Research On Bifurcation And Pattern Self-organization Of A Discrete Reaction-diffusion Predator-prey System

Predator-prey system as an important part of the ecosystem,the research of it revealing the basic nature has a key role.In this paper,we reveal the complicated diverse soace-time self-organized structure in the ecosystem by analyzing the bifurcation and pattern self-organization of discrete predator-prey system.By establishing the time-discrete reaction-diffusion system we study the Neimark-Sacker bifurcation with time symmetry-breaking;by establishing the coupled map lattice model of reaction self-diffusion system and reaction self-and cross-diffusion system we study the pattern self-organization with spatiotemporal symmetry-breaking.The following results are stated:(1)Neimark-Sacker bifurcation occurs in discrete systems when the time-discrete predator-prey system satisfies the fixed point stability conditions and the Neimark-Sacker bifurcation theorem.Numerical simulations show the Neimark-Sacker bifurcation diagram,the largest Lyapunov exponent plot and various attractor phase diagrams for the time-discrete system.Neimark-Sacker bifurcation process can lead to nonlinear behavior of periodic orbitals,invariant rings,chaotic attractors and chaos,and is also accompanied by period doubling bifurcation in the process of chaos.(2)In the reaction self-diffusion predator system based on the coupled map lattice,reating pure Turing patterns are generated under the action of pure Turing instability mechanism,such as patterns of gap,point-gap,point and mosaic;Neimark-Sacker Turing patterns of spatio-temporal oscillations produce under the Neimark-Sacker Turing instability mechanism,such as patterns of maze,spiral,arc,and ring.There is a complex process of transformation between the various types of patches with the change of diffusion parameters,revealing the complexity and diversity of the pattern of the spatiotemporal predator-prey system.(3)In the reaction self-and cross-diffusion predator system based on the coupled map lattice,the Turing patterns and structural changes are more abundant than the reaction self-diffusion system.Pure Turing instability results in gap,striped,labyrinth and irregular point patterns self-organization;Neimark-Sacker-Turing instability leads to self-organized patterns of curly,spiraled,ring.In the sensitivity analysis of Turing pattern,the Turing pattern showed strong sensitivity to the initial conditions and parameters conditions,which proves the existence of spatiotemporal chaos in the pattern formation.(4)In order to explore the role of cross diffusion in the formation of Turing patterns,cross diffusion coefficients were added based on the typical patterns of reaction self-diffusion model in the reaction self-diffusion-cross diffusion system,to reveal the role of cross diffusion in the formation and transformation of complex patterns by a large number of numerical simulations.The increases of predator cross-diffusion coefficient more easily lead to the appearance of regular Turing patterns,and the increases of cross-diffusion coefficient of the prey will result in the formation of irregular patterns or homogeneous state.This paper is based on a predator-prey system studied by Guin with a rate-dependent functional response function and a predator with alternative food sources.The coupled map lattice model of spatio-temporal discreteness can better describe the population with short life and nonoverlapping generations.Numerical simulations of bifurcation diagrams and Turing patterns of discrete predator systems reveal the complexity and diversity of ecosystems.