Mutualism as a stabilizing effect on the population densities of two interacting species
DOI:
https://doi.org/10.17268/sel.mat.2025.02.05Keywords:
Mutualism, conditioned interaction, consumer-resources model, Allee effect, Dulac function, periodic solutionAbstract
Mathematical models are a very useful tool to understand, describe or predict the population dynamics of species interacting. Ecologists and mathematicians have extensively studied the predator-prey, victim-exploiter, competition and mutualistic relationships. However, mutualism between species has not received the same attention as the other ecological interactions. In this work, we exclude periodic solutions of three types of systems by the construction of Dulac functions. These systems can be used to describe the population dynamics of mutualistic species. The system type I includes a wide variety of mutualistic models in which both the intrinsic rate of increase and the carrying capacity of each species increase by the interaction between species.
In particular, the system type I can be applied to exclude periodic solutions of models with conditioned interactions such that mutualism occurs at low population densities and competition occurs at high population densities. The system type II includes mutualistic models that describe a consumer-resources interaction. In these models, it is assumed that the net change of benefitscosts due to the interaction depends on the densities of the recipient species and the partner one.
The system type III describes mutualistic models in which the per capita growth rate of each species is affected by a weak Allee effect. We also apply the results of this work to models mentioned in a historical list of mutualistic models provided in [1]. From the results obtained, we conclude that mutualism leads to the exclusion of periodic behaviors in the population dynamics of interacting species. Therefore, the population densities of the mutualistic species converge to an equilibrium point. Then, when the population densities oscillate, the oscillatory behaviors are transient. These results are relevant since the dynamics of mutualistic species has not been deeply characterized and the discussion about the existence of sustained oscillatory behavior in mutualistic species is relevant from an ecological perspective.
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