Analysis of Asymmetrical Rock-Paper-Scissors (RPS) Competition Model in Heterogeneous Environments

Abstract

Asymmetrical rock-paper-scissors (RPS) competition refers to a dynamical system where species interact in a cyclic dominance pattern, but with unequal strengths or advantages. In the RPS models, three species interact such that each one outcompetes another, creating asymmetrical interactions that prevent any single species from dominating indefinitely. This competitive asymmetry phenomenon can arise due to the differences in reproduction rates, survival probabilities, or environmental factors that favour one species over another. Thus, understanding the factors driving community dynamics and coexistence mechanisms of multiple species is an important goal in ecology. In this paper, we examine the RPS competition framework extended along an environmental gradient. We formulate an extended system of nonlinear differential equations to capture asymmetrical RPS interactions under varying environmental conditions. Analytical techniques, including the evaluation of equilibrium points and linear stability analysis, are employed to determine stability and long-term dynamics of the system. In addition, numerical simulations are conducted to explore dynamical behaviours such as distinct steady states, limit cycles and heteroclinic cycles across environmental gradients. Our results reveal a variety of ecological outcomes driven by the interplay of asymmetrical RPS interactions and varying environmental conditions. Specifically, we demonstrate the emergence of distinct steady states, including scenarios where only a single-species persists, two-species coexistence, as well as the occurrence of multi-species coexistence. Additionally, stable limit cycles can occur, which indicate periodic fluctuations in species abundances. We also discover the presence of heteroclinic cycles, suggesting transient dynamics where species sequentially dominate before eventually returning to a similar state, and alternative stable states, where the final community composition depends on initial conditions. Ecologically, these findings highlight that the joint influences of RPS competitive asymmetries and environmental heterogeneity can lead to different community structures and ecological dynamics.