The Stability Analysis of the diseased predator-prey model incorporating migration in the contaminated environment

We proposed and analyzed a predator–prey model. The disease effects in predator due to pollution in environment, as well the immigration factor effected is discussed. We assumed that, the population are divided into three parts prey, susceptible predator and infected predator. Firstly, the existence, uniqueness and bounded-ness of the solution of the model are discussed. Secondly, we studied the existence and local stability of all equilibrium points. Furthermore, some of the Sufficient conditions of the global stability of the positive equilibrium are established using suitable Lyapunov functions. Finally, those theoretical results are demonstrated with numerical simulations.


Introduction
To study the dynamical behavior of a phenomenon, the mathematical modeling is used as an effective tool to describe and analyze this phenomenon. Around 1800, the British Economist Malthus formulated a single species model and subsequently modified by Verhulst [1].
In the beginning of the twentieth century several attempts have been made to predict the evolution and existence of species mathematically. Indeed, the first major attempt in this direction was due to the well-known classical Lotka and Volterra [2,3]. They proposed the prey-predator model in 1927. They also describe the continuous Lotka-Volterra model by ordinary differential equations. Further the delay differential equations is widely used to characterize the dynamics of biological systems.
During the last three decades, the relationship between the predator and their prey is studied and its very crucial component of study in ecology. The prey-predator interaction is prominent and significant area of research in applied mathematical modeling and population dynamics. Venturino [4] investigated the long-term behavior in predator-prey model which assuming the epidemics occurred in prey population and can be transmitted by the contact of predators. Mathematical ecology and mathematical epidemiology are two different fields in the study of biology and applied mathematics. The combination of these fields are studied which termed as an ecoepidemiology. Many authors have studied eco-epidemiological models and considered infection in prey population only. Later, other authors such as Kant and Kumar [5] formulated and studied a predator-prey model with migrating prey and disease infection in both species. In [6], Haque and Venturino analyzed the prey-predator model by considering a Holling-Tanner functional response. They also investigated some bifurcations around the disease-free equilibrium. Recently, many authors have proposed and discussed eco-epidemiological models with some assumptions (for instance, [7][8][9][10]). They considered prey-predator model with infection in prey population only. Naji and Mustafa [9] discussed the dynamics of an eco-epidemiological model with nonlinear incidence rate. On the other hand, there are another category papers in literature, in which the authors consider the eco-epidemiological models where the disease spreads in predator population [11][12][13][14]. 3 easily the prey compared to the healthy predator. Thus, we assume that searching coefficient of the healthy predator for prey is greater than that of infected predator.
H4. The functional response of the predator to the prey is assumed to be of Lotka-Volterra type.

H5.
Prey population has migration rates as 1 . It is a natural factor that healthy predator is stronger as compared with the infected predator and therefore we neglected the probability of migration of infected predator.
H6. It has been assumed that infected predator recovers with rate 2 .

Boundedness of the solution.
Since all the parameters are non-negative and the interaction functions are continuously differentiable the right hand side of system (1) is a smooth function of variables ( , 1 , 2 , ) in the positive octant, Furthermore, it is easy to prove that Ω is an invariant set. In addition, it is easy to verify that, all the interaction functions are globally Lipschitz and then the system (1) has a unique solution. Now we will prove the boundedness of the system (1).
3.2 Theorem: All solutions of system (1) which initiate in ℜ + 4 are uniformly bounded.

Global stability analysis
In this section, the region of global stability (basin of attraction) of each equilibrium points of system (1) is presented as shown in the following theorems.

Theorem (5.1):
Assume that, the vanishing equilibrium point 0 is locally asymptotically stable in ℜ + 4 . Then it is a globally asymptotically stable provided that the following conditions hold Now, by doing some algebraic manipulation and using the condition (36), we get Consequently, due to condition above 0 < 0 is negative definite and hence 0 is Lyapunov function with respect to 0 . Thus 0 is a globally asymptotically stable and the proof is complete. ∎

Theorem (5.2):
Assume that the equilibrium point 1 is locally asymptotically. Then it is a globally asymptotically stable in the subregion of ℜ + 4 provided that Now, by doing some algebraic manipulations and using the condition (38), we get Consequently, due to the condition above 1 < 0 is negative definite and hence 1 is Lyapunov function with respect to 1 in the region that satisfies the given condition.
Where the symbol is given in the proof. Consequently, due to condition above 2 < 0 is negative definite and hence 2 is Lyapunov function with respect to 2 in the region that satisfies the given condition. Thus 2 is a globally asymptotically stable and the proof is complete. ∎

Theorem (5.4):
Assume that the equilibrium point 3 is locally asymptotically stable. Then it is a globally asymptotically stable in the sub region of ℜ + 4 that satisfied the following conditions Furthermore by taking the derivative with respect to the time and simplifying the resulting terms, we get that Consequently by using (46) conditions we get that Obviously, 5 is negative definite and hence 5 is Layapunov function with respect to 5 . So 5 is globally asymptotically stable in the sub region that satisfies the given condition. ∎

Numerical Simulation
To      Trajectories of E(t)

CONCLUSIONS AND DISCUSSION
In this paper, we consider a predator-prey model with modified Leslie-Gower and Holling type-II functional response. We discuss the structure of nonnegative equilibria and their local stability.
Migration has been allowed among prey and healthy predator population. It is also remarkable that Holling type-II functional responses are more frequently used as compare to other functional responses. By the above discussion, we can note that each of the functional responses are useful and have their specific importance in ecology. However, in the present study we have considered Holling type-II functional response.
Finally, to complete our understanding to the global dynamical behavior of system (1), numerical simulation is used using hypothetical set of parameters values given by Eq. (48) and (49). In the following, the obtained numerical simulation results are summarized.
1. The trajectory of system (1) approaches asymptotically to positive equilibrium point starting from different initial points using the data Eq. (48), which indicates to existence of globally asymptotically stable positive equilibrium point.
2. Increasing the inhibition rate of disease or disease death rate above a specific value leads to extinction in predator species due to the lack in their food. Further increasing at least one of these parameters causes extinction in the infected prey specie and the trajectory of system (1) approaches asymptotically to free equilibrium point. Otherwise, the system still persists at a positive equilibrium point. 3. We observed that migration of prey ( 1 ) plays a leading role in the existence and stability of equilibria of systems (1).