Joint Effects of Symmetric and Asymmetric Dispersal on the Complex Dynamics of a Multiple Interactions Type System

Abstract This paper extends a multiple interactions type ecological system by incorporating both symmetric and asymmetric dispersal mechanisms. Using this partial differential equations model, we numerically show different spatio-temporal dynamics of interacting species. We then examine the dynamical behaviours of the system for a wide range of parameter values, thereby permitting an investigation of the interplay between local dynamics due to mutualist-resource-competitor-exploiter interactions on the one hand and dispersal rate and its (a)symmetry, on the other. To do this, a co-dimension one bifurcation analysis is performed using the magnitude of competitor and resource species interactions under various dispersal scenarios. In the absence of dispersal, our analysis by varying the competitive pressure uncovers some bifurcational changes in dynamics such as supercritical Hopf and transcritical bifurcations. The interactions between these two local bifurcations result in intriguing outcomes, namely coexistence steady states, stable limit cycles and alternative stable states. Inclusion of low (or moderate) levels of symmetric dispersal into the system modifies the response of this ecological community towards the change in competitive pressure and mediates distinct multiple species coexistence steady states. As symmetric dispersal increases to rapid migration levels, a greater stabilising impact on the dynamics of the system is observed and this situation enhances the likelihood of species diversity. In the case of asymmetric dispersal, alternative stable states phenomenon has been found to be more pronounced due to the emergence of different bistable attractors, which include two coexisting stable limit cycles. Owing to this reason, we predict sensitivity of dynamics to small perturbations in initial densities of mutualist-resource-competitor-exploiter species. It is also unveiled that the (in)direct effects exerted by resource species can interact with dispersal asymmetry to engender subcritical Hopf, period-doubling and limit point bifurcations of cycles: these phenomena also exhibit complex dynamical behaviours such as alternative stable states or unstable dynamics, which can stabilise or destabilise this multi-species community.