Effects of seasonal hunting and food quantity variations on the Lotka-Volterra model

The Lotka-Volterra model is instrumental for understanding the dynamic interactions between predator and prey populations, especially as human activities like habitat destruction, pollution, and climate change rapidly alter living environments, making it more important than ever to understand the underlying mechanisms that drive population changes and to predict and mitigate the impacts of human interventions on wildlife populations. In this study, we investigate how periodic hunting and variations in food quantity impact the classical Hamiltonian Lotka-Volterra model. We do this by modeling variations in the prey reproduction rate with a periodically varying coefficient. We aim to understand how the system responds to these periodic disturbances and to identify the conditions under which the population sizes undergo significant oscillations. Our findings suggest that specific frequencies of hunting or food quantity variations can drive populations out of equilibrium to dangerously low levels, increasing the risk of extinction. The analysis is based on perturbation methods, primarily addressing the 1:1 resonance and using action-angle variables to simplify the system into new canonical coordinates. The results have significant implications for understanding and managing biological systems, offering insights that can aid in preserving species by identifying critical hunting thresholds and frequencies. We also give estimates for the time period of the Lotka-Volterra model in the vicinity of the nontrivial equilibrium and very far from the equilibrium.