A diffusive mutualistic model in a heterogeneous and periodically evolving environment

In the current paper, a two-species mutualistic model defined in a heterogeneous and periodically evolving domain is comprehensively investigated to explore some new dynamical phenomena induced by the domain evolution. By assuming that the evolution of domain is uniform and isotropic, the mutualistic model is converted from the evolving domain to a fixed domain. Initially, the ecological reproduction index, which depends on the evolution rate of the domain and spatial heterogeneity, is defined and its monotonicity is discussed in detail. Then, by employing the monotone iteration method and suitable upper and lower solutions, the asymptotic behaviors of the solutions are explored. Furthermore, the impact of the domain evolution rate on the persistence and extinction of mutualistic species is analyzed. Finally, numerical simulations are performed to illustrate our analytical results.