Long time behavior of a Lotka–Volterra competition system with two dynamical resources and density-dependent motility

In this paper, we consider a Lotka–Volterra competition system with two dynamical resources and density-dependent motility under the homogeneous Neumann boundary condition. Here, we put the two competing species into a predator–prey system, and assume that the two competing species as predators can feed on different preys and that the preys as resources admit temporal dynamics including spatial movement, intrinsic birth–death kinetics and loss due to predation. When the distributions of prey’s resources can be homogeneous, by using some proper Lyapunov functionals and applying LaSalle’s invariant principle, we obtain that the solution can converge to the positive steady state exponentially or to the competitive exclusion steady states algebraically as time goes to infinity. Our finding shows that the consideration of temporal dynamics on the resources can lead to the coexistence of two competitors in some parameter conditions regardless of their dispersal rates. When the distributions of prey’s resources are spatially heterogeneous, we conduct several numerical simulations in different combinations of dispersal strategy and the distributions of prey’s resources, and we show that the non-random dispersal and heterogeneous distributions of prey’s resources can affect the fates of two competitors.