Narrow Results

This study presents a comprehensive model of predator–prey interactions within a toxic environment, with a particular focus on the effect of toxicant compounds on the development of populations. By incorporating environmental disturbances, the dynamics of the model are investigated to enhance the system’s authenticity. Analytical explanations have been provided for the deterministic system solutions, including positivity, uniform boundedness and persistence. The deterministic portion of the investigation entails a comprehensive examination of occurrence and stability criteria pertaining to every possible equlibria. The bifurcation studies conducted on the system exhibit the appearance of local bifurcations, including transcritical, saddle-node and Hopf bifurcations. Moreover, these evaluations establish the parametric region in which Bautin, Bogdanov–Takens and cusp bifurcation occur. Under a relevant selection of parametric values, the suggested system has the capacity to manifest a wide range of dynamic phenomena, such as bi-stable behavior, emergence of limit cycles, and presence of homoclinic loops. Furthermore, in a stochastic environment, the use of Lyapunov functions explains the existence of a global positive solution. It has additionally been argued that the proposed system exhibits ultimate stochastic boundedness. Subsequently, specific and adequate criteria demonstrate the eradication of both species as well as the long-term survival of prey communities. We have also investigated the impact of the exogenous input rate of toxic substances and the coefficient of toxic substances in both species on the behavior of the whole system, both in deterministic and stochastic scenarios. Theoretical findings have been confirmed by various numerical investigations.

A mathematical model describing the dynamics of toxin producing phytoplankton–zooplankton interaction with instantaneous nutrient recycling is proposed. We have explored the dynamics of plankton ecosystem with multiple delays; one due to gestation period in the growth of phytoplankton population and second due to the delay in toxin liberated by TPP. It is established that a sequence of Hopf bifurcations occurs at the interior equilibrium as the delay increases through its critical value. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined using the theory of normal form and center manifold. Meanwhile, effect of toxin on the stability of delayed plankton system is also established numerically. Finally, numerical simulations are carried out to support and supplement the analytical findings.