| In nature,the living areas of species are always changing and this change has a great impact on the reproduction of species,the spread of diseases,etc.There are two kinds of reasons for the change of area:one is caused by species’own behavior such as migration and invasion,which is called free boundary problem;the other is caused by external factors such as environment and climate.In this paper,we focus on the evolution of species’habitats and discuss the free boundary and periodic evolution domain of a class of cooperating models.The specific work is as follows.The free boundary problem of a class of two-species model is studied in a one-dimensional weak heterogeneous environment.We firstly derive the local existence and uniqueness of the solution by the contracting mapping principle and the L~p theory for parabolic partial differential equations.Then the uniform estimate of the solution is obtained,and its global existence and uniqueness follows in consequence.We use the upper and lower solution methods to give the spreading-vanishing dichotomy,i.e.,two reciprocal populations can spread to the right half-space and survive,or fail to spread successfully and eventually die out.Finally,a comparison principle is established and a criterion for spreading-vanishing is given.The effect of domain evolution on the persistence or extinction of species is understood by studying a class of three-species cooperating models on periodically evolving domain.To overcome the analytical difficulties caused by convective and dilution terms arising from regional evolution,we assume that evolution is periodic and isotropic,and transform the model by Lagrangian transformations into a class of reaction-diffusion equation models over an initial domain with diffusion coefficients and reaction terms that depend on the domain evolution rate.First,we linearize the model at the point of equilibrium and study the related periodic parabolic eigenvalue problem to obtain the dependence of the principal eigenvalue on the domain evolution rate and other properties.Then,the principal eigenvalues are considered as threshold parameters,and the existence and stability of the positive periodic solutions of the model are investigated using the upper and lower solution method,the comparison principle,the theory of the proposed singly increasing system,and the a priori estimation theory of the parabolic equation.The model is averaged over a period as the inverse of the square of the domain evolution rate ρ.The results show that the effect of periodic evolving domain on the persistence of species is negative if (?)＞1,while if (?)＜1,it is positive.Moreover,there is no effect if (?)=1.Finally,numerical simulations are used to confirm the theoretical analyses’ findings. |
