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.

1. Introduction

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 [5] 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 [6] 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 [7].

Incorporating the sex of prey in a classical Lotka-Volterra model, Liu et al. [1] considered the following sex-structure model:

()

where u, v, and w are the population densities of the male prey, the female prey, and the predator species respectively. The parameters D₁, D₂, and D₃ are their mortality rates, b₁ and b₂ are the birth rates of the male prey and the female prey, c₁ and c₂ are the predation rate and the conversion rate of the predators, and k and c₃ are the intraspecific competition rates of the prey and predators. All the parameters in model (1.1) are positive.

If β = b₂ − D₂ > 0, then two obvious nonnegative equilibria of model (1.1) are u₀ = (0,0, 0) and u₁ = (u₁, v₁, 0), 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 R plays an important role in determining the stability of u₁ and

[1]. To be precise, u₁ is locally asymptotically stable if R < 0, while

is locally asymptotically stable if R > 0. This shows that a uniform coexistence state exists and is stable when the intrinsic growth rate β of the female prey is larger than the critical value kD₃/c₂.

Taking account of the inhomogeneous distribution of the prey and the predator in different spatial locations within a fixed bounded domain Ω at any given time, and the natural tendency of each species to diffuse to areas of smaller population concentration, Liu and Zhou in [8] investigated the following weakly coupled reaction-diffusion system:

()

where η is the outward unit normal vector of the boundary ∂Ω which is smooth, ∂_η = ∂/∂_η. The homogeneous Neumann boundary condition indicates that the predator-prey system is self-contained with zero population flux across the boundary. The constants d₁, d₂, and d₃, called diffusion coefficients, are positive, and the initial values u₀(x), v₀(x), and w₀(x) are nonnegative smooth functions which are not identically zero. Liu and Zhou in [8] 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 Δ(d₃w + d₄uw) for the predators, where d₄ > 0, 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 [13] 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 d₄ = 0, 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

Let u = (u, v, w) ^T, Φ(u) = (d₁u, d₂v, d₃w + d₄uw) ^T, and G(u) = (G₁(u), G₂(u), G₃(u)) ^T. Then the stationary problem of (1.5) can be written as

()

In this section, we study the linearization of (2.1) at

and calculate the fixed point index.

Similar to [14, 15], let 0 = μ₁ < μ₂ < μ₃ < μ₄⋯ be the eigenvalues of the operator −Δ on Ω with the homogeneous Neumann boundary condition, and let E(μ_i) be the eigenspace corresponding to μ_i in H¹(Ω). Let {ϕ_ij : j = 1,2, …, dim E(μ_i)} be the orthonormal basis of E(μ_i), X = [H¹(Ω)] ³, and X_ij = {cϕ_ij : c ∈ ℝ³}. Then

()

Let

, Y⁺ = {u ∈ Y : u, v, w > 0 on

, and B(C) = {u ∈ Y : C⁻¹ < u, v, w < C on

for C > 0. Since det Φ_u(u) = d₁d₂(d₃ + d₄u) > 0 for all nonnegative u,

exists and

is positive. Hence, u is a positive solution to (2.1) if and only if

()

where (I − Δ) ⁻¹ is the inverse of I − Δ under homogeneous Neumann boundary conditions.

Further, we note that

and λ is an eigenvalue of

if and only if, for some i ≥ 1, it is an eigenvalue of the matrix

()

Writing

()

we see that if H(μ_i) ≠ 0, then for each integer 1 ≤ j ≤ dim E(μ_i), the number of negative eigenvalues of

on X_ij is odd if and only if H(μ_i) < 0. As a consequence, we have the following proposition.

Proposition 2.1 (see [16].)Suppose that, for all i ≥ 1, H(μ_i) ≠ 0. Then

()

where

()

To facilitate our computation of

, we need to determine the sign of H(μ_i). 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 d₄, we will concentrate on the dependence of H(μ_i) on d₄. 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 𝒞 on d₄. Let , , and be the three roots of 𝒞(d₄; μ) = 0 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, a₁ < 0 if

()

In the following, we restrict our attention to

. In this range, a₁ < 0 and C₁(d₄) < 0 for all sufficiently large d₄. Notice that

()

and a₁ < 0 < a₃. A continuity argument shows that, when d₄ is large,

is real and negative. Furthermore, as

and

are real and positive, and

()

Thus we have the following proposition.

Proposition 2.2. Assume that (H1) and (H2) hold. Then there exists a positive number such that, when , the three roots , , of 𝒞(d₄; μ) = 0 are all real and satisfy (2.13). Moreover, for all ,

()

Remark 2.3. Proposition 2.2 gives a criterion for the instability of when and the cross-diffusion coefficient d₄ is large enough. We further check conditions (H1) and (H2). Let the parameters d₁, d₂, d₃, b₁, D₁, k, c₁, c₂, and c₃ be fixed. Condition (H1) is equivalent to β > kD₃/c₂, and condition (H2) is equivalent to β > 2kc₁D₃/γ₁ for some γ₁ : = c₁c₂ − kc₃ > 0. Notice that 2c₁/γ₁ > 1/c₂, so there exists an unbounded region , such that for any (D₃, β) ∈ U₁, is an unstable equilibrium with respect to (1.5) when and the cross-diffusion coefficient d₄ 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 d₁, d₂, and d₃ 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 [17].)Let a and b be positive constants. Assume that φ, ψ ∈ C¹([a, +∞)), ψ(t) ≥ 0, and φ is bounded from below. If φ′(t)≤−bψ(t) and ψ′(t) is bounded in [a, +∞), then lim _t→∞ψ(t) = 0.

Theorem 3.2. Let the parameters d₁, d₂, d₃, b₁, D₁, D₃, k, c₁, c₂, c₃, and β be fixed positive constants that satisfy (H1) and

()

Let (u, v, w) be a positive solution of (1.4). Then

()

Proof. Notice from [8] that is uniformly and locally asymptotically stable in the sense of [18]. We only need to prove the global stability of . Define

()

where

. Obviously, V₁(u, v, w) is nonnegative and V₁(u, v, w) = 0 if and only if

. The time derivative of V₁(u, v, w) 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 δ such that

, we have

()

Thus,

()

Similarly to [19, Theorem 2.1], we can prove that the solution (u, v, w) is bounded, and so are the derivatives of

by the equations in (1.4). Using Lemma 3.1, we have

()

By the fact that V₁(u, v, w) is decreasing for t ≥ 0, 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 γ₂ < 0, it is easy to verify that −c₁/γ₂ > k/c₂. Hence, there exists an unbounded region

()

such that for any (D₃, β) ∈ U₂,

is the unique positive steady state with respect to (1.4).

Remark 3.4. From Remarks 2.3 and 3.3, there exists an unbounded region

()

such that for any (D₃, β) ∈ U₃, cross-diffusion can destabilize the uniform equilibrium

of (1.5) when

and d₄ is sufficiently large.

4. A Priori Estimates

In the following, the generic constants C, C_*, and so forth, will depend on the domain Ω and the dimension N. However, as Ω and the dimension N are fixed, we will not mention the dependence explicitly. Also, for convenience, we will write Λ instead of the collective constants (b₁, D₁, D₃, c₁, c₂, c₃, k, β). The main purpose of this section is to give a priori positive upper and lower bounds for the positive solutions to (2.1) when R > 0. For this, we will cite the following two results.

Lemma 4.1 (Harnack’s inequality [20]). Let be a positive solution to Δw(x) + c(x)w(x) = 0, where , satisfying the homogeneous Neumann boundary condition. Then there exists a positive constant C_* which depends only on ∥c∥_∞ such that .

Lemma 4.2 (maximum principle [21]). Let g ∈ C(Ω × ℝ¹) and , j = 1,2, …, N.

(i)
If satisfies
()
and , then g(x₀, w(x₀)) ≥ 0.
(ii)
If satisfies
()
and , then g(x₀, w(x₀)) ≤ 0.

Theorem 4.3 (upper bound). Let d and d^* be two fixed positive constants. Assume that d_i ≥ d, i = 1,2, 3, and 0 ≤ d₄ ≤ d^*. Then every possible positive solution (u, v, w) of (2.1) satisfies

()

Proof. A direct application of the maximum principle to (2.1) gives v ≤ β/k on . Let . Using the maximum principle again, we have b₁v(x₀) ≥ u(x₀)[D₁ + ku(x₀) + kv(x₀) + c₁w(x₀)]. Thus, ku(x₀)v(x₀) ≤ b₁v(x₀) and u(x₀) ≤ b₁/k.

Define φ = d₃w + d₄uw; then, φ satisfies

()

Let

. By Lemma 4.2, we have

()

It follows that

()

Hence,

()

for any d₃ ≥ d and 0 ≤ d₄ ≤ d^*.

Turning now to the lower bound, we first need some preliminary results.

Lemma 4.4. Let d_ij be positive constants, i = 1,2, 3,4, j = 1,2, …, and let (u_j, v_j, w_j) be the corresponding positive solution of (2.1) with d_i = d_ij. If (u_j, v_j, w_j)→(u^*, v^*, w^*) uniformly on as j → ∞ and (u^*, v^*, w^*) is a constant vector, then (u^*, v^*, w^*) must satisfy

()

Moreover, if u^*, v^*, and w^* are positive constants, then

Proof. It is easy to see that for all j,

()

If G₁(u^*, v^*, w^*) > 0, then G₁(u_j, v_j, w_j) > 0 when j is large since (u_j, v_j, w_j)→(u^*, v^*, w^*). This is impossible. Similarly, G₁(u^*, v^*, w^*) < 0 is impossible. Therefore, G₁(u^*, v^*, w^*) = 0. The same argument shows that β − ku^* − kv^* − c₁w^* = 0 and −D₃ + c₂u^* + c₂v^* − c₃w^* = 0. Consequently,

Lemma 4.5. The system

()

has a unique positive constant steady state (u₁, v₁) which is globally asymptotically stable, where u₁ and v₁ are given by (1.2).

Proof. Let

()

where ρ = (b₁ − ku₁)/k = b₁(b₁ + D₁)/[k(b₁ + D₁ + β)] > 0, and let (u, v) be a positive solution of (4.10). Then a direct computation gives

()

and dV₂/dt = 0 holds if and only if (u, v) = (u₁, v₁). By Lemma 3.1, we can conclude that (u₁, v₁) is globally asymptotically stable.

Theorem 4.6 (lower bound). Let d and d^* be two fixed positive constants. Assume that d_i ≥ d, i = 1,2, 3, and 0 ≤ d₄ ≤ d^*. Then there exists a positive constant C = C(Λ, d, d^*), such that every possible positive solution (u, v, w) of (2.1) satisfies

()

Proof. If the conclusion does not hold, then there exists a sequence with d_1j, d_2j, d_3j ≥ d and 0 ≤ d_4j ≤ d^* such that the corresponding positive solution (u_j, v_j, w_j) of (2.1) satisfies

()

Moreover, we assume that d_ij → d_i ∈ [d, ∞] for i = 1,2, 3, and d_4j → d₄ ∈ [0, d^*]. By Theorem 4.3 and the standard regularity theory for the elliptic equations, we may also assume that (u_j, v_j, w_j)→(u, v, w) in

for some nonnegative functions u, v, w. It is easy to see that (u, v, w) also satisfies estimate (4.3), and min

. Moreover, we observe that, if d₁, d₂, d₃ < ∞, then (u, v, w) satisfies (2.1).

Next we derive a contradiction for all possible cases.

Firstly, we consider the case d₁, d₂, d₃ < ∞.

(1)
In view of (2.1), implies v = 0 on from the Harnack inequality. In this case, by the strong maximum principle and the Hopf boundary lemma, it follows that u = w = 0 on . This is a contradiction to Lemma 4.4. Thus, .
(2)
If , we denote . By the maximum principle we have b₁v(x₀) ≤ u(x₀)[D₁ + ku(x₀) + kv(x₀) + c₁w(x₀)] = 0, and so . This is a contradiction to . Thus .
(3)
If , let φ = d₃w + d₄uw. Then and φ satisfies

()

The Harnack inequality shows that

implies φ = 0 on

. Hence, w = 0 on

. From Lemma 4.5, we have (u, v) = (u₁, v₁). 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

and

. Moreover,

satisfies

()

Since

, by the strong maximum principle and the Hopf boundary lemma, we find that

. Applying the maximum principle again, we have −D₃ + c₂u₁ + c₂v₁ = 0. Thus, u₁ + v₁ = D₃/c₂. Noting that u₁ + v₁ = β/k in (1.2), it follows that c₂β − kD₃ = 0, which is a contradiction to the condition R = c₂β − kD₃ > 0.

Next, we consider the remaining cases.

Integrating by parts, we obtain that

()

for j = 1,2, …. Moreover, (u, v, w) satisfies

()

If d₁ = ∞, then u satisfies

()

Hence, u is a constant. If u = 0, from (4.19), we have in turn that v = w = 0. This contradicts Lemma 4.4. So, u is a positive constant and either

If d₂ < ∞. In the case of , similarly to the arguments of (1), we have v = 0 on . This contradicts the first equation of (4.19). Thus and . Note that w satisfies

()

If d₃ < ∞, the Harnack inequality implies that w = 0 on . If d₃ = ∞, then w ia a constant. Since , so w = 0 on . Therefore, by Lemma 4.5, (u, v, w) = (u₁, v₁, 0). Similarly to the arguments of (3), we arrive at c₂β − kD₃ = 0, 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 d₄, with the other parameters d₁, d₂, d₃, b₁, D₁, D₃, k, c₁, c₂, c₃, and β suitably fixed. Our main result is as follows.

Theorem 5.1. Let the parameters d₁, d₂, d₃, b₁, D₁, D₃, k, c₁, c₂, c₃, and β be fixed such that (H1), (H2), and (H3) hold. Let be given by the limit (2.13). If for some n ≥ 2 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 θ ∈ [0,1], define Φ(θ; u) = (d₁u, d₂v, d₃w + θd₄uw) ^T and consider the problem

()

Then u is a positive nonconstant solution of (2.1) if and only if it is such a solution of (5.2) for θ = 1. It is obvious that

is the unique constant positive solution of (5.2) for any 0 ≤ θ ≤ 1. As we observed in Section 2, for any 0 ≤ θ ≤ 1, u is a positive solution of (5.2) if and only if

()

It is obvious that F(1; u) = F(u). Theorem 3.2 shows that F(0; u) = 0 has only the positive solution

in Y⁺. By a direct computation,

()

In particular,

()

where D = diag (d₁, d₂, d₃) 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 i ≥ 1, 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 C such that, for all 0 ≤ θ ≤ 1, the positive solutions of (2.1) satisfy 1/C < u, v, w < C. Therefore, F(θ; u) ≠ 0 on ∂B(C) for all 0 ≤ θ ≤ 1. By the homotopy invariance of the topological degree,

()

On the other hand, by our supposition, both equations F(1; u) = 0 and F(0; u) = 0 have only the positive solution

in B(C). 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 d₄ > 0 is small.

Theorem 6.1. If the parameters d₁, d₃, d₄, b₁, D₁, D₃, k, c₁, c₂, c₃, and β satisfy (H1), (H3), and

()

then the problem (2.1) has no nonconstant positive solution.

Proof. Assume that (u, v, w) 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 0 = −I₃ − I₄, where

()

Applying (H3) and (H4), it is easy to prove that I₃ ≥ 0 and I₄ ≥ 0. This implies that

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 d₁ and d₃ is sufficiently large; that is, unlimitedly increasing one of the diffusion rates d₁ and d₃ will eventually wipe out all nonconstant solutions of (2.1). However, Theorem 6.1 does not tell us the effect of the diffusion rate d₂ on the stationary problem (2.1). Using the similar arguments in Section 2, we can find that d₂ does not cause instability of . Therefore, we conjecture that the problem (2.1) has no nonconstant positive solutions if d₂ is sufficiently large.

Remark 6.3. Theorems 5.1 and 6.1 seem to indicate that diffusion tends to suppress pattern formation, while cross-diffusion seems to help create patterns.

7. Discussion

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 [22] in the predator-prey model

()

where d_ij, i, j = 1, 2, 3, i ≠ j, are cross-diffusion coefficients. Using the method in Section 2 of this paper, we have investigated the stability of

for each d_ij, respectively. We found that either d₁₃ or d₂₃ does not cause instability of

, while each of the other d_ij can induce the instability of

. For d₃₂ ≠ 0, 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 d₁₂ ≠ 0 or d₂₁ ≠ 0, 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 [23], 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.

Acknowledgments

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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Instability Induced by Cross-Diffusion in a Predator-Prey Model with Sex Structure

Abstract

1. Introduction

2. Local Stability Analysis and Fixed Point Index of

3. Global Asymptotic Stability of for (1.4)

4. A Priori Estimates

5. Existence of Stationary Patterns for the Model (1.5)

6. Nonexistence of Nonconstant Positive Solution of (2.1)

7. Discussion

Acknowledgments

References

Citing Literature

References

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Instability Induced by Cross-Diffusion in a Predator-Prey Model with Sex Structure

Abstract

1. Introduction

2. Local Stability Analysis and Fixed Point Index of

3. Global Asymptotic Stability of for (1.4)

4. A Priori Estimates

5. Existence of Stationary Patterns for the Model (1.5)

6. Nonexistence of Nonconstant Positive Solution of (2.1)

7. Discussion

Acknowledgments

References

Citing Literature

References

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