Dynamics Of Stochastic Functional Differential Equations

Functional differential equations(FDEs)arise from evolution phenomena in physical processes and biological systems,in which time-delay is used for mathematical modelling to describe the dynamical influence from the past.Random phenomena are common in nature,and complex systems in science and engineering are often affected by random factors.As mathematical models of complex systems under the influence of random factors,stochastic FDEs have a strong application background and practical significance,and have been widely concerned and developed rapidly in recent decades.Stochastic systems are not only closely related to the dynamical behavior of deterministic systems,but also put forward their own conditions and requirements,which are different from deterministic systems.In particular,the change of dynamical behaviors caused by noise is one of the most difficult problems.By using theory of FDEs,stochastic dynamical systems,and infinite-dimensional dynamical systems,this dissertation is devoted to the dynamical behaviors of several kinds of stochastic FDEs,including the existence,uniqueness,regularity,and structure of random attractors,the existence of invariant measures,the existence and exponential stability of stationary solutions,the stationary distribution and long-time behavior of a stochastic predator-prey model and an epidemic model.This dissertation consists of five chapters.In Chapter 1,the background of dynamics of stochastic FDEs is briefly introduced,some recent results on invariant measures and random attractors of stochastic FDEs as well as the dynamics of biological population models are provided.In the meantime,our main contribution of this dissertation is summarized and some preliminaries are provided as well.Chapter 2 is devoted to a general stochastic delay differential equation with infinite dimensional diffusions in a Hilbert space.We not only investigate the existence of invariant measures with either Wiener process or Lévy jump process,but also obtain the existence of a pullback attractor under Wiener process.In particular,we prove the existence of a non-trivial stationary solution which is exponentially stable and is generated by the composition of a random variable and the Wiener shift.At last,examples of reaction-diffusion equations with delay and noise are provided to illustrate our results.Chapter 3 is devoted to a stochastic semilinear degenerate parabolic equation with delay in a bounded domain in RN and the nonlinearity satisfying an arbitrary polynomial growth condition.The random dynamical system generated by the equation is shown to have a random attractor in C([-τ,0],LP(O)∩D01(O,σ)),which is a compact and invariant tempered set and attracts every tempered random subset of C([-τ,0],L2(O))in the topology of C([-τ,0],Lp(O)).In a particular case,the random attractor consists of singleton sets(i.e.,a random fixed point),which generates an exponentially stable non-trivial stationary solution.This theoretical result improves some recent ones for stochastic semilinear degenerate parabolic equations.In Chapter 4,we investigate the dynamical behaviors of a stochastic predatorprey model with a general functional response and the random factors acting on both the intrinsic growth rates and the intra-specific interaction rates.First,the stochastic boundedness of the solution to the stochastic model is derived.Then conditions for the persistence and extinction of the two species are established in terms of two newly introduced thresholds λ1 and λ2.In particular,the existence of a stationary distribution of the system and weak convergence of the boundary process are investigated as well.Finally,some numerical simulations are performed to illustrate our theoretical results and to show that appropriate intensities of white noises may make the predator and prey population fluctuate around their deterministic steady-state values;but too large intensities of white noises may make the predator and/or prey population go to extinction.In Chapter 5,we investigate a stochastic regime-switching susceptible-infectedsusceptible epidemic model with nonlinear incidence rate and Lévy jumps.By defining a threshold λ,it is proved that if λ>0,the disease is persistent and there is a stationary distribution.And when λ<0,we show that the disease goes to extinction and the susceptible population weakly converges to a boundary distribution.Moreover,the existence of the stationary distribution is obtained and some numerical simulations are performed to illustrate our results.It is very interesting to notice that Lévy noise can suppress disease outbreak,that the disease of a regime-switching model may have the opportunity to persist eventually even if it is extinct in one regime,and that the spread of the disease can be controlled by way of suitable protection measures to reduce the value of transmission rate of disease when susceptible individuals contact with infected individuals.