Dynamics analysis of disturbance propagation in ecosystem with proportional migration based on epidemic model

In order to investigate the propagation dynamics of ecological disturbances within ecological networks, we conceptualize ecological disturbances as infectious diseases and employ complex network theory to analyze ecosystems. In this approach, the transmission process of ecological interference is abstracted as the spread of infectious diseases on complex networks. Based on the principles of infectious disease models, a disturbance propagation model for ecological networks is constructed and analyzed. In this paper, species within the ecosystem are abstracted as nodes in a complex network, where connections between nodes signify predator-prey relationships. When the ecosystem is under attack, a small number of species are initially disturbed, and this disturbance spreads among the ecosystem through the food chain. Given that species possess self-recovery capabilities, some species will return to a stable state over time after being disturbed. Consequently, any species within the ecosystem can be in one of three states at a given time: undisturbed, disturbed, or recovered. By establishing and analyzing the disturbance propagation dynamics, we determine the basic reproduction number and its influencing factors, and assess the stability of the disease-free equilibrium and the endemic equilibrium. The results demonstrate that when the basic reproduction number is less than 1, the system exhibits only a disease-free equilibrium, which is globally stable. When the basic reproduction number exceeds 1, an endemic equilibrium exists, the disease-free equilibrium becomes unstable, while the endemic equilibrium is globally stable. The basic reproduction number is associated with the topological structure of the food web, the probability of disturbance propagation, and the probability of species recovery. Subsequently, we validate the conclusions of the theorem using the actual food web data of 85 species from a pine forest in Otago, New Zealand. Finally, we consider protection measures for species from a human-intervention perspective, treating species protection as species immunity. Through theoretical derivation and numerical simulation, we find that the active immunization strategy is the most effective. This is because it effectively targets and protects neighbor nodes with medium and high degrees (i.e., highly connected species), thereby inhibiting the cascade of disturbance more efficiently than random or targeted strategies.