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A spatiotemporal discrete predator-prey system with Allee effect is investigated to learn its Neimark-Sacker-Turing instability and pattern formation. Based on the occurrence of stable homogeneous stationary states, conditions for Neimark-Sacker bifurcation and Turing instability are determined. Numerical simulations reveal that Neimark-Sacker bifurcation triggers a route to chaos, with the emergence of invariant closed curves, periodic orbits, and chaotic attractors. The occurrence of Turing instability on these three typical dynamical behaviors leads to the formation of heterogeneous patterns. Under the effects of Neimark-Sacker-Turing instability, pattern evolution process is sensitive to tiny changes of initial conditions, suggesting the occurrence of spatiotemporal chaos. With application of deterministic initial conditions, transient symmetrical patterns are observed, demonstrating that ordered structures can exist in chaotic processes. Moreover, when local kinetics of the system goes further on the route to chaos, the speed of symmetry breaking becomes faster, leading to more fragmented and more disordered patterns at the same evolution time. The rich spatiotemporal complexity provides new comprehension on predator-prey coexistence in the ways of spatiotemporal chaos.

Since the pioneering work of Lotka and Volterra on population dynamics [

The researchers have developed various reaction-diffusion predator-prey models in order to study the pattern formation of predator-prey systems in varying circumstances, such as different functional responses, growth forms, and population motions. In literature, it is found that many types of functional responses can contribute greatly to the spatiotemporal complexity of predator-prey systems, such as Beddington-DeAngelis type, Leslie-Gower type, Holling type IV, and ratio-dependent type functional response. Upadhyay et al. discovered that zooplankton-phytoplankton systems with Holling type IV functional response have a common mechanism as wave of chaos, which can result in the self-organization of complex patterns [

Different growth forms of prey can also exert influences on pattern formation. Generally, the most common growth forms include logistic form, exponential form, and the form with Allee effect. Many works have considered continuous-time models for the predator-prey systems with Allee effect. Numerical results in Rao and Kang [

Moreover, many researchers further found that the population motions play a key role in spatiotemporal pattern formation of predator-prey systems. As important natural phenomena, cross-diffusion and convection of the predator and prey are often considered and studied in spatially extended predator-prey systems. Ghorai and Poria found that the cross-diffusion supports the formation of a wide variety of spatial and spatiotemporal patterns [

The nonlinear mechanism of pattern formation, Turing instability, can date back to the pioneering work of Turing who studied a reaction-diffusion model which included two chemical species, an activator and an inhibitor, in 1952 [

Most of the reaction-diffusion predator-prey models developed in literature are time-and space-continuous. Recently, a new type of mathematical model was introduced into biology to study the pattern formation, i.e., coupled map lattices (CMLs). In the 1980s, the CML was used by Kaneko to explore the spatial-temporal structure of coupled logistic map, he found very complex spatiotemporal dynamics, including frozen chaos, defect turbulence, spatiotemporal intermittency, and fully developed spatiotemporal chaos [

On the basis of former research works, we aim at studying the complex dynamics of a space- and time-discrete predator-prey system under the influence of Allee effect with the application of CML. For the discrete predator-prey system considered, four properties should be specified. First, the habitat where predator and prey populations dwell is distinctly patchy or even fragmented [

In this research, we investigate a predator-prey system with Allee effect and Michaelis-Menten type functional response. Allee effect, which was introduced by as well as named after Allee, refers to a positive correlation between population density and the per capita growth rate [

The Michaelis-Menten type reaction-diffusion predator-prey system with Allee effect in prey can be described as follows [

Considering dimensionless variables with the following scaling:

The CML model of spatiotemporal discrete predator-prey system is developed based on discretizing (

Via discretizing the nonspatial part of (

Equations (

A homogeneous stationary state of the spatiotemporal discrete predator-prey system is a state where the prey and predator densities are homogeneous in space and simultaneously keep steady in time. Its stability determines whether the predator-prey system can stably stay at the corresponding homogeneous stationary state under some external disturbances. Generally, the stationary states of the discrete predator-prey system satisfy the following conditions: i.e.,

For convenience of the following analysis, (

Mathematically, the homogeneous stationary states of the discrete predator-prey system are exactly equivalent to the fixed points of the map (

In order to determine the stability of the fixed points, the method of Jacobian matrix is often applied. The Jacobian matrix associated with map (

Substitute the value of each fixed point into (

Substituting

Similarly for the fixed points

As for the fixed points

According to

Besides the homogeneous stationary states, the discrete predator-prey can also generate many other complex spatially homogeneous states, which always keep oscillating with time. To determine the transition from homogeneous stationary states to homogeneous oscillating states, bifurcation analysis is one of the most reliable and efficient ways. It should be noticed that such transition in the spatiotemporal discrete predator-prey system is mathematically equivalent to the transition from the fixed point to other attractors in the map (

When map (

Under the satisfaction of conditions (

The map (

When Neimark-Sacker bifurcation occurs, the modulus of the eigenvalues (

On the basis of map (

The second-order and third-order partial derivatives of

According to the bifurcation theorem [

Based on the above calculations, the occurrence of Neimark-Sacker bifurcation in the discrete predator-prey system needs the satisfaction of conditions (

When spatially heterogeneous perturbations occur at the homogeneous states, the spatiotemporal discrete predator-prey system may converge to heterogeneous states [

Based on the calculations in former approach [

Substituting (

Let

The Turing instability emerges when system (

According to previous research works [

The purpose of numerical simulations is to demonstrate the Neimark-Sacker bifurcation as well as pattern formation under the influence of Neimark-Sacker-Turing instability. For such demonstration, the parametric conditions should be provided firstly. Based on former research work of Rao and Kang [

Under the above provided parametric conditions, the fixed point of the discrete predator-prey system is determined as (0.6538, 02972), and the critical point for Neimark-Sacker bifurcation is calculated as

Figure

(a) Neimark-Sacker bifurcation diagram and (b) maximum Lyapunov exponent corresponding to (a).

Three typical dynamical behaviors on the route to chaos induced by the Neimark-Sacker bifurcation, (a) invariant closed curve, (b) period-31 orbit, and (c) chaotic attractor.

When heterogeneous perturbations occur on the spatially homogeneous states, the discrete predator-prey system may experience Turing instability and the system dynamics converges to stable spatially heterogeneous states. Under the previous given parametric conditions, the critical point for Turing bifurcation is overlapping with the Neimark-Sacker bifurcation critical point. Via calculation, we find the

Figures

Pattern formation resulting from spatially symmetry breaking on invariant closed curves and starting with stochastic initial condition.

Pattern formation resulting from spatially symmetry breaking on period-31 orbit and starting with stochastic initial condition.

Pattern formation resulting from spatially symmetry breaking on chaotic attractor and starting with stochastic initial condition.

As exhibited in Figures

In order to exhibit the dynamical trajectory of pattern evolution, Figure

Phase portraits and wave diagrams corresponding to the change of state variables in lattice (100, 100) of Figures

To explore the nonlinear characteristics of pattern evolution process in Figures

Sensitive analysis of pattern evolution process to variations of initial conditions (graphs a–c) and parameter

The above sensitivity analysis suggests that change of initial conditions may lead to the self-organization of different patterns. Figures

Pattern formation resulting from spatially symmetry breaking on invariant closed curves and starting with deterministic initial condition.

Pattern formation resulting from spatially symmetry breaking on period-31 orbit and starting with deterministic initial condition.

Pattern formation resulting from spatially symmetry breaking on chaotic attractor and starting with deterministic initial condition.

The decrease of symmetry in Figures

Figure

Phase portraits corresponding to the change of state variables in lattice (100, 100) of Figures

The simulation results in Figures

Various transient strongly symmetrical patterns when Neimark-Sacker-Turing instability occurs on the front part of route to chaos.

This research investigates the spatiotemporal dynamics of a space- and time-discrete predator-prey system with Allee effect with the application of CML model. The development of the CML model in the present research is based on two types of discrete models in literature. The first is the type derived by directly discretizing partial differential equations with finite difference scheme, for example, the exponential discrete Lotka-Volterra model proposed by Li et al. [

In the research, we find new nonlinear characteristics for the system dynamics and pattern formation under Neimark-Sacker-Turing instability. Summarizing the results obtained in theoretical analysis and numerical simulations, the concluding remarks of this research can be made as the following.

The system can undergo Neimark-Sacker bifurcation inducing a route to chaos, on which three typical dynamical behaviors emerge, including invariant closed curves, periodic orbits, and chaotic attractors.

Neimark-Sacker-Turing instability triggers spatial symmetry breaking on invariant closed curves, periodic orbits, and chaotic attractors and leads to the formation of heterogeneous patterns, which keep oscillating property all along.

Different initial conditions result in diverse pattern evolution processes, revealing the multistability of the predator-prey system.

Under the effect of Turing instability, the invariant closed curves and periodic orbits change to more complex chaotic attractors. Moreover, spatiotemporal chaos occurs when Neimark-Sacker instability and Turing instability both take place.

With the application of deterministic initial conditions, transient symmetrical patterns are observed and the symmetry breaking increases as time continuously progresses. Moreover, when local kinetics of the system goes further on the route to chaos, the speed of symmetry breaking becomes faster, leading to more fragmented and more disordered patterns at the same evolution time.

Under Neimark-Sacker-Turing instability, the pattern evolution process is disordered and chaotic. However, at transient times, the predator-prey pattern may exhibit some symmetrical property. This demonstrates that ordered structures can exist in chaotic processes.

Since pattern evolution processes are sensitive to variations of initial conditions and parameter

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no financial and nonfinancial conflicts of interest for this research work. They confirm that the received funding mentioned in Acknowledgments will not lead to any conflicts of interest regarding the publication of this manuscript.

The authors would like to acknowledge with great gratitude the support of the National Major Science and Technology Program for Water Pollution Control and Treatment (nos. 2017ZX07101-002 and 2015ZX07203-011) and the Fundamental Research Funds for the Central Universities (no. JB2017069).