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.
This paper describes the examination of the following diffusive ratio-dependent predator-prey system with disease in the prey:where is a bounded region with a smooth boundary; the coefficients , and are positive constants; the initial functions , , and are not identically zero in ; , , and represent the densities of susceptible prey, infected prey, and predator, respectively. Predator preys only on infected prey . is the intrinsic growth rate of susceptible prey ; and 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) . Also in , 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  and references therein.
Xiao and Chen  suggested the following nondimensionalized model, which is a nonspatial version of (1):where , , and denote the population density of the susceptible prey, infected prey, and predator, respectively. The initial conditions are .
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 . Since then, these models have been mathematically studied for both the spatially homogeneous [5–8] and spatially inhomogeneous cases [9–11]. The actual evidence and justification for the models have also been studied [12–15]. For more background on the model, we refer the reader to .
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.
2. Asymptotic Behavior of Solutions
First it should be noted that, according to the results obtained by Pao , the solution of (1) is unique and continuous for all positive time in . Moreover, because , , and is assumed, , , and on for all .
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.
For , , consider the following eigenvalue problem:Recall that problem (4) has eigenvalues and eigenfunctions such that and , where . Furthermore, the eigenfunction of (4) corresponding to the eigenvalue is unique and positive. Throughout this paper, is denoted as the principal eigenvalue of the eigenvalue problem (4) corresponding to the unique positive principal eigenfunction . In addition, is simply denoted as . It is known that is a strictly increasing function, given that and imply that .
Consider the single equation:where is a bounded connected domain in with a smooth boundary. Assume that the function satisfies the following:(H1) is -function in , where .(H2) is -function in with for all .(H3) on for some positive constant .
Theorem 1. (i) The nonnegative solution of (5) satisfies for all .
(ii) If , then (5) has no positive solutions. Moreover, the trivial solution is globally asymptotically stable.
(iii) If , then (5) has a unique positive solution which is globally asymptotically stable. In this case, the trivial solution is unstable.
Obviously, if , then system (3) only has one semitrivial solution , where is the unique solution to
Now the stability of the trivial solution and the semitrivial solution is investigated.
Theorem 2. Let be a positive solution of (1):(i)If , then as .(ii)If , , and , then as .
Proof. We only prove (ii). Since According to Theorem 1(iii) and the comparison principle Let be sufficiently small positive constant such that Then there exists a such that for all . Thus Since , according to Theorem 1(ii) and the comparison principle, it follows that uniformly as . Since , it follows that uniformly as , again according to Theorem 1(ii). Thus, there exists a such that , for all . Thus, the following is obtained:Since , Theorem 1(iii) and the comparison principle yield where is the positive unique solution of under a homogeneous Dirichlet boundary condition. Thus, by using the continuity for , it is concluded that as .
3. 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 and . A wedge is said to be a cone if . For , define for some 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 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 , where is the identity map. If is an isolated fixed point of , then the fixed point index of at in is defined by , where is a small open neighborhood of in .
Theorem 3. Assume that is invertible on :(i)If has property on , then .(ii)If does not have property on , then , where is the sum of multiplicities of all the eigenvalues of which are greater than 1.
Lemma 4. If , then any nonnegative solution of system (3) has an a priori bound where , .
Proof. Consider the following elliptic problem:Let be a positive solution of (3). Then the comparison property of eigenvalues yields ; hence the unique positive solution to (14) exists. It is easy to show that is an upper solution of (14). It follows that in . Since is a positive solution of (3), holds in . Hence is satisfied for and is a positive lower solution of (14); therefore, in .
Next, multiplying the second equation in (3) by , and adding it to the first equation in (3), we obtain Now, take and consider the equation Then, , and so we obtain .
Finally, consider the following elliptic boundary value problem:For a positive solution of (3), holds according to the comparison property of eigenvalues. Hence, there exists a unique positive solution of (17), which is less than or equal to . Moreover, becomes a positive lower solution of (17) since therefore in .
Remark 5. Note that if , then . Thus one cannot expect a coexistence of predator under the assumption .
Note that the given growth rates in (3) are not defined at . To overcome this, the approach of Kuang and Beretta is adopted . More precisely, sincethe domain of may be extended to so that becomes a trivial solution of (3).
The following notation will be used throughout this paper.
Notation 1. (i) Consider .
(ii) Consider .
(iii) is the positive cone of .
(iv) Consider .
(v) Consider and .
(vi) Consider .
For , , , the positive and compact operator is defined by where is taken sufficiently large such that . Since each coordinate of is a positive and compact operator by the definition at , is also a positive and compact operator. Thus (3) has a positive solution if and only if has a positive fixed point.
The following lemma can be represented similarly to the proof of Lemma 4.6 in  if a homotopy is defined as above.
Lemma 6. Consider , where is defined in Notation 1.
Next, the index value of is calculated at . Since the Fréchet derivative of is not defined at , the difficulty is overcome by using an perturbation of . More precisely, define , where is defined as
Note that gives a well-definedness of since this index value does not depend on the particular choice of perturbation.
Lemma 7. If , .
Proof. If it is shown that , which is independent of , then the proof will be completed by the preceding argument of this lemma. Note that and . Define Then, similar to Lemma 3.3 , it can be shown that (i) is invertible on and (ii) has property . Hence, according to Theorem 3(i).
Next, indices are computed for the semitrivial solutions to model (3).
Lemma 8. Assume that . If , then .
Proof. First we observe that and . is denoted byFirst, let . For , we have . However, since , which is always true, we have . Similarly, can be obtained since . Thus, satisfies . Since is the unique solution of (14), . Hence, and is invertible on .
Next, since holds, the spectral radius of , say , is greater than . Thus, the Krein-Rutman theorem gives that an eigenvalue of is and the corresponding eigenfunction exists in . If we consider , then and . This shows that has property . Hence according to Theorem 3(i). Therefore, it is concluded that .
The following notation is used in the next lemma for the computation of the index of semitrivial solutions when exactly one of the species is absent.
Notation 2. For a sufficiently small ,(i) on and ,(ii).
Here, can be taken such that , but .
Note that it is not necessary to consider and since there is no positive semitrivial solution of the forms and for according to the assumptions of model (3). In the case of , it is easy to see that and by a strong maximum principle. Also if , then , 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 in the slice . The desired result is obtained by using the homotopy invariance of the index for another positive and compact operator , which is defined by for , , .
Lemma 9. Assume that If , then .
Proof. Note that if has no fixed points in , then holds by using the homotopy invariance of the index.
First, it is shown that there are no fixed points of in for all and a sufficiently small . Suppose, for the purpose of contradiction, that there are some fixed points of in for any . Then for and , there exist the sequences and fixed points of such that and . Here, by a choice of . Hence is contained in , such that and as .
Since as , using the comparison property of eigenvalues. This is a contradiction for large values of . Thus, it is possible to use the homotopy invariance in the slice for a sufficiently small .
Now let us proceed to complete the proof. For , gives , , and . Here, and follow from a choice of and a strong maximum principle, respectively. Hence, has only one fixed point in . Thus, . To calculate , we can use as in the proof of Lemma 8. Hence, using the assumptions and , we obtain therefore .
Using the above lemmas, sufficient conditions are provided for the existence of positive solutions of system (3).
Theorem 10. Assume and . If , then (3) has a positive solution.
Proof. Using the additivity of the index, Hence, there exists a positive solution in .
Remark 11. (i) If , then (3) has a trivial solution.
(ii) Assume . If or , then there are no nonnegative nonzero solutions of (3).
(iii) Assume . If or , then there are no nonnegative nonzero solutions of (3).
4. 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 predator-hunting 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the anonymous referees for careful reading and valuable comments which have helped to improve the presentation of this paper. This work was supported by a Korea University Grant.
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