# Classification of global dynamics of competition models with nonlocal dispersals I: Symmetric kernels

## TL;DR

This paper classifies the global dynamics of two-species competition models with symmetric nonlocal dispersals, revealing conditions for stability, existence of steady states, and convergence behaviors, including extensions to heterogeneous coefficients and mixed dispersal strategies.

## Contribution

It provides a complete classification of the global dynamics for symmetric nonlocal dispersal competition models, including new results on stability, steady states, and extensions to variable coefficients and mixed dispersal strategies.

## Key findings

- Existence of infinitely many steady states when both semi-trivial states are stable.
- Solutions with non-negative, nontrivial initial data converge to a steady state.
- Generalization to models with location-dependent coefficients and mixed dispersal strategies.

## Abstract

In this paper, we gives a complete classification of the global dynamics of two- species Lotka-Volterra competition models with nonlocal dispersals: where K, P represent nonlocal operators, under the assumptions that the nonlo- cal operators are symmetric, the models admit two semi-trivial steady states and 0<bc<1. In particular, when both semi-trivial steady states are locally stable, it is proved that there exist infinitely many steady states and the solution with non- negative and nontrivial initial data converges to some steady state. Furthermore, we generalize these results to the case that competition coefficients are location-dependent and dispersal strategies are mixture of local and nonlocal dispersals.