Spreading Speed And Acceleration Propagation Of Nonlocal Dispersal Equations

This paper is devoted to the investigation of spatial propagation in nonlocal dispersal equation which is an important kind of evolution equation,including trav-eling wave solution,spreading speed and acceleration propagation.The following three problems are considered:(i)how does asymmetric dispersal kernel influence the signs of spreading speeds,(ii)how does different types of initial data influ-ence spreading speeds,(iii)whether the acceleration propagation has transferability property.Finally,we have completed the following four results.Firstly,the spreading speed is studied in nonlocal dispersal equation.Here the dispersal kernel is asymmetric and "light-tailed",and the initial data belongs to two types of exponentially bounded functions.For the first type of initial data,the asymmetry property of dispersal kernel can determine the signs of spreading speeds.Furthermore,there are three important influences of asymmetric dispersal kernel on nonlocal dispersal equation:(i)it can determine the spatial propagation directions of solutions;(ii)it may influence the stabilities of equilibrium states,(iii)it may also influence some monotone property of the solutions.For the second type of initial data,the spreading speed depends on the exponentially decaying rate of initial data,more precisely7,it increases with the decrease of such exponentially decaying rate.This result shows the diversity property of spreading speed in nonlocal dispersal equation.From these results,nonlocal dispersal equation and reaction-diffusion equation have not only similarity but also their own speciality in the aspect of spreading speed.On the one hand,if the dispersal kernel is symmetric,then such two equations have a similar property.On the other hand,the asymmetry property of dispersal kernel can change the signs of spreading speeds,which further change many properties of nonlocal dispersal equation.In addition,we construct a new lower solution and apply a new technique—"forwards-backwards spreading" technique.All the methods used here are applicable not only to nonlocal dispersal equations,but also to classical reaction-diffusion equations.Secondly,the spreading speed is considered in a nonlocal dispersal epidemic model,which is widely applied to the study of epidemics with oral-faecal transmis-sion.This part will continue to study the influences on spreading speed of asym-metric dispersal kernel and two types of exponentially bounded initial data.For the first type of initial data,the spreading speeds are two constants.Furthermore,their signs are determined by the number of elements in some set,which is essentially influenced by two dispersal kernels,Therefore,a relationship between the signs of spreading speeds and the dispersal kernels is established.For the second type of initial data,the spreading speed is decreasing with respect to the exponentially de-caying rate of initial data,further,its minimum value coincides with the spreading speed for the first type of initial value.These results also provide some guidance for better control of the spatial propagation of epidemics.Thirdly,the acceleration propagation of nonlocal dispersal cooperative system is studied.A phenomenon is found that the components which propagate by acceler-ating can accelerate the spatial propagation of other components.This phenomenon is called the transferability property of acceleration propagation.Furthermore,the spatial propagation of all species in this system is mainly determined by the maximum function of dispersal kernel.From it,a more surprising phenomenon is found:the spatial propagation of every species is accelerated by each other.Then it gives us a new and deep understanding of the cooperation property in the spatial propagation,which is usually attended in the population growth.Finally,the transferability property of acceleration propagation is considered in reaction-diffusion cooperative system.This paper mainly considers two types of initial data,namely,the exponentially unbounded and the partial exponentially unbounded.For the first type of initial da.ta,the solution propagates by accelerating.As we know,this is the first research work on acceleration propagation in reaction-diffusion system.For the second type of initial data,all components of the solution propagate also by accelerating,although they may have different types of initial data.Indeed,the cooperation property leads to that the spatial propagation of the components with exponentially bounded initial data is accelerated by those with the exponentially unbounded.This result shows that the transferability property of acceleration propagation also exists in reaction-diffusion cooperative system.