Dynamics of Two Species with Density-Dependent Interactions and Application to Mutualism

Ecological interactions shape the dynamics of natural populations in the wild. Density-dependent processes are widespread and may change the respective effects of populations on one another, for instance by shifting interactions from mutualistic to parasitic relationships. Here, we develop a general deterministic model of two interacting populations, assuming density-dependent costs and benefits for the interacting individuals within and between species. This framework aims at generalizing pre-existing population dynamics models involving competition, predation, mutualism and parasitism, by allowing ecological interactions to transition when the respective densities of interacting species change. Through ordinary differential equations and phase portrait analysis, we derive general principles governing these systems, identifying constraints on the organization of equilibria and sufficient conditions for the emergence of certain dynamic behaviors. In particular, we show that equilibrium indices alternate along isoclines under broad geometric assumptions, and that limit cycles can arise when interactions include mutualistic and parasitic phases, while they cannot be generated locally in strictly mutualistic regions where the relevant interaction signs remain fixed. This framework provides a general approach for characterizing the population dynamics of interacting species and highlights the effect of the density dependent transitions in ecological interactions.