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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 240432, 18 pages
Instability Induced by Cross-Diffusion in a Predator-Prey Model with Sex Structure
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Received 26 July 2011; Accepted 15 January 2012
Academic Editor: Junjie Wei
Copyright © 2012 Shengmao Fu and Lina Zhang. 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.
In this paper, we consider a cross-diffusion predator-prey model with sex structure. We prove that cross-diffusion can destabilize a uniform positive equilibrium which is stable for the ODE system and for the weakly coupled reaction-diffusion system. As a result, we find that stationary patterns arise solely from the effect of cross-diffusion.
Sex ratio means the comparison between the number of male and female in the species. The sex ratio is generally regarded as 1 : 1. But for wildlife, the sex ratio of species varies with the category, environment condition, community behavior, orientation and heredity, and so forth. The animal’s sex ratio in the different life history stages may vary with different animals. Take a bird as an example: the number of the older males is larger than that of the older females with the increase in age, which is contrary to the case of the mammal where the number of the older females is larger than that of the older males with the increase in age [1, 2]. Sex ratio is the basis of analyzing the dynamic state of different species, the variation of which has a huge influence on the dynamic state of the species [1–7]. Abrahams and Dill  have provided evidence that male and female guppies forage differently in the presence of predators and that sexual differences in the energetic equivalence of the risk of predation exist. Eubanks and Miller  have found that female Gladicosa pulchra (Lycosidae) wolf spiders climb trees significantly more often than males in the presence of forest floor predators. It is found that the sex of voles affects the risk of predation by mammals and female voles are more easily predated than male voles in .
Incorporating the sex of prey in a classical Lotka-Volterra model, Liu et al.  considered the following sex-structure model: where , , and are the population densities of the male prey, the female prey, and the predator species respectively. The parameters , , and are their mortality rates, and are the birth rates of the male prey and the female prey, and are the predation rate and the conversion rate of the predators, and and are the intraspecific competition rates of the prey and predators. All the parameters in model (1.1) are positive.
If , then two obvious nonnegative equilibria of model (1.1) are and , where Moreover, model (1.1) has a positive equilibrium if and only if In this case the positive equilibrium is uniquely given by , where It turns out that plays an important role in determining the stability of and . To be precise, is locally asymptotically stable if , while is locally asymptotically stable if . This shows that a uniform coexistence state exists and is stable when the intrinsic growth rate of the female prey is larger than the critical value .
Taking account of the inhomogeneous distribution of the prey and the predator in different spatial locations within a fixed bounded domain at any given time, and the natural tendency of each species to diffuse to areas of smaller population concentration, Liu and Zhou in  investigated the following weakly coupled reaction-diffusion system: where is the outward unit normal vector of the boundary which is smooth, . The homogeneous Neumann boundary condition indicates that the predator-prey system is self-contained with zero population flux across the boundary. The constants , , and , called diffusion coefficients, are positive, and the initial values , , and are nonnegative smooth functions which are not identically zero. Liu and Zhou in  found that the nonnegative constant steady states have the same stability properties as the ODE model (1.1). Therefore, Turing instability cannot occur for this reaction-diffusion system.
However, in model (1.4), only diffusion of each individual species is taken into account. In some cases, the reality is that the female prey is easily predated because of physiological factor, while the male prey can congregate and form a huge group to protect itself from the attack of the predators [7, 9]; therefore, the predators tend to keep away from their male prey. Similarly as in [10–12], we model this by the cross-diffusion term for the predators, where , called the cross-diffusion coefficient. Thus, the cross-diffusion system that we will study is the following: To our knowledge, only a few works investigated the effect of cross-diffusion on population structure and dynamics in the above model. Recently, H. Xu and S. Xu in  investigated the global existence of solutions for the corresponding full SKT model of (1.5) when the space dimension is less than ten.
An interesting feature of (1.5) is that the interaction between the predators and the male prey gives rise to a cross-diffusion term. The resulting mathematical model is a strongly coupled system of three equations which is mathematically much more complex than those considered earlier. In this paper, we will show that cross-diffusion can destabilize the uniform equilibrium which is stable for models (1.1) and (1.4). Moreover, we will demonstrate that the nonlinear dispersive force can give rise to a spatial segregation of these species.
Our paper is organized as follows. In Section 2, we analyze the local stability of for (1.5) and calculate the fixed point index, which is important for our later discussions on the existence of nonconstant positive steady states. In Section 3, we prove global asymptotic stability of with , that is, when no cross-diffusion occurs in the model. This implies that cross-diffusion has a destabilizing effect. In Section 4, we establish a priori upper and lower bounds for all possible positive steady states of (1.5). In Section 5, we study the global existence of nonconstant positive steady states of (1.5) for suitable values of the parameters. This is done by using the Leray-Schauder degree theory and the results obtained in Sections 2, 3, and 4. In Section 6, we discuss the nonexistence of nonconstant positive steady states of (1.5). In the last section, we give a brief discussion about our model.
2. Local Stability Analysis and Fixed Point Index of
Similar to [14, 15], let be the eigenvalues of the operator on with the homogeneous Neumann boundary condition, and let be the eigenspace corresponding to in . Let be the orthonormal basis of , , and . Then
Let , on , and on for . Since for all nonnegative , exists and is positive. Hence, is a positive solution to (2.1) if and only if
where is the inverse of under homogeneous Neumann boundary conditions.
Further, we note that and is an eigenvalue of if and only if, for some , it is an eigenvalue of the matrix
Writing we see that if , then for each integer , the number of negative eigenvalues of on is odd if and only if . As a consequence, we have the following proposition.
Proposition 2.1 (see ). Suppose that, for all , . Then where
To facilitate our computation of , we need to determine the sign of . In particular, as the aim of this paper is to study the existence of stationary patterns of (2.1) with respect to the cross-diffusion coefficient , we will concentrate on the dependence of on . At this point, we note that . Since is positive, we will need only to consider . By we have where
Notice that ; thus . We consider the dependence of on . Let , , and be the three roots of with . It follows that from . Thus, among , , at least one is real and negative, and the product of the other two is positive.
Consider the following limits:
Therefore, if In the following, we restrict our attention to . In this range, and for all sufficiently large . Notice that
and . A continuity argument shows that, when is large, is real and negative. Furthermore, as , and are real and positive, and Thus we have the following proposition.
Remark 2.3. Proposition 2.2 gives a criterion for the instability of when and the cross-diffusion coefficient is large enough. We further check conditions (H1) and (H2). Let the parameters , , , , , , , , and be fixed. Condition (H1) is equivalent to , and condition (H2) is equivalent to for some . Notice that , so there exists an unbounded region , such that for any , is an unstable equilibrium with respect to (1.5) when and the cross-diffusion coefficient is sufficiently large.
3. Global Asymptotic Stability of for (1.4)
The aim of this section is to prove Theorem 3.2 which shows that model (1.4) has no nonconstant positive steady state no matter what the diffusion coefficients , , and are; in other words, diffusion alone (without cross-diffusion) cannot drive instability and cannot generate patterns for this predator-prey model. For this, we will make use of the following result.
Lemma 3.1 (see ). Let and be positive constants. Assume that , , and is bounded from below. If and is bounded in , then .
Proof. Notice from  that is uniformly and locally asymptotically stable in the sense of . We only need to prove the global stability of . Define where , . Obviously, is nonnegative and if and only if . The time derivative of for the system (1.4) satisfies where If the matrix is positive definite, then the quadratic form is positive definite. A direct calculation shows that the matrix is positive definite if (H3) holds. Meanwhile, for every such that , we have Thus, Similarly to [19, Theorem 2.1], we can prove that the solution is bounded, and so are the derivatives of by the equations in (1.4). Using Lemma 3.1, we have By the fact that is decreasing for , it is obvious that is globally asymptotically stable, and the proof of Theorem 3.2 is completed.
Remark 3.3. Notice that condition (H3) is equivalent to If , it is easy to verify that . Hence, there exists an unbounded region such that for any , is the unique positive steady state with respect to (1.4).
4. A Priori Estimates
In the following, the generic constants , , and so forth, will depend on the domain and the dimension . However, as and the dimension are fixed, we will not mention the dependence explicitly. Also, for convenience, we will write instead of the collective constants . The main purpose of this section is to give a priori positive upper and lower bounds for the positive solutions to (2.1) when . For this, we will cite the following two results.
Lemma 4.1 (Harnack’s inequality ). Let be a positive solution to , where , satisfying the homogeneous Neumann boundary condition. Then there exists a positive constant which depends only on such that .
Lemma 4.2 (maximum principle ). Let and , . (i)If satisfies and , then .(ii) If satisfies and , then .
Theorem 4.3 (upper bound). Let and be two fixed positive constants. Assume that , and . Then every possible positive solution of (2.1) satisfies
Proof. A direct application of the maximum principle to (2.1) gives on . Let . Using the maximum principle again, we have . Thus, and .
Define ; then, satisfiesLet . By Lemma 4.2, we have It follows that Hence, for any and .
Turning now to the lower bound, we first need some preliminary results.
Lemma 4.4. Let be positive constants, , , and let be the corresponding positive solution of (2.1) with . If uniformly on as and is a constant vector, then must satisfy Moreover, if , , and are positive constants, then .
Proof. It is easy to see that for all , If , then when is large since . This is impossible. Similarly, is impossible. Therefore, . The same argument shows that and . Consequently, .
Lemma 4.5. The system has a unique positive constant steady state which is globally asymptotically stable, where and are given by (1.2).
Theorem 4.6 (lower bound). Let and be two fixed positive constants. Assume that , , and . Then there exists a positive constant , such that every possible positive solution of (2.1) satisfies
Proof. If the conclusion does not hold, then there exists a sequence with and such that the corresponding positive solution of (2.1) satisfies
Moreover, we assume that for , and . By Theorem 4.3 and the standard regularity theory for the elliptic equations, we may also assume that in for some nonnegative functions . It is easy to see that also satisfies estimate (4.3), and min. Moreover, we observe that, if , then satisfies (2.1).
Next we derive a contradiction for all possible cases.
Firstly, we consider the case .
(1)In view of (2.1), implies on from the Harnack inequality. In this case, by the strong maximum principle and the Hopf boundary lemma, it follows that on . This is a contradiction to Lemma 4.4. Thus, .(2) If , we denote . By the maximum principle we have , and so . This is a contradiction to . Thus .(3) If , let . Then and satisfies The Harnack inequality shows that implies on . Hence, on . From Lemma 4.5, we have . Define ; then, satisfies Similarly to the above, we can prove that there exists a subsequence of , denoted by itself, and a nonnegative function , such that in and . Moreover, satisfies Since , by the strong maximum principle and the Hopf boundary lemma, we find that on . Applying the maximum principle again, we have . Thus, . Noting that in (1.2), it follows that , which is a contradiction to the condition .
Next, we consider the remaining cases.
Integrating by parts, we obtain thatfor . Moreover, satisfies
If , then satisfiesHence, is a constant. If , from (4.19), we have in turn that . This contradicts Lemma 4.4. So, is a positive constant and either or .
If . In the case of , similarly to the arguments of (1), we have on . This contradicts the first equation of (4.19). Thus and . Note that satisfies
If , the Harnack inequality implies that on . If , then ia a constant. Since , so on . Therefore, by Lemma 4.5, . Similarly to the arguments of (3), we arrive at , which is a contradiction.
Similarly, we can derive contradictions for all the other cases.
5. Existence of Stationary Patterns for the Model (1.5)
In this section we discuss the existence of nonconstant positive solutions to (2.1). These solutions are obtained for large cross-diffusion coefficient , with the other parameters , , , , , , , , , , and suitably fixed. Our main result is as follows.
Theorem 5.1. Let the parameters , , , , , , , , , , and be fixed such that (H1), (H2), and (H3) hold. Let be given by the limit (2.13). If for some and the sum is odd, then there exists a positive constant such that (2.1) has at least one nonconstant positive solution for .
Proof. By Proposition 2.2 and our assumption on , there exists a positive constant such that (2.14) holds if , and
We will prove that for any , (2.1) has at least one nonconstant positive solution. The proof, which is by contradiction, is based on the homotopy invariance of the topological degree.
Suppose on the contrary that the assertion is not true for some . In the following we fix .
For , define and consider the problem Then is a positive nonconstant solution of (2.1) if and only if it is such a solution of (5.2) for . It is obvious that is the unique constant positive solution of (5.2) for any . As we observed in Section 2, for any , is a positive solution of (5.2) if and only if It is obvious that . Theorem 3.2 shows that has only the positive solution in . By a direct computation, In particular, where and From (2.5) and (2.9) we see that In view of (2.14) and (5.1), it follows that Therefore, zero is not an eigenvalue of the matrix for all , and Thanks to Proposition 2.1, we have Similarly, we can easily show that Now, by Theorems 4.3 and 4.6, there exists a positive constant such that, for all , the positive solutions of (2.1) satisfy . Therefore, on for all . By the homotopy invariance of the topological degree, On the other hand, by our supposition, both equations and have only the positive solution in . Hence, by (5.10) and (5.11), we have This contradicts (5.12), and thus we complete the proof of Theorem 5.1.
Remark 5.2. Assume that all the conditions hold in Theorem 5.1. Theorem 3.2 shows that is a globally asymptotically stable equilibrium for the system (1.4). However, Theorem 5.1 implies that the cross-diffusion system (1.5) has at least one nonconstant positive steady state. Our results demonstrate that stationary patterns can be found due to the emergence of cross-diffusion.
6. Nonexistence of Nonconstant Positive Solution of (2.1)
In this section, we discuss the nonexistence of nonconstant positive solution of (2.1) when the cross-diffusion coefficient is small.
Proof. Assume that is a positive solution of (2.1). Let , . Multiplying the equations of (2.1) by , , and , respectively, and integrating by parts, as in the proof of Theorem 3.2, we obtain , where Applying (H3) and (H4), it is easy to prove that and . This implies that on and the proof is complete.
Remark 6.2. Theorem 6.1 shows that the problem (2.1) has no nonconstant positive solutions if one of and is sufficiently large; that is, unlimitedly increasing one of the diffusion rates and will eventually wipe out all nonconstant solutions of (2.1). However, Theorem 6.1 does not tell us the effect of the diffusion rate on the stationary problem (2.1). Using the similar arguments in Section 2, we can find that does not cause instability of . Therefore, we conjecture that the problem (2.1) has no nonconstant positive solutions if is sufficiently large.
In this paper, we have introduced a more realistic mathematical model for a diffusive predator-prey system where the prey has a sex structure comprising male and female members. In this model, we model the tendency of the predators to keep away from the male prey by a cross-diffusion. As a result, our model is a strongly coupled cross-diffusion system, which is mathematically more complex than systems used to model sex-structured predator-prey behavior hitherto [1, 8]. What is noteworthy about this model is that, as the cross-diffusion term arises, it is precisely this cross-diffusion that destabilizes the uniform positive equilibrium and gives rise to stationary patterns for the model. Indeed, stationary patterns do not arise for the ODE (spatially independent) model, nor the PDE model without cross-diffusion.
In fact, one can see that this particular cross-diffusion term is also significant from the mathematical point of view. The following system represents the general form of SKT-type cross-diffusion  in the predator-prey model where , , , are cross-diffusion coefficients. Using the method in Section 2 of this paper, we have investigated the stability of for each , respectively. We found that either or does not cause instability of , while each of the other can induce the instability of . For , we can obtain the similar conclusions as Theorem 5.1 by the same mathematical treatment in this paper. Unfortunately, the existence of nonconstant positive steady states has not been obtained when or , because we cannot establish a priori lower bounds for all possible positive steady states of (7.1).
On the other hand, as pointed out in , a Lotka-Volterra-type model can be regarded as a local approximation to a nonlinear system. In the present paper, we only consider the case that the interaction terms on the right-hand side of (1.5) are linear. For the nonlinear case, it will be extremely difficult to analyze positive steady states.
In this paper, we do not discuss the stability and the number of the nonconstant positive solutions. We will consider them in the coming papers.
This work was supported by the National Natural Science Foundation of China (no. 11061031, 11101334), the NSF of Gansu Province (096RJZA118), the Fundamental Research Funds for the Gansu University, and NWNU-KJCXGC-03-47, 61 Foundations.
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