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.

Toxic or allelopathic compounds liberated by toxin-producing phytoplankton (TPP) acts as a strong mediator in plankton dynamics. On an analysis of a set of phytoplankton biomass data that have been collected by our group in the northwest part of the Bay of Bengal, and by analysis of a three-component mathematical model under a constant as well as a stochastic environment, we explore the role of toxin-allelopathy in determining the dynamic behavior of the competing phytoplankton species. The overall results, based on analytical and numerical wings, demonstrate that toxin-allelopathy due to the TPP promotes a stable co-existence of those competitive phytoplankton that would otherwise exhibit competitive exclusion of the weak species. Our study suggests that TPP might be a potential candidate for maintaining the co-existence and diversity of competing phytoplankton species.

In this paper, the issue on the optimal harvesting of fish catching after eliminating toxin during the fish aquaculture is studied. Taking the aquaculture of bighead carp and silver carp as an example, considering the characteristics of the growth and the reproduction of prymnesiacee, and taking prymnesiacee toxin as the pollutant source, a harvesting model of fish aquaculture is built. The finite-time stability of the system is discussed. While the fish aquaculture and the elimination of the algae toxin targeted as pollutant source can be carried out simultaneously, an optimal harvesting method is made by the Pontryagin maximum principle, from which a general algorithm of the optimal harvesting solution can be obtained. The stimulation shows the effectiveness of the result.

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.

A simple model of phytoplankton-zooplankton interaction with a periodic input nutrient is presented. The model is then used to study a nutrient-plankton interaction with a toxic substance that inhibits the growth rate of phytoplankton. The effects of the toxin upon the existence, magnitude, and stability of the periodic solutions are discussed.