Theory and Applications of Periodic Solutions and Almost Periodic SolutionsView this Special Issue
Pattern Formation in a Cross-Diffusive Holling-Tanner Model
We present a theoretical analysis of the processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with self- as well as cross-diffusion in a Holling-Tanner predator-prey model; the sufficient conditions for the Turing instability with zero-flux boundary conditions are obtained; Hopf and Turing bifurcation in a spatial domain is presented, too. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by self- as well as cross-diffusion in the model, and find that the model dynamics exhibits a cross-diffusion controlled formation growth not only to spots, but also to strips, holes, and stripes-spots replication. And the methods and results in the present paper may be useful for the research of the pattern formation in the cross-diffusive model.
A fundamental goal of theoretical ecology is to understand the interactions of individual organisms with each other and with the environment and to determine the distribution of populations and the structure of communities. Empirical evidence suggests that the spatial scale and structure of environment can influence population interactions . The study of complex population dynamics is nearly as old as population ecology, starting with the pioneering work of Lotka and Volterra, a simple model of interacting species that still bears their joint names [2, 3].
And the predator-prey system models such a phenomenon, pursuit-evasion, predators pursuing prey and prey escaping from the predators . In other words, in nature, there is a tendency that the preys would keep away from predators and the escape velocity of the preys may be taken as proportional to the dispersive velocity of the predators. In the same manner, there is a tendency that the predators would get closer to the preys and the chase velocity of predators may be considered to be proportional to the dispersive velocity of the preys . Keeping these in view, cross-diffusion arises, which was proposed first by Kerner  and first applied in competitive population system by Shigesada et al. . From the pioneering work of Turing , spatially continuous models formulated as reaction-diffusion equations have been intensively used to describe spatiotemporal dynamics and to investigate mechanisms for pattern formation . And the appearance and evolution of these patterns have been a focus of recent research activity across several disciplines .
In recent years, there has been considerable interest to investigate the stability behavior of a predator-prey system by taking into account the effect of self- as well as cross-diffusion [11–25]. Cross-diffusion expresses the population fluxes of one species due to the presence of the other species.
Furthermore, in , the authors gave a numerical study of pattern formation in the Holling-Tanner model with self- and cross-diffusion as the form with the following nonzero initial conditions: and zero-flux boundary conditions: where , represent population densities of prey and predator, respectively, is the intrinsic growth rate or biotic potential of the prey, is the maximal predator per capita consumption rate, that is, the maximum number of prey that can be eaten by a predator in each time unit, and is the number of preys necessary to achieve one-half of the maximum rate . And the nonnegative constants, and , called cross-diffusion coefficients, express the respective population fluxes of the prey and predators resulting from the presence of the other species, respectively. and give the size of the system in the directions of and , respectively. In (1.3), is the outward unit normal vector of the boundary , which we will assume is smooth. The main reason for choosing such boundary conditions is that we are interested in the self-organization of pattern; zero-flux conditions imply no external input [11, 27].
Biologically, cross diffusion implies countertransport . And the induced cross-diffusion rate in (1.1) represents the tendency of the prey to keep away from its predators and represents the tendency of the predator to chase its prey. The cross-diffusion coefficients, and , may be positive or negative. Positive cross-diffusion coefficient denotes that one species tends to move in the direction of lower concentration of another species while negative cross diffusion expresses the population fluxes of one species in the direction of higher concentration of the other species .
In , the authors found that, for the equal self-diffusion coefficients (i.e., the coefficients of and in (1.1) are both equal to 1), only spots pattern could be obtained. In addition, they indicated that cross diffusion may have an effect on the distribution of the species, that is, it may lead the species to be isolated.
There comes a question: besides spots, does model (1.1) exhibit any other pattern replication controlled by cross-diffusion?
In this paper, based on the results of , we mainly focus on the effect of self- as well as cross-diffusion on pattern formation in the two-species Holling-Tanner predator-prey model. In the next section, we give the sufficient conditions for the Turing instability with zero-flux boundary conditions. And, by using the bifurcation theory, we give the Hopf and Turing bifurcation analysis of the model. Then, we present and discuss the results of complex, not simple, pattern formations via numerical simulations, which are followed by the last section, that is, concluding remarks.
2. Linear Stability Analysis
In the absence of diffusion, model (1.1) has two equilibrium solutions in the positive quadrant. One equilibrium point is given by . In the phase plane, is a saddle . Another equilibrium point depends on the parameters and and is given by where .
And in the presence of diffusion, we will introduce small perturbations and , where . The zero-flux boundary conditions (1.3) imply that no external input is imposed from outside. To study the effect of diffusion on the model system, we have considered the linearized form of the system as follows: where
Set . And define as the diffusion matrix, and the determinant is .
Having substituted and with (2.2), we obtain where .
Correspondingly, and arise as the solution of the following equation: where
Theorem 2.1. If , and , then the uniform steady state of model (1.1) is globally asymptotically stable.
Proof. For global stability of nonspatial model of (1.1), we select a Liapunov function:
where . Then
Substituting the value of and from the nonspatial model of (1.1), we obtained
Noting that , one can obtain If holds, .
Next, we select the Liapunov function for model (1.1) (two dimensional with diffusion case): so
Using Green's first identity in the plane And considering the zero-flux boundary conditions (1.3), one can show that then where Obviously, and . The proof is complete.
And from , we know that the equilibrium is Turing unstable if it is an asymptotically stable equilibrium of model (1.1) without self and cross diffusion but is unstable with respect to solutions of model (1.1). Hence, Turing instability sets in when the condition either or is violated, which subject to the conditions and . It is evident that the condition is not violated when the requirement is met. Hence, only violation of condition gives rise to Turing instability. Then the condition for Turing instability is given by
Thus, a sufficient condition for Turing instability is that is negative. Therefore,
Summarizing the previous discussions, we can obtain the following theorem.
Theorem 2.2. The equilibrium of model (1.1) is Turing instability if and .
It is easy to see that the minimum of occurs at , where The critical wavenumber of the first perturbations to grow is found by evaluating from (2.22).
Figure 2 shows that the linear stability analysis yields the bifurcation diagram with , , and . Turing and Hopf lines intersect at the Turing-Hopf bifurcation point and separate the parametric space into four domains. On domain I, located above all two bifurcation lines, the steady state is the only stable solution of the model. Domain II is the region of pure Turing instability. In domain III, which is located above all two bifurcation lines, both Hopf and Turing instability occur. And domain IV is the region of pure Hopf instability.
3. Pattern Formation
In this section, we performed extensive numerical simulations of the spatially extended model (1.1) in 2-dimension spaces, and the qualitative results are shown here. All our numerical simulations employ the zero-flux boundary conditions with a system size of , with discretized through and , with . Other parameters are fixed as , , and . The numerical integration of (1.1) was performed by means of forward Euler integration, with a time step of and spatial resolution and using the standard five-point approximation for the 2D Laplacian with the zero-flux boundary conditions [30, 31]. More precisely, the concentrations at the moment at the mesh position are given by with the Laplacian defined by
Initially, the entire system is placed in the stationary state , and the propagation velocity of the initial perturbation is thus on the order of space units per time unit. And the system is then integrated for 100000 or 300 000 time steps and some images saved. After the initial period during which the perturbation spreads, the system goes either into a time dependent state, or to an essentially steady state (time independent).
In the numerical simulations, different types of dynamics are observed, and it is found that the distributions of predator and prey are always of the same type. Consequently, we can restrict our analysis of pattern formation to one distribution. In this section, we show the distribution of prey , for instance.
Firstly, we show the pattern formation for the parameters located in domain II (cf. Figure 2); the region of pure Turing instability occurs while Hopf stability occurs. We have performed a large number of simulations and found that there only exhibits spots pattern in this domain. As an example, we show the time evolution of spots pattern of prey at 0, 10000, 30000, and 300000 iteration for in Figure 3. In this case, one can see that for model (1.1), the random initial distribution (cf. Figure 3(a)) leads to the formation of regular spots except for apparently stable defects (cf. Figure 3(d)).
Next, we show the pattern for the parameters located in domain III (cf. Figure 2), both Hopf and Turing instability occur. The model dynamics exhibits spatiotemporal complexity of pattern formation. In Figure 4, we show four typical types of patterns we obtained via numerical simulation.
Pattern is time independent and consists primarily of stripes. Pattern is time independent and consists of stripes and spots. Pattern is time dependent: a few of the stripes without apparent decay, but the remainder of the spots pattern remains time independent. Pattern is time independent and consists of black spots on a white background, that is, isolated zones with low-population densities. Baurmann et al.  called this type pattern “cold spots” and von Hardenberg et al.  called it “holes.” In this paper, we adopt the name “holes.”
4. Concluding Remarks
In summary, we have investigated a cross-diffusive Holling-Tanner predator-prey model with equal self-diffusive coefficients. Based on the bifurcation analysis (Hopf and Turing), we give the spatial pattern formation via numerical simulation, that is, the evolution process of the system near the coexistence equilibrium point .
In contrast to the results in , we find that the model dynamics exhibits a cross-diffusion controlled formation growth not only to spots (in , Sun et al. claimed that the spots pattern is the only pattern of the model), but also to stripes, holes, and stripes-spots replication. That is to say, the pattern formation of the Holling-Tanner predator-prey model is not simple, but rich and complex.
On the other hand, in the predator-prey model, predators will tend to gravitate toward higher concentrations of prey while prey will preferentially move toward regions where predators are rare. Models of predator-prey systems with cross diffusion have been extensively analyzed in the literature, though often with respect to their mathematical properties rather than to provide insight into the kinds of patterns that can emerge . And the methods and results in the present paper may be useful for the research of the pattern formation in the cross-diffusive predator-prey model.
The authors would like to thank the anonymous referee for the very helpful suggestions and comments, which led to improvements of their original paper. This research was supported by the Natural Science Foundation of Zhejiang Province (LY12A01014 and LQ12A01009), the National Science Foundation of China (11226172), and the National Basic Research Program of China (2012CB426510).
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