Abstract
Environmental fluctuations and toxin-producing phytoplankton are crucial factors affecting marine ecosystems. In this paper, we propose a stochastic phytoplankton-toxin phytoplankton–zooplankton model to study the effect of environmental fluctuations on extinction and persistence of the population. The results show that large environmental fluctuations may lead to the extinction of the population, and small environmental fluctuation can keep population weakly persistent in the mean. We also find that the noise-induced extinction of one phytoplankton population may lead to the density increase of the other phytoplankton population in two competitive phytoplankton populations. By constructing appropriate Lyapunov functions, we obtain the sufficient conditions for the existence of an ergodic stationary distribution of the model. Finally, numerical simulations are carried out to support our main results.
Highlights
Plankton includes plants and animals that float along at the mercy of the sea’s tides and currents
A few studies about the effect of environmental fluctuations on aquatic ecosystems have been carried out, it is worth noting that in this paper, we split the phytoplankton, identifying the toxic-producing subpopulation, in other words, the considered system includes three species: phytoplankton, toxic phytoplankton, and zooplankton; we investigate the effect of the environmental fluctuations on this system
3 Existence and uniqueness of globally positive solution we show that there is a unique local positive solution for the system (1.3) and we prove that this solution is global by constructing a suitable Lyapunov function
Summary
Plankton includes plants and animals that float along at the mercy of the sea’s tides and currents. Mukhopadhyay et al [11] investigated a nutrient-plankton model in an aquatic environment in the context of phytoplankton bloom and they observed that the zooplankton populations try to avoid the areas where toxin-producing phytoplankton density is high. 3, we prove the existence of the global positive solution and establish the threshold between weakly persistence in the mean and extinction in Sect. (ii) Population x(t) is said to be weakly persistent in the mean if there exists a constant. (iii) Population x(t) is said to be strongly persistent in the mean if there exists a constant. (iv) Population x(t) is said to be persistent in the mean if there exists a constant N > 0 such that limt→+∞. Proof, we know that is extinct, the population T(t) is weakly persistent in the mean. From (4.3), it is easy to see that ln Z(t) < e t m + σ32 t + G(t)
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