Positive Coexistence of Steady States for a Diffusive Ratio-Dependent Predator-Prey Model with an Infected Prey

We examine a diffusive ratio-dependent predator-prey system with disease in the prey under homogeneous Dirichlet boundary conditions with a hostile environment at its boundary. We investigate the positive coexistence of three interacting species (susceptible prey, infected prey, and predator) and provide nonexistence conditions of positive solutions to the system. In addition, the global stability of the trivial and semitrivial solutions to the system is studied. Furthermore, the biological interpretation based on the result is also presented. The methods are employed from a comparison argument for the elliptic problem as well as the fixed-point theory as applied to a positive cone on a Banach space.


Introduction
This paper describes the examination of the following diffusive ratio-dependent predator-prey system with disease in the prey: (, ) = V (, ) =  (, ) = 0 on Ω × (0, ∞) ,  (, 0) = ũ0 () ≥ 0 in Ω, V (, 0) = Ṽ0 () ≥ 0,  (, 0) = w0 () ≥ 0, where Ω ⊆ R  is a bounded region with a smooth boundary; the coefficients ,  1 ,  2 , , and  are positive constants; the initial functions ũ0 , Ṽ0 , and w0 are not identically zero in Ω; , V, and  represent the densities of susceptible prey, infected prey, and predator, respectively.Predator  preys only on infected prey V.  is the intrinsic growth rate of susceptible prey ;  2 and  1 are the death rates of the infected prey and predator;  is known as the half saturation parameter;  is the predation coefficient;  represents the efficiency at which consumed prey is converted into predator births.The homogeneous Dirichlet boundary condition describes a hostile boundary environment at the boundary of the region under investigation.
Model (1) is based on the following assumptions: (a) Prey consists of two classes: susceptible prey and infected prey.
(b) Only susceptible prey can reproduce themselves according to logistic law and infected prey, together with susceptible prey, contributes to population growth.
(c) Disease can only be spread among the prey and is not inherited.
(d) Predators only prey on infected prey.
Assumption (d) is in accordance with the fact that the infected prey is less active and can be caught more easily, or the behavior of the prey individuals is modified such that they live in parts of the habitat which are accessible to the predator (fish and aquatic snails staying close to water surface, snails staying on the top of the vegetation rather than under the plant cover) [1].Also in [2], the authors indicated that wolf attacks on moose are more often successful if the moose is heavily infected by "Echinococcus granulosus." Additional background information pertaining to model (1) may be obtained from [3] and references therein.
Over the last three decades, predator-prey models have been studied extensively by many researchers.Among these, the ratio-dependent predator-prey models, in which the per capita predator growth rate depends on a function of the ratio of prey to predator abundance, have been proposed by Arditi and Ginzburg [4].Since then, these models have been mathematically studied for both the spatially homogeneous [5][6][7][8] and spatially inhomogeneous cases [9][10][11].The actual evidence and justification for the models have also been studied [12][13][14][15].For more background on the model, we refer the reader to [16].
The goal of this study is to investigate the asymptotic behavior of positive solutions for (1) and the positive solutions to the steady-state system of (1): We say that system (3) has a positive solution (, V, ) if () > 0, V() > 0, and () > 0 for all  ∈ Ω.The existence of positive solutions (, V, ) to system (3) is called positive coexistence.
The paper is organized as follows.In Section 2, the result for the global stability at semitrivial solution is derived.Sufficient conditions for the existence of positive solutions to system (3) are provided in Section 3, and the biological interpretations are briefly stated in Section 4.
In this section, the global stability of the trivial and semitrivial solutions of system (1) are examined.
Our discussion is based on the following well-known fact about an eigenvalue problem.
The following theorem is a consequence of the main results of [18].In addition, one can also refer to [17,19].

Positive Steady-State Solutions
In this section, the sufficient conditions for the existence of positive solutions of (3) are given by using fixed point index theory.
A fixed point index theorem is provided to calculate the index of certain operators for semitrivial solutions of the system.
Let  be a real Banach space and  ⊂  a closed convex set. becomes a total wedge if  ⊂  for all  ≥ 0 and  −  = .A wedge is said to be a cone if  ∩ (−) = {0}.For  ∈ , define   = { ∈  :  +  ∈  for some  > 0} and   = { ∈   : − ∈   }.Then   is a wedge containing , , and −, whereas   is a closed subspace of  containing .Let  be a compact linear operator on  which satisfies (  ) ⊂   .We say that  has property  on   if there is a  ∈ (0, 1) and a  ∈   \   such that  −  ∈   .Let  :  →  be a compact operator with a fixed point  ∈  and  Fréchet differentiable at .Let  =   () be the Fréchet derivative of  at , in which case  maps   into itself.For an open subset  ⊂ , define index  (, ) = index(, , ) = deg  ( − , , 0), where  is the identity map.If  is an isolated fixed point of , then the fixed point index of  at  in  is defined by index  (, ) = index(, , ) = index(, (), ), where () is a small open neighborhood of  in .
The following theorem can be obtained from the results of [18,20,21].
Remark 5. Note that if  ≤  1 , then () ≤ 0. Thus one cannot expect a coexistence of predator  under the assumption  ≤  1 .
The following lemma can be represented similarly to the proof of Lemma 4.6 in [22] if a homotopy is defined as above.Lemma 6.Consider (A,   ) = 1, where   is defined in Notation 1.
Next, indices are computed for the semitrivial solutions to model (3).
Note that it is not necessary to consider  1 () and  2 () since there is no positive semitrivial solution of the forms (0, V, ) and (, 0, ) for V,  > 0 according to the assumptions of model (3).In the case of  ≡ 0, it is easy to see that V ≡ 0 and  ≡ 0 by a strong maximum principle.Also if V ≡ 0, then  ≡ 0, because predators only prey on infected prey according to our assumption.Thus, system (3) cannot be maintained when the prey  becomes extinct from a biological point of view.Hence, within the category that ensures the subsistence of the prey , it is possible to obtain conditions that guarantee the existence of positive solutions.
Next, we calculate the index value of A in the slice  3 ().The desired result is obtained by using the homotopy invariance of the index for another positive and compact operator Â 1 , 2 , 3 :   → , which is defined by for If  1 ( 2 −  0 ) < 0, then (A,  3 ()) = 0.
Before this section is concluded, the nonexistence conditions of the positive solutions of system (3) are mentioned in Remark 11.

Biological Interpretation
This paper examined a time-independent predator-prey system with ratio-dependent functional responses incorporating susceptible prey, infected prey, and predators under homogeneous Dirichlet boundary conditions.The existence and nonexistence of positive solutions to the system were investigated.
First, sufficient conditions were provided under which the system has positive solutions, such that all three species (susceptible prey, infected prey, and predator) are able to coexist in a region with a hostile environment at its boundary.The criterion for positive coexistence is significantly influenced by the sign of the principal eigenvalues of certain types of Schrödinger operators.The results indicated that either a large predation rate or high conversion rate together with a low predator death rate as well as a low death rate of infected prey facilitates the coexistence of the system.
Next, the nonexistence results for the positive coexistence states provided the conditions for both total extinction and the extinction of predators with the coexistence of susceptible and infected prey.We observed that total extinction can occur when a susceptible low-density prey species forces infected prey and predators to become extinct, and although the density of the susceptible prey is not low, low predatorhunting or a low conversion rate may lead to the extinction of infected prey, thereby resulting in the disappearance of a predator.Our model also indicates that the extinction of a predator can occur regardless of the density of infected prey, provided the predator has a sufficiently low maximal growth rate and a high death rate.
3. Assume that  −  is invertible on   : (i) If  has property  on   , then   (, ) = 0. (ii) If  does not have property  on   , then   (, ) = (−1)  , where  is the sum of multiplicities of all the eigenvalues of  which are greater than 1.