The global stability of coexisting equilibria for three models of mutualism

Survival and stationary distribution of a stochastic facultative mutualism model with distributed delays and strong kernels

To investigate the roles of both coupling noises and distributed delays with strong kernels, a novel delayed stochastic two-species facultative mutualism model is established, in where the strong kernels indicate that the maximum influence on the growth rate response at some time is due to population densities at the previous time, and the saturation effect is also incorporated because the facultative capacity of each species is finite and their interspecific mutualism should be upper bounded in real life. We first transfer the two-species stochastic model with strong kernels into an equivalent six-dimensional model through a linear chain technique. Later, sufficient conditions for the extinction exponentially, persistence in the mean, permanent in time average and stationary distribution are respectively obtained. Finally, numerical simulations are supplied to support our theoretical results. Our analytical results show that the coupling noise intensities play an important role in the long-time behaviors while the strong kernels are independent of the above asymptotic properties.

Global stability of the coexistence equilibrium for a general class of models of facultative mutualism

Many models of mutualism have been proposed and studied individually. In this paper, we develop a general class of models of facultative mutualism that covers many of such published models. Using mild assumptions on the growth and self-limiting functions, we establish necessary and sufficient conditions on the boundedness of model solutions and prove the global stability of a unique coexistence equilibrium whenever it exists. These results allow for a greater flexibility in the way each mutualist species can be modelled and avoid the need to analyse any single model of mutualism in isolation. Our generalization also allows each of the mutualists to be subject to a weak Allee effect. Moreover, we find that if one of the interacting species is subject to a strong Allee effect, then the mutualism can overcome it and cause a unique coexistence equilibrium to be globally stable.

Global stability results for models of commensalism

We analyze the global stability of the coexisting equilibria for several models of commensalism, first by devising a procedure to modify several Lyapunov functionals which were introduced earlier for corresponding models of mutualism, further confirming their usefulness. It is seen that commensalism promotes global stability, in connection with higher-order self-limiting terms which prevent unboundedness. We then use the theory of asymptotically autonomous systems to prove global stability results for models of commensalism which are subject to Allee effects, finding that commensalisms of appropriate strength can overcome the influence of strong Allee effects.