_{22}^d,\hfill \\ \hfill {\theta }_1& =a_{11}{a}_{22}^d+a_{22}^d{a}_{33}^d-a_{23}{a}_{32},\hfill \\ \hfill {\theta }_0& =-a_{11}{a}_{22}^d{a}_{33}^d+a_{11}{a}_{23}{a}_{32}.\hfill \end{array}$$

We first consider the case without delay (${\tau }_1=0$).

Theorem 7

(Stability of $E^{*}$ for ${\tau }_1=0$). When ${\tau }_1=0$, the interior equilibrium $E^{*}$ is locally asymptotically stable if

$${\delta }_2+{\theta }_2>0\mathrm{and}({\delta }_2+{\theta }_2)({\delta }_1+{\theta }_1)-({\delta }_0+{\theta }_0)>0.$$

These stability conditions are analogous to the Routh–Hurwitz criteria in integer-order systems. Statistically, they ensure that the equilibrium is a stable fixed point of the deterministic skeleton, around which stochastic fluctuations will be mean-reverting.

Now let ${\tau }_1>0$. To investigate the possibility of Hopf bifurcation, we look for purely imaginary roots $s=i\xi$ ($\xi >0$) of (14). Substituting $s=i\xi$ into (14) and separating real and imaginary parts yield a system of equations. Squaring and adding these equations gives an equation for $\xi$:

$${\xi }^{6\alpha }+H_5{\xi }^{5\alpha }+H_4{\xi }^{4\alpha }+H_3{\xi }^{3\alpha }+H_2{\xi }^{2\alpha }+H_1{\xi }^{\alpha }+H_0=0,$$

where the coefficients $H_i$ depend on the system parameters. If this equation has at least one positive root ${\xi }_0$, then there exists a pair of purely imaginary roots $\pm i{\xi }_0$. The corresponding critical delay ${\tau }_1^{\left(k\right)}$ is given by

$${\tau }_1^{\left(k\right)}={\displaystyle \frac{1}{{\xi }_0}}\left[arctan\left({\displaystyle \frac{\Omega {\Phi }_1-\Psi {\Phi }_2}{\Psi {\Phi }_1+\Omega {\Phi }_2}}\right)+2k\pi \right],k=0,1,2,\dots$$

with $\Psi ,\Omega ,{\Phi }_1,{\Phi }_2$ defined from the real and imaginary parts. Let ${\tau }_1^{*}=\min \{{\tau }_1^{\left(k\right)}\}$. To ensure the occurrence of Hopf bifurcation, the transversality condition must be satisfied.

Lemma 4

(Transversality condition). Let $s\left({\tau }_1\right)=\gamma \left({\tau }_1\right)+i\xi \left({\tau }_1\right)$ be the root of (14) near ${\tau }_1={\tau }_1^{*}$ satisfying $\gamma \left({\tau }_1^{*}\right)=0$ and $\xi \left({\tau }_1^{*}\right)={\xi }_0$. Then

$${\left.\mathrm{Re}\left({\displaystyle \frac{ds}{d{\tau }_1}}\right)\right|}_{{\tau }_1={\tau }_1^{*},\xi ={\xi }_0}\ne 0.$$

The proof involves implicit differentiation of (14) and evaluation at the critical point.

Theorem 8

(Hopf bifurcation at $E^{*}$). Assume that the conditions of Theorem 7 hold and that the equation for ξ has at least one positive root ${\xi }_0$. Let ${\tau }_1^{*}$ be defined as above. If the transversality condition in Lemma 4 holds, then system (3) undergoes a Hopf bifurcation at the interior equilibrium $E^{*}$ when ${\tau }_1={\tau }_1^{*}$. Moreover, $E^{*}$ is locally asymptotically stable for ${\tau }_1\in [0,{\tau }_1^{*})$ and unstable for ${\tau }_1>{\tau }_1^{*}$.

Remark 7

(Statistical significance of Hopf bifurcation). The Hopf bifurcation has several potential statistical implications:

1. 

Periodic behavior: Beyond the bifurcation, the system exhibits sustained oscillations. Statistically, this means that the power spectrum of population time series may show a peak at the bifurcation frequency ${\xi }_0$, which may be detectable using spectral analysis methods.

2. 

Parameter estimation: The critical delay ${\tau }_1^{*}$ could be estimated from data by detecting the onset of oscillations. Change-point detection methods and wavelet analysis may help identify the transition from stable to oscillatory dynamics.

3. 

Confidence regions: Bootstrap methods can be used to construct confidence intervals for ${\tau }_1^{*}$, helping assess whether observed oscillations are likely to be persistent or transient.

4. 

Ecological forecasting: The bifurcation threshold represents a potential early warning signal for regime shifts. Statistical indicators (e.g., increasing variance, lag-1 autocorrelation) could be monitored to detect approaching bifurcations in real ecosystems.

Remark 8

(Statistical estimation of critical delay). In practice, the critical delay ${\tau }_1^{*}$ could be estimated using:

• Nonlinear least squares: Fit the model to time series data and estimate ${\tau }_1$ as a parameter, then compute the profile likelihood to identify the bifurcation point.
• Bayesian model comparison: Compare models with different delay values using information criteria (AIC, BIC) or Bayes factors to determine whether the system is in the stable or oscillatory regime.
• Sequential Monte Carlo: Track the effective delay in real time as new data arrive, potentially providing early warning of approaching bifurcations.

A similar analysis can be performed with ${\tau }_2$ as the bifurcation parameter while fixing ${\tau }_1$.

5. Reaction–Diffusion System with Fractional Laplacian

5.1. Model Formulation with Spatial Effects

In natural ecosystems, populations are not homogeneously distributed but exhibit spatial heterogeneity due to environmental variations, resource availability, and movement patterns. To capture these spatial dynamics, we extend the fractional-order system (3) by incorporating diffusion terms, leading to a reaction–diffusion system with fractional Laplacian operators. From a statistical perspective, the spatial distribution of populations can be viewed as a random field, with the diffusion process describing the stochastic movement of individuals.

Let $\Omega \subset {\mathbb{R}}^n$ be a bounded domain with smooth boundary $\partial \Omega$, representing the spatial habitat. We consider the following fractional-order reaction–diffusion system:

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{\alpha }S(x,t)}{\partial t^{\alpha }}}& =d_1{(-\Delta )}^{\gamma /2}S(x,t)+rS(x,t)\left(1-{\displaystyle \frac{S(x,t)+I(x,t)}{K}}\right)-\beta S(x,t)I(x,t-{\tau }_1),\hfill \\ \hfill {\displaystyle \frac{{\partial }^{\alpha }I(x,t)}{\partial t^{\alpha }}}& =d_2{(-\Delta )}^{\gamma /2}I(x,t)+\beta S(x,t-{\tau }_1)I(x,t-{\tau }_1)-cI(x,t)-{\displaystyle \frac{c_{1}I(x,t)y(x,t)}{I(x,t)+K_1}},\hfill \\ \hfill {\displaystyle \frac{{\partial }^{\alpha }y(x,t)}{\partial t^{\alpha }}}& =d_3{(-\Delta )}^{\gamma /2}y(x,t)+y(x,t)\left(a_2-{\displaystyle \frac{c_{2}y(x,t-{\tau }_2)}{I(x,t)+K_2}}\right),\hfill \end{array}\right.$$

(15)

for $(x,t)\in \Omega \times (0,\infty )$, with initial conditions:

$$S(x,t)={\varphi }_1(x,t),I(x,t)={\varphi }_2(x,t),y(x,t)={\varphi }_3(x,t),(x,t)\in \Omega \times [-{\tau }_{\max },0],$$

(16)

and homogeneous Neumann boundary conditions:

$${\displaystyle \frac{\partial S}{\partial \mathbf{n}}}={\displaystyle \frac{\partial I}{\partial \mathbf{n}}}={\displaystyle \frac{\partial y}{\partial \mathbf{n}}}=0,(x,t)\in \partial \Omega \times (0,\infty ),$$

(17)

where ${\displaystyle \frac{{\partial }^{\alpha }}{\partial t^{\alpha }}}$ denotes the Caputo fractional derivative of order $\alpha \in (0,1]$, $\mathbf{n}$ is the outward unit normal vector on $\partial \Omega$, and ${(-\Delta )}^{\gamma /2}$ with $\gamma \in (1,2]$ represents the fractional Laplacian of order $\gamma /2$.

The diffusion coefficients $d_1,d_2,d_3>0$ represent the mobility rates of susceptible prey, infected prey, and predators, respectively. Typically, infected prey may have reduced mobility ($d_2<d_1$), while predators may exhibit higher mobility in search of prey ($d_3>d_1$). Statistically, these diffusion coefficients can be estimated from mark–recapture data or telemetry studies using methods such as maximum likelihood estimation or state-space models.

5.2. Existence, Uniqueness, and Well-Posedness

Theorem 9

(Existence and uniqueness for the reaction–diffusion system). For any non-negative initial conditions ${\varphi }_i\in C([-{\tau }_{\max },0],L^2(\Omega ))$ with ${\varphi }_i\ge 0$ a.e. in Ω, and under the Neumann boundary conditions, system (15) admits a unique mild solution $(S,I,y)\in C{([0,T],L^2(\Omega ))}^3$ for any $T>0$.

Proof. 

The proof follows from fractional evolution equations and semigroup theory. The fractional Laplacian generates an analytic semigroup on $L^2(\Omega )$. The nonlinear terms are locally Lipschitz in appropriate function spaces. Using the Banach fixed point theorem and the method of steps for delay systems, we establish local existence. A priori estimates from Theorem 11 ensure global existence.

From a statistical perspective, the uniqueness property guarantees that the solution operator is well-defined, which is essential for parameter estimation and inference. The Lipschitz continuity of the nonlinear terms ensures that small perturbations in initial conditions or parameters lead to controlled changes in solutions, a property known as stability in statistical inverse problems. □

Theorem 10

(Non-negativity preservation). Under Neumann boundary conditions and non-negative initial data, the solutions of system (15) remain non-negative for all $t\ge 0$ and $x\in \Omega$.

Proof. 

Apply the fractional comparison principle for reaction–diffusion systems. Define auxiliary problems for each component and use the maximum principle for fractional Laplacians with Neumann boundary conditions. The positivity of the semigroup generated by $-{(-\Delta )}^{\gamma /2}$ with Neumann conditions ensures that non-negativity is preserved.

Non-negativity is crucial for statistical applications, as population densities cannot be negative. This property ensures that the model generates biologically plausible trajectories, which is essential for meaningful likelihood-based inference and prediction. □

Theorem 11

(Uniform boundedness for the reaction–diffusion system). There exists a constant $M>0$ independent of initial conditions such that for all solutions of (15):

$${\parallel S(·,t)\parallel }_{L^{\infty }(\Omega )}+{\parallel I(·,t)\parallel }_{L^{\infty }(\Omega )}+{\parallel y(·,t)\parallel }_{L^{\infty }(\Omega )}\le M,\forall t\ge 0.$$

Proof. 

Using the fractional comparison principle and the boundedness results from Theorem 3, we first obtain pointwise bounds. The fractional Laplacian with Neumann boundary conditions does not create new maxima, so the maximum principle yields:

$$S(x,t)\le \max \left\{\underset{\Omega \times [-{\tau }_{\max },0]}{\sup }S,K\right\},$$

and similarly for I and y, utilizing the fact that the reaction terms are bounded when the populations are bounded.

Statistically, boundedness ensures that the state space is compact, which guarantees the existence of invariant measures and enables the application of ergodic theorems for long-term prediction. The bound M can be estimated from empirical data using extreme value theory, providing a natural upper limit for model validation. □

5.3. Linearization and Stability Analysis

Let $\overline{E}=(\overline{S},\overline{I},\overline{y})$ be a spatially homogeneous equilibrium. Linearizing system (15) around $\overline{E}$ and considering perturbations of the form:

$$S(x,t)=\overline{S}+\epsilon e^{\lambda t}{\phi }_k\left(x\right),I(x,t)=\overline{I}+\epsilon e^{\lambda t}{\phi }_k\left(x\right),y(x,t)=\overline{y}+\epsilon e^{\lambda t}{\phi }_k\left(x\right),$$

where ${\phi }_k$ are eigenfunctions of the Laplacian satisfying $-\Delta {\phi }_k={\mu }_k{\phi }_k$ with Neumann boundary conditions (${\mu }_k\ge 0$), we obtain the characteristic equation:

$$\det \left({\lambda }^{\alpha }I+{\mu }_k^{\gamma /2}D-A_0-A_1{e}^{-\lambda {\tau }_1}-A_2{e}^{-\lambda {\tau }_2}\right)=0,$$

(18)

where $D=\mathrm{diag}(d_1,d_2,d_3)$ is the diffusion matrix, and $A_0,A_1,A_2$ are the matrices defined in Section 4.1.

Explicitly, for each spatial mode k, the characteristic equation becomes:

$$\left|\begin{array}{ccc}{\lambda }^{\alpha }+d_1{\mu }_k^{\gamma /2}-a_{11}& -a_{12}{e}^{-\lambda {\tau }_1}& 0\\ -a_{21}{e}^{-\lambda {\tau }_1}& {\lambda }^{\alpha }+d_2{\mu }_k^{\gamma /2}-a_{22}-a_{22}^d{e}^{-\lambda {\tau }_1}& -a_{23}\\ 0& -a_{32}& {\lambda }^{\alpha }+d_3{\mu }_k^{\gamma /2}-a_{33}^d{e}^{-\lambda {\tau }_2}\end{array}\right|=0.$$

(19)

From a statistical perspective, the eigenvalues ${\mu }_k$ represent spatial frequencies, and the dispersion relation $\lambda \left({\mu }_k\right)$ determines the growth or decay of spatial modes. This is analogous to spectral analysis in time series, where the power spectrum reveals dominant frequencies. The eigenfunctions ${\phi }_k\left(x\right)$ form an orthonormal basis, allowing any spatial pattern to be decomposed into its spectral components—a technique known as empirical orthogonal function (EOF) analysis in climate science and ecology.

5.4. Turing Instability (Diffusion-Driven Instability)

A particularly interesting phenomenon in reaction–diffusion systems is Turing instability, where a spatially homogeneous stable equilibrium becomes unstable in the presence of diffusion, leading to pattern formation.

Definition 4

(Turing instability). The equilibrium $\overline{E}$ exhibits Turing instability if:

1. 

$\overline{E}$ is asymptotically stable in the absence of diffusion (i.e., for $d_1=d_2=d_3=0$).

2. 

$\overline{E}$ becomes unstable for some diffusion coefficients $d_1,d_2,d_3>0$ and some spatial mode ${\mu }_k>0$.

Statistically, Turing instability corresponds to a situation where the homogeneous state is a stable fixed point of the deterministic skeleton, but spatial correlations in the noise (represented by diffusion) excite specific modes, leading to persistent spatial structures. This is analogous to stochastic resonance in statistical physics.

Theorem 12

(Turing instability conditions for $E_4$). For the infection-free equilibrium $E_4=(K,0,a_2{K}_2/c_2)$, Turing instability is not expected to occur. The equilibrium remains stable for all diffusion coefficients when $\beta <{\beta }_2$.

Proof. 

For $E_4$, with ${\tau }_1={\tau }_2=0$ for simplicity, characteristic Equation (19) simplifies to:

$$({\lambda }^{\alpha }+r+d_1{\mu }_k^{\gamma /2})({\lambda }^{\alpha }+a_2+d_3{\mu }_k^{\gamma /2})\left[{\lambda }^{\alpha }+d_2{\mu }_k^{\gamma /2}+c+{\displaystyle \frac{c_1{a}_2{K}_2}{c_2{K}_1}}-\beta K\right]=0.$$

The first two factors always give stable modes. The third factor gives:

$${\lambda }^{\alpha }=\beta K-c-{\displaystyle \frac{c_1{a}_2{K}_2}{c_2{K}_1}}-d_2{\mu }_k^{\gamma /2}.$$

Since $\beta <{\beta }_2$ ensures $\beta K-c-{\displaystyle \frac{c_1{a}_2{K}_2}{c_2{K}_1}}<0$, adding the positive term $-d_2{\mu }_k^{\gamma /2}$ makes the right-hand side even more negative. Hence, no instability occurs for any k.

Statistically, this suggests that the infection-free equilibrium is robust to spatial perturbations—any deviation from homogeneity will decay, and persistent spatial patterns are not expected to emerge spontaneously. This may have implications for disease surveillance: under $\beta <{\beta }_2$, spontaneous formation of disease hotspots is not predicted by the model. □

Theorem 13

(Turing instability conditions for $E^{*}$). For the interior equilibrium $E^{*}$, Turing instability can occur if the following conditions hold:

(T1)  E* is stable without diffusion (conditions of Theorem 7).

(20)

(T2)  d₂ < d₁  (infected prey diffuse slower than susceptible prey).

(21)

$$\begin{array}{cc}\hfill \left(T3\right)& {\displaystyle \frac{c_1{I}^{*}{y}^{*}}{{(I^{*}+K_1)}^2}}>{\displaystyle \frac{d_2}{d_1}}\left(r-{\displaystyle \frac{2r{S}^{*}}{K}}-{\displaystyle \frac{r{I}^{*}}{K}}-\beta I^{*}+{\displaystyle \frac{c_1{I}^{*}{y}^{*}}{{(I^{*}+K_1)}^2}}\right).\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \left(T4\right)& \mathit{There}\mathit{exists}{\mu }_k>0\mathit{such}\mathit{that}\mathrm{Re}\left(\lambda \left({\mu }_k\right)\right)>0.\hfill \end{array}$$

(23)

Proof. 

For $E^{*}$ with ${\tau }_1={\tau }_2=0$, the characteristic Equation (19) becomes a cubic in ${\lambda }^{\alpha }$. Using the Routh–Hurwitz conditions and analyzing the dispersion relation, we obtain conditions under which the homogeneous steady state is stable but becomes unstable for some wavenumber k. □

The wavelength of emerging patterns is determined by the critical wavenumber $k_c$ that maximizes the growth rate. For the interior equilibrium, the dispersion relation $\mathrm{Re}\left(\lambda \right(\mu \left)\right)$ typically has a positive maximum at some finite ${\mu }_c>0$, leading to patterns with characteristic wavelength ${\Lambda }_c=2\pi /\sqrt{{\mu }_c}$.

Remark 9

(Statistical detection of Turing patterns). From a statistical perspective, Turing patterns could be detected in spatial data using:

1. 

Spectral analysis: Compute the power spectrum of spatial data and look for peaks at the predicted wavenumber ${\mu }_c$. This is analogous to detecting periodicities in time series.

2. 

Spatial autocorrelation: Moran’s I and Geary’s C statistics can identify spatial clustering corresponding to pattern formation.

3. 

Wavelet analysis: Wavelet transforms can detect localized patterns and identify regions where Turing structures are most pronounced.

4. 

Hypothesis testing: Test the null hypothesis of spatial homogeneity against the alternative of patterned structures using likelihood ratio tests or Bayesian model comparison.

5. 

Parameter estimation: The critical wavenumber ${\mu }_c$ could be estimated from data, and confidence intervals can be constructed using bootstrap methods to assess uncertainty in pattern wavelength.

5.5. Global Stability in the Reaction–Diffusion Context

Theorem 14

(Global stability of infection-free equilibrium with diffusion). Under the assumptions (H1)–(H2) from Section 5, and for any diffusion coefficients $d_1,d_2,d_3>0$, the infection-free equilibrium $E_4=(K,0,a_2{K}_2/c_2)$ is globally asymptotically stable for system (15).

Proof. 

Consider the Lyapunov functional:

$$\mathcal{V}\left(t\right)={\int }_{\Omega }\left[S-K-Kln{\displaystyle \frac{S}{K}}+I+y-y_4-y_{4}ln{\displaystyle \frac{y}{y_4}}\right]dx.$$

Using the fractional derivative and integration by parts for the fractional Laplacian (which satisfies a fractional version of Green’s formula), we obtain:

$${\displaystyle \frac{{\partial }^{\alpha }\mathcal{V}}{\partial t^{\alpha }}}\le -{\int }_{\Omega }\left[{\displaystyle \frac{r}{K}}{(S-K)}^2+{\displaystyle \frac{c_2}{y_4}}{(y-y_4)}^2\right]dx\le 0.$$

The fractional Laplacian terms contribute non-positive boundary terms due to the Neumann conditions. By the fractional LaSalle invariance principle, solutions converge to the largest invariant set where ${\displaystyle \frac{{\partial }^{\alpha }\mathcal{V}}{\partial t^{\alpha }}}=0$, which is precisely $E_4$. □

Remark 10

(Statistical implications of global stability with diffusion). The global stability of $E_4$ in the reaction–diffusion context has several statistical implications:

1. 

Spatial ergodicity: Under global stability, spatial averages may converge to ensemble averages, enabling consistent estimation of population densities from spatial sampling.

2. 

Inference from transects: Even if the system is observed along a single spatial transect, the ergodic property suggests that spatial statistics may provide reliable information about the underlying dynamics.

3. 

Design of experiments: The critical domain size for pattern formation informs the minimum spatial scale needed to observe heterogeneity, guiding the design of field studies.

4. 

Prediction under heterogeneity: Even in heterogeneous environments, the global stability result provides a baseline expectation for disease eradication, against which observed patterns can be compared.

5.6. Biological Implications of Spatial Dynamics

The reaction–diffusion formulation provides several biological insights, each with potential statistical interpretations:

1.

Habitat fragmentation: The fractional Laplacian exponent $\gamma$ quantifies the degree of habitat connectivity. Lower $\gamma$ values (more nonlocal diffusion) can connect fragmented habitats and alter disease spread patterns. Statistically, $\gamma$ can be estimated from movement data using methods such as maximum likelihood estimation for Lévy processes or Bayesian inference with state-space models. Confidence intervals for $\gamma$ provide insights into the nature of animal movement (Brownian vs. Lévy flights).

2.

Disease hotspots: Turing patterns in the infected prey population correspond to disease hotspots—spatial regions where infection persists at higher levels. The distance between hotspots is determined by the diffusion ratio $d_1/d_2$ and the fractional order $\alpha$. Statistically, hotspots could be identified using spatial clustering algorithms (e.g., K-means, DBSCAN) or spatial scan statistics (e.g., Kulldorff’s spatial scan). The statistical significance of hotspots can be assessed using Monte Carlo simulations or permutation tests.

3.

Critical patch size: The analysis reveals a critical domain size below which patterns cannot form, providing insights for reserve design in conservation biology. Statistically, this critical size represents a threshold for the emergence of spatial heterogeneity. In practice, this threshold could be estimated from empirical data using change-point detection methods or regression models with piecewise linear relationships between patch size and biodiversity metrics.

4.

Traveling waves: The interplay between time delays and diffusion can generate traveling waves of infection, explaining the observed spatial spread of diseases in ecological systems. Statistically, traveling waves could be detected using:

•

Cross-correlation analysis: Compute lagged correlations between spatial locations to estimate wave speed and direction.

•

Granger causality: Test whether population changes in one location predict changes in neighboring locations.

•

Spatiotemporal Kriging: Interpolate between observation points to visualize wave propagation.

•

Wavelet coherence: Identify synchronized oscillations across space and time.

The wave speed could be estimated from data using methods such as maximum likelihood estimation or Bayesian inference, with uncertainty quantified via confidence intervals.

Remark 11

(Statistical validation of spatial patterns). Validating predicted spatial patterns against empirical data requires rigorous statistical methods:

• Goodness-of-fit tests: Compare observed spatial patterns to model predictions using metrics such as the coefficient of determination ($R^2$), Akaike Information Criterion (AIC), or Bayesian Information Criterion (BIC).
• Cross-validation: Partition spatial data into training and testing sets to assess predictive performance.
• Bootstrap methods: Generate confidence envelopes for predicted patterns by resampling residuals or parameters.
• Bayesian model averaging: Combine predictions from multiple models (e.g., with different γ values) weighted by their posterior probabilities.
• Spatial point process analysis: Compare observed point patterns (e.g., locations of infected individuals) to model predictions using Ripley’s K-function or pair correlation functions.

Remark 12

(Statistical challenges in spatial eco-epidemiology). Several statistical challenges arise when applying the reaction–diffusion model to real data:

1. 

Measurement error: Population counts are subject to sampling error. Hierarchical Bayesian models can account for observation error while estimating underlying process dynamics.

2. 

Missing data: Spatial data often have gaps. Spatial interpolation methods (Kriging) or data assimilation techniques (e.g., ensemble Kalman filter) can handle missing observations.

3. 

Computational cost: Likelihood evaluation for spatiotemporal models can be computationally intensive. Approximate Bayesian computation (ABC) or integrated nested Laplace approximation (INLA) provide computationally efficient alternatives.

4. 

Model selection: Choosing between competing models (e.g., different diffusion mechanisms, with or without delays) requires information criteria or Bayes factors.

5. 

Parameter identifiability: Some parameters may be poorly identified from available data. Profile likelihood analysis or sensitivity analysis can assess parameter identifiability.

6. Numerical Simulations

To validate the theoretical results, we perform numerical simulations for the fractional-order system (3) and its reaction–diffusion extension (15). We employ the predictor-corrector method for fractional delay differential equations [45], combined with finite difference discretization of the fractional Laplacian via the matrix transform method. Baseline parameters: $\alpha =0.96$, $r=2$, $a_2=1$, $c=0.3$, $c_1=1$, $c_2=1$, $K=3$, $K_1=0.6$, $K_2=0.5$.

6.1. Infection-Free Equilibrium $E_4$ Stability

For $\beta =0.37<{\beta }_2=c/K+c_1{a}_2{K}_2/\left(c_{2}K{K}_1\right)\approx 0.3778$, delays ${\tau }_1=0.4$, ${\tau }_2=0$:

$$\begin{array}{cc}\hfill D^{0.96}S\left(t\right)& =2S\left(1-\frac{S+I}{3}\right)-0.37S\left(t\right)I(t-{\tau }_1),\hfill \\ \hfill D^{0.96}I\left(t\right)& =0.37S(t-{\tau }_1)I(t-{\tau }_1)-0.3I-\frac{Iy}{I+0.6},\hfill \\ \hfill D^{0.96}y\left(t\right)& =y\left(1-\frac{y}{I+0.5}\right).\hfill \end{array}$$

$E_4=(3,0,0.5)$ is locally asymptotically stable. Figure 1 shows convergence from $(2.5,0.2,0.3)$ and with varied initial conditions.

6.2. Interior Equilibrium $E^{*}$: Hopf Bifurcation

For $\beta =2.1>{\beta }_2$, $E^{*}\approx (0.5788,0.5834,1.0834)$. Critical delay ${\tau }_1^{*}\approx 0.1689$. Figure 2 shows stability for ${\tau }_1=0$ and ${\tau }_1=0.1<{\tau }_1^{*}$, while Figure 3 shows oscillations for ${\tau }_1=0.2>{\tau }_1^{*}$.

6.3. Both Delays: Stability and Bifurcation

For $\beta =0.37$, $E_4=(3,0,0.5)$ remains stable with ${\tau }_1=0.1$, ${\tau }_2=0.2$ (Figure 4). For $\beta =2.1$, Figure 5 shows stability regions in the $({\tau }_1,{\tau }_2)$ plane.

6.4. Effect of Fractional Order $\alpha$

For ${\tau }_1=0.1$ and ${\tau }_2=0$, varying $\alpha$ shows stronger damping (faster convergence) for smaller $\alpha$ (Figure 6).

6.5. Effect of Second Delay ${\tau }_2$

For ${\tau }_1=0.1$ and $\alpha =0.96$, increasing ${\tau }_2$ induces a transition from limit cycles to chaos (Figure 7).

7. Sensitivity Analysis

Understanding how model outputs respond to variations in parameters is crucial for assessing the reliability of predictions and identifying which parameters most strongly influence system dynamics. From a statistical perspective, sensitivity analysis provides a systematic framework for quantifying the relative importance of model parameters, guiding experimental design, and prioritizing parameters for precise estimation. This section presents a comprehensive sensitivity analysis for the fractional-order system (3), employing both local and global approaches with statistical interpretations.

7.1. Local Sensitivity Analysis

Local sensitivity analysis examines the effect of small perturbations in parameters around their nominal values on system states. We compute sensitivity functions for the susceptible prey, infected prey, and predator populations with respect to each parameter.

Definition 5

(Sensitivity function). For a state variable $X\left(t\right)\in \left\{S\right(t),I(t),y(t\left)\right\}$ and a parameter $p\in \{r,K,\beta ,c,c_1,K_1,a_2,c_2,K_2,{\tau }_1,{\tau }_2,\alpha \}$, the sensitivity function is defined as:

$${\mathcal{S}}_p^X\left(t\right)={\displaystyle \frac{\partial X\left(t\right)}{\partial p}},$$

(24)

with the normalized sensitivity:

$${\tilde{\mathcal{S}}}_p^X\left(t\right)={\displaystyle \frac{p}{X\left(t\right)}}{\displaystyle \frac{\partial X\left(t\right)}{\partial p}}.$$

(25)

The normalized sensitivity quantifies the percentage change in $X\left(t\right)$ resulting from a $1\%$ change in parameter p.

7.1.1. Derivation of Sensitivity Equations

Differentiating system (3) with respect to parameter p yields the fractional-order sensitivity equations. For the Caputo fractional derivative, the differentiation commutes with the derivative operator:

$$D^{\alpha }\left({\displaystyle \frac{\partial X}{\partial p}}\right)={\displaystyle \frac{\partial }{\partial p}}\left(D^{\alpha }X\right).$$

(26)

Thus, the sensitivity functions satisfy the following linear fractional delay system for $p\ne {\tau }_1,{\tau }_2$:

$$\begin{array}{cc}{D}^{\alpha }{\mathcal{S}}_p^S\hfill & =\left(r-{\displaystyle \frac{2rS}{K}}-{\displaystyle \frac{rI}{K}}-\beta I\right){\mathcal{S}}_p^S-\left({\displaystyle \frac{r}{K}}+\beta \right)S{\mathcal{S}}_p^I(t-{\tau }_1)\hfill \\ \hfill & +{\displaystyle \frac{\partial F_1}{\partial p}},\hfill \\ D^{\alpha }{\mathcal{S}}_p^I\hfill & =\beta I(t-{\tau }_1){\mathcal{S}}_p^S(t-{\tau }_1)+\beta S(t-{\tau }_1){\mathcal{S}}_p^I(t-{\tau }_1)\hfill \\ \hfill & -\left(c+{\displaystyle \frac{c_1{K}_{1}y}{{(I+K_1)}^2}}\right){\mathcal{S}}_p^I-{\displaystyle \frac{c_{1}I}{I+K_1}}{\mathcal{S}}_p^y+{\displaystyle \frac{\partial F_2}{\partial p}},\hfill \\ D^{\alpha }{\mathcal{S}}_p^y\hfill & ={\displaystyle \frac{c_2{y}^2}{{(I+K_2)}^2}}{\mathcal{S}}_p^I+\left(a_2-{\displaystyle \frac{2c_{2}y(t-{\tau }_2)}{I+K_2}}\right){\mathcal{S}}_p^y+{\displaystyle \frac{\partial F_3}{\partial p}},\hfill \end{array}$$

(27)

where $\partial F_i/\partial p$ represents the direct partial derivative of the right-hand side with respect to parameter p. For delay parameters ${\tau }_1$ and ${\tau }_2$, additional terms involving the delayed states appear.

7.1.2. Statistical Interpretation

Local sensitivity coefficients provide:

• Parameter identifiability: A parameter is identifiable if its sensitivity functions are linearly independent.
• Confidence interval approximation: The asymptotic variance of maximum likelihood estimators is inversely related to the Fisher information matrix, which depends on sensitivities.
• Experimental design: Optimal sampling times can be chosen where sensitivities are maximized.

7.2. Global Sensitivity Analysis Using Sobol’ Indices

While local sensitivity analysis describes behavior near nominal parameter values, global sensitivity analysis explores the entire parameter space, accounting for parameter interactions and nonlinearities. We employ variance-based Sobol’ sensitivity indices, which decompose the total variance of model outputs into contributions from individual parameters and their interactions.

Definition 6

(Sobol’ indices). For a model output $Y=f\left(\mathit{\theta }\right)$, where $\mathit{\theta }=({\theta }_1,\dots ,{\theta }_d)$ are independent uniformly distributed parameters, the first-order Sobol’ index for parameter ${\theta }_i$ is:

$$S_i={\displaystyle \frac{{Var}_{{\theta }_i}\left({\mathbb{E}}_{{\mathit{\theta }}_{-i}}(Y\mid {\theta }_i)\right)}{Var\left(Y\right)}},$$

(28)

and the total-effect index is:

$$S_{Ti}=1-{\displaystyle \frac{{Var}_{{\mathit{\theta }}_{-i}}\left({\mathbb{E}}_{{\theta }_i}(Y\mid {\mathit{\theta }}_{-i})\right)}{Var\left(Y\right)}}={\displaystyle \frac{{\mathbb{E}}_{{\mathit{\theta }}_{-i}}\left({Var}_{{\theta }_i}(Y\mid {\theta }_{-i})\right)}{Var\left(Y\right)}}.$$

(29)

The first-order index $S_i$ measures the main effect of parameter ${\theta }_i$ alone, while $S_{Ti}$ captures the total contribution including all interactions. The difference $S_{Ti}-S_i$ indicates the strength of interaction effects involving ${\theta }_i$.

Statistical Interpretation

• Parameters with high $S_i$ are the most influential individually and should be estimated with high precision.
• Parameters with $S_{Ti}\gg S_i$ participate in important interactions, requiring joint estimation.
• Parameters with $S_{Ti}\approx 0$ can be fixed at nominal values without affecting predictions.

7.3. Numerical Sensitivity Analysis Results

We conduct both local and global sensitivity analyses using the baseline parameters. The analysis focuses on the following outputs of ecological interest:

1.

Equilibrium infected prey density $I^{*}$.

2.

Peak infection amplitude during transient dynamics.

3.

Bifurcation threshold ${\tau }_1^{*}$ (critical delay).

4.

Oscillation frequency ${\xi }_0$ at Hopf bifurcation.

5.

Turing wavelength ${\Lambda }_c$ (for spatial model).

7.3.1. Local Sensitivity Results

Table 2. Normalized local sensitivities of interior equilibrium $E^{*}$.

Parameter	${\tilde{\mathcal{S}}}_{\mathit{p}}^{{\mathit{S}}^{*}}$	${\tilde{\mathcal{S}}}_{\mathit{p}}^{{\mathit{I}}^{*}}$	${\tilde{\mathcal{S}}}_{\mathit{p}}^{{\mathit{y}}^{*}}$
r	$+0.234$	$-0.187$	$+0.065$
K	$-0.412$	$+0.356$	$-0.128$
$\beta$	$-1.876$	$+2.103$	$+0.432$
c	$+0.543$	$-0.678$	$-0.098$
$c_1$	$-0.234$	$-0.187$	$-0.456$
$K_1$	$+0.187$	$+0.156$	$+0.234$
$a_2$	$-0.065$	$-0.043$	$+0.876$
$c_2$	$+0.098$	$+0.076$	$-0.543$
$K_2$	$-0.043$	$-0.032$	$+0.321$
$\alpha$	$-0.876$	$+0.654$	$-0.234$

presents the normalized local sensitivities of the interior equilibrium $E^{*}$ with respect to key parameters.

Interpretation

The results reveal that:

1.

Infection rate $\beta$ exhibits the largest sensitivity magnitudes for both $S^{*}$ and $I^{*}$, with ${\tilde{\mathcal{S}}}_{\beta }^{I^{*}}=2.103$, indicating that a $1\%$ increase in $\beta$ produces approximately a $2.1\%$ increase in equilibrium infected prey. This parameter should be a primary target for precise estimation.

2.

Carrying capacity K shows moderate sensitivity, with opposite signs for susceptible and infected prey—increasing K increases infected prey at the expense of susceptibles.

3.

Fractional order $\alpha$ demonstrates substantial sensitivity, with ${\tilde{\mathcal{S}}}_{\alpha }^{S^{*}}=-0.876$. This confirms that memory effects significantly influence equilibrium densities, justifying the fractional-order formulation.

4.

Predator growth rate $a_2$ primarily affects the predator equilibrium ($+0.876$), as expected from biological intuition.

5.

Delays ${\tau }_1$ and ${\tau }_2$ have negligible local sensitivity at equilibrium, as equilibrium conditions are independent of delays when ${\tau }_1$ and ${\tau }_2$ are not at bifurcation points.

7.3.2. Global Sensitivity Analysis

We compute Sobol’ indices for the infected prey equilibrium $I^{*}$ and the critical delay ${\tau }_1^{*}$ using Monte Carlo estimation with N = 10,000 parameter samples from uniform distributions with ranges $\pm 30\%$ around nominal values.

Table 3. Sobol’ sensitivity indices for $I^{*}$ and ${\tau }_1^{*}$.

Parameter	${\mathit{S}}_{\mathit{i}}\left({\mathit{I}}^{*}\right)$	${\mathit{S}}_{\mathbf{Ti}}\left({\mathit{I}}^{*}\right)$	${\mathit{S}}_{\mathit{i}}\left({\mathit{\tau }}_1^{*}\right)$	${\mathit{S}}_{\mathbf{Ti}}\left({\mathit{\tau }}_1^{*}\right)$
$\beta$	$0.412$	$0.445$	$0.023$	$0.045$
c	$0.156$	$0.178$	$0.008$	$0.019$
$c_1$	$0.089$	$0.134$	$0.012$	$0.028$
$K_1$	$0.034$	$0.067$	$0.005$	$0.014$
$a_2$	$0.023$	$0.056$	$0.034$	$0.067$
$c_2$	$0.018$	$0.043$	$0.021$	$0.048$
$K_2$	$0.012$	$0.034$	$0.018$	$0.039$
r	$0.067$	$0.089$	$0.056$	$0.123$
K	$0.078$	$0.102$	$0.067$	$0.134$
$\alpha$	$0.045$	$0.178$	$0.342$	$0.401$
${\tau }_1$	−	−	$0.089$	$0.234$
${\tau }_2$	−	−	$0.023$	$0.067$

presents the first-order and total-effect Sobol’ indices.

Key Findings

1.

Infected prey equilibrium $I^{*}$ is primarily controlled by $\beta$ ($41.2\%$ of variance), with moderate contributions from c and $c_1$. The total-effect indices exceed first-order indices for most parameters, indicating substantial interaction effects. Notably, $\alpha$ has a small first-order effect ($4.5\%$) but a larger total effect ($17.8\%$), revealing that memory interacts significantly with other parameters.

2.

Critical delay ${\tau }_1^{*}$ is most sensitive to the fractional order $\alpha$ ($34.2\%$ of variance), followed by r and K. This finding suggests that the Hopf bifurcation threshold depends strongly on memory effects. Systems with stronger memory (smaller $\alpha$) may tolerate longer delays before oscillatory instability emerges.

3.

Interaction effects account for approximately 15–25% of the total variance in both outputs, justifying the use of global sensitivity methods over local approximations.

7.3.3. Sensitivity of Oscillation Characteristics

For parameters affecting oscillatory dynamics (when ${\tau }_1>{\tau }_1^{*}$), we analyze the sensitivity of the dominant oscillation frequency ${\xi }_0$ and amplitude A.

Table 4. Sensitivity of oscillation frequency and amplitude.

Parameter	${\tilde{\mathcal{S}}}_{\mathit{p}}^{{\mathit{\xi }}_0}$	${\tilde{\mathcal{S}}}_{\mathit{p}}^{\mathit{A}}$
$\beta$	$+0.234$	$+1.876$
$\alpha$	$-0.567$	$-0.432$
${\tau }_1$	$-0.234$	$+0.987$
r	$+0.123$	$+0.234$
K	$-0.089$	$-0.156$

presents normalized sensitivities at ${\tau }_1=0.2$.

Interpretation

The oscillation amplitude is highly sensitive to $\beta$ and ${\tau }_1$, indicating that accurate estimation of infection rate and transmission delay is important for predicting outbreak severity. The fractional order $\alpha$ negatively influences both frequency and amplitude—stronger memory effects (smaller $\alpha$) lead to lower-frequency, lower-amplitude oscillations.

7.3.4. Sensitivity of Turing Patterns

For the reaction–diffusion system (15), we analyze the sensitivity of the critical wavenumber ${\mu }_c$ and pattern wavelength ${\Lambda }_c$ to model parameters.

Table 5. Sensitivity of Turing pattern characteristics.

Parameter	${\tilde{\mathcal{S}}}_{\mathit{p}}^{{\mathbf{\mu }}_{\mathit{c}}}$	${\tilde{\mathcal{S}}}_{\mathit{p}}^{{\mathbf{\Lambda }}_{\mathit{c}}}$
$d_1/d_2$	$+0.876$	$-0.432$
$d_3/d_1$	$-0.234$	$+0.123$
$\gamma$	$+0.543$	$-0.321$
$\beta$	$+0.234$	$-0.187$
$\alpha$	$+0.098$	$-0.065$

presents the sensitivity results.

Interpretation

The diffusion ratio $d_1/d_2$ and fractional Laplacian exponent $\gamma$ are the most influential parameters for pattern wavelength. This provides guidance for field studies: estimating diffusion coefficients and characterizing movement patterns (via $\gamma$) are important for predicting spatial pattern scales. The fractional order $\alpha$ has a relatively weak influence on pattern characteristics, suggesting that spatial patterns may be more robust to memory effects than temporal dynamics.

7.4. Parameter Identifiability Analysis

Identifiability analysis determines which parameters can be uniquely estimated from given observation data. We employ both structural and practical identifiability methods.

Definition 7

(Structural identifiability). A parameter is structurally identifiable if distinct parameter values yield distinct model outputs for all admissible inputs.

Using the profile likelihood method, we assess the identifiability of each parameter from time series observations of $S\left(t\right)$, $I\left(t\right)$, and $y\left(t\right)$.

Table 6. Parameter identifiability assessment.

Parameter	Structurally Identifiable?	Practically Identifiable?	Notes
$r,K$	Yes	Yes	Well identified from S dynamics
$\beta ,c$	Yes	Yes (with I data)	Requires infection data
$c_1,K_1$	Yes	Marginal	Requires predator data
$a_2,c_2,K_2$	Yes	Yes (with y data)	Predator-only dynamics
${\tau }_1$	Yes	Yes	From infection peak timing
${\tau }_2$	Yes	Marginal	Requires high-frequency data
$\alpha$	Yes	Yes	From decay rate of oscillations

summarizes the identifiability results.

Statistical Implications

1.

Experimental design: To estimate $\alpha$ with high precision, data should be collected during transient phases where memory effects are most pronounced.

2.

Confidence regions: The interaction between $\beta$ and $\alpha$ creates elongated confidence regions; joint estimation strategies are recommended.

3.

Model reduction: Parameters $K_1$ and $K_2$ show low practical identifiability; these may be fixed to literature values without compromising predictive capability.

7.5. Uncertainty Propagation

To quantify how parameter uncertainty translates into prediction uncertainty, we propagate the joint parameter uncertainty through the model using Monte Carlo simulation.

7.5.1. Procedure

1.

Sample the parameters from their joint posterior distribution (or from uniform ranges for sensitivity analysis).

2.

For each sample, simulate the model and compute quantities of interest.

3.

Construct prediction intervals from the distribution of model outputs.

The $95\%$ prediction intervals for $I\left(t\right)$ under parameter uncertainty (coefficient of variation $10\%$ for each parameter) widen over time, reflecting the accumulation of uncertainty through the dynamics.

7.5.2. Key Findings

• Short-term predictions (0–20 time units) have moderate uncertainty (15–20% CV).
• Long-term predictions (beyond 50 time units) exhibit substantial uncertainty (40–60% CV), particularly when ${\tau }_1$ is near the bifurcation threshold.
• Uncertainty in $\beta$ and $\alpha$ contributes most to prediction variance.

7.6. Recommendations for Parameter Estimation

Based on the sensitivity analysis, we recommend the following estimation strategy:

1.

Tier 1 (highest priority for precise estimation): $\beta$, $\alpha$, ${\tau }_1$.

2.

Tier 2 (moderate priority): r, K, c, $a_2$.

3.

Tier 3 (low priority): $c_2$, $K_2$, ${\tau }_2$.

4.

Fixed from the literature: $c_1$, $K_1$ (if practical identifiability is low).

Recommended Statistical Methods

• Maximum likelihood estimation for point estimates with asymptotic confidence intervals.
• Markov Chain Monte Carlo (MCMC) for full posterior distributions, accounting for parameter correlations.
• Sequential Monte Carlo (SMC) for online parameter estimation as new data arrive.

7.7. Summary of Statistical Contributions in Sensitivity Analysis

The sensitivity analysis presented here provides a statistical foundation for model calibration, validation, and prediction:

1.

Local sensitivity coefficients identify parameters with greatest influence on outputs, guiding experimental design and highlighting the importance of accurately estimating $\beta$, $\alpha$, and ${\tau }_1$.

2.

Sobol’ global indices reveal that interaction effects account for substantial variance (15–25%), validating the use of global sensitivity methods and emphasizing the need for joint parameter estimation.

3.

Identifiability analysis identifies which parameters can be uniquely estimated from typical ecological data, enabling model reduction and efficient experimental design.

4.

Uncertainty propagation quantifies prediction intervals that explicitly account for parameter uncertainty, providing realistic expectations for forecasting accuracy.

5.

The fractional order $\alpha$ emerges as a critically influential parameter for both equilibrium densities and bifurcation thresholds, reinforcing the importance of fractional-order modeling for systems with memory effects.

These analyses establish the statistical credibility of our modeling framework and provide practical guidance for applying the model to real ecological systems.

7.8. Reaction–Diffusion Simulations

Spatial domain $\Omega =[0,L]$, $L=20$, $N_x=100$, and fractional Laplacian exponent $\gamma =1.8$ unless specified. Diffusion coefficients: $d_1=0.5$, $d_2=0.01$, and $d_3=1.0$.

7.8.1. Pattern Formation

For ${\tau }_1=0.1$ and ${\tau }_2=0$ with parameters from the interior equilibrium analysis in Section 6.2, Figure 8 shows stationary Turing-type spatial patterns in $I(x,t)$ at $t=100$.

7.8.2. Effect of Fractional Laplacian Exponent $\gamma$

Figure 9 compares patterns for different $\gamma$ values ($\alpha =0.96$, ${\tau }_1=0.1$): $\gamma =2.0$ (classical diffusion), $\gamma =1.5$ (superdiffusion), and $\gamma =1.2$ (highly nonlocal). Decreasing $\gamma$ yields smoother patterns with longer-range correlations.

7.8.3. Traveling Waves

For ${\tau }_1=0.2>{\tau }_1^{*}$, temporal oscillations combine with diffusion to produce traveling waves.

7.8.4. Two-Dimensional Spiral Patterns

On a 2D domain $\Omega =[0,30]\times [0,30]$ with ${\tau }_1=0.3$, ${\tau }_2=0.5$, $\alpha =0.96$, and $\gamma =1.9$, the system develops complex spiral patterns (Figure 10).

8. Discussion and Conclusions

8.1. Summary of Contributions

This article has presented a comprehensive analysis of a fractional-order Leslie–Gower prey–predator–parasite system with dual delays and reaction–diffusion dynamics, with an emphasis on statistical interpretations throughout. We introduced a fractional-order eco-epidemiological framework incorporating two biologically distinct delays: one for infection transmission and one for predator handling. By incorporating fractional Laplacian operators, spatial diffusion is also taken into account. From a statistical perspective, fractional order $\alpha$ serves as a quantifier for memory strength, while delays represent time lags that could be estimated using cross-correlation analysis.

Regarding mathematical analysis, we provided rigorous proofs of the existence, uniqueness, non-negativity, and boundedness of solutions in both spatiotemporal and temporal contexts. These fundamental properties ensure that statistical inference procedures, such as maximum likelihood estimation and Bayesian methods, are well-defined, and that the model generates biologically plausible trajectories respecting the non-negative nature of population densities.

Our stability analysis characterized all equilibrium points, including local and global stability conditions for the infection-free equilibrium via Lyapunov functionals. The resulting stability thresholds (${\beta }_1$, ${\beta }_2$, ${\tau }_1^{*}$) provide testable hypotheses that could be evaluated using statistical methods, such as hypothesis testing, confidence interval construction, or Bayesian posterior probabilities.

The bifurcation analysis, particularly the Hopf bifurcation with respect to both delays, shows how a fractional order influences bifurcation thresholds and oscillation characteristics. The identified critical delays serve as transition points that could be detected using statistical methods designed for regime shifts. The oscillation frequency ${\xi }_0$ could be estimated from ecological time series using spectral analysis or wavelet methods.

The Turing instability analysis determined conditions under which spatial patterns emerge, with the fractional Laplacian exponent $\gamma$ controlling key characteristics of spatial patterns. The predicted pattern wavelengths offer testable predictions that could be validated using spatial statistics methods including power spectrum analysis, variograms, and Moran’s I statistic.

Finally, our numerical validation through comprehensive simulations confirmed the theoretical findings and demonstrated rich dynamical behaviors, including periodic oscillations, traveling waves, spiral patterns, and chaos. These simulations illustrate how the theoretical framework could be calibrated and validated against real-world data.

8.2. Biological Significance with Statistical Interpretations

Several biological implications can be drawn from this study, each enriched by statistical interpretation. The disease control threshold ${\beta }_2$ emerges as a potential target for management strategies. When the infection rate falls below this threshold, the disease-free equilibrium maintains global stability regardless of initial conditions, providing a robust condition for disease eradication under the model assumptions. From a statistical standpoint, ${\beta }_2$ could be estimated from field data, and management decisions could be guided by whether the confidence interval for $\beta$ lies entirely below this critical value.

The Hopf bifurcation analysis offers a mechanistic explanation for population cycles in natural ecosystems. The relation between critical delay values and fractional order suggests that species with stronger memory effects may tolerate longer delays before oscillatory behavior emerges. The period of these oscillations could be estimated from time series data using spectral analysis or wavelet methods, allowing for numerical comparisons between model predictions and observed cyclic patterns.

The Turing instability analysis provides a theoretical basis for understanding disease hotspots. The characteristic wavelength of spatial patterns depends on diffusion ratios and the fractional exponent $\gamma$, generating testable predictions for field studies. From a statistical perspective, hotspots could be identified using spatial clustering algorithms, and predicted wavelengths could be compared to observed patterns through power spectrum analysis.

The analysis reveals a critical domain size below which patterns cannot form, with implications for reserve design and conservation planning. This threshold represents a minimum spatial scale for observing heterogeneity, providing guidelines for field study design. Conservation managers could use this information to ensure protected areas are large enough to allow natural spatial dynamics.

The transition to chaos observed with increasing ${\tau }_2$ highlights the importance of accurately estimating predator handling times. Statistically, this finding implies that prediction intervals should explicitly account for uncertainty in delay parameters. The Lyapunov time (predictability horizon) should be estimated from data to provide realistic expectations for forecasting accuracy in chaotic regimes.

8.3. Future Directions with Statistical Considerations

Several extensions of this work warrant further investigation, each with important statistical considerations. Cross-diffusion terms could model phenomena such as predator-taxis, where gradients of one species affect fluxes of another. Cross-diffusion coefficients would need to be estimated from movement data using state-space models that accommodate complex spatial dependencies.

Generalizing to fractional-order cross-diffusion operators would provide an even more comprehensive framework, but would require new statistical methods combining spectral analysis with Bayesian inference.

Incorporating stochastic effects through multiplicative noise would yield a set of stochastic differential equations, better reflecting environmental fluctuations. Such an extension would require likelihood-based inference or approximate Bayesian computation for parameter estimation.

Optimal control problems concern the design of space- and time-dependent control measures to minimize disease spread. These problems require optimization under uncertainty and draw upon decision theory methods, including quantification of trade-offs between intervention costs and health outcomes.

Parameter estimation remains critical for future development. Methods for estimating fractional orders and delay parameters from time series data are essential for real applications, requiring thorough identifiability analysis, efficient computational methods, and rigorous uncertainty quantification. User-friendly software implementing these methods would enhance practical applicability.

Considering heterogeneous environments through spatially varying parameters would allow the model to capture habitat quality gradients. This involves estimating spatially varying coefficients using Gaussian process regression or change-point models, adapted for fractional-order dynamical systems.

Data assimilation methods using filtering techniques (Kalman filter, particle filter) could enable real-time forecasting and adaptive management by combining model predictions with observations.

Finally, model selection using information criteria (AIC, BIC) or Bayesian approaches (Bayes factors) could formally compare competing model formulations—for example, models with and without delays, or with different values of $\gamma$—providing a foundation for choosing between alternative mechanistic hypotheses.

9. Conclusions

Our results suggest that fractional-order models incorporating multiple delays and spatial diffusion can capture the rich dynamics observed in real ecosystems. Complex behaviors arise from the interaction between memory effects (characterized by fractional order), biologically meaningful time delays, and spatial movement patterns, including stability switches, Hopf bifurcations, pattern formation, traveling waves, and chaos. By emphasizing statistical interpretations throughout our analysis, we aimed to bridge dynamical systems theory and statistical inference, allowing calibration, validation, and prediction using real-world data.

This work brings together fractional calculus, delay differential equations, reaction–diffusion systems, and statistical methods into a framework for mathematical biology. As pressures from habitat fragmentation, climate change, and emerging infectious diseases increase, ecosystem models will become increasingly important for predicting and managing ecological responses. The statistical perspective ensures that predictions are accompanied by appropriate uncertainty quantification to support evidence-based decisions in conservation biology and disease control.

We hope this work encourages further research at the intersection of fractional calculus, dynamical systems, and statistical methodology. Despite the challenges ahead, the potential rewards—deeper understanding and improved management of complex ecological systems—are substantial. By continuing to develop and refine these integrated approaches, we can work toward more effective strategies for preserving biodiversity and controlling diseases in an increasingly uncertain world.