Research On Spatiotemporal Dynamics Of Biological System With Time Delay

The development and evolution of most biological systems have two notable characteristics:one is the existence of time delay,that is,the evolution of the system not only depends on the current state,but also relates to the past state;another one is the existence of space diffusion,that is,both humans and other organisms live in the spatial environment with diffusion.Therefore,the spatiotemporal phenomena of the evolution of biological system can be described more accurately by the reaction-diffusion dynamics model with time delay.In this thesis,we established some predator-prey and infectious disease diffusion models with time delay for several typical biological systems.By studying the spatiotemporal dynamics of the systems,we have found some interesting dynamical behaviors including Hopf bifurcations,Turing bifurcations(Turing instability)and abundant Turing patterns like spot,labyrinth and black-eye.The stability and periodicity of periodic solutions of Hopf bifurcation are further studied,and the final epidemic size after the outbreak of infectious disease is calculated.The results of this research explain the reasons of the periodic oscillations and spatial patterns in some biological systems,which provide some theoretical foundation for revealing the evolution law and development processes of predator-prey system and infectious disease,and have certain reference value for predicting the subsequent evolution law.The main research contents are summarized as follows:(1)The background and significance of research of several typical biological systems are introduced,then the research progress of modeling and analysis methods of biological system are summarized,and the basic concepts and methods used in this thesis are given.Moreover,the main works of this thesis are drawn.(2)We propose a delayed predator-prey diffusion model with additional food supply,and give the existence conditions of Hopf bifurcation.Furthermore,by using the center manifold theorem and the normal form theory,the direction of bifurcation,stability,periodicity and amplitude of bifurcated periodic solutions are given.The spatial periodic solutions are illustrated by numerical simulation.The research results show that time delay can make the predator and prey continue to coexist in the form of periodic oscillation.(3)We establish a delayed predator-prey diffusion model considering the predator population with infectious diseases.By using the distribution of the root of the characteristic equation,the delay conditions for the instability of the system and the diffusion conditions for the Turing instability are given.The numerical simulations depict two different black-eye patterns.These research results show that time delay affects the stability of the coexisted predator and prey,and diffusion affects the spatial distribution of predator and prey under the predator population with infectious diseases condition.(4)We establish an SEI epidemic diffusion model with incubation delay.By analyzing the distribution of the eigenvalues of the transcendental characteristic equation,the conditions of the existence of Hopf bifurcation and Turing bifurcation are given.The numerical simulations show that the number of the infected population presents periodic oscillation phenomenon over time and the evolution process of Turing pattern of the infected population.The research results indicate that the introduction of incubation delay can give rise the periodic solutions in the system,while diffusion can induce the aggregation phenomena of the infected population in different forms,such as spot pattern,labyrinth pattern,black-eye pattern and so on.These research results can provide theoretical guidance for prevention and control of infectious disease.(5)Based on the idea of age structure,an SIR network epidemic model with preventive rewiring and infectious period delay is derived by rigorous mathematical derivation,then the basic reproduction number and the final epidemic size are theoretically analyzed.The numerical and stochastic simulations show that preventive rewiring inhibits the spread of infectious disease,and postpone the outbreak of infectious disease,while the extension of the infectious period promotes disease transmission.These research results can provide new method for modeling biological system with time delay and theoretical basis for prevention and control of infectious disease.