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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 167856, 12 pages
Effect of Diffusion and Cross-Diffusion in a Predator-Prey Model with a Transmissible Disease in the Predator Species
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Received 16 September 2013; Accepted 27 February 2014; Published 3 June 2014
Academic Editor: Francisco Solis
Copyright © 2014 Guohong Zhang and Xiaoli Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study a Lotka-Volterra type predator-prey model with a transmissible disease in the predator population. We concentrate on the effect of diffusion and cross-diffusion on the emergence of stationary patterns. We first show that both self-diffusion and cross-diffusion can not cause Turing instability from the disease-free equilibria. Then we find that the endemic equilibrium remains linearly stable for the reaction diffusion system without cross-diffusion, while it becomes linearly unstable when cross-diffusion also plays a role in the reaction-diffusion system; hence, the instability is driven solely from the effect of cross-diffusion. Furthermore, we derive some results for the existence and nonexistence of nonconstant stationary solutions when the diffusion rate of a certain species is small or large.
The study of the dynamic relationship between predator and prey has long been one of the most important themes in population dynamics because of its universal existence in nature and many different phenomena have been observed (see [1–13] and references therein). At the same time, since species need to interact with the environment, they are always subject to diseases in the natural world. So it is necessary and interesting to combine demographic as well as epidemic aspects in the standard classical population models. This type of systems is now known as ecoepidemic model.
In fact, the importance of disease influence on the dynamics of plant as well as animal populations has been recognized and several such studies are reviewed in a number of recent publications. However, most of the previous researches on ecoepidemic models assume that the distribution of the predators and prey is homogeneous, which leads to the ODE system (see [14–23] and references therein). As we know, both predators and prey have the natural tendency to diffuse to areas of smaller population concentration. At the same time, some prey species always congregate and form a huge group to protect themselves from the attack of infected predator. So it is important to take into account the inhomogeneous distribution of the predators and prey within a fixed bounded domain and consider the effect of diffusion and cross-diffusion.
In order to construct the corresponding reaction-diffusion type model, we first propose the following assumptions, which are proper in biological background.The disease spreads among the predator species only by contact and the disease incidence follows the simple law of mass action.In the absence of predators, the prey population grows logistically with the intrinsic growth rate and carrying capacity , in which measures intraspecific competition of the prey.The sound predator population has no other food sources, and represents natural mortality. The infected predator population cannot recover and their total death rate encompasses natural and disease-related mortality. The conversion factor of a consumed prey into a sound or infected predator is .The sound and infected predators hunt the prey with different searching efficiencies, denoted, respectively, by and , with . This is due to the fact that sound predators are more efficient to catch the prey than the infected ones, weakened by the infection.Both predators and prey have the natural tendency to diffuse to areas of smaller population concentration and the natural dispersive forces of movements of the prey, sound predators, and infected predators are , , and , respectively.The prey species congregate and form a huge group to protect themselves from the attack of infected predator.
With the above assumptions, our model takes the following form, in which all parameters are assumed to be positive: where is a bounded domain in ( is an integer) with a smooth boundary and is the outward unit rector on . The homogeneous Neumann boundary condition indicates that there is zero population flux across the boundary. In the diffusion terms, the constant , which is usually termed self-diffusion coefficient, represents the natural dispersive force of movement of an individual. The constant could be referred to as cross-diffusion pressure, which describes a mutual interference between individuals.
In fact, it is easy to see that the infected predator diffuses with flux: As , the part of the flux is directed toward the decreasing population density of the prey , which means that the prey species congregate and form a huge group to protect themselves from the attack of infected predator. We remark that this kind of nonlinear diffusion was first introduced by Shigesada et al.  and has been used in different type of population models [25–28]. We also point out that the corresponding ODE system of (1) with delay has been studied by , and they mainly investigate the stability and bifurcations related to the two most important equilibria of the ecoepidemic system, namely, the endemic equilibrium and the disease-free one.
Since the first example of stationary patterns in a predator-prey system arising solely from the effect of cross-diffusion is introduced by Pang and Wang , recently, more attention has been given to investigate the effect of cross-diffusion in reaction-diffusion systems; see, for example, [31–36] and references therein. Here we point out that, to our knowledge, there is little work about ecoepidemic models with diffusion and cross-diffusion was discussed.
In our work here, one of the main purposes is to study the existence of positive stationary solutions of (1) by using degree theory, which are the positive solutions of Hence we are interested in nonconstant positive solutions of (3), which correspond to coexistence states of prey and predators. For convenience, we denote . By a direct computation, we can show that (3) has a semitrivial constant steady state if and has a positive constant steady state , where provided that Here we remark that the semitrivial constant steady state and the positive constant steady state are also called disease-free equilibrium and endemic equilibrium, respectively, in endemic models.
The rest of this paper is organized as follows. In Section 2, we will investigate the stability of disease-free equilibrium and the endemic equilibrium and show that the cross-diffusion destabilizes a uniform equilibrium which is stable for the kinetic and self-diffusion reaction systems. In Section 3, a priori upper bounds and lower bounds for the nonconstant positive solutions of (3) are given. In Section 4, we study nonexistence of nonconstant positive solutions of model (3) when considering only the self-diffusion. Finally, in Section 5, we investigate the existence of the nonconstant positive solutions of (3) by using the Leray-Schauder degree theory, which explains why shrub ecosystem generates patterns.
2. Stability Analysis of the Constant Solutions and
In order to study the stability of the constant steady states and of (1), we first set up the following notation.
Notation 1. Consider the following.(i) are the eigenvalues of in under homogeneous Neumann boundary condition.(ii) is the set of eigenfunctions corresponding to .(iii), where are orthonormal basis of for .(iv) on , and so .
Now, we first consider system (1) without cross-diffusion and introduce the following system: Obviously, system (6) has the same disease-free equilibrium and endemic equilibrium with system (1). From (6), we can get the following theorem.
Proof. (i) For simplicity, throughout this paper, we denote
By a direct calculation, we obtain
The linearization of (6) at can be expressed by
According to Notation 1, is invariant under the operator , and is an eigenvalue of this operator on if and only if it is an eigenvalue of the matrix .
A direct calculation shows that the characteristic polynomial of can be given by It follows from (11) that, if , the corresponding eigenvalues have negative real parts for all , so we know that is locally asymptotically stable.
(ii) Since , it follows from (7) that The linearization of (6) at can be expressed by where the matrix is defined in (10). Direct calculation shows that the characteristic polynomial of is given by where It is easy to see that , , and are positive.
Notice that Then by the Routh-Hurwitz criterion, we know that, for each , all the three roots , , and of characteristic equation have negative real parts. Now we can prove that there exists a positive constant such that In fact, let ; then we have Note that as . It follows that Using the Routh-Hurwitz criterion again, we can see that all the three roots , , and of equation have negative real parts. Thus, there exists a positive constant such that
By continuity, we know that there exists such that the three roots , , and of satisfy which implies that Let then , and (17) holds for . Thus the proof is completed by Theorem of Henry .
Remark 2. From Theorem 1, we can see that if only the free diffusion is introduced to the corresponding ODE system of (1), the uniform positive stationary solution is also locally stable, which means that only self-diffusion cannot induce Turing instability.
We now consider the effect of the cross-diffusion and introduce the following theorem, which give the necessary conditions for the existence of nonconstant positive solution of system (3).
Theorem 3. Consider the following.(i)If and , the disease-free equilibrium of system (1) is locally asymptotically stable.(ii)Assume that (5) holds and in (1). Suppose that and , where is given in (31). If and for some , where and are defined in (34), there exists a positive constant such that the uniform stationary solution of (1) is unstable when .
Proof. (i) For simplicity, we denote that . Then the linearized system of system (1) at is
By some calculations, the characteristic polynomial of can be given by
It is easy to see that, if , all the corresponding eigenvalues of have negative real parts for all , which implies that is locally asymptotically stable.
(ii) The linearized system of system (1) at is where By some calculations, the characteristic polynomial of can be given by where Let , , and be the three roots of ; then . In order to have at least one , it is sufficient to prove that .
In the following we will find out the conditions such that . Let and let , , and be the three roots of with . Notice that and . Then . Thus, one of the three roots , , and is real and negative, and the product of the other two is positive.
Consider the following limits: It is easy to see that and .
Note that It follows that equation has two strictly positive solutions when the following conditions hold: By a continuity argument, we know that, when is large enough, is real and negative, and and are real and positive as . Furthermore, we have So there exists a positive number such that, when , the following hold: Since and for some , we have when . Thus we know that , and the proof is completed.
3. A Priori Estimates to the Positive Solution of (3)
Lemma 4 (maximum principle). Suppose that .(i)Assume that and satisfiesIf , then .(ii)Assume that and satisfiesIf , then .
Next, we state the second lemma which is due to Lin et al. .
Lemma 5 (Harnack inequality). Assume that . Let and satisfy Then there exists a positive constant , depending only on such that
Our results are the following theorems.
Theorem 6 (upper bounds). Any positive solution of (3) satisfies where
Proof. Let such that . Then by Lemma 4, it is clear that
Define ; then satisfies
Let such that . Then, by Lemma 4, we can get
So, by the definition of , we have
Let ; then . Define ; then satisfies Let such that . Then, by using Lemma 4 again, we can obtain which implies It follows that Then we obtain the three upper bounds in (40).
Theorem 7. There exist three positive constants (depending on , ), (depending on ), and (depending on , , ) such that any positive solution of (3) satisfies
Proof. It is easy to see that satisfies where and . By (40), we know that So by Lemma 5, we know that the first two inequalities of (52) hold. Define ; we have where . By (40), we know that Then Lemma 5 yields for some positive constant , and The proof is completed.
Theorem 8 (lower bounds). Let , , , , and be fixed positive constants. Assume that where and are given in (47) and (51). Then there exists a positive constant such that, when and , any positive solution of (3) satisfies
Proof. Suppose that (59) fails. Then there exist sequences with and such that the corresponding positive solutions of (3) satisfy
By a direct application of the maximum principle to the first equation of (3), we can obtain . Integrating by parts, we obtain that
for . By the standard regularity theorem for the elliptic equations, we know that there exists a subsequence of , which we will still denote by , and three nonnegative functions such that
By (60), we know that
Furthermore, we assume that . Let in (61); we obtain
Now, we consider the following three cases, respectively.
Case 1 (). Note that , as . Then we know that Integrating the differential equation for over by parts, we have which is a contradiction.
Case 2 (, on ). By using Hopf boundary lemma, we know on . Then and satisfy the following equation: Let . It follows from Lemma 4 and (67) that that is, By using the assumption , we know that Integrating the differential equation for over by parts, we have which is a contradiction.
Case 3 (, , and on ). By using Hopf boundary lemma, we know and on .
Then and satisfy the following equation: Let . It follows from Lemma 4 and (72) that that is, By using the assumption , we know that Integrating the differential equation for over by parts, we have which is a contradiction. The proof is completed.
4. Nonexistence of Nonconstant Positive Solution of System (3) without Cross-Diffusion
In order to discuss the effect of cross-diffusion on the existence of nonconstant positive solution of system (3), we first give a nonexistence result when the cross-diffusion term is absent, which shows that the cross-diffusion coefficients do play important roles. The mathematical technique to be employed here is the energy method.
Theorem 9. Suppose that and , where is given in Notation 1. There exist positive constants and , depending on , , such that (3) has no nonconstant positive solution provided that and . Furthermore, one has
Proof. Assume that is a positive solution of (3) with . Multiplying the th equation of (3) by and integrating the results over by parts, we have Then it follows from (78) that By Cauchy inequality with , we can get from (79) that On the other hand, applying Poincaré inequality, we know that Then, by assumption, we can choose a sufficiently small positive constant such that So by taking we can conclude that, when , (3) has only the positive constant solution for . The proof is completed.
5. Existence of Nonconstant Positive Solution of System (3)
From Theorem 9 we know that, when the cross-diffusion is absent, (3) has no nonconstant positive solution under some conditions. In the following, we will discuss the effect of cross-diffusion on the existence of nonconstant positive solution of system (3) for certain values of diffusion coefficient , while the other parameters are fixed.
Our main findings are the following theorem, which shows that the presence of cross-diffusion creates nonhomogeneous solution.
Theorem 10. Let , , and be fixed and satisfy (33) and (58), and let and be defined in (34). If and for some and the sum is odd, then there exists a positive constant such that, if , (3) admits at least one nonconstant positive solution.
In order to prove the above theorem by using Leray-Schauder theory, we start with some preliminary results. Throughout this section, Notation 1 and defined in Section 2 will be used again. Define the set
where and is given in Theorem 8. Then we will look for nonconstant positive solutions of (3) in the set . Let . Then (3) can be written as
Noting that the determinant of is positive for all , we know that exists and is positive. Then, is a positive solution to (85) if and only if
where is the inverse of in with the no-flux boundary condition. As is a compact perturbation of the identity operator, for any , the Leray-Schauder degree deg is well defined if on . Note that
If is invertible, the index of at