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 and Michaelis-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.
In recent decade, predator-prey systems are significant in mathematical biology and applications [1–3]. 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 [6–8] 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 [9–11] 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 and predator . In (1a), is susceptible prey, and 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 , and 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, Infected prey has a death rate and is a death rate of predators. Prey and predators are inhabited in a bounded region with the smooth boundary , where is a spatial dimension number. is the Laplace operator and represents diffusion of individuals from a high density region to low density one. Zhang et al.  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 time-dependent 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 and initial valuesBesides, 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.
In this section, some notations and basic well-known results are introduced. For any , the linear eigenvalue problem [13, 14],has an infinite sequence of eigenvalues. Let be the principal eigenvalue of (9) which is simple and its eigenfunction does not change sign in . We denote by for simplicity and the eigenfunction corresponding to is denoted by . with being not identically equal to implies .
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 For , we define We always assume that . Let be a compact linear operator on . We say that has property on if there exists and such that . 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 the fixed point index of at relative to
Proposition 1 (see [16–18]). Assume that is invertible on . Then (i)if has property on , then ;(ii)if does not have property on , then , where is the sum of multiplicities of all eigenvalues of which is greater than
For a linear operator , we denote by the spectral radius of .
Now, we introduce the degree calculations, which was introduced by Dancer and Du in . Assume that and are ordered Banach spaces with positive cones and , respectively. Let and Then is an ordered Banach space with positive cone Let be an open set in containing and be completely continuous operators, Denote by a general element in with and Let be defined by Define
Theorem 3 (see ). Suppose that is relatively open and bounded and Suppose that extends to a continuously differentiable mapping of a neighborhood of into , is dense in , and Then the following are true: (i) for small, if, for any , the spectral radius and is not an eigenvalue of corresponding to a positive eigenvector;(ii) for small, if, for any , the spectral radius
For convenience, we introduce some notations. For , we denote , , , and and extend these notations to real-valued functions. If is partially ordered vector space, we denote its positive cone by . For , , let . Assume and its norm is defined as .
Next, the existence of the local solution for (9) will be proved. First, we give out the following lemma.
Lemma 4. For every , Cauchy problem (9) has a unique maximal local solutionwhich satisfies the following Duhamel formula for :where is a operator and the closure of in . Moreover, if then .
Proof. Since the operator is the closure of in , it generates an analytic, condensed, strong continuous operator semigroup . Moreover, we observe that is locally Lipschitz on a bounded set . Via Theorem in , we complete this proof.
Proof. To prove for and , we consider the following system:with boundary and initial conditions After multiplying (14a), (14b), and (14c) by , , and , respectively, and integrating by parts on the domain , we obtain Hence, for , Consequently, Now, we can know that is a solution of (1a), (1b), and (1c), and according to Lemma 4, we finally get that
Next, we will prove the global existence of the positive solution for (1a), (1b), and (1c). Via Lemmas 4 and 5, we only need to show that the solution of (1a), (1b), and (1c) is uniformly bounded, that is, dissipation.
Proof. In view of Lemma 5, we only prove the upper bound of (1a), (1b), and (1c) for . From (1a), we haveWe get , by maximum principle, where
Let . From (1b) we have Hence, by maximum principle we have that By using the same method, we consider ; then from (1c) we have that Thus, the maximum principle shows that
Remark 7. From Theorem 6, we know that the upper bound of the solution of (1a), (1b), and (1c) is independent of the maximal time , which implies that the solution of (1a), (1b), and (1c) is a global solution.
Proof. (i) First, we observe that solution of (1a), (1b), and (1c) satisfiesLet be a sufficiently small positive constant such that Since , we see that uniformly as by using comparison principle for elliptic problems. Then there exists such that for all Therefore, we havewhich concludes that uniformly as . Similarly, there exists such that for all . Further, since is monotone increasing with respect to the variable , we havewhich concludes that as
(ii) Let be a sufficiently small positive constant such that Since , we get from (28) and comparison theorem thatThen, there exists such that for all Thus, we haveSince satisfies (a) and , we conclude that uniformly as Similarly, we can see that uniformly as
Besides, there exists such that for all Hence, we haveSince satisfies (b), we get thatThus, from (32), (35), and the arbitrariness of , we can conclude that as We finish this proof.
5. 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.
Theorem 10. Any nonnegative solution of (2) has a priori bounds:where
Proof. From the first equation of (2), we haveand we get by maximum principle. Let with . From (1b), we have Thus, by maximum principle, we get We set with , and then we have By using maximum principle again, we obtain Therefore, we complete this proof.
Now, we introduce some notations: , where , , where , , according to Theorem 10.Taking we can define a positive and compact operator byIt is obvious that having a positive fixed point in interior of implies that (2) has a positive solution.
Lemma 11. Assume that , then one has(i),(ii)
Proof. (i) It is easy to see that has no fixed point on , so is well defined. For , we define a positive and compact operator by and then For each , a fixed point is a solution of the following problem:Form Theorem 10 has no fixed point and is well defined. Hence, Note that (46) has only the trivial solution when . Set Let for some Obviously,
Hence, has no nontrivial kernel in . Since , it is easy to see that by Proposition 2. It implies that has no property Therefore, by Proposition 1, we get the first result of Lemma 11.
(ii) It is obvious that . Let ; then Assume that for some , and thenSince and , from (51) and (52), we know that If with , then from (50), which contradicts with the assumption. Thus So has nontrivial kernel in
For , by Proposition 2, we see that , and is the principle eigenvalue of with a corresponding eigenfunction . Since , we see that Set , and then Thus, has the property By Proposition 1 we have
Now, we give out the index of the semitrivial solution of (2).
Lemma 12. Assume that , and then one has(i)if , then ;(ii)if , then
Proof. Let ; then Let for some ThenSince , and . So we get that Hence, from the first equation of (54) we haveIf , then by Proposition 2. On the other hand, , and we get a contradiction. Hence, Then has no nontrivial kernel in .
(i) Since , by Proposition 1, we have which is an eigenvalue of with a corresponding eigenfunction Since , we have Set , and thenHence, has the property on resulting in according to Proposition 1.
(ii) First, we prove that does not have the property on
Since , we have On the contrary, we suppose that has the property on Then there exist and such that SoThen is an eigenvalue of the operator , which contradicts So does not have the property on By Proposition 2, we have , where is the sum of the multiplicities of all eigenvalues of . Next, we will prove that
Assume that is an eigenvalue of with corresponding eigenfunction , and then ; that is,Since and , it follows from the third equation (58) that
We suppose that , and it follows from the second equation of (58) and Proposition 2 that which contradicts , and so
If , it follows from the first equation of (58) that Thus, we get the contradiction and this implies that Hence, , which implies that has no eigenvalue being greater than ; that is, , and We complete the proof.
Next, we consider the other semitrivial solutions of (2). Let us consider the following three subsystems:It is obvious that (63) has trivial solution and that (62) only has a trivial solution and a semitrivial solution if and only if For subsystem (61), a lot of papers [15, 22] and references wherein have discussed systems in this case, so we directly give out some results about (61).
Theorem 13. Equation (61) has a positive solution with if and only if and Moreover, if and , then the positive solution satisfies and , where is the unique positive solution of the following equations:
Lemma 14. Assume , and then one has the following:(i)if , for any , then ;(ii)if , for any , then .
Proof. Let Define Then and we are in the framework to use Theorem 3. We choose a neighborhood of such that
Now, with if and only if By the results in , we have For any , a direct calculation shows that Notice that, by our choice of , so it follows from the maximum principle that is -positive in the sense of  with Hence, and is the only eigenvalue corresponding to a positive eigenvector. By a simple calculation, we can easily show that if and only if and if and only if . Therefore, by Theorem 3 and the discussion above, we have Again since the degree does not depend on the particular choice of and , means and . Thus, we complete this proof.
From the discussion above, we can obtain the existence of positive solutions of (2).
Proof. Since , we obtain and from Lemma 11. Hence, we haveSince , we have from Lemma 12. Moreover, Theorem 13 implies Let Since and by Theorem 13, we have and thus from Lemma 14. Therefore, So (71) holds. We complete this proof.
Next, we give out the result of nonexistence of positive solutions.
Theorem 16. If any one of the following conditions holds, then (2) has no positive solution:(i);(ii) and
Proof. Assume that (2) has positive solution , and then satisfy the following three equations, respectively:(i) Since in , we get from (72) that By Proposition 2 and comparison property of principle eigenvalue, this contradicts Then (73) and (74) have and
(ii) Since in , we get from (73) that by Proposition 2 and comparison property of principle eigenvalue, we get a contradiction
The authors declare that they have no competing interests.
Thanks are due to reviewers’ comments and suggestions for improving this paper. This work is supported by Grants 61563033, 11563005, and 71363043 from the National Natural Science Foundation of China and 20161BAB201010, 20151BAB212011, and 20151BAB201021 from the Natural Science Foundation of Jiangxi Province.
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