Stability Analysis Of Discrete-time Multi-patch Model With Multiple Dispersal

In the increasingly diverse society,multi-patch models appear everywhere,such as transportation networks,social networks,neural networks,multi-patch population models and so on.It should be noted that it is more close to the realistic situation that each component can dispersal among these patches.For instance,in multi-patch predator-prey model,both predators and preys could disperse among these patches.Furthermore,in recent years,discrete-time systems have received much attentions since it is necessary to discretize these models into their discrete-time version when we compute or simulate some practical models by computer.Also,for most differential equations,especially for high-dimensional multi-patch models described by differential equations,their analytic solutions cannot be derived easily.In this case,some numerical methods should be used to discretize them to get their approximate numerical solution.Since many applications of practical systems are based on their stability,it is interesting to study the stability of discrete-time multi-patch model with multiple dispersal.On the one hand,time delay is ubiquitous in our lives,and it may make systems unstable.In the second chapter,the stability of discrete-time delayed multi-patch model with multiple dispersal will be investigated.By combining multi-digraph theory and Lyapunov functional method,two kinds of sufficient criteria are given to ensure the exponential stability of the considered model.These stability criteria show the close relationship with the dispersal topology of each component,dispersal strength and the upper and lower bounds of the time delay.Then the derived results are applied to study the stability of multi-patch discrete-time predator-prey model where both predators and preys dispersal among the multiple patches.The numerical simulation is also presented to show the rationality.On the other hand,in our daily life,stochastic perturbations are inevitably met,and they can not only make a stable system unstable,but stabilize an unstable system.In the third chapter,stochastic factor is considered.By utilizing Lyapunov method,graph-theoretic technique and stochastic analysis technique,several sufficient criteria are given to promise the asymptotic stability in probability of discrete-time stochastic multi-patch model with multiple dispersal.These criteria show that under required dispersal topology,if the dispersal strength and the strength of stochastic perturbation are small enough,the considered model will remain stable.Based on the derived results,a class of discrete-time stochastic multi-patch oscillator model is considered.The numerical simulation is given at thesame time.