A Nonlocal Diffusion Problem With Free Boundaries In Spatial Heterogeneous Environment

Population invasion will not only break the original ecological balance,but also fundamentally change and destroy the ecological landscape,and then pose a threat to biodiversity,and even bring unpredictable impact on the development of human society.By using the method of mathematical modeling,biological mathematicians simulate the performance of alien species in the new living environment,so that we can clearly understand and predict the spread of alien species.During the spread of species,the boundary of the population expanding outwards is often changed,which promotes the introduction of free boundary conditions.A growing number of researchers have found that nonlocal diffusion can describe the diffusion phenomenon of species more accurately compared with local diffusion.Therefore,it is more practical to consider a nonlocal diffusion problem with free boundariesThis paper studies a nonlocal diffusion problem with free boundaries in spatial heterogeneous environment.First of all,we use the contraction mapping principle and extension method to obtain the global existence and uniqueness of the solution.After the comparison principle is established,through the study of the principal eigenvalue of the corresponding problem,we get the spreading-vanishing dichotomy and the corresponding criteria.These results show that if the intrinsic growth rate is greater than or equal to the diffusion rate of species,propagation phenomenon will always happen,if the intrinsic growth rate is less than the diffusion rate of species,when the species with larger initial living environment also can expand successfully,if the original living environment is small,the species to be successful requires larger expansion ability.As we consider the problem with a nonlocal item in the reaction item,we will change this problem into the system problem,then use iterative method to get the long-time behavior of the solution of the transmission case.