LAUSR

In this study, we consider the environmental stochastic effects on the phytoplankton–zooplankton dynamics. The model used in our investigation is the model on interactions between algae and Daphnia with the environmental stochastic constraint. The role of environmental random constraint of the oscillations dynamics on the phytoplankton–zooplankton interactions has been studied. We estimate mathematically and numerically the threshold value of the intensity for noise generating a transition from the coexistence to the extinction. Our study has shown that environmental stochastic noise can destroy the limit cycle attractor shown in previous work on the deterministic model, and thus drive the system toward extinction of populations. We consider only the situation where the quantity of copper concentration is sufficient for the survival of the populations of the ecosystem. Depending on the copper concentration and on which populations the environmental constraints are applied, the present study reveals that stochastic environmental constraints have positive and negative effects on the life of Daphnia and algae populations. The stochastic environmental constraint provides other necessary nutrients, and we observe the extinction of the populations with the increases in the noise intensity. Both populations extinct when $$\sigma _1\ge 20$$σ1≥20. One finds that when the noise is applied to the Scenedesmus populations, the Daphnia populations extinct for $$\sigma _1\ge 10$$σ1≥10, while only the Scenedesmus populations survive with the increase in time. When the environmental constraint noise is applied to both populations, the Daphnia populations survive until $$\sigma _1=\sigma _2\ge 20$$σ1=σ2≥20, while after this, one observes the extinction of both populations.