On an advection–reaction–diffusion competition system with double free boundaries modeling invasion and competition of Aedes Albopictus and Aedes Aegypti mosquitoes

Abstract

Based on the invasion of the Aedes albopictus mosquitoes and the competition between Ae. albopictus and Ae. aegypti mosquitoes in the United States, we consider an advection–reaction–diffusion competition system with two free boundaries consisting of an invasive species (Ae. albopictus) with density u and a local species (Ae. aegypti) with density v in which u invades the environment with leftward front x=g(t) and rightward front x=h(t). In the case that the competition between the two species is strong-weak and species v wins over species u, the solution (u,v) converges uniformly to the semi-positive equilibrium (0,1), while the two fronts satisfy that limt→∞⁡(g(t),h(t))=(g∞,h∞)⊂R. In the case that the competition between the two species is weak, we show that when the advection coefficients are less than fixed thresholds there are two scenarios for the long time behavior of solutions: (i) when the initial habitat h0<π(4−ν12)−1 and the initial value of u is sufficiently small, the solution (u,v) converges uniformly to the semi-positive equilibrium (0,1) with the two fronts (g∞,h∞)⊂R; (ii) when the initial habitat h0≥π(4−ν12)−1, the solution (u,v) converges locally uniformly to the interior equilibrium with the two fronts (g∞,h∞)=R. In addition, we propose an upper bound and a lower bound for the asymptotic spreading speeds of the leftward and rightward fronts. Numerical simulations are also provided to confirm our theoretical results.