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

Abstract

In this paper, we give a complete classification of the global dynamics of two-species Lotka–Volterra competition models with nonlocal dispersals: $$\begin{aligned} {\left\{ \begin{array}{ll} u_t= d \mathcal {K}[u] +u(m(x)- u- c v) &{}\text {in } \Omega \times [0,\infty ), v_t= D \mathcal {P}[v] +v(M(x)-b u- v) &{}\text {in } \Omega \times [0,\infty ), u(x,0)=u_0(x),~v(x,0)=v_0(x) &{}\text {in } \Omega , \end{array}\right. } \end{aligned}$$ where $$\mathcal {K}$$ , $$\mathcal {P}$$ represent nonlocal operators, under the assumptions that the nonlocal operators are symmetric, the models admit two semi-trivial steady states and $$0<bc\le 1$$ . 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 nonnegative and nontrivial initial data converges to some steady state in $$C(\bar{\Omega })\times C(\bar{\Omega })$$ . Furthermore, we generalize these results to the case that competition coefficients are location-dependent and dispersal strategies are mixture of local and nonlocal dispersals.