Analysis of Non-Monotone Dynamical Systems: Bifurcation Theory of Mutualism and Travelling Wave Solutions in Predator-Prey Models and Viral Transmission Models

Abstract

We investigate predator–prey models which take mutualism between two different predators and within one predator species, and within one predator species with logistic population growth respectively. Local and global dynamics are obtained for the model systems. By a detailed bifurcation analysis, we investigate the dependence of predation dynamics on cooperative predation and prove that mutualism may enhance survival of mutualist predators in a severe condition and break the competitive exclusion principle. We further provide quantitative information about how the cooperative predation (mutualism) may (i) establish multiple stability switches on the positive equilibrium; (ii) generate backward bifurcation on equilibria; (iii) induce supercritical or subcritical Hopf bifurcations; and (iv) establish bi-stability phenomenon between the predator-free equilibrium and a positive equilibrium (or a limit cycle). We further investigate traveling wave solutions of a diffusive predator-prey model which takes into consideration hunting cooperation as well as diffusive viral infection model with time delay. The sublinearity condition is violated for the function of cooperative predation. When the basic reproduction number for the diffusion-free model is greater than one, we find a critical wave speed below which no positive traveling wave solution shall exist. If the wave speed exceeds this critical value, we prove the existence of a positive traveling wave solution connecting the predator-free equilibrium to the unique positive equilibrium under a technical assumption of weak cooperative predation. The key idea of the proof contains two major steps: (i) we construct a suitable pentahedron and find inside it a trajectory connecting the predator-free equilibrium; and (ii) we construct a suitable Lyapunov function and use LaSalle invariance principle to prove that the trajectory also connects the positive equilibrium. For the diffusive viral infection model with time delay, we first establish the existence result by considering a perturbed system along with the construction of suitable upper and lower solutions so that Schauder fixed point theorem can be applied. Next, we use a limiting argument together with Lyapunov functional techniques to find a traveling wave solution which connects the infection-free equilibrium and the endemic equilibrium. We also propose open problems related to traveling wave solutions in cooperative predation.