A nonautonomous model for the effect of rarity value on the dynamics of a predator–prey system with variable harvesting

Abstract

The human inclination to value rarity contributes to the exploitation of many wildlife species by driving up their market value when their numbers are scarce. This research investigates a model to explore how the value attached to rarity impacts the harvesting of predator species and subsequently influences the dynamics of predator–prey interactions in an ecological system. To broaden the model's scope, it's extended to its nonautonomous form by allowing the harvesting rate of predator species and the degree of neutrality in consumers to respond for the rarity of predator species to change over time. Both the autonomous and nonautonomous systems are explored through analytical and numerical investigation. Numerical findings reveal that as the prey population's growth rate gradually increases, the unforced system shifts from stable boundary equilibrium (harvesting absent) to stable interior equilibrium, then to oscillatory interior equilibrium, followed by period doubling oscillations leading to chaotic behavior. The autonomous system exhibits limit cycle oscillations around the boundary equilibrium when predator species are minimally harvested. Moderate harvesting stabilizes the system, while excessive harvesting induces instability. However, the chaotic oscillations can be eliminated through period‐halving route upon decreasing the consumers' response to predator rarity. Furthermore, the seasonally forced model demonstrates periodic solutions, higher order periodic solutions, and chaotic dynamics contingent upon different settings of prey growth rate, the strength of seasonality in predator harvesting, and the consumers' response to predator rarity.