STOCHASTICMODELLINGOFPREDATOR-PREYDYNAMICSINA THREE-PATCHECOSYSTEMWITHOPTIMALHARVESTING

Abstract

Predator-prey interactions play a pivotal role in shaping ecological dynamics, and understand ing these interactions is critical for sustainable resource management and effective conservation. Existing literature predominantly focused on deterministic models incorporating optimum har vesting policy involving two ecosystems or less. Moreover, while stochastic models had been employed to account for randomness and uncertainty in ecological interactions, these studies were often limited to single-patch ecosystems. The stochastic dynamics of predator prey models involving more than two ecosystems had been given little attention in literature yet they are crit ical in conservation and resource management. Therefore, this research developed a stochastic predator-prey model with optimum harvesting for three patches namely “cages” which are within a lake, containing a predator-prey system involving large Nile perch as predators and smaller fish as prey. The dynamics of the prey population could transfer from one cage to the other. The study aims to investigate how randomness and harvesting controls affect population stability and sus tainability. Stability analysis of the deterministic part was carried out in order to study the long term behaviour of solutions around equilibrium points. Stability analysis of the stochastic model was done using stochastic Lyapunov function method assessing its impact on the system dynam ics. Numerical analysis was done to explain the analytic approach. From the results, when ei < 1 (ei is the predator’s efficiency in converting prey into a new predator), the lyapunov function, V(t), stays bounded indicating stochastic stability, and when ei > 1, V(t) grows without bound indicating that the system is unstable. An optimal control problem was formulated to derive harvesting functions that maximize resource utility while maintaining a sustainable ecosystem employing the Pontryagin’s Maximum Principle. From Numerical simulation, prey populations remained viable when the harvesting rates were maintained below ν1 = 0.02, ν2 = 0.02, and ν3 =0.02, and noise intensities were controlled at σ = 0.10, and σ = 0.90. The findings high light the impact of human activities, particularly harvesting, on ecosystem balance. They also contribute to conservation biology, fisheries management, and mathematical ecology, providing insights for sustainable resource management and effective conservation strategies.