Positive Solutions of a Diffusive Predator-Prey System including Disease for Prey and Equipped with Dirichlet Boundary Condition

We study a three-dimensional system of a diffusive predator-prey model including disease spread for prey and with Dirichlet boundary condition andMichaelis-Menten functional response. By semigroup method, we are able to achieve existence of a global solution of this system. Extinction of this system is established by spectral method. By using bifurcation theory and fixed point index theory, we obtain existence and nonexistence of inhomogeneous positive solutions of this system in steady state.


Introduction
In recent decade, predator-prey systems are significant in mathematical biology and applications [1][2][3].These systems are usually described by ordinary differential equation (ODEs).Besides, epidemic models have been deeply studied by differential equations in [4,5], based on the pioneer work of the classical SIR model by Kermack and Mckendrick.However, epidemic disease can spread within a species except for interactions among species.For this case, some ecoepidemiological models were introduced in [6][7][8] to study how disease affects the dynamic of predator-prey systems.Since we live in the real world, spatial factor plays an important role in populations dynamic.Moreover, space could induce some significant phenomena different from the corresponding ODE systems.At present, a lot of literature has showed that predator-prey systems with spatial diffusion can produce a variety of pattern formation by Turing bifurcation [9][10][11] and references wherein.In this paper, we are interested in the following model having Holling-type II functional response with disease in the prey: There are two species in the model: prey (+V) and predator ().In (1a),  is susceptible prey, and V denotes infected prey.Only susceptible prey  is capable of reproducing and its reproducing rate is .However, the infected prey does possess the capability of reproducing and still contributes to population growth of susceptible prey  according to the logistic growth law.The disease is transmitted only within prey in the form of V, and  ∈ (0, 1) is called the transmission coefficient.The disease is not genetically inherited.The infected populations do not recover or become immune.Infected individuals are less active and can be caught more easily by predator resulting in increasing survival of susceptible prey.Catching rate of predators is a ratio-dependent Michaelis-Menten functional response function; that is, V/( + V).
Infected prey has a death rate  > 0 and  > 0 is a death rate of predators.Prey and predators are inhabited in a bounded region Ω ⊆ R  with the smooth boundary Ω, where  is a spatial dimension number.Δ is the Laplace operator and represents diffusion of individuals from a high density region to low density one.Zhang et al. [12] have studied (1a) under the homogeneous Neumann boundary condition and obtain permanence and stability.
However, few author, as far as we know, investigated mathematical property of (1a) equipped with Dirichlet boundary condition, such as existence of global timedependent positive solution and inhomogeneous positive solutions in steady state.To this end, in this paper we make the first attempt to fill this gap and study existence of a global positive solution (1a) with homogeneous Dirichlet boundary conditions: which imply that predator and prey die out on the boundary Ω and initial values Besides, we first attempt to investigate existence and nonexistence of inhomogeneous positive solutions of (1a), (1b), and (1c) in steady state as follows: The remainder of the paper is organized as follows: in Section 2, some necessary notations and theoretical results are introduced; in Section 3, we examine existence of a global positive solution to (1a), (1b), and (1c); in Section 4 we discuss extinction of system (1a), (1b), and (1c); existence and nonexistence of inhomogeneous positive solutions are discussed by bifurcation theory and degree theory in Section 5.
We denote this unique positive solution by   and   < .
Next, the fixed point index theory, which is used later, is introduced.Let  be a real Banach space. is called a wedge if  is a closed convex set and  ⊂  for all  ≥ 0. For  ∈ , we define We always assume that  =  − .Let  :   →   be a compact linear operator on .We say that  has property  on   if there exists  ∈ (0, 1) and  ∈   \   such that (1 − ) ∈   .Let  :  →  be a compact operator with a fixed point  ∈  and  is Fréchet differentiable at .Let  =   () be the Fréchet derivative of  at .Then  maps   into itself.We denote by index  (, ) the fixed point index of  at  relative to .
For a linear operator , we denote by () the spectral radius of .
Similarly, there exists    > 0 such that V(, ) ≤  for all  >    .Further, since V/( + V) is monotone increasing with respect to the variable V, we have which concludes that (, ) → 0 as  → ∞.

Existence and Nonexistence of Positive Solutions of (2)
To prove the existence of positive solution of (2) by using fixed point index theory, we need a priori estimate for nonnegative solutions of (1a), (1b), and (1c).So we first give the following theorem.
(ii) First, we prove that  does not have the property  on  (  ,0,0) .