Analysis of an Ecoepidemiological Model with Prey Refuges

An ecoepidemiological system with prey refuges and disease in prey is proposed. Bilinear incidence and Holling III functional response are used to model the contact process and the predation process, respectively. We will study the stability behavior of the basic system from a local to a global perspective. Permanence of the considered system is also investigated.


Introduction
Ecoepidemiology is the branch of biomathematics that understands the dynamics of disease spread on the predator-prey system.Modeling researches on such ecoepidemiological issues have received much attention recently 1-13 .Anderson and May 1 investigated a preypredator model with prey infection and observed destabilization due to the spread of infectious diseases within animal and plant communities.Chattopadhyay analyzed predatorprey system with disease in the prey 5 and applied the ecoepidemiological study to the Pelicans at risk in the Salton Sea 6 .Bairagi et al. 13 made a comparative study on the role of prey infection in the stability aspects of a predator-prey system with several functional responses.An ecoepidemiological model with prey harvesting and predator switching was investigated by Bhattacharyya and Mukhopadhyay 14 .Kooi et al. 15 studied stabilization and complex dynamics in a predator-prey system with disease in predator.Most of the above-mentioned studies focused on the role of disease in regulating the dynamical consequences of the interacting populations concerned, such as disease-induced stabilization and destabilization of population states 13 .In fact, the dynamical consequences of the predator-prey model can be determined by much ecological effect, such as the Allee effect and prey refuge.Theoretical research With the previously mentioned assumptions, the generalized predator-prey system with prey refuges and disease in prey can be represented by the following equations: , it is easy to show that the set G is the positively invariable set of system 2.1 .

The Positivity and Boundedness
Theorem 3.1.All solutions of system 2.1 initiating R 3 are positive and ultimately bounded.
Proof.Let S t , I t , Y t be one of the solutions of system 2.1 .
Integrating 2.1 with initial conditions S 0 , I 0 , Y 0 , we have

3.1
Hence all solutions starting in R 3 remain in R 3 for all t ≥ 0.
Next, we will prove the boundedness of the solutions.
Because dS/dt ≤ rS 1 − S/K , then we have Let W eS eI Y , then we obtain that Hence, we have

3.5
Thus, all curves of system 2.1 will enter the following region:

The Equilibrium Point
All equilibrium points of system 2.1 can be obtained by solving the following equations: These points are as follows: 1 the trivial equilibrium point E 0 0, 0, 0 , 2 the equilibrium point E K K, 0, 0 , 3 the predator-extinction equilibrium point E S, I, 0 , 4 the disease-free equilibrium point E * S * , 0, Y * , where It is clear to show that the disease-free equilibrium point E * S * , 0, Y * has its ecological meaning when R 1 > 1.

The Stability Property
In this section, we will study the local and global stability of the equilibrium points of system 2.1 .
then one has the following.
Proof.The Jacobian matrix of system 2.1 at the trivial equilibrium point E 0 0, 0, 0 is Clearly, the trivial equilibrium point E 0 0, 0, 0 is unstable.
The Jacobian matrix of system 2.1 at the equilibrium point E K K, 0, 0 is According to the theorem about the local stability, the local stability of the equilibrium point E K K, 0, 0 is determined only by the sign of ec Therefore, if R 0 < 1 and R 1 < 1, the equilibrium point E K K, 0, 0 is locally asymptotically stable.
Next, we will prove the global stability of the equilibrium point E K K, 0, 0 .Defining the Lyapunov function V eI Y , then we obtain that According to the LaSalle invariable set theorem, lim t → ∞ I t 0, lim t → ∞ Y t 0. Thus, the limit equation of system 2.1 is Clearly, the equilibrium point S K is globally asymptotically stable.
According to the limit system theorem, if R 0 < 1 and R 1 < 1, the equilibrium point E K K, 0, 0 is globally asymptotically stable.
Setting R 0 > 1, the Jacobian matrix of system 2.1 at the predator-extinction equilibrium point E S, I, 0 is The characteristic equation of system 2.1 at the predator-extinction equilibrium point E S, I, 0 is where

5.7
According to the Routh-Hurwitz rule, the predator-extinction equilibrium point E S, I, 0 is locally asymptotically stable when R 1 < 1.
Next, we will prove the global stability of the predator-extinction equilibrium point E S, I, 0 .
Defining the Lyapunov function V Y , then we have

Journal of Applied Mathematics
Thus, Φ t 0 on the set D , if and only if dS/dt 0 and dI/dt 0, then S S M , I I M .
If S I 0, then Φ S, I 0. If S S M , then I I M .If S < S M , then dI/dt < 0; that is, Therefore, the maximum value of the function Φ t is obtained at the point K, 0 or According to the LaSalle invariable set theorem, lim t → ∞ Y t 0. Thus, the limit equation of system 2.1 is

5.10
According to the results of the appendix section, if R 0 > 1, the equilibrium point S M , I M is globally asymptotically stable.
According to the limit system theorem, if R 0 > 1 and R 2 < 1, the predator-extinction equilibrium point E S, I, 0 is globally asymptotically stable.
then the disease-free equilibrium point E * S * , 0, I * is nonnegative, 1, then the disease-free equilibrium point E * S * , 0, I * is locally asymptotically stable; 2 < 1, then the disease-free equilibrium point E * S * , 0, I * is globally asymptotically stable; then the disease-free equilibrium point E * S * , 0, I * is unstable.
Proof.Assuming R 1 > 1, the Jacobian matrix of system 2.1 at the disease-free equilibrium point where

5.12
The characteristic equation of system 2.1 at the disease-free equilibrium point
Next, we will study the global stability of the disease-free equilibrium point Defining the Lyapunov function V I, then we obtain Thus, if R 0 < 1, dV/dt ≤ 0 and dV/dt 0 if and only if

Journal of Applied Mathematics
According to the LaSalle invariable set theorem, I t → 0 when t → ∞.The limit system of system 2.1 is

5.16
Clearly, the equilibrium points of the system 5.16 are in which S * and Y * are similar as the equilibria expression of system 2.1 .
It is easy to show that the equilibrium point E * has its ecological meaning when R 1 > 1.
According to the Routh-Hurwitz rule, the equilibrium point Again, the Jacobian matrix of system 5.16 at the equilibrium point where

5.19
The characteristic equation of system 5.16 at the equilibrium point E * is According to the above study, if R * 2 < 1, then a 11 < 0. Hence, if R * 2 < 1, then the equilibrium point E * is locally asymptotically stable in the region D by the Routh-Hurwitz rule.
It is easy to note that the globally asymptotically stability of the equilibrium point E * implies that there is no close orbit in the region D for the considered system.Let

5.21
and rewrite S, Y , and t into S, Y , and t, then system 5.16 becomes as follows:

5.22
where Thus, the positive equilibrium point E * S * , Y * of system 5.16 becomes the positive equilibrium point E P S P , Y P of system 5.22 , where

5.23
Considering the Dulac function B S, Y S −2 Y n−1 , then we have

5.24
In order to prove the global stability, we will prove only that there exists a real number n such that Φ S, n ≤ 0.
Therefore, the function Φ S, n has the maximum value at the point S − A 2 n 1 − γ 2 /3A 3 and Φ 0, n − A 0 n < 0.
Hence, there exists only one real number n, such that

5.25
Thus, we will prove only that there exists A 2 n 1 − γ 2 > 0, such that where

5.27
It is easy to show that if p − A 0 0, then Δ 0. According to Shengjin's distinguishing means, the cubic equation has one negative real root and two positively real roots.
If 2A 3 S 3 p A 2 S 2 p − A 0 < 0, the cubic equation has three roots which are not equal.According to the Descartes rule of signs, the cubic equation has two positively real roots and one negatively real root at most.Therefore, the cubic equation has at least one positively real root.That is to say, there exists a number n, such that A 2 n 1 − γ 2 > 0.
Furthermore, we obtain that According to the Bendixson-Dulac theorem, there does not exist the limit cycle for the limit system.
Hence, the equilibrium point E * is globally asymptotically stable.Therefore, if R 0 < 1 and R * 2 < 1, then the equilibrium point E * is globally asymptotically stable according to the limit system theorem.
Proof.Considering the average Lyapunov function V S, I, Y S α 1 I α 2 Y α 3 , where α i i 1, 2, 3 is positive, then in the region R 3 , we have 6.1 In order to prove the permanence of system 2.1 , we only indicate the following results: the function Ψ S, I, Y > 0 for all boundary equilibrium points.
By simple computation,