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Turing instability and stationary patterns in a predator-prey systems with nonlinear cross-diffusions
© Wen; licensee Springer. 2013
Received: 6 September 2012
Accepted: 16 June 2013
Published: 1 July 2013
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).
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].
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 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
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 2.1 Suppose that , . Then the positive equilibrium of (1.1) is uniformly asymptotically stable.
It is easy to verify that if .
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 .
Obviously, if , then and so . Thus, . It follows from Routh-Hurwitz criterion that the two roots , of have both negative real parts for all .
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.
for some , whereas it is uniformly asymptotically stable if .
Lemma 2.3 Suppose that , . Then the positive equilibrium of (1.1) is uniformly asymptotically stable for disappeared cross-diffusion.
Lemma 2.4 Suppose that , , . Then the positive equilibrium of (1.1) is uniformly asymptotically stable.
We thus have the following result.
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 2.6 Assume 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).
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 2.7 Assume 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
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 , .
and , then .
and , then .
Lemma 3.2 (Harnack inequality )
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)
This completes the proof. □
Theorem 3.4 (Lower bound)
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 4.1 Let , . Then there exists a positive constant such that problem (1.1) has no non-constant positive steady state provided that .
where ϵ is the arbitrary small positive constant arising from Young’s inequality.
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
where C is a positive constant whose existence is guaranteed by Theorems 3.3 and 3.4.
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 .
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.
Then we have the following result.
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.
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.
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 .
from Theorem 4.1.
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 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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