Impact of carrying capacity on the dynamics of a discrete-time plant-herbivore system

This paper investigates the complex dynamics of a discrete-time plant-herbivore model obtained by extending an existing framework through the incorporation of a logistic growth term for the plant population, reflecting resource limitation such as nutrients, space, and light. The addition of carrying capacity introduces intraspecific competition among plants, significantly enriching the system’s behavior. We conduct a rigorous mathematical analysis to establish the existence and local stability of all biologically feasible fixed points. In particular, boundedness of solutions is proved, and in the case of saddle points, stable and unstable manifolds are explicitly computed. These results clarify the conditions under which plant and herbivore populations can persist or collapse. We further establish the occurrence of a transcritical bifurcation at the boundary equilibrium. Using bifurcation theory, we demonstrate that the system experiences both period-doubling and Neimark–Sacker bifurcations at the positive fixed point. Notably, our analysis reveals that the inclusion of logistic growth leads to a cascade of period-doubling bifurcations, ultimately resulting in chaotic dynamics, a phenomenon not reported in the original model. From a biological perspective, this suggests that resource limitation can induce irregular population fluctuations, making long-term prediction of species abundances difficult. Numerical simulations, including bifurcation diagrams, phase portraits, and Lyapunov exponent computations, are presented to support the theoretical results. This study highlights the critical role of resource limitation in ecological modeling and demonstrates how simple biologically realistic modifications can produce complex, unpredictable dynamics.