Fractional order prey-predator model with infected predators in the presence of competition and toxicity

In this paper, we propose a fractional-order prey-predator model with reserved area in the presence of the toxicity and competition. We prove di(cid:27)erent mathematical results like existence, uniqueness, non negativity and boundedness of the solution for our model. Further, we discuss the local and global stability of these equilibria. Finally, we perform numerical simulations to prove our results.


Introduction
Each population within an ecosystem does not exist in isolation, and there must be some relationships between these different populations [3].The relationship between them is divided into types: mutualism, parasitism, competition and predation.The dynamic relationship between predator and prey is long established and will remain among the crucial topics in ecology and mathematical ecology because of its universal existence and its importance [30].
In recent years, the fractional calculations have developed rapidly and have shown broad application prospects in many areas.Useful results can be obtained by extracting a dynamic behaviour of biological systems presented by a mathematical model of integer derivatives.However, most biological systems also have memory.In this case, the modelling in fractional order, unlike the classical mode.The existence of the memory is taken into account.The fractional derivative of a biological process at a point is affected by all the information and behaviour of the model at all previous times, while the classical derivative at a point is only affected by information from the local neighbourhood of that point.For this reason, many papers study the theory of differential fractional equations [19,21,23,24,26].The effort applied for harvesting in the unreserved area, the susceptible predator populations, the infected predator populations, respectively r 1 , r 2 The growth rates of fish population inside reserved and the unserved areas q 1 , q 2 The catchability coefficient in the unreserved area and the predator species σ 1 , σ 2 Migration rate from unreserved area to reserved area and reserved area to unreserved area n 1 , n 2 The competition coefficients γ The strength of intra-specific between prey and infected predator δ The disease transmission coefficient β The search rate of the prey toward susceptible predator µ The death rate of susceptible predator η The death rate of infected predator α Saturation constant while susceptible predators attack the prey σ The conversion rate of susceptible predator due to prey ux 2 , vy 2  The reduction terms, in the unreserved area and reserved area respectively, where u and v the coefficients of toxicity βxS α +x The functional response of feeding prey by susceptible predator Not long ago, many researchers began to study fractional biological models [1,10,20,25].In article [14], a dynamic system modelling a prey-predator with harvest area and reserve for prey in the presence of competition and toxicity.In article [25], let us introduce a fractional prey-predator model with two types of susceptible and infected predators.In our paper it has been supposed that the prey are divided into two areas reserved and free and reserved zone, as well as the predators are divided into two categories, susceptible and infected predators.Now the basic model based on [14,25] is governed by the following fractional system (Fig. 1): where D α is in the sense of Caputo fractional derivative and 0 < α ≤ 1 defined by [22]: Where f is defined by : f The detailed description of the model (1.1) is illustrated in the following schema: the explanation an Units of this parameters and variables given by the tables (Tabs. 1 and 2): From [7], if there is no migration of fish population from reserved area to unreserved area (σ 2 = 0) and (r 1 − σ 1 − q 1 E 1 < 0), we find that D α x < 0. Similarly, if there is no migration of fish population from unreserved area to reserved area (σ 1 = 0) and r 2 − σ 2 < 0, then D α y < 0.
If σβ − µ − q 2 E 2 < 0, then D α S < 0. So, finally we conclude that: Our paper is organized as follows.In the following section, we prove the of the existence and uniqueness solutions of the system (1.1), in Section 3, we show the boundedness and positivity of the solutions.Number per unit of fishing effort, σ 1 , σ 2 , r 1 , r 2 , µ, η, β, u, v, δ, n 1 , n 2 Per day γ, σ Constant In Section 4, we study the existence and stability of all the equilibria of our model (1.1).Finally, we present the numerical simulations to study the stability of the equilibria.

Basic properties and equilibria
Theorem 2.1.The sufficient condition for the existence and uniqueness of the solution of system (1.1) in the region Ω × [t 0 , T ] with initial conditions X(0) = (x(0), y(0), S(0), I(0)) and t ∈ [t 0 , T ] is: Proof.Let X = (x, y, S, I) T and X = (x , y , S , I ) T the system (1.1) can be is written in this form: where To prove the global existence and uniqueness of system (1.1), consider the region Ω × [t 0 , T ], where Ω = (x, y, S, I) ∈ R 4 : max {|x|, |y|, |S|, |I|} ≤ M, M > 0 .For any X, X ∈ Ω: Thus, F (X) satisfies the Lipschitz's condition [13] with respect to X. Now, we describe the uniform boundedness of the solutions of the system (1.1).
Lemma 2.2.The set Ω = (x, y, S, I) ∈ R 4 is a region of attraction for all solutions initiating in the interior of the positive octant, where Proof.We pose w = x + y Applying the theory of fractional inequality [22] we get: , Now, we find the positive equilibria, then we study their local stability.We denote the function on the right hand side of the system (1.1) by F i (x, y, S, I), for i = 1, . . ., 4.

Numerical simulations
To show the influence of the parameter α on our fractional order model, we take the different values of α in numerical simulations of the curves x(t), y(t), S(t) and I(t) that are shown in Figures 2-5.These figures show that the system (1.1) reaches the equilibrium state for the different values of α.These results show the effectiveness of Theorems 2.4-2.7.As we can see, numerical solutions are permanently dependent on the fractional order derivative α and the model reaches the equilibrium point more rapidly by reducing α.In other words, the model approaches the steady state more quickly when the memory factor effect is increased.
We observe from simulations, the effect of reducing the order of the time derivative can be observed.As the fractional order α decreases, the system (with Caputo derivative) stabilizes more quickly.It is the largest "memory" of the system of past states, the greater the damping of the oscillations in the dynamics of the system.The simulations show that, even with fairly moderate reductions in α, the amplitude of the population density oscillations is greatly delayed.

Conclusion
In this paper, we investigated a Dynamics of the fractional order prey-predator model in the presence of competition and toxicity using the Caputo fractional derivative.We have established the existence and boundedness of the solutions.After calculating the equilibrium of our model under certain conditions, we have analyzed the local stability using Matiginon's conditions [17].Global stability has been studied using Lyapunov functions.From our numerical results, we can observe that the different values of α have no effect on the stability of equilibria but have an effect on the time necessary to achieve equilibrium states.These variations are verified in the numerical simulations illustrated in the Figures 2-5, as the curves x, y, S and I that converge towards the equilibrium points.Finally, we can conclude that the memory effect of the fractional order derivative affects the dynamics of our proposed system.

Figure 2 .
Figure 2. Solution curves corresponding to the set values parameters of the system (1.1) of equilibrium P 1 with different values of α.

Figure 3 .
Figure 3. Solution curves corresponding to the set values parameters of the system (1.1) of equilibrium P 2 with different values of α.

Figure 4 .
Figure 4. Solution curves corresponding to the set values parameters of the system (1.1) of equilibrium P 3 with different values of α.

Figure 5 .
Figure 5. Solution curves corresponding to the set values parameters of the system (1.1) of equilibrium P 4 with different values of α.

Table 1 .
Variables and parameters descriptions.

Table 2 .
Units of variables and parameters.