MODELING PLANKTON INTERACTION WITH ENVIRONMENTAL STOCHASTICITY

Abstract

This study formulated two mathematical models to understand the complex interactions between phytoplankton and zooplankton populations in aquatic ecosystems by considering environmental stochasticity. In aquatic ecosystems, we observe interactions between non-toxic-producing phytoplankton (NTP), toxin-producing phytoplankton (TPP) and zooplankton. Focusing on NTP, TPP, and zooplankton, the research develops two stochastic differential equation (SDE) models by incorporating environmental noise using two distinct stochastic perturbation techniques. The first model introduces white noise directly into the growth equations of each species, while the second model applies stochastic perturbations around equilibrium points. The primary objectives are to derive conditions for existence, uniqueness and positivity and assess the populations’ asymptotic behaviour and persistence under random environmental fluctuations. By defining a suitable Lyapunov function, we determine conditions for the existence and uniqueness of solutions. We analyze the asymptotic dynamics of models by determining the conditions for stochastic persistence and bounds using Itô’s formula. Moreover, stochastic perturbations around an equilibrium point are discussed. Using the Euler-Maruyama numerical method, we explore the complex behaviour of stochastic systems. We numerically simulate the first stochastic system. Deterministic systems fail to capture the sudden extinction events that stochastic models can demonstrate. Results reveal that low-intensity noise supports species coexistence and enhances plankton persistence, while high-intensity noise may lead to system destabilization and the formation of new equilibria. The analysis also highlights how mutual interference, avoidance, and toxin effects impact system stability. Including stochastic elements provides deeper understanding of how environmental fluctuations influence population dynamics and stability. Our results emphasize the critical role of environmental stochasticity in shaping population dynamics, revealing sudden changes and extinction events that deterministic approaches cannot capture.