Search Results (286)

In this paper, an eco-epidemiological model with a weak Allee effect and prey disease dynamics is discussed. Mathematical features such as non-negativity, boundedness of solutions, and local stability of the feasible equilibria are discussed. Additionally, the transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation are proven using Sotomayor’s theorem and Poincare–Andronov–Hopf theorems. In addition, the correctness of the theoretical analysis is verified by numerical simulation. The numerical simulation results show that the eco-epidemiological model with a weak Allee effect has complex dynamics. If the prey population is not affected by disease, the predator becomes extinct due to a lack of food. Under low infection rates, all populations are maintained in a coexistent state. The Allee effect does not influence this coexistence. At high infection rates, if the prey population is not affected by the Allee effect, the infected prey is found to coexist in an oscillatory state. The predator population and the susceptible prey population will be extinct. If the prey population is affected by the Allee effect, all species will be extinct. Full article

We explore chaos, local dynamics, codimension-one, and codimension-two bifurcations of an asymmetric discrete predator–prey model. More precisely, for all the model’s parameters, it is proved that the model has two boundary fixed points and a trivial fixed point, and also under parametric conditions, it has an interior fixed point. We then constructed the linearized system at these fixed points. We explored the local behavior at equilibria by the linear stability theory. By the series of affine transformations, the center manifold theorem, and bifurcation theory, we investigated the detailed codimensions-one and two bifurcations at equilibria and examined that at boundary fixed points, no flip bifurcation exists. Furthermore, at the interior fixed point, it is proved that the discrete model exhibits codimension-one bifurcations like Neimark–Sacker and flip bifurcations, but fold bifurcation does not exist at this point. Next, for deeper understanding of the complex dynamics of the model, we also studied the codimension-two bifurcation at an interior fixed point and proved that the model exhibits the codimension-two 1:2, 1:3, and 1:4 strong resonances bifurcations. We then investigated the existence of chaos due to the appearance of codimension-one bifurcations like Neimark–Sacker and flip bifurcations by OGY and hybrid control strategies, respectively. The theoretical results are also interpreted biologically. Finally, theoretical findings are confirmed numerically. Full article

In this paper, a class of two-delay predator–prey models with coefficient-dependent delay is studied. It examines the combined effect of fear-induced delay and post-predation biomass conversion delay on the stability of predator–prey systems. By analyzing the distribution of roots of the characteristic equation, the stability conditions for the internal equilibrium and the existence criteria for Hopf bifurcations are derived. Utilizing normal form theory and the central manifold theorem, the direction of Hopf bifurcations and the stability of periodic solutions are calculated. Finally, numerical simulations are conducted to verify the theoretical findings. This research reveals that varying delays can destabilize the predator–prey system, reflecting the dynamic complexity of real-world ecosystems more realistically. Full article

We survey developments in the use of entropy maximization for applying the Gibbs Canonical Ensemble to finite situations. Biological insights are invoked along with physical considerations. In the game-theoretic approach to entropy maximization, the interpretation of the two player roles as predator and prey provides a well-justified and symmetric analysis. The main focus is placed on the Lagrange multiplier approach. Using natural physical units with Planck’s constant set to unity, it is recognized that energy has the dimensions of inverse time. Thus, the conjugate Lagrange multiplier, traditionally related to absolute temperature, is now taken with time units and oriented to follow the Arrow of Time. In quantum optics, where energy levels are bounded above and below, artificial singularities involving negative temperatures are eliminated. In a biological model where species compete in an environment with a fixed carrying capacity, use of the Canonical Ensemble solves an instance of Eigen’s phenomenological rate equations. The Lagrange multiplier emerges as a statistical measure of the ecological age. Adding a weak inequality on an order parameter for the entropy maximization, the phase transition from initial unconstrained growth to constrained growth at the carrying capacity is described, without recourse to a thermodynamic limit for the finite system. Full article

This paper presents and examines a discrete-time predator–prey model of the Leslie type, integrating a Holling type IV functional response for analysis. The mathematical analysis succinctly identifies fixed points and evaluates their local stability within the model. The study employs the normal form approach and bifurcation theory to explore codimension-one and two bifurcation behaviors for this model. The primary conclusions are substantiated by a combination of rigorous theoretical analysis and meticulous computational simulations. Additionally, utilizing fractal basin boundaries, periodicity variations, and Lyapunov exponent distributions within two-parameter spaces, we observe a mode-locking structure akin to Arnold tongues. These periods are arranged in a Farey tree sequence and embedded within quasi-periodic/chaotic regions. These findings enhance comprehension of bifurcation cascade emergence and structural patterns in diverse biological systems with discrete dynamics. Full article

This study examines the impact of fear effects and cooperative hunting strategies in the context of intraguild predation food webs. The presented model includes a shared prey species with logistic growth and assumes that both the intraguild prey and intraguild predator draw their sustenance from the same resource. Using a Lyapunov function enables the system’s global stability to be proven. The impacts of key parameters on system stability are determined through bifurcation analysis. Numerical simulations show that even slight increases in the intensity of fear have drastic impacts on intraguild prey populations and, at higher levels, populations may go extinct. In addition, shifts in the parameter of cooperative hunting have a profound impact on the survival of the intraguild prey. Full article

Neural Ordinary Differential Equations (Neural ODEs) have emerged as a promising approach for learning the continuous-time behaviour of dynamical systems from data. However, Neural ODEs are black-box models, posing challenges in interpreting and understanding their decision-making processes. This raises concerns about their application in critical domains such as healthcare and autonomous systems. To address this challenge and provide insight into the decision-making process of Neural ODEs, we introduce the eXplainable Neural ODE (XNODE) framework, a suite of eXplainable Artificial Intelligence (XAI) techniques specifically designed for Neural ODEs. Drawing inspiration from classical visualisation methods for differential equations, including time series, state space, and vector field plots, XNODE aims to offer intuitive insights into model behaviour. Although relatively simple, these techniques are intended to furnish researchers with a deeper understanding of the underlying mathematical tools, thereby serving as a practical guide for interpreting results obtained with Neural ODEs. The effectiveness of XNODE is verified through case studies involving a Resistor–Capacitor (RC) circuit, the Lotka–Volterra predator-prey dynamics, and a chemical reaction. The proposed XNODE suite offers a more nuanced perspective for cases where low Mean Squared Error values are obtained, which initially suggests successful learning of the data dynamics. This reveals that a low training error does not necessarily equate to comprehensive understanding or accurate modelling of the underlying data dynamics. Full article

A continuous point of a trajectory for an ordinary differential equation can be viewed as a special impulsive point; i.e., the pulsed proportional change rate and the instantaneous increment for the prey and predator populations can be taken as 0. By considering the variation multiple pulse intervention effects (i.e., several indefinite continuous points are regarded as impulsive points), an impulsive predator–prey model for characterizing chemical and biological control processes at different fixed times is first proposed. Our modeling approach can describe all possible realistic situations, and all of the traditional models are some special cases of our model. Due to the complexity of our modeling approach, it is essential to examine the dynamical properties of the periodic solutions using new methods. For example, we investigate the permanence of the system by constructing two uniform lower impulsive comparison systems, indicating the mathematical (or biological) essence of the permanence of our system; furthermore, the existence and global attractiveness of the pest-present periodic solution is analyzed by constructing an impulsive comparison system for a norm $V\left(t\right)$, which has not been addressed to date. Based on the implicit function theorem, the bifurcation of the pest-present periodic solution of the system is investigated under certain conditions, which is more rigorous than the corresponding traditional proving method. In addition, by employing the variational method, the eigenvalues of the Jacobian matrix at the fixed point corresponding to the pest-free periodic solution are determined, resulting in a sufficient condition for its local stability, and the threshold condition for the global attractiveness of the pest-free periodic solution is provided in terms of an indicator $R_a$. Finally, the sensitivity of indicator $R_a$ and bifurcations with respect to several key parameters are determined through numerical simulations, and then the switch-like transitions among two coexisting attractors show that varying dosages of insecticide applications and the numbers of natural enemies released are crucial. Full article

Seasonality is a complex force in nature that affects multiple processes in wild animal populations. Animal mass migration refers to the migration of a large number of animals from a certain distance due to breeding, foraging, climate change or other reasons. In this work, an impulsive predator–prey model with a seasonally mass migrating prey population is constructed. The predator–extinction boundary periodic solution of system (3) is proved to be globally asymptotically stable. System (3) is also proved to be permanent. Our results provide a theoretical reference for biodiversity protection management. Full article

In this work, we present a novel four-species periodic diffusive predator–prey model, which incorporates delay and feedback control mechanisms, marking substantial progress in ecological modeling. This model offers a more realistic and detailed portrayal of the intricate dynamics of predator–prey interactions. Our primary objective is to establish the existence of a periodic solution for this new model, which depends only on time variables and is independent of spatial variables (we refer to it as a spatially homogeneous periodic solution). By employing the comparison theorem and the fixed point theorem tailored for delay differential equations, we derive a set of sufficient conditions that guarantee the emergence of such a solution. This analytical framework lays a solid mathematical foundation for understanding the periodic behaviors exhibited by predator–prey systems with delayed and feedback-regulated interactions. Moreover, we explore the global asymptotic stability of the aforementioned periodic solution. We organically combine Lyapunov stability theory, upper and lower solution techniques for partial differential equations with delay, and the squeezing theorem for limits to formulate additional sufficient conditions that ensure the stability of the periodic solution. This stability analysis is vital for forecasting the long-term outcomes of predator–prey interactions and evaluating the model’s resilience against disturbances. To validate our theoretical findings, we undertake a series of numerical simulations. These simulations not only corroborate our analytical results but also further elucidate the dynamic behaviors of the four-species predator–prey model. Our research enhances our understanding of the complex interactions within ecological systems and carries significant implications for the conservation and management of biological populations. Full article

We modified the work of Wang and Zou, where both the costs and benefits of fear effects were considered, and a constant time delay was used to represent the biomass conversion time from prey to predator. In our work, we assumed that the digestion delay is not a constant, but rather follows a specific distribution. The delay was modeled using a general kernel function, and a more general functional response function was also employed. Then, we established an integral–differential model with distributed time delays. We show that there exists a delay-dependent threshold that determines the system’s dynamics and the presence of coexistence equilibrium. In the absence of coexistence equilibrium, both populations tend toward extinction, or only the prey population survives. Conversely, when coexistence equilibrium exists, the system persists. Four kernel functions were considered to explore the effect of fear levels and time delays on population dynamics. We found that an increase in the fear level of the prey may alter the system dynamics from periodic oscillations to stability. Furthermore, our findings indicate that a fear effect-related functional response has great influence in shaping the model’s dynamics. These results indicate that ignoring time delay or fear effects, or the inappropriate use of kernel functions, may lead to inaccurate prediction results of the model. We want to point out that, when we investigate a pair of purely imaginary roots of the characteristic equation at the coexistence equilibrium, we just need to consider one of them due to the symmetry. Full article

In aquaculture practices, fish are mostly protected from lethal actions of predators. However, sub-lethal effects can be challenging to prevent, as they may be associated with chemical cues signaling predation risk that easily dissolve and spread in water, serving as potential stressors. These cues originate from predators, stressed or injured prey releasing blood, a conspecific alarm substance (CAS), and/or other bodily fluids. In this study, we simulated a small-scale net cage system and assessed the feeding and growth of Nile tilapia exposed chronically to a CAS. Nile tilapia, an invasive species in many aquatic systems, frequently coexist freely alongside those cultivated in cages. Consequently, caged tilapia may regularly be exposed to a CAS, potentially leading to chronic stress and impacting growth and development. Fish were exposed daily to either a CAS or a control vehicle (distilled water) for 45 days (one fish per cage). Fish in both conditions exhibited similar increases in body mass, weight gain, and length over time and displayed an allometric negative growth profile, indicating that the CAS did not affect the length–weight relationship as well. Specific and relative growth rates, condition factor, body axes, food intake, and feeding conversion efficiency were also unaffected by the CAS over time. This body of evidence suggests that the CAS did not act as a chronic stressor for caged Nile tilapia and a possible explanation is habituation. Full article

In this work, we investigate the theoretical properties of a generalized coupled system of finite-delay fractional differential equations involving Caputo derivatives. We establish rigorous criteria to ensure the existence and uniqueness of solutions under appropriate assumptions on the problem parameters and constituent functions, employing contraction mapping principles and Schauder’s fixed-point theorem. Then, we examine the Ulam–Hyers stability of the proposed system. To illustrate the main findings, three examples are provided. Moreover, we provide numerical solutions using the Adams–Bashforth–Moulton method. The practical significance of our results is demonstrated through illustrative examples, highlighting applications in predator–prey dynamics and control systems. Full article

In this paper, the local and global structure of positive solutions for a general predator–prey model in a multi-dimension with ratio-dependent predator influence and prey taxis is investigated. By analyzing the corresponding characteristic equation, we first obtain the local stability conditions of the positive equilibrium caused by prey taxis. Secondly, taking the prey-taxis coefficient as a bifurcation parameter, we obtain the local structure of the positive solution by resorting to an abstract bifurcation theorem, and then extend the local solution branch to a global one. Finally, the local stability of such bifurcating positive solutions is discussed by the method of the perturbation of simple eigenvalues and spectrum theory. The results indicate that attractive prey taxis can stabilize positive equilibrium and inhibits the emergence of spatial patterns, while repulsive prey taxis can lead to Turing instability and induces the emergence of spatial patterns. Full article

Submerged macrophytes can profoundly influence interactions between aquatic predators and their prey due to changes in foraging efficiencies, pursuit time and swimming behaviors of predator–prey participants. Water hyacinth, Eichhornia crassipes (Mart.) Solms-Laub. (Pontederiaceae), is the most widely distributed of the aquatic invasive weeds in South Africa. This invasive weed contributes to changes in physicochemical (turbidity, temperature and water column stratification) and biological (total chlorophyll-a (Chl-a) concentrations and species composition and distribution of vertebrates and invertebrates) variables within freshwater systems of the region. The current study assessed the influence of varying levels of water hyacinth cover (0, 25, 50 and 100% treatments) on the total Chl-a concentration, size structure of the phytoplankton community and the strength of the interaction between a predatory notonectid, Enithares sobria, and zooplankton using a short-term 10-day long mesocosm study. There were no significant differences in selected physicochemical (temperature, dissolved oxygen, total nitrogen and total phosphate) variables in these different treatments over the duration of this study (ANOVA; p > 0.05 in all cases). Results of this study indicate that treatment had a significant effect on total Chl-a concentrations and total zooplankton abundances. The increased surface cover of water hyacinth contributed to a significant reduction in total Chl-a concentrations and a significant increase in total zooplankton abundances (ANCOVA; p < 0.05 in both cases). The increased habitat complexity conferred by the water hyacinth root system provided refugia for zooplankton. The decline in total Chl-a concentration and the size structure of the phytoplankton community under elevated levels of water hyacinth cover can therefore probably be related to both the unfavorable light environment conferred by the plant cover and the increased grazing activity of zooplankton. The presence of the water hyacinth thus suppressed a predator–prey cascade at the base of the food web. Water hyacinth may, therefore, have important implications for the plankton food web dynamics of freshwater systems by reducing food availability (Chl-a), changing energy flow and alternating the strength of interactions between predators and their prey. Full article