Classification of the spreading behaviors of a two-species diffusion-competition system with free boundaries

Abstract

<p>In this paper, we revisit the spreading behavior of two invasive species modelled by a diffusion-competition system with two free boundaries in a radially symmetric setting, where the reaction terms depict a weak-strong competition scenario. Our previous work (Du and Wu in Cal Var PDE 57:52, 2018) proves that from certain initial states, the two species develop into a "chase-and-run coexistence" state, namely the front of the weak species <i>v</i> propagates at a fast speed and that of the strong species <i>u</i> propagates at a slow speed, with their population masses largely segregated. Subsequent numerical simulations in Khan et al. (J Math Biol 83:23, 2021) suggest that for all possible initial states, only four different types of long-time dynamical behaviours can be observed: (1) chase-and-run coexistence, (2) vanishing of <i>u</i> with <i>v</i> spreading successfully, (3) vanishing of <i>v</i> with <i>u</i> spreading successfully, and (4) vanishing of both species. In this paper, we rigorously prove that, as the initial states vary, there are exactly five types of long-time dynamical behaviors: apart from the four mentioned above, there exists a fifth case, where both species spread successfully and their spreading fronts are kept within a finite distance to each other all the time. We conjecture that this new case can happen only when a parameter takes an exceptional value, which is why it has eluded the numerical observations of Khan et al. (J Math Biol 83:23, 2021) .</p>