Analysis On Properties Of Some Stochastic Population Models

Stochastic di?erential equations developed rapidly and have been widely appliedin system science,automatic control,medicine,biology,economics etc. As far asbiomathematics is concerned, there are lots of conclusions about deterministic populationmodels, but up to now, not many research results are obtained on stochastic populationmodels. We mainly consider dynamical properties of the solutions of some stochasticpopulation models. The main results are as follows:1. Optimal harvest of some stochastic single-species logistic models are studied. It iswell-known that logistic model is a very fundamental population model. We consider theoptimal harvesting policy of the corresponding stochastic logistic models: when whitenoise is introduced, by virtue of the existing results, the optimal harvesting policy of astochastic autonomous logistic model is analyzed; when white noise and color noise areboth considered, we obtain the optimal harvesting policy of a stochastic non-autonomouslogistic model by ergodic properties.2. Two stochastic two-species competitive population models are proposed and in-vestigated, including a stochastic modified Lotka-Volterra competitive model for two-species and a stochastic two-species Gilpin-Ayala competition model. The existence,uniqueness of global solutions,stochastic boundedness,stochastic permanence,extinctionand other asymptotic properties of the corresponding population models are obtained byLyapunov functions and basic theory of stochastic di?erential equations.3. A stochastic predator-prey system with Holling II functional response is proposedand considered firstly,the existence,uniqueness of global solutions,stochastic perma-nence and extinction of the stochastic model are studied. In particular, we obtain that thestochastic model is persistent in mean by the comparison theorem of stochastic di?eren-tial equations. And when white noise and color noise are both introduced, a stochasticratio-dependent predator-prey model under regime switching is considered here. Thenthe dynamical properties of a two-species stochastic predator–prey model is investigated,by which we obtain the optimal harvest policy of its stochastic autonomous prey-predatormodel. Finally a stochastic prey-predator model with Leslie-Gower and Holling-type IIschemes under regime switching is considered, and the optimal harvesting policy of thepopulation prey is obtained. 4. Two stochastic two-species mutualism models are proposed and studied. Theexistence,uniqueness of global solutions are considered, moreover, we show that thesolutions are stochastically bounded, uniformly continuous and globally attractive. Espe-cially, under some conditions, we conclude that the stochastic models are stochasticallypermanent and persistent in mean. Su?cient conditions of extinction of stochastic sys-tems are obtained.5. A stochastic n-species Lotka-Volterra model is investigated. Stochastic bounded-ness,stochastic permanence and other asymptotic properties of the stochastic model arestudied. The definition of almost surely stochastic permanence is proposed here, more-over, we consider the relationship of stochastic permanence and almost surely stochasticpermanence: stochastic permanence can imply almost surely stochastic permanence; viceis not true by a counterexample.