Dynamical Analysis of a Fractional Order Prey-Predator Model in Crowley-Martin Functional Response with Prey Harvesting †

: In this paper, we investiage a fractional-order predator-prey model incorporating prey harvesting. In a non-delayed model, the functional response of Crowley-Martin has been studied. We ﬁrst prove the existence, uniqueness, non-negativity and boundedness of the solutions for the proposed model. Furthermore, analyze the existence of various equilibrium points to examine the local asymptotically stable properties, and use the suitable Lyapunov function to study the globally asymptotic stability. Finally, some numerical simulations are veriﬁed for the analytic results.


Introduction
Fractional Calculus (FC) is a general field that attempts to understand real-world phenomena.A non-integer sequence is modeled with derivatives and is a field of differentiation and integrations are performed with non-integer order derivatives.The memory impact and conserved relevant physical properties are the benefits of fractional derivatives.The predator prey models developed by Lotka and Voltera are considered early developments in contemporary mathematical eology in coupled system of non-linear differential equations [1,2].Since Kermack-Mckendrick's pioneering work on SIRS, epidemiological models have attarcted much interest from researchers [3].In general, there are two main types of mathematical models: ecological and epidemiological models.The relations between populations of a certain community are explored in ecological models [4].Epidemiology models are studies of how illnesses spread between humans and animals.This article's major objective is to investigate how infection affects prey during prey harvesting in a predator-prey system.Here, we examined the local and global stabilities of the equilibrium points of this system, as well as the boundedness and positivity of the solution [5,6].An eco-epidemiological predator-prey system with disease affecting only prey species and harvesting of both susceptible and infected prey has been taken into consideration by Bhattacharya et al. [7,8].Agnihotri and Gakkhar studied a prey-predator system with disease affecting both species and only the prey species being harvested [9,10].The SIRS-type models are mathematically similar to models of the s-called geometric Brownian motion (GBM): both predict an exponential growth, of infected individuals or of the value of the process.The state of dead individuals in the SIRS models is akin to the so-called resetting in the GBM model [11,12].The very relevant model of GBM with resetting was considered recently in refs.Ecologists, economists, and those involved in natural resource management have been interested in the studies on harvesting in predator-prey systems for some time.Few people have specifically included a harvested parameter in a predator-prey-parasite model and analysed the system's response to it [13].In this article, we examine the function of harvesting in an eco-epidemiological system where susceptible and diseased prey are harvested together [14,15].This study applies the Caputo fractional derivative and the harvesting rate to the predator-prey model.This author's major objective is to investigate the effects of prey infection and prey harvesting in a predator-prey system.Here, we examined the boundedness, positivity, local and global stabilities, and stability of the system's equilibrium points.

Model Formulation
The model explains how the diseased prey system interact with harvesting, which results in the following set of equations.The dynamical prey and predator mathematical model was investigated using the proposed model, The variables l = L K , m = M K , n = N K and the dimension time t = λKT can be changed in order to reduce the number of parameters in the system (1).We apply the following transformations According to the above transformations the Equation ( 1) can be rewritten in the following non-dimensional form In the system,we have taken fractional-order derivative α to model (2) with restore the fractional-order Caputo derivative.Then, the model ( 2) is take into the following form subject to the initial conditions l(0) ≥ 0, m(0) ≥ 0, n(0) ≥ 0.

Existence and Uniqueness of the Solutions
In this Section, boundedness of solution of the system (3) has been examined.The fractional-order system as follows: Theorem 1.The fractional order system (3) has a unique solution, for the non-negative initial conditions.
Proof.A sufficient condition for the solutions of system (3) in the region χ × (0, T] where, where, Hence, the solution of the system (3) exist and unique.

Equilibrium Points and Stability Analysis
In this section, the system (3) have the following possible equilibrium points: (i) E 0 (0, 0, 0) is the trivial equilibirum point.
(iv) E 3 ( l, m, 0) is the predator free equilibrium point, where l = s 4 + θ 2 and (v) The interior equilibrium point E * (l * , m * , n * ).Where, ans l * is the unique positive root of the quadratic equation ).Now, we want to calculate Jacobian matrix for local stability analysis around different equilibrium points.The Jacobian matrix at an arbitrary point (l, m, n) is given by where, , Theorem 2. The trivial equilibrium point E 0 (0, 0, 0) of a system (3) is stable if s 1 < θ 1 , otherwise it is unstable.
Theorem 3. The infected free and predator free equilibrium point E 1 ( Proof.The Jacobian matrix of the system (3) at an equilibrium point E 1 is given by Hence, the infected prey and predator free equilibrium point E 1 is stable.
Theorem 5.The equilibrium point E 3 of a system (3) is locally asymptotically stable if s 6 > c(s 2 + s 5 ).
Proof.The Jacobian matrix at E 3 is given by where Here, the characteristic equation of the above Jacobian matrix is where, E = −m 11 − m 33 , F = −m 21 m 12 + m 33 m 11 , G = m 12 m 21 m 33 .According to Routh-Hurwitz criteria [16], E > 0, G > 0 and EF − G > 0. Hence, E 3 is locally asymptotically stable.
Theorem 6.The endemic equilibrium point E * of system (3) is locally asymptotically stable .

Numerical Analysis
In this section, we present some numerical simulation results for Caputo-sense fractional-order eco-epidemic models.To accomplish this, we use Diethelm et al.'s predictorcorrector approach to solve the defined model.theparameter values are chosen as s 1 = 0.5, s 2 = 0.15, s 3 = 0.2, s 4 = 0.1, s 5 = 0.4, s 6 = 0.1, c = 0.5, ζ = 0.5, η = 0.3 and different values of α = 1 and then the equilibrium point E 4 (0.794787, 0.0476298, 0.343099) is unstable (see Figure 1).Fixing the derivative α as a variable.Here we consider the derivative value as α = 0.92, the effect of the predator harvesting θ 1 on the evolution of the three species and clearly influences the final size of the three populations are shown in Figure 2.

Conclusions
In this study, we investigated a fractional-order derivative-based model of a threespecies food web.Each equilibrium point's local stability in our proposed fractional-order system has also been examined.The suggested mathematical model's numerical simulation results show that the proposed system changes from unstable to stable as the order of the fractional derivative's value, α, goes from 0 to 1.It is obvious that for different values of α in the range 0 < α < 1, the unstable system with integer-order α = 1 becomes a stable system.For the interior equilibrium point, when the derivative of α = 1, the system becomes unstable, and if we change the order to fractional order, α = 0.92, the system becomes stable.When the susceptible prey population harvesting rate increases, then the infected prey population harvesting rate decreases in the fractional order derivative.Since the susceptible prey population is inversely proportional to the infected prey population in the system.Consequently, the derivative of fractional-order alpha makes an important contribution to the proposed system's dynamical stability.

Figure 2 .
Figure 2. It is varying the harvesting rate of susceptible prey,infected prey and predator population with different values of θ 1 = 0, 0.1, 0.2 for the fractional order derivative α = 0.92.

Table 1 .
Biological representation of the system (1) parameters.