Spatial Propagation Dynamics Of Several Classes Of Diffusive Models

In recent years,reaction-diffusion models have attracted extensive attentions in various fields such as epidemiology,population ecology and chemistry for their rich and practical backgrounds.Particulary,the traveling waves and the spreading speeds have been widely studied since they precisely describe the partially spatialtemporal characteristics,which further explain many interesting natural phenomena and provide powerfully theoretical guidance.However,with the new problems emerging in technology and medicine,there are still many influencing factors that need to be explored,such as the role of latency and vaccination in a diffusive epidemic model,the impact of fast and slow progression on disease propagation and the influence of chemoattractant on moving population.Based on the importance of propagation dynamics,we select several reaction-diffusion models with important theoretical significance and backgrounds to study their propagation properties.Firstly,we study the propagation properties in a diffusive epidemic model with latency,vaccination and general nonlinear incidence.When the initial condition satisfies proper exponential decaying behavior,we present the spatial expansion feature of the infected.Different leftward and rightward spreading speeds are obtained with respect to different decaying initial values.Moreover,the convergence in the sense of compact open topology is also studied when the spreading speeds are finite.Finally,we show that the minimal spreading speed is the minimal wave speed of traveling wave solutions,which also presents the precisely asymptotic behavior of traveling wave solutions for the infected branch at the disease-free side.Due to the threshold,the vaccination does reduce the spreading ability of the disease.Secondly,we consider the propagation properties of SEIR model with fast and slow progression.We firstly use the local Lp-estimates and combine with the dual arguments to obtain the uniform boundedness of the solution for the initial value problem.When the basic reproduction number of the corresponding kinetic system is larger than one,we construct the upper solution of linearized system and apply the comparison principle of cooperative parabolic system to prove the upper bounds of spreading speed.To get the lower bounds,we introduce the principal eigenvalue of corresponding weak coupled elliptic system,the generalized eigenvalue problem and the uniform persistence theory.When the diffusion rates are the same,it is proven that the solution locally uniformly converges to the positive steady state by the corresponding entire solutions and the Lyapunov like argument.When the basic reproduction number of the corresponding kinetic system is less than one,the epidemic dies out uniformly.Finally,some numerical examples are provided to explain the main results and explore possible generalization.Thirdly,we investigate the speed selection mechanism of the monostable cholera model and the speed sign of the bistable integrodifference competition system.These systems generate monotonic semiflows,and we mainly apply the comparison principle combining with the method of upper-lower solution to analyze the wave speeds.For the cholera model,we theoretically prove that the nonlinear selection is realized if there is a fast decaying lower solution;and if there is a slowly decaying upper solution,then the linear selection is realized.Then by selecting smooth functions as the upper and lower solutions in a specific model,the parameter threshold which determines the linear or nonlinear selection is obtained.For the bistable integrodifference competition system,with specific kernel functions,we transfer our model into a coupled localized differential system.Sufficient conditions with symmetry are obtained on the propagation directions of the wavefronts.From these conditions,we may find that the relative competitiveness is monotone with respect to some parameters.Finally,we present the existence of traveling waves for the Keller-Segel chemotaxis system with logistic source.By constructing the suitable upper(lower)solution and applying Schauder’s fixed point theorem,we obtain that parabolic-parabolic(elliptic)system admits traveling waves under the new condition.Under this new condition,we show that the minimal wave speed of traveling wave for our system is linearly selected.Therefore,the chemoattractant do not speed up or slow down the spatial spreading of the moving population in some cases.