A Stage-structured Continuous-/Discrete-time Population Model: Persistence And Spatial Spread

Population persistence and spatial propagation are of great importance in spatial ecology.The life cycle of many species in nature includes several distinct stages(such as seeds(larvae),juveniles,adults,etc.),and the species invade new habitats through the dispersal of organisms in their early life stages(such as plant seeds,shellfish larvae).We developed a stage-structured continuous/discrete-time hybrid model to describe the spatiotemporal dynamics of such species,in which a reaction-diffusion equation describes the random movement of dispersal individuals,while two difference equations describe the growth of non-dispersal individuals.Through theoretical analysis and numerical simulation,we studied the population persistence and spatial propagation.The thesis consists of six chapters.In Chapter 1,we introduced biological background,significance,the current research situation at home and abroad,the main contents of this thesis,and the basic theoretical knowledge of mathematics needed.In Chapter 2,we constructed the impulsive reaction-diffusion population model with stage-structure continuous-/discrete-time.We first considered the case that the population does not dispersal,and obtained the condition that the system has a unique positive constant equilibrium solution,and showed that the condition for the existence of positive constant equilibrium solution is necessary for the growth and dispersal of the population in space.In Chapter 3,we studied the spreading speed of the population in the unbounded domain under the condition of the existence of the positive constant equilibrium solution of the system.In order to understand the dynamic behavior of the population,we discussed the monotone and non-monotone cases of the population birth function,respectively.When the population birth function is monotone,we linearized the model at the origin and obtained the accurate formula for spreading speed in terms of model parameters,and then verify that the nonlinear model and the linearized model have the same spreading speed(i.e.linear determinacy).When the population birth function is non-monotone,we introduced two monotone birth functions and constructed auxiliary system to study the population spreading speed.Numerical simulations verify and complement the theoretical results.In Chapter 4,we discussed the existence of traveling wave solutions of the model and showed that the spreading speed of the population can be characterized as the slowest speed of a class of traveling wave solution.Due to the solution mapping of the system does not have compactness and the population birth function is monotone,the solution mapping of the system satisfied the condition of weak compactness.So we obtain the existence of traveling wave solutions of the system.When the population birth function is non-monotone,we obtain the existence of traveling wave solutions of the system by the asymptotic fixed point theorem.Numerical simulation is used to verify the theoretical analysis of traveling wave result.In Chapter 5,we studied the critical domain size for the sustainable survival of the population in a bounded domain with a death boundary,that is,the minimum interval length to ensure the population persistence.When the population birth function is monotone,we linearized the model at the origin and used the model parameters to give an explicit formula for critical domain size.When the population birth function is nonmonotone,we constructed two auxiliary systems with monotone birth rate functions to study the critical domain size.Finally,Numerical simulations verify and complement the theoretical results.In Chapter 6,we summarized the main conclusions of this thesis,and discussed the shortcomings of the research work and future research directions.