Propagation Dynamics Of Nonlocal Dispersal Equations In Inhomogeneous Environments

As the development of science and technology as well as mathematical tools,the understanding and description of natural phenomena become more and more accurate.The nonlocal diffusion equation thereupon has been proposed in the fields of ecology,materials science and epidemiology,and further got numerous experts and scholars of all subjects interested.It is the propagation dynamics of the nonlocal diffusion equation that has attracted much attention of mathematicians and biologists.Specifically,the propagation dynamics of the nonlocal diffusion equation mainly consist of traveling wave solutions,entire solutions and the asymptotic spreading speeds.In particular,influenced by factors such as temperature,humidity,climate and human society,the natural environment is complex and changeable.Therefore,it is essential to consider the inhomogeneity of the environment in the theoretical study.This paper is devoted to the propagation dynamics of several nonlocal diffusion equations in inhomogeneous environments.Firstly,we consider the propagation phenomena of bistable traveling waves for a nonlocal dispersal equation in locally disturbed environments.The locally disturbed environment means that the inhomogeneous nonlinearity consists of two spatially independent bistable nonlinearities connected by a spatially dependent nonlinearity defined on a bounded domain.We establish the existence and uniqueness of the entire solution connecting two traveling wave solutions pertaining to the different nonlinearities by upper and lower solutions and a "squeezing" technique,and show that such an entire solutions is Lyapunov stable.Our results indicate that the bistable traveling wave of the equation with one homogeneous nonlinearity approaching from infinity,after going through the transition region,and as time goes to infinity it converges to the other bistable traveling wave prescribed by the homogeneous nonlinearity on the other side.Secondly,we derive the existence of multi-type forced waves to nonlocal dispersal KPP equations with shifting habitats.The shifting habitat means that the intrinsic growth rate depends on time and space by the form of moving coordinates(?=x-ct).The severity of the environment is usually described by the asymptotic behaviors near negative infinity of the intrinsic growth rate,which is nondecreasing along the positive spatial direction.We find that for different severity of the environment with shifting speed in different intervals,there exist multi-type forced traveling waves.Specifically,applying upper and lower solutions with monotone iteration methods,we derive that:(i)there exist not only monotone but non-monotone forced waves,and(ii)even at the same wave speed coinciding with the shifting speed of habitats,more than one forced waves can be found.Thirdly,we investigate the asymptotic propagation of a nonlocal dispersal population model with shifting habitats.In particular,we verify that the invading speed of the species is determined by the speed c of the shifting habitat edge and the behaviors near infinity of the species' growth rate which is nondecreasing along the positive spatial direction.In the case where the species declines near the negative infinity,we conclude that:extinction occurs if c>c*(?);while c<c*(?),spreading happens with a leftward speed min{c;c*(?)} and a rightward speed c*(?),where c*(?)is the minimum KPP traveling wave speed associated to the species' growth rate at the positive infinity.The same scenario will play out for the case where the species' growth rate is zero at negative infinity.In the case where the species still grows near negative infinity,we show that the species always survives "by moving"with the rightward spreading speed being either c*(?)or c*(-?)and the leftward spreading speed being one of c*(?),c*(?)and c,where c*(-?)is the minimum KPP traveling wave speed corresponding to the growth rate at the negative infinity.In addition,we use some numeric simulations to present and explain the theoretical results.Finally,we study the propagation of bistable plane waves in environments with an obstacle.Here,the environment with an obstacle is indeed an exterior domain,which refers to the complement of some compact set K in N dimensions.Specifically,we use the plane traveling wave solution of bistable nonlocal diffusion equation to construct the suitable upper and lower solutions,and apply the comparison principle to obtain the entire solution which uniformly converges to the plane traveling wave solution in space as time goes to negative infinity.By taking advantage of this property,the uniqueness of the entire solution is established.Furthermore,the long time behavior of the entire solution depends on the geometry property of K:If K is a compacted convex set,then the entire solution,after goes through the obstacle K,converges uniformly to the same plane traveling wave solution as time goes to positive infinity;otherwise,the entire solution can not recover its profile,instead,it converges uniformly to some nonconstant steady-state solution as time goes to positive infinity in any bounded subset.Finally,the entire solution still converges to the plane traveling wave solution in any direction with the distance away from K becoming infinity.Our results indicate that a plane traveling wave solution coming from infinity continue to propagate after going through an obstacle,which causes disturbance on the profile of the plane traveling wave solution,and whether the profile can recover uniformly in space depends on the geometry of the obstacle.