No abstract

In this paper, we investigate the dynamics of an improved Leslie-Gower predator–prey model which is characterized by the reduction of the prey growth rate due to fear of the predator (i.e., antipredator behavior). The value of this study lies in two aspects: mathematically, (i) it provides the existence and the stability of the positive equilibrium; (ii) it gives the existence of the Hopf bifurcation and limit cycle; and (iii) it shows the mechanisms of the fear factor and the prey refuge on the level of the positive equilibrium. Biologically, we find that the influence of the fear factor is complex: (i) increasing the level of fear can cause the level of the population density to decrease and the prey to become extinct; (ii) the effect of the cost of fear on the stability of the positive equilibrium is rich and complex: it can either destabilize the stability and benefit the emergency of the periodic behavior or stabilize the system by excluding the existence of periodic solutions; (iii) with a fixed level of fear, the prey refuge is beneficial to the coexistence of the prey and the predator, and with the increase of the level of the prey refuge, the positive equilibrium may change from stable spiral sink to unstable spiral source to stable spiral sink. These results may enrich the dynamics of the predator–prey systems.

In this paper, the global stability of a virus dynamics model with intracellular delay, Crowley-Martin functional response of the infection rate, and CTL immune response is studied. By constructing suitable Lyapunov functions and using LaSalles invariance principle, the global dynamics is established; it is proved that if the basic reproductive number, R 0 , is less than or equal to one, the infection-free equilibrium is globally asymptotically stable; if R 0 is more than one, and if immune response reproductive number, R 0 , is less than one, the immune-free equilibrium is globally asymptotically stable, and if R 0 is more than one, the endemic equilibrium is globally asymptotically stable.

a b s t r a c tIn this paper, an n-species strongly coupled cooperating diffusive system is considered in a bounded smooth domain, subject to homogeneous Neumann boundary conditions. Employing the method of energy estimates, we obtain some conditions on the diffusion matrix and inter-specific cooperatives to ensure the global existence and uniform boundedness of a nonnegative solution. The globally asymptotical stability of the constant positive steady state is also discussed. As a consequence, all the results hold true for multispecies Lotka-Volterra type competition model and prey-predator model.