Complexity

Volume 2019, Article ID 3148323, 19 pages

https://doi.org/10.1155/2019/3148323

## Exploring Spatiotemporal Complexity of a Predator-Prey System with Migration and Diffusion by a Three-Chain Coupled Map Lattice

Research Center for Engineering Ecology and Nonlinear Science, North China Electric Power University, Beijing 102206, China

Correspondence should be addressed to Huayong Zhang; nc.ude.upecn@sneecr

Received 12 January 2019; Accepted 3 March 2019; Published 5 May 2019

Academic Editor: Eric Campos-Canton

Copyright © 2019 Tousheng Huang et al. 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.

#### Abstract

The topic of utilizing coupled map lattice to investigate complex spatiotemporal dynamics has attracted a lot of interest. For exploring the spatiotemporal complexity of a predator-prey system with migration and diffusion, a new three-chain coupled map lattice model is developed in this research. Based on Turing instability analysis, pattern formation conditions for the predator-prey system are derived. Via numerical simulation, rich Turing patterns are found with subtle self-organized structures under diffusion-driven and migration-driven mechanisms. With the variation of migration rates, the predator-prey system exhibits a gradual dynamical transition from diffusion-driven patterns to migration-driven patterns. Moreover, new results, the self-organization of non-Turing patterns, are also revealed. We find that even in the cases where the nonspatial predator-prey system reaches collapse, the migration can still drive pattern self-organization. These non-Turing patterns suggest many new possible ways for the coexistence of predator and prey in space, under the effects of migration and diffusion.

#### 1. Introduction

Ecological systems are basically characterized by the interactions between populations and natural environment [1]. Among all of ecological interactions, predation is a significant type of interaction which occurs over a wide range of spatial and temporal domains [2, 3]. Studies on predator-prey systems which have played an important role in ecological science can date back to the pioneering work of Lotka and Volterra in the early 1920s [4, 5]. Over the last one hundred years, rich results have been obtained on the complex dynamics of predator-prey systems [6, 7].

With the application of Turing’s theory, the pattern formation of predator-prey systems has become a hot topic, attracting attention of ecologists and mathematical biologists [8–11]. For example, Yi et al. [12] investigated a diffusive predator-prey system with Holling type-II functional response and obtained spatiotemporal patterns which provide theoretical evidence to complex spatiotemporal dynamics, i.e., existence of loops of spatially nonhomogeneous periodic orbits and steady state solutions. Shi and Ruan [13], Song et al. [14], Song and Tang [15], and Cai et al. [16] explored Turing patterns in predator-prey systems, demonstrating that the system undergoing Turing-Hopf bifurcation can exhibit not only stationary Turing patterns but also temporal periodic patterns. Cai et al. [17] and Wang et al. [18] investigated Turing patterns as a result of disease spread, determining a special range for the disease parameters to control the diseases via studying the characteristics of the Turing patterns.

The pattern formation of predator-prey systems is always related to spatial movement of predator and prey populations [19]. Recently, research on mathematical biology has shown that diffusion and migration, the most two important spatial movements in nature, hold a crucial influence on spatiotemporal population dynamics [20]. For diffusion, it is based on the assumption that the motion of individuals of a given population is random and isotropic [2, 20]. In the diffusion circumstance, Wang et al. [21] found complex dynamic patterns in a ratio-dependent predator-prey system, including chaos patterns and spiral waves which better explain the dynamics of aquatic communities in marine environment. Different from diffusion, migration means that the individuals can exhibit a correlated motion towards a certain direction instead of random motion [2, 20]. This kind of motion may often lead to the self-organization of traveling spatial patterns in predator-prey systems [20].

In previous research on the pattern formation of predator-prey systems, investigations mainly focused on the cases with population diffusion [2]. Recently, the researchers have been interested in the spatial heterogeneities resulting from migration [20]. Sun et al. [2] revealed that migration can lead to transformation of Turing pattern into traveling pattern. Later, Liu [20] analyzed a predator-prey system with Holling-III type functional response combined with both diffusion and migration terms, detecting that the effects of migration on the wave behavior would lead to oscillation frequency increasing with wave number. Moreover, Zhang et al. [22] revealed that the combining effects of diffusion and migration can account for the dynamical complexity of predator-prey systems.

Mathematical modeling is one of the most effective methods to study the dynamics of predator-prey systems. As widely recognized, reaction-diffusion models have been well developed for revealing the predator-prey patterns [2, 3, 20]. So far, most researchers have focused on the reaction-diffusion predator-prey models with continuous time and space. However, for the cases of patchy environment or fragmented habitat, discrete predator-prey models may be more reasonable and adequate [23, 24]. For example, Mistro et al. [23] considered that patchy and discontinuous environmental properties should be described more adequately by a space-discrete model. They found that all typical invasion scenarios observed in continuous models can be observed as well in the discrete model and that very few artifacts of the regular lattice structure observed in the discrete model cannot be found in continuous models. Recently, a few researchers further developed a new type of spatially extended predator-prey model that is given by coupled map lattice (CML) [23, 24]. It is found that the CMLs can describe rich spatiotemporal complexity and produce new attractive results of predator-prey systems [24–27].

A CML is characterized by discrete time, discrete space, and continuous states. In the 1980s, the CML was firstly developed to explore the spatiotemporal structure of coupled logistic map by Kaneko. Abundant spatiotemporal dynamics was revealed, including frozen chaos, spatiotemporal intermittency, defect turbulence, and fully developed spatiotemporal chaos [28–30]. Based on Kaneko’s work, Mistro et al. [23], Rodrigues et al. [27], and Punithan et al. [31] proposed CML models for the reaction-diffusion predator-prey systems. Their research discovered new dynamical behaviors and provided reliable prediction for the predator-prey dynamics. Lately, Huang et al. [24, 25] further developed CML models through discretizing continuous reaction-diffusion model and detected a surprising variety of spatiotemporal patterns, demonstrating that the nonlinear mechanisms of CML can effectively capture the dynamical complexity of predator-prey systems [31, 32].

In this research, we intend to explore the spatiotemporal complexity of a predator-prey system with migration and diffusion by applying CML model. In such a system, there actually exist three distinctly different ecological processes: “reaction”, diffusion, and migration. Based on previous thoughts for developing CML model [30, 33] and the ecological reality, we consider the three processes as separate stages. Corresponding to the three processes in the predator-prey system, a new three-chain CML model which holds ecological satisfaction can be developed. However, to the best of our knowledge, there is still scarce work which developed three-chain CML model to explore dynamical behaviors of the spatially extended predator-prey systems.

With the application of the three-chain CML model, new nonlinear characteristics of the predator-prey patterns may be found under the effects of both diffusion and migration. This work is organized as follows. In Section 2, we propose a three-chain CML model to describe the ratio-dependent predator-prey system with diffusion and migration and give the results of stability analysis. In Section 3, analysis of Turing instability of the predator-prey system is made to determine the pattern formation conditions. Section 4 provides numerical simulation results to demonstrate the spatiotemporal complexity. In Section 5, discussion and conclusions are described.

#### 2. Development of the Three-Chain Coupled Lattice Map Model

In this research, we focus attention on the dynamics of a spatiotemporal ratio-dependent predator-prey system, which has received great attention among theoretical and mathematical biologists [1, 21]. As described in literature, the governing equations of the nonspatial ratio-dependent predator-prey model take the following form [21]: where* N* and* P* stand for prey and predator density, respectively;* r* stands for maximal growth rate of the prey;* K* describes the carrying capacity for the prey population; *α* is the capture rate of predator on prey;* h* is the handling time; *γ* is the conversion efficiency from prey to predator;* μ* is the predator mortality rate.

In order to reduce the number of parameters involved in the model, the equations in (1a) and (1b) are simplified via nondimensionalization. Here, taking then we obtain the following simplified differential equations:

The goal of the present study is to explore the spatiotemporal dynamics of the predator-prey system with migration and diffusion. Considering diffusion and migration of predator and prey populations based on (3a) and (3b), a spatiotemporal predator-prey model can be described aswhere is the usual Laplacian operator in two-dimensional space; is the advection operator, meaning that the individuals migrate along one direction, i.e., direction. One-directional migration of the populations is a common phenomenon in nature. For example, for the two-year-old hatchery-reared progeny of spawning brown trout in the case of the R. Imas, south-western Norway, Jonsson et al. [34] found that migratory direction of juvenile fish and rivet fish population is always against and follows water current, respectively.* D*_{1} and* D*_{2} are prey and predator diffusion coefficients;* C*_{1} and* C*_{2} are the corresponding migration coefficients.

The CML model is then developed through discretizing system (4a) and (4b) [23]. At first, time interval and space interval for discretization are given. Generally, the dynamics of predator-prey system can be observed by a particular time scale, which can be defined by the generation span of predator and prey populations and measures the regeneration time of both populations. In this study, we denote the time scale on which predator-prey dynamics is observed as parameter* τ* and apply it as the time interval for discretization of system (4a) and (4b). The space interval used for discretization represents the space scale on which spatial movements of predator and prey take place. Its value can be defined by the maximum size of dwelling sites of predator and prey individuals. In this study, the space interval is denoted as parameter

*l*.

Considering a two-dimensional rectangular lattice which includes grid cells for establishing the CML model. The length of each grid cell is exactly the value of* l*. Each grid cell represents one site where predator and prey individuals dwell and is ascribed to two numbers, i.e., the prey density and the predator density. The prey and predator densities in each site change with time in course of the system dynamics, due to local inter- and intraspecific interactions as well as migration and diffusion between different sites [23]. Here, we use symbol* m* (, and* N* describes the set of positive integers) to represent the time increasing with discrete iterations. With given initial time* t*_{0}, symbol* m* means the* m*th iteration and refers to the time* t*_{0}+*m τ*. On this basis, two discrete state variables,

*u*

_{(i,j,m)}and (

*i*, ), are defined, representing the prey density and the predator density in the (

*i*,

*j*) site at iteration

*m*(for simplification we also use time

*m*to refer to the time

*t*

_{0}+

*m*).

*τ*At each discrete step from iteration* m* to* m* + 1, the dynamics of predator and prey consists of three parts, reaction, diffusion, and migration. In literature, when CML is applied to study the predator-prey systems, the “reaction” between predator and prey and the diffusion of both populations are often regarded as two distinctly different stages [23, 27, 31]. Hassell et al. [35] argued that with such processes of segregating and corresponding operators splitting, counter-intuitive results, such as the production of negative local population densities at some extreme cases [35], can be avoided. Moreover, Hassell et al. suggested that the processes of segregating and operators splitting are actually more biologically sensible. Mathematically, splitting of diffusion terms and advection terms in an advection-diffusion partial differential equation is a canonical example of operator splitting [36]. It is faster to compute the solution of the splitting terms separately than to compute the solution directly when they are treated together [33]. Based on previous research on the CML, we further consider the migration of populations as another different stage in this research, split the terms of reaction, diffusion, and migration, and solve them separately.

Such consideration of treating reaction, diffusion, and migration processes as three separate stages is based on following ecological facts. First, migration and diffusion may often take place when the populations and individuals seek for new habitats and necessary life conditions, whereas the reaction process mainly occurs when the populations settle down in stable habitats. Therefore, the reaction stage can be considered as a separate stage different from diffusion and migration. Second, migration and diffusion represent directional and unidirectional population motions which show difference in spatial scale. The population migration often takes place in a huge spatial scale; e.g., the migration of fish populations may cross over thousand miles, whereas the population diffusion may merely take place in the spatial scale of a local habitat. Third, the population migration and diffusion may occur in different stages; specifically, phenomenon where populations migrate first and then randomly spread indeed exists in nature. In the case of large herbivores, the population often migrates between discrete home ranges as a means of enhancing access to high quality food and/or reducing the risk of predation [37]. After settling down in new habitats, individuals of the population move randomly in the home range to acquire enough food. Simultaneously, since the predator is dependent on the prey to survive, the motion of the predator population that feeds on the herbivores also follows the same way. Based on the above descriptions, it is reasonable to consider the processes of reaction, diffusion, and migration as separate stages. Moreover, the sequence of the three stages can be considered as migration, diffusion, and reaction, respectively.

Therefore, in the CML of predator-prey systems with migration and diffusion, three parts of population dynamics are modeled as distinctly different stages, (a) the migration stage, (b) the diffusion stage, and (c) the “reaction” stage. The first stage is obtained by discretizing the population migrating dynamics with time interval* τ* and space interval

*l*. Then we obtain where is the discrete advection operator, i.e., where

*represents the variables of*

*ψ**u*and

*v*. Likewise, the diffusion stage can be described by the following equations: where is the discrete Laplacian operator, i.e., For the last “reaction” stage, we discretize the nonspatial system (3a) and (3b) and obtain in which

Equations (5a), (5b), (5c), (5d), (5e), (5f), (5g), (5h), (5i), and (5j) give the expression of the three-chain CML model which describes a spatiotemporally discrete ratio-dependent type predator-prey system with migration and diffusion. Such a CML can be viewed as a discretization of the continuous reaction-advection-diffusion predator-prey model. In the migration stage, the predator and prey individuals move from cell (*i*−1,* j*) to cell (*i*,* j*), i.e., along the negative* x* direction, and the corresponding migration rates are* C*_{1} and* C*_{2}, respectively. This stage describes the process that populations migrate between different habitats. Then the system enters the diffusion stage; the predator and prey individuals disperse to four adjacent cells with diffusion rates as* D*_{1} and* D*_{2}, demonstrating predator and prey moving randomly in local habitat which may be fragmented. In the last stage, the predator and prey populations react under the predation relation described by (5i) and (5j). From the point view of ecological significance, the CML model can be regarded as an alternative to the corresponding continuous reaction-advection-diffusion model. All the parameters used in the CML should be positive. Moreover, for ensuring nonnegativity of* u* and* v* and convergence of the CML, the parameter values should be provided to make* τ*C

_{i}

*/l*and

*τ*D_{i}

*/l*

^{2}(

*i*= 1, 2) less than 0.5 [26].

In order to explore the spatiotemporal complexity of the discrete predator-prey system, firstly the nonspatial dynamics of system (5a), (5b), (5c), (5d), (5e), (5f), (5g), (5h), (5i), and (5j) should be investigated. Let* C*_{i} and* D*_{i} (*i* = 1, 2) be zero; the equations of the CML model become

It should be noticed that the dynamical behaviors of (6a) and (6b) represent the homogeneous states of the spatiotemporally discrete predator-prey system. One of the important dynamical behaviors is the fixed point, which represents the homogeneous stationary state of the system. Via the definition of fixed point of difference equations, two fixed points of (6a) and (6b) can be obtained as follows:

(*u*_{1},* v*_{1}) means the extinction of the predator, and system (6a) and (6b) will finally converge to the state with prey reaching carrying capacity. (*u*_{2},* v*_{2}) describes the coexistence of the predator and prey. The stability of this fixed point indicates whether predator and prey can stably coexist. To determine the stability of (*u*_{2},* v*_{2}), Jacobian matrix is applied. The Jacobian matrix associated with system (6a) and (6b) at any point is calculated as

Substituting the value of fixed points into Jacobian matrix (8) and then calculating the eigenvalues of the matrix, from the values of the two eigenvalues, the stability of the two fixed points is determined. If the modules of the two eigenvalues are both smaller than one, the corresponding fixed point is stable; if the modulus of one of the eigenvalues is larger than one, the corresponding fixed point is unstable. After calculation, the two eigenvalues of (*u*_{1},* v*_{1}) are 1*- τR* and 1+

*(1*

*τ**-Q*). In the case where positive (

*u*

_{2},

*v*

_{2}) exists, the condition 1−

*R*/

*S*<

*Q*< 1 must hold. Therefore, we have 1+

*(1*

*τ**-Q*) > 1, and the fixed point (

*u*

_{1},

*v*

_{1}) is unstable. The two eigenvalues of (

*u*

_{2},

*v*

_{2}) are found as whereAccording to the above determination, we find that (

*u*

_{2},

*v*

_{2}) is stable if

The stability of homogeneous stationary state reveals whether the system can resist external disturbances and stay on the stable state. The above stability analysis suggests that the predator-prey system may reach stable homogeneous stationary state. When the conditions in (11) are satisfied, the predator and prey populations in the system can stably coexist and the system will eventually evolve to a state where the predator and prey densities remain at* u*_{2} and* v*_{2} in entire space and do not change with time. Even when homogeneous perturbations exert influences, the system will turn back to this state. However, when the conditions in (11) are not in satisfaction, the predator-prey system will collapse at the end; i.e., the predator population will be extinct and the prey population will grow to its carrying capacity.

#### 3. Turing Instability Analysis

In the three-chain CML developed above, the combination effects of diffusion and migration may destabilize a spatially homogeneous stationary state of the predator-prey system that would be stable in the absence of population motions in space, i.e., triggering the Turing instability [38], which leads to spatial pattern self-organization in the discrete predator-prey system.

To perform Turing instability analysis, spatially heterogeneous perturbations are introduced to linearize the discrete predator-prey system around (*u*_{2},* v*_{2}). We assume where and . Substituting (12) into (5a), (5b), (5c), (5d), (5e), (5f), (5g), (5h), (5i), and (5j) directly yieldsLinearizing (13e) and (13f) around (*u*_{2},* v*_{2}), we obtainwhere Simplifying (14a) and (14b) by applying (13a)-(13d) leads to which is equivalent to

Since the discrete operators and are commuting operators. Let , and we can verify that , , and are also commuting, i.e., Hence, the three operators have a group of common eigenfunctions. Moreover, to determine the eigenvalues of , , and , we consider the following equations: with periodic boundary condition. As per the method described in Bai and Zhang [39], the eigenvalues of the three operators are solved and described as where the symbols and represent wavenumbers, , and .

Assuming that is the common eigenfunction corresponding to the eigenvalues , , and and using to multiply (17a) and (17b), we obtainSumming (24a) and (24b) for all of* i* and* j* and letting then we have

Rewrite (24a) and (24b) in the following form:

in which

The two eigenvalues of the system (25) are determined as

Since (25) describes the dynamics of spatially heterogeneous perturbations integrating all the sites, when (25) converges, the discrete predator-prey system will go back to (*u*_{2},* v*_{2}), whereas when divergence of (27) occurs, Turing instability occurs and lead to the self-organization of Turing patterns. Hence, the occurrence conditions for Turing instability can be described as [25, 26, 32]

#### 4. Numerical Simulations

The theoretical results given in previous sections reveal that the spatiotemporally discrete predator-prey system can produce Turing patterns with parameter values satisfying conditions (11) and (28). Nevertheless, it is difficult to theoretically obtain the dynamical solutions of the discrete system. Thus, there is necessity to perform numerical simulations, the most reliable approach with satisfactory accuracy and efficiency, for comprehensively investigating the self-organized patterns of the discrete predator-prey system with feasible parameters.

In the numerical simulations, it is found that the discrete predator-prey system can exhibit two types of self-organized patterns. The first type is Turing patterns, which are self-organized under the mechanism of Turing instability, and the occurrence conditions for Turing patterns are provided in Section 3. The other type is non-Turing patterns, the emergence of which does not satisfy the Turing instability conditions. Moreover, the non-Turing patterns merely appear when the fixed point (*u*_{2},* v*_{2}) is unstable, i.e., the conditions in (11) are unsatisfied.

##### 4.1. Turing Patterns

Firstly, numerical simulations are performed to demonstrate the Turing pattern formation based on the theoretical results. In the simulations, parameter values are provided based on former research works [19, 20, 24]. Accordingly, feasible parameter ranges/values are given as follows:* Q *= 0.05~0.92,* R *= 0.05~1.4,* S *= 0.5~1.48,* τ *= 0.01 or 0.05,

*l*= 0.25 or 0.5,

*C*

_{1}= 0.001~0.5,

*C*

_{2}= 0.01~0.5,

*D*

_{1}= 0.002~0.1,

*D*

_{2}= 0.2; unless otherwise specified, for all patterns we give

*n*= 200,

*m*= 45000. The initial condition for simulating pattern formation is set by perturbing (

*u*

_{2},

*v*

_{2}) using small random spatially heterogeneous perturbations. Since predator and prey patterns always show similar configuration, the results of prey patterns are merely demonstrated. Simultaneously, it should be noticed that the migration direction in all pattern graphs is set to be from up to down.

As heterogeneous perturbations take place on the spatially homogeneous states, the discrete predator-prey system may experience Turing instability and the system dynamics converges to stable spatially heterogeneous states. To demonstrate the Turing instability and pattern formation conditions, curves of* L*(), which is defined as , are plotted in Figure 1. It is shown that the curves of* L*() appear like a “V” form, descending at the beginning and then rising with increase of the wavenumber . Moreover, as the values of parameters* R* and* Q* decrease, or the value of parameter* S* grows, curves of* L*() will go up over the critical line* L*() = 1. With given parametric conditions in Figure 1, when* S* > 0.98,* R* < 0.6, and* Q* < 0.65, respectively, the maximum value of* L*() is larger than one and Turing instability occurs, leading to the emergence of Turing patterns in the discrete predator-prey system.