Event-triggered impulse control on reaction–diffusion Gilpin–Ayala competition model with multiple stationary solutions

Abstract

In previous literature, the parameter restrictions for the reaction–diffusion Gilpin–Ayala competition ecosystem were limited to the control of the first eigenvalue of the Laplace operator. This paper considers extending the parameter to the control of the second eigenvalue. However, exceeding the first eigenvalue results in a loss of compactness in the energy functional of the determined ecosystem. Consequently, this paper discusses the orthogonal decomposition of a given Sobolev space and variational techniques, obtaining the existence of multiple equilibrium points. The presence of multiple equilibrium points is a common phenomenon in real-world ecosystems and implies that the system cannot be globally stable under normal circumstances. However, this study develops an effective event-triggered impulse mechanism (ETIM) that achieves global stability for the positive equilibrium solution in the ecosystem. Notably, the newly designed ETIM not only eliminates Zeno behaviour but also reduces the cost of impulse control. This is achieved by removing some unnecessary impulses in the fixed impulsive instants mechanism associated with the ecosystem in previous literature. Finally, a numerical simulation validates the effectiveness of the proposed method.