Research On Chaotic Pattern Dynamics Of A Spatiotemporal Discrete Reaction-diffusion Predator-prey System

The investigations on ecological spatiotemporal complexity of reaction-diffusion predator-prey system is a hot topic in study of contemporary ecology.In this paper,based on the coupled map lattice model,we run analysis of nonlinear theory and numerical simulations of the discrete reation-diffusion predator-prey system.By application of coupled map lattice model,reaction-diffusion system and reaction-diffusion-migration system are established,then we investigate the nonlinear dynamic behaviours and self-organized strcutures of spatial pattern,and the results reveal compexity and variety of spatiotemporal self-organizing structures in ecosystems.The following results are stated:(1)In the ratio-dependent reaction-diffusion system,we investigate the Neirmark-Sacker-Turing instability and conditions for pattern formation.Conditions for Neimark-Sacker bifurcation and Turing instability are determined based on the occurrence of stable homogeneous stationary states.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 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.(2)In order to investigate spatiotemporal complexity of a reaction-diffusion-migration predator-prey system,a new coupled map lattice model is developed.Based on the analysis of Turing instability,conditions for pattern formation are determined.Meanwhile,via numerical simulations,rich Turing patterns with subtle self-organized structures under diffusion-driven and migration-driven mechanisms are explored.With the variation of migration rates,the predator-prey system exhibits a gradual dynamical transition from diffusion-driven patterns to migration-driven patterns.Moreover,the formation of non-Turing patterns suggest many new possible ways for the coexistence of predator and prey in space,under the effects of migration and diffusion.(3)Apply three-chain CML to investigate a reaction-diffusion-migration system for understanding the effect of migration in pattern self-organization on the route to chaos triggered by Neimark-Sacker bifurcation with growth rate as bifurcation parameter.Based on conditions of the Neimark-Sacker bifurcation where we choose growth rate as bifurcation parameter,and results of numerical simulation,it is demonstrated that Neimark-Sacker starts a route to chaos where the predator-prey dynamic behaviors emerge,including an invariant closed curve,periodic orbits,flip process and chaotic attractors.With the variation of growth rate,banded patterns,spiral patterns and irregular patterns transform on the route to chaos which Neimark-Sacker starts.Moreover,it also discovers that the effects of predator-migration and prey-migration on the pattern formation and transformation on the route to chaos are different in the reaction-diffusion-migration system.In this paper,we apply the coupled map lattice model to investigate the spatiotemporal complexity of reaction-diffusion predator-prey system and reacton-diffusion-migration predator-prey system,so that we can better describe the population with dynamical characteristic as fractured habitats and nonoverlapping generations.Via analysis of nonlinear theory of discrete predator-prey system,we exhibit the bifurcation behaviors of them.And via numerical simulations,we reveal dynamic behaviors under the effect of Neimark-Sacker-Turing instability and new nonline feature of pattern formation.The research further promotes the understanding of the the effects of population migration on process for pattern formation and transformation on the route to chaos in a spatiotemporal discrete reaction-diffusion-migration predator-prey system.