Abstract
In this paper, we propose and study a stochastic two-species cooperation model with functional response in a polluted environment. We first perform the survival analysis and establish sufficient conditions for extinction, weak persistence, and stochastic permanence. Then we further perform the survival analysis based on the temporal average of population size and derive sufficient conditions for the strong persistence in the mean and weak persistence in the mean. Finally, we present numerical simulations to justify the theoretical results.
Highlights
With rapid development of industries and agriculture, a mass of toxicants has been emitted into the environment, such as the industrial wastewater, domestic sewage, and other contaminants
In the classical cooperation model the mutualism effects are described by a bilinear function, that is, xi response to xj is assumed to be increasingly monotonic, an inherent assumption meaning that the more xj there exist in the environment, the better off the xi
5 Numerical simulations we present several specific examples to justify our theoretical results based on the Milstein method, which is mentioned by Higham [36]
Summary
With rapid development of industries and agriculture, a mass of toxicants has been emitted into the environment, such as the industrial wastewater, domestic sewage, and other contaminants. It is important to estimate the environmental toxicity, which requires quantitative estimates for the survival risk of species in a polluted environment This motivates scholars to utilize the mathematical models to assess the effects of toxicants on various ecosystems. In the classical cooperation model the mutualism effects are described by a bilinear function, that is, xi response to xj is assumed to be increasingly monotonic, an inherent assumption meaning that the more xj there exist in the environment, the better off the xi. A coupling between species and toxicant is formulated by assuming that the intrinsic growth rate of the ith population, ri0 – ri1C0(t), is a linear function of concentration of toxicant present in the organism. A stochastic two-species cooperation model in polluted environment is derived as follows:. The limitation of the model is discussed in the last section
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