Propagation Dynamics Of Two Kinds Of Classical Biological Mathematical Models

In this thesis,we study the propagation dynamics of two kinds of classical biomath-ematical models:a delayed non-local reaction diffusion equation and the Lotka-Volterra reaction diffusion model with periodic coefficients.The main contents are as follows:The first chapter briefly introduces the research background and significance of this thesis,the mathematical symbols and the preliminary knowledge used throughout of this thesis.In the second chapter,we first determine an interval of values of the bistable wave speed for delayed non-local reaction diffusion equations by the theory of asymptotic speed-s of spread for monotone semiflows.Then we determine the relationship between the bistable wave speed and wave speeds of the upper and lower solutions by using the com-parison principle and the globally asymptotic stability of the bistable travelling wave.Then we obtain a novel general condition for determining the speed sign by constructing a new upper and lower solution.Finally,we obtain the explicit conditions for determining the bistable wave speed sign by taking three kinds of specific kernel functions.In the third chapter,we study the existence,uniqueness and stability of the time-periodic bistable traveling wave solution of Lotka-Volterra reaction diffusion model by applying the theory related to the monotone dynamical system.The obtained results improve that in the known literatures.In the fourth chapter,we study the bistable wave speed of the time-periodic bistable traveling wave of Lotka-Volterra reaction diffusion model.Firstly,the upper and lower bounds of the time-periodic bistable wave speed are determined by using the theory of asymptotic spreading speeds for monotone semiflows,and the relationship between the bistable wave speed and the monostable wave speed is given.Then,by using the comparison principle and the global asymptotic stability of the bistable traveling wave,the relationship between the bistable wave speed and wave speed of the upper(lower)solution is established.According to the decay rate of bistable traveling wave near two stable equilibrium points,new upper and lower solutions are constructed,thus the general conditions for determineing the sign of bistable wave speed are established,and the explicit conditions for determining the sign of bistable wave speed are obtained.The numerical simulation results are consistent,with the analysis results.In the fifth chapter,we summarize the main work of this thesis,illuminate the main innovations,and put forward the problem that can be further studied.