The Study Of Dynamics Behavior For Some Mathematical Models Arising From Biology

Mathematical biology is a embranchment of biology which study the develop-ment law of species. Scholars have studied, explained, predicted these laws by setting up deterministic mathematical models all the time. However, population is inevitably affected by all kind environments and outburst calamities like white noise and L(?)vy noise in real world. Then Stochastic differential equations can depict reality problem more accurately. This thesis, which consists of five chapters, is mainly concerned with the well-posedness and dynamical behaviors for some types of stochastic population modelsIn Chapter 1, the backgrounds of our study are introduced and the relevant pre-liminaries are presented. We first recall backgrounds of the mutualism model by May [61] and research progress of the concerned stochastic differential equations. Then we present some preliminaries used throughout this thesis, such as the stochastic in-tegration w.r.t. Brown motion, the stochastic integration w.r.t. L(?)vy jump, Wiener integration w.r.t. multi-fractional Brown motion as well as some useful inequalities.In Chapter 2, the long-time behaviours about a stochastic mutualism model driv-en by white noises are investigated. First, we refer to some ideas from Mao, Xuerong et.al and Jiang Daqing et.al [48,58,62,70,132] to obtain existence and uniqueness of the global solution by using upper-sub solution method and comparison theorem for SODE with white noise. Then, we discuss the stochastically ultimate boundedness, uniformly Holder-continuous and stochastic permanence, persistence in mean and ex-tinction together with Ito formula, Chebyshev’s inequality, moment inequality. Most important of all, we get the sufficient conditions of persistence in mean and extinction.In Chapter 3, we carry out on stochastic non-autonomous mutualistic model driv-en by L(?)vy noises. In this chapter we aim to extend the conclusions about stochas-tic mutualistic model driven by white noises to the stochastic stochastic mutualistic model driven by the L(?)vy jump noises. Because of L(?)vy jump noises leading to non-continuity of solution, some method in chapter 2 will be unsuited. In section 3.4, we obtain a critical lemma to cover this gap by using Laws of Strong Large Numbers, Exponential martingale inequality and theory of measure.-vii-Chapter 4 is devoted to a mutualism model with double free boundaries and main-ly has the following results:Firstly we give the local existence and uniqueness of a classical solution by straightening boundaries. Then we obtain the monotonicity of double boundaries and dichotomy. Furthermore, we give the the existence of global fast solution and global slow solution. Our results show that when a1a2 > 1 and the initial data is sufficiently small the global fast solution exist or the solution is grow up, and the global slow solution is possible if a1a2 < 1 and the initial data is suit-ably large. In Section 4.5,4.6, we study how advection term affects the asymptotic spreading or vanish. Section 4.5 is devoted to sufficient conditions for the invasive species to vanish or spread, a spreading-vanishing dichotomy will be given. Some rough estimates of the spreading speed are also given in Section 4.6In Chapter 5, we study a non-autonomous Gilpin-Ayala equation driven by multi-fractional Brown motion. The main difficulty in studying multi-fractional Brown mo-tion is that we cannot apply stochastic calculus developed by Ito since MFBM is nei-ther a Markov process nor a semi-martingale when h(t)>2/1. We prove the existence of solution by using semi-martingale approximation method and Malliavin calculus.