Propagation Dynamics Of A Class Of Population Evolution Systems With Seasonality

Biological invasion,a common ecological phenomenon,has become a hot spot in the current international multidisciplinary intersection,which attracts the attention of scholars in many fields of science,including mathematicians.Consider the complex life cycle of the population and the varied nature of the environment,the invasion process present a rich pattern of space-time propagation.It is meaningful to describe these patterns mathematically to understand the invasion phenomenon.This thesis is devoted to modeling the invasive species with yearly breeding and maturation,and the model turn out to be periodic reaction-diffusion equations with periodic time delay and nonlocal term.Moreover,the influence of seasonal characteristics,dispersal schema and strong Allee effect on the propagation dynamics of the species can be analyzed mathematically by the model.Firstly,the biological characters of the interested alien species are described mathematically.Based on an age-structured population models,by classifying the population into two stages-matured species and immatured species,the time periodic population models are established.In such a way,the periodic time delay is introduced which means the critical age of being adult.With the seasonal characteristics,the Poincaré map is reduced to a special phase space.Furthermore,the iterative system defined by the reduced Poincaré map plays the key role in analyzing the models.Secondly,under the monostable structure,when the diffusion schema is local dispersal,the propagation dynamics of the iterative system defined by the Poincaré mapping is studied,including the existence of spreading speed,and its coincidence with the minimal wave speed.Through the variational characterization of spreading speed,it is found that the influence of the factors,such as the seasonal maturity and death on the spreading speed is complex.In particular,if the mature season is longer(shorter)than the breeding season,then the spreading speed can be slowed down(speed up).Further,the existence of traveling waves without monotonicity and compactness assumptions is proved by using the Schauder fixed point theorem and constructing upper and lower solutions.Also,the uniqueness of traveling waves up to a translation is proved by estimating the exact decay speed of the wave profiles.Thirdly,under the monostable structure,when the diffusion schema is nonlocal dispersal,the hidden relationship between dispersal kernel and the spreading speed is discovered: the spreading speed is finite if and only if dispersal kernel is exponentially bounded,the spreading speed is infinite if and only if dispersal kernel is exponentially unbounded.The latter case gives rises to an accelerating propagation.Further,the level set is proved to locate in between two exponential functions by applying induction method and constructing upper and lower solutions.The analysis is highly depend on the relationship between diffusion kernel and linear operator’s kernel of the Poincaré map.Fourthly,under the bistable structure induced by strong Allee effect,the existence of bistable waves are obtained by applying monotone semiflow theory.Invoking upper and lower solutions and appealing the squeezing idea,the uniqueness,Lyapunov stability,and globally exponentially stability of bistable waves are proved.Finally,the propagation dynamics of the iterative system are returned to the model systems.The results firstly be extended to the mature equation,then to the immature equation.For the latter step,an integral equation derived from the conservation law of ecological evolution plays the key role.