Table of Links Abstract and 1. Introduction Abstract and 1. Introduction The well-posed global solution


Nontrivial stationary solution
3.1 Spectral theory of integro-differential operator
3.2 The existence, uniqueness and stability of nontrivial stationary solution


The sharp criteria for persistence or extinction


The limiting behaviors of solutions with respect to advection


Numerical simulations


Discussion, Statements and Declarations, Acknowledgement, and References The well-posed global solution The well-posed global solution The well-posed global solution Nontrivial stationary solution
3.1 Spectral theory of integro-differential operator
3.2 The existence, uniqueness and stability of nontrivial stationary solution Nontrivial stationary solution Nontrivial stationary solution 3.1 Spectral theory of integro-differential operator 3.1 Spectral theory of integro-differential operator 3.2 The existence, uniqueness and stability of nontrivial stationary solution 3.2 The existence, uniqueness and stability of nontrivial stationary solution The sharp criteria for persistence or extinction The sharp criteria for persistence or extinction The sharp criteria for persistence or extinction The limiting behaviors of solutions with respect to advection The limiting behaviors of solutions with respect to advection The limiting behaviors of solutions with respect to advection Numerical simulations Numerical simulations Numerical simulations Discussion, Statements and Declarations, Acknowledgement, and References Discussion, Statements and Declarations, Acknowledgement, and References Discussion, Statements and Declarations, Acknowledgement, and References 3 Nontrivial stationary solution 3.1 Spectral theory of integro-differential operator The eigenvalue problem is an important tool to study the longtime behaviors of elliptic and parabolic problems, especially the sign of principal eigenvalue of the linearized operator can determine the existence and stability of nontrivial stationary solution, even the persistence of species, see [20, 32, 44, 45]. And some standard properties have shown in [32]. 3.2 The existence, uniqueness and stability of nontrivial stationary solution Firstly, we shall the strong maximal principle for the stationary problem. Notice that (3.5) (or (3.6)) has only one trivial solution u = 0.In the following, we still only consider the details for (3.5). Hence, we obtain a contradiction, and so the theorem holds. In summary, the results about the existence/nonexistence, uniqueness of nontrivial solutions of (3.5) and the stability of nontrivial/trivial solution are as follows. 4 The sharp criteria for persistence or extinction Authors:
(1) Yaobin Tang, School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, P. R. China;
(2) Binxiang Dai, School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, P. R. China (bxdai@csu.edu.cn). Authors: Authors: (1) Yaobin Tang, School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, P. R. China; (2) Binxiang Dai, School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, P. R. China (bxdai@csu.edu.cn). This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv available on arxiv

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