Stability and Oscillatory Dynamics in Fractional Ecological Models Using a Crossing Boundary Framework

Abstract

In this study, we delve into the dynamics of plant, pollinator, and herbivore interactions using fractional models, which effectively capture memory effects and intricate temporal dependencies that conventional integer-order models often overlook. A significant result is the numerical evidence of oscillatory dynamics induced by variations in the predator mortality parameter. Our findings reveal that the dynamics and stability of the system depend on the fractional order. In the classical case, a Hopf bifurcation emerges, accompanied by a limit cycle, in agreement with the existing literature. Moreover, analysis across different fractional orders shows similar behavior when a pair of complex eigenvalues crosses the Matignon sector, inducing a change in the equilibrium’s stability and producing oscillatory patterns. These insights offer valuable information on the parameters that drive ecosystem dynamics and contribute to a more comprehensive understanding of fractional system stability in ecological modeling.