In this paper, we study a strongly coupled reaction-diffusion system which describes two interacting species in prey-predator ecosystem with nonlinear cross-diffusions and Holling type-II functional response. By a linear stability analysis, we establish some stability conditions of constant positive equilibrium for the ODE and PDE systems. In particular, it is shown that Turing instability can be induced by the presence of cross-diffusion. Furthermore, based on Leray-Schauder degree theory, the existence of non-constant positive steady state is investigated. Our results indicate that the model has no non-constant positive steady state with no cross-diffusion, while large cross-diffusion effect of the first species is helpful to the appearance of Turing instability as well as non-constant positive steady state (stationary patterns).
Keywords:cross-diffusion; Holling type-II functional response; Turing instability; non-constant positive steady state; stationary patterns
Let Ω be a bounded domain in with smooth boundary ∂Ω. In this paper, we are interested in a strongly coupled reaction-diffusion equations with Holling type-II functional response
where ν is the unit outward normal to ∂Ω. The two unknown functions and represent the spatial distribution densities of the prey and predator, respectively. The constants , , ( ), K, m and θ are all positive, and , are nonnegative functions which are not identically zero. Moreover, is the diffusion rate of the two species, expresses the self-diffusion effect, is called the cross-diffusion coefficient, K accounts for the carrying capacity of the prey, θ is the death rate of the predator, and m can be regarded as the measure of the interaction strength between the two species. In this model, the prey u and the predator v diffuse with fluxes
respectively. The cross-diffusion terms and can be explained that the prey keeps away from the predator while the predator moves away from a large group of prey. For more detailed biological meaning of the parameters, one can make some reference to [1-3].
The ODE system of (1.1)
has been extensively studied in the existing literature; see, for example, [4-6]. The known results mainly focused on the existence and uniqueness of a limit cycle. In , Rosenzweig argued that enrichment of the environment (larger carrying capacity K) leads to destabilizing of the coexistence equilibrium, which is the so-called paradox of enrichment. Cheng  first proved the uniqueness of limit cycle. Hsu and Shi  discussed the relaxation oscillator profile of the unique limit cycle and found that (1.2) has a periodic orbit if m is larger than a threshold value.
In mathematical biology, the classical prey-predator model (ODE system) reflects only population changes due to predation in a situation where predator and prey densities are not spatially dependent. It does not take into account either the fact that population is usually not homogeneously distributed, or the fact that predators and preys naturally develop strategies for survival. Both of these considerations involve diffusion processes which can be quite intricate as different concentration levels of predators and preys caused by different population movements. Such movements can be determined by the concentration of the same species (diffusion) and that of other species (cross-diffusion). In view of this, Shigesada, Kawasaki and Teramoto first proposed a strongly coupled reaction-diffusion model with Lotka-Volterra type reaction term (SKT model) to describe spatial segregation of interacting population species in one-dimensional space . Since then the two-species SKT competing system and its overall behaviors continue to be of great interest in literature to both mathematical analysis and real-life modeling [7-10]. For the studies on biological models, since each model has rich and interesting properties and often describes complex biological process, it is very difficult to get some general conclusions for a class of mathematical models. So research in mathematical biology has often been performed by investigating a specific model, the focus of which is to discuss the influences of parameters on the behavior of species in the ecosystem. Thus, more and more attention has been recently focused on three or multi-species systems and the SKT model in any space dimension due to their more complicated patterns, and the SKT models with other types of reaction terms have also been proposed and investigated [11-19]. The obtained results mainly relate to the stability analysis of constant positive steady states and the existence of non-constant positive steady states (stationary patterns) [9,10,12-21], Turing instability [22,23], and the global existence of non-negative time-dependent solutions [7,8,11,24].
The role of diffusion in the modeling of many biological processes has been extensively studied. Starting with Turing’s seminal work , diffusion and cross diffusion have been observed as causes of the spontaneous emergence of ordered structures, called patterns, in a variety of nonequilibrium situations. Diffusion-driven instability, also called Turing instability, has also been verified empirically in some chemical and biological models [26-28]. For the system with cross-diffusion, we can know that this kind of cross-diffusion may be helpful to create non-constant positive steady-state solutions for the predator-prey system, for example [9,10,16]. Recently, the authors of  discussed a two-species Holling-Tanner model with simple linear cross-diffusion
and showed that under some parameters the positive equilibrium is stable for a diffusion system while unstable for a cross-diffusion system, which implies that cross-diffusion can induce the Turing instability of the uniform equilibrium. In , Xie investigated a class of strongly coupled prey-predator models with four Holling-type functional responses:
The results indicated that diffusion and cross-diffusion in these models cannot drive Turing instability. However, diffusion and cross-diffusion can still create non-constant positive solutions for the models.
As for reaction-diffusion system of (1.2), the diffusive predator-prey equations with no self- and cross-diffusion ( in (1.1)) under Neumann boundary value conditions have also been investigated (see, for example, [29-33]). Ko and Ryu  obtained some results on the global stability of the constant steady state solutions and the existence of at least one non-constant equilibrium solution. Medvinsky et al. used this model as a simple mathematical model to investigate the pattern formation of a phytoplankton-zooplankton system, and their numerical studies show a rich spectrum of spatiotemporal patterns. The discussion in  shows this system possesses complex spatiotemporal dynamics via a sequence of bifurcation of spatial nonhomogeneous periodic orbits and spatial nonhomogeneous steady state solutions. In , Peng and Shi proved the non-existence of non-constant positive steady state solutions. Recently, the existence, multiplicity and stability of positive solutions for the weakly coupled equations in (1.1) with Dirichlet boundary conditions were investigated in .
From the above introductions, one can learn that few studies have been conducted into the occurrence of Turing instability for a strongly coupled reaction-diffusion system with nonlinear cross-diffusion terms in the literature. Motivated by a series of pioneering works such as [9,10,16], we are interested in the instability induced by cross-diffusion and the stationary patterns of strongly coupled model (1.1). The aim of this paper is to discuss Turing instability and establish the existence of non-constant positive steady states of system (1.1). The methods we employed are the classical linearization method and the Leray-Schauder degree theory. However, while performing a priori estimates and stability analysis, we must try a new method and techniques to solve difficulties caused by nonlinear cross-diffusion terms and . Nonlinear cross-diffusion terms also add complexity of computation of characteristic equations. Moreover, this paper focuses on the influence of nonlinear cross-diffusion terms on the appearance of Turing instability, and the discussion shows that large cross-diffusion coefficient of the first species is helpful to the appearance of Turing instability as well as non-constant positive steady state.
The paper is organized as follows. In Section 2, we discuss the stability of a positive equilibrium point for ODE and PDE systems and then obtain sufficient conditions of the appearance of Turing pattern. The results imply that cross-diffusion has a destabilizing effect, which is helpful to the occurrence of Turing instability. In Section 3, we obtain a priori upper and lower bounds for the positive steady states problem of (1.1) in order to calculate the topological degree. In Section 4, the non-existence of non-constant positive steady state for (1.1) with vanished cross-diffusions is discussed. In Section 5, we establish the global existence of non-constant positive steady state of (1.1) for suitable values of cross-diffusion coefficient and then show that large cross-diffusion effect can create non-constant positive steady states.
2 Turing instability driven by cross-diffusion
Denote . It is known from  that problem (1.1) has a unique positive equilibrium
if and only if
Moreover, problem (1.1) has a trivial equilibrium and a semi-trivial equilibrium .
We first investigate the stability of positive equilibrium for a reaction-diffusion system.
Lemma 2.1Suppose that , . Then the positive equilibrium of (1.1) is uniformly asymptotically stable.
Proof For simplicity, we denote and
Then problem (1.1) can be rewritten as
The linearization of problem (2.2) at the positive equilibrium is
where , . Here
It is easy to verify that if .
Let be a set of eigenpairs for −Δ in Ω with no flux boundary condition, where , and let be the eigenspace corresponding to in , let , , be an orthonormal basis of . Let
Then we can do the following decomposition:
For each , is invariant under the operator . Then problem (2.3) has a non-trivial solution of the form ( is a constant vector) if and only if is an eigenpair for the matrix .
The characteristic equation of the matrix is given by
Obviously, if , then and so . Thus, . It follows from Routh-Hurwitz criterion that the two roots , of have both negative real parts for all .
In order to obtain the local stability of , we need to prove that there exists a positive constant δ such that
Let , then
Notice that as . We can calculate that
By Routh-Hurwitz criterion, the two roots , of have both negative real parts. Then we can conclude that there exists a positive constant such that . By continuity, we see that there exists such that the two roots of satisfy for all . Then for all . Let
Then (2.5) holds true. The theorem is thus proved. □
Similarly, we can also learn, by the proof of Lemma 2.1, a series of stability results about the positive equilibrium for problem (1.1) with different cross-diffusion cases.
Lemma 2.2Suppose that , , . The positive equilibrium of (1.1) is unstable if and
for some , whereas it is uniformly asymptotically stable if .
Lemma 2.3Suppose that , . Then the positive equilibrium of (1.1) is uniformly asymptotically stable for disappeared cross-diffusion.
Lemma 2.4Suppose that , , . Then the positive equilibrium of (1.1) is uniformly asymptotically stable.
Now we consider the case when , . For simplicity, denote
We thus have the following result.
Lemma 2.5Suppose that , , . The positive equilibrium of (1.1) is unstable if , and
for some , whereas it is uniformly asymptotically stable if , or and .
Now we consider the corresponding ODE system. Let be a positive solution of (1.2). It is easy to show that and are both well posed. Similar to the proof of Lemma 2.1, we can get the following stability result.
Lemma 2.6Assume that . The positive equilibrium point of (1.2) is locally asymptotically stable. In particular, is globally asymptotically stable if .
Proof According to the proof of Lemma 2.1, we can easily obtain local asymptotical stability of for ODE system (1.2).
Define the following Lyapunov function:
where ρ is a positive constant to be determined. Obviously, , and if . We compute the derivative of for system (1.2):
It is easy to demonstrate that if . On the other hand, we can choose and then . Then we get
By the Lyapunov-LaSalle invariance principle , is globally asymptotically stable. So the proof of Lemma 2.6 is completed. □
Based on the above discussion, we now can establish some sufficient conditions for the occurrence of Turing instability induced by cross-diffusion. Our main result in this section is the following theorem.
Theorem 2.7Assume that . The stability of the constant equilibrium is stable for the ODE dynamics (1.2) while unstable for the PDE dynamics (1.1) if one of the following two conditions is fulfilled:
(C1) , , and for some ,
(C2) , , , and for some .
Remark 2.8 The Turing instability refers to ‘diffusion driven instability’, i.e., the stability of the constant equilibrium changing from stable for the ODE dynamics, to unstable for the PDE dynamics. Lemma 2.4 and Theorem 2.7 imply that cross-diffusion has a destabilizing effect, which is helpful to the occurrence of Turing instability. Moreover, we can see that sufficiently large cross-diffusion can guarantee and , even and under a proper parameter condition. So large cross-diffusion effect can induce Turing instability.
3 Prior bounds for the positive steady states of the PDE system
The corresponding steady state problem of (1.1) is
In this section, we give a priori positive upper and lower bounds for positive solutions to the elliptic system (3.1). For this, we need to make use of the following two results.
Lemma 3.1 (Maximum principle )
Let and , .
(1) If satisfies
and , then .
(2) If satisfies
and , then .
Lemma 3.2 (Harnack inequality )
Let be a positive solution to with subject to the homogeneous Neumann boundary condition. Then there exists a positive constant such that
In this paper, we assume that the classical solution is in . The results of upper and lower bounds can be stated as follows.
Theorem 3.3 (Upper bound)
For any positive classical solutionwof (3.1), there exist two positive constants , , such that
Proof Problem (3.1) can be rewritten as
Let be a point such that . Applying Lemma 3.1 to the first equation in (3.2) yields and
Denote . Let be a point such that . Since
from Lemma 3.1, we can obtain and
This completes the proof. □
Theorem 3.4 (Lower bound)
Suppose that . For any positive classical solutionwof (3.1), there exists a positive constant , , such that
Proof Since the inequalities
Harnack inequality in Lemma 3.2 shows that there exist two positive constants , , such that
By the same way, we have
On the other hand, by integrating the second equation in (3.1), we have , which implies that there exists a point such that , i.e.,
Now we need to prove v has a positive lower bound. Suppose on the contrary that does not hold. Then there exists a sequence with such that the corresponding nonnegative solution of (3.1) with satisfies
We may assume, by passing to a subsequence if necessary, that as ,
By (3.1) and the regularity theory of elliptic equations, we can conclude that for any . Then, for , by Sobolev embedding theorem, we have . It follows, by passing to a subsequence if necessary, that converges uniformly to the nonnegative function in as . Then
By (3.3), we note that . Moreover, since
Multiplying the above equation by and then integrating the resulting equation over Ω, we can obtain
Thus, and then as . At the same time, we consider the integral equation
However, since as , we can conclude that is positive or negative as n is large enough. It is a contradiction. □
4 Non-existence of non-constant positive steady states
The aim of this section is to investigate the non-existence of non-constant positive steady states of problem (1.1) with no cross-diffusion.
Theorem 4.1Let , . Then there exists a positive constant such that problem (1.1) has no non-constant positive steady state provided that .
Proof For any , denote . Assume that is a positive solution of (3.1) with . Multiplying the two equations in (3.1) by and , respectively, and integrating the results over Ω by parts, we can obtain
where ϵ is the arbitrary small positive constant arising from Young’s inequality.
Similar to the proof of Lemma 3.3 and Lemma 3.4, we can conclude that
It follows from the Poincaré inequality that
we may choose ϵ sufficiently small and sufficiently large such that , . Thus, we can conclude that , . Then the proof is completed. □
5 Existence of non-constant positive steady states
In this section, we shall use the Leray-Schauder degree theory to develop a general setting to establish the existence of stationary patterns for system (1.1). Denote
where C is a positive constant whose existence is guaranteed by Theorems 3.3 and 3.4.
Since the determinant is positive for all non-negative w, exists and is positive, thus w is a positive solution of system (3.1) if and only if
where is the inverse of in X, subject to the homogeneous Neumann boundary condition. Since is a compact perturbation of the identity operator, the Leray-Schauder degree is well defined if for any . Further, we calculate
We recall that if does not have any pure imaginary or zero eigenvalue, the index of the operator Ψ at the fixed point is defined as , where r is the total number of eigenvalues of with negative real parts (counting multiplicities). Then the degree is equal to the sum of the indexes over all solutions to equation in , provided that on .
In order to calculate r, we employ the eigenspaces of −Δ. Using the decomposition (2.4) we investigate the eigenvalues of matrix . First, we know is invariant under for each and each , i.e., , for any . Hence, μ is an eigenvalue of on if and only if it is an eigenvalue of the matrix
So is invertible if and only if, for any , the matrix is non-singular. Denote
We notice that if , then for each , the number of negative eigenvalues of on is odd if and only if . In conclusion, we have the following result.
Lemma 5.1Assume that, for each , the matrix is non-singular. Then
According to the above lemma, we should consider the sign of in order to calculate . Since
and , we only need to consider the sign of . A direct calculation shows
Let and be the two roots of with . Then
So the signs of and are identical. Perform the following limits:
Then we have the following result.
Lemma 5.2Assume that . Then there exists a positive constant such that for any , the two roots , of are all real and satisfy
Moreover, we can conclude that
Now we establish the global existence of non-constant positive solution to (3.1) with respect to the cross-diffusion coefficients , as the other parameters are all fixed positive constants.
Theorem 5.3Assume that the parameters , , , , , K, Mandθare all fixed and satisfy , and
Let be given by the limit in (5.2). If for some and the sum is odd, then there exists a positive constant such that, if , problem (1.1) has at least one non-constant positive steady state.
Proof By Lemma 5.2, there exists a positive constant such that, if , (5.3) holds and
We will prove that for any , (1.1) has at least one non-constant positive steady state. The proof will be fulfilled by contradiction. Suppose on the contrary that the assertion is not true for some . Let be fixed as .
For , define
and then consider the problem
Then w is a non-constant positive steady state of (1.1) if and only if it is a non-constant positive solution of problem (5.6) for . It is obvious that is the unique constant positive solution of (5.6) for any . From (5.1), we know that for any , w is a positive solution of problem (5.6) if and only if
It is obvious that . Theorem 4.1 indicates that only has the constant positive solution in . A direct calculation shows that
Here . Moreover, we already know that
For , by (5.3), (5.5) and (5.7), we have
Thus, 0 is not an eigenvalue of the matrix for all , and
is odd. It follows from Lemma 5.1 that
For , we have
from Theorem 4.1.
On the other hand, by Theorems 3.3 and 3.4, there exists a positive constant M such that for all , the positive solution of (5.6) satisfies and on . By the homotopy invariance of the topological degree, we can obtain
Now, by our supposition, both equations and have only the constant positive solution in . Thus, by (5.8) and (5.9),
which contradicts (5.10). The proof is completed. □
Remark 5.4 Condition (5.4) may be fulfilled if m is much larger than K, and K is rather small in comparison with m and θ. Moreover, the conclusion in Theorem 5.3 coincides with the discussion in Section 2. So we know that large cross-diffusion effect is helpful to the formation of stationary patterns.
Remark 5.5 The results of Theorems 2.7, 4.1 and 5.3 show that large cross-diffusion effect of the first species can create not only Turing patterns but also stationary patterns (non-constant positive steady states).
The author declares that she has no competing interests.
The author is supported by the Tianyuan Youth Foundation of NSFC (No. 11026067) and the National Natural Science Foundation of China (No. 11201204). The author appreciates the referee for the helpful comments and suggestions.
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