Study Of Single-Species Logistic Model In Polluted Environments

Environmental pollution by various industries and other activities of human isone of the most important socio-ecological problems. The presence of variety oftoxicant in the environment has a serious impact on the survival of the exposedpopulations. Research on single-species Logistic model in a polluted environment isvery meaningful. We have very profound understanding of the living conditions ofpopulation in polluted environment. On the other hand, the natural growth of most ofpopulations is always affected inevitably by some random disturbance， so thesurvival analysis of single-species stochastic population models in pollutedenvironments is particularly important。Thresholds for survival and extinction arevery important for assessing the risk of mortality for populations in pollutedenvironments. In general, the survival threshold is determined by the populationintrinsic growth rate, the mean stress measure in organisms, stress response rate andintensity of the white noise. The persistence and extinction for the dynamic model ofspecies in polluted environments have gotten a lot of results. This results not onlyhas a wide range of theoretical biology, but also has important practical value. In thispaper, we will introduce the development conditions and the research results ofLogistic model in a polluted environment, and give the evaluation of research results.Development of Logistic model in a polluted environment follows from thedeterministic model to stochastic model of evolution. In the study of pollution model,it’s very important to get the sufficient conditions for extinction,(stochastic) weaklypersistence in the mean,(stochastic) strongly persistence in the mean and (stochastic)persistence of the population, and obtain the acute thresholds between localextinction and (stochastic) weakly persistence in the mean. At the end of the article,we emphatically introduce the survival analysis of stochastic Logistic model withMarkovian switching in polluted environments. From the threshold theorem we canfind the presence of white noise in the environment of population growth is negative, Wile for colored noise, the stationary distribution of Markov chains plays a key roleon population stochastic weakly persistence in the mean and local extinction. Inconclusion, both white noise and colored noise will affect the survival and extinctionof population. The main contents are summarized as follows:The first section is the introduction of this paper.In the second section, we first introduce the research significance of single-Spec-ies population model. Then we introduce the establishment and ecological ignificanceof Logistic population model.In the third section, we introduce the research background of Logistic populationmodel in polluted environments. Then we introduce the establishment and ecologicalsignificance of Logistic population model in polluted environments.In the fourth section, we first introduce the research significance and the establish-ment of stochastic Logistic model in polluted environments. Then we introduce thebasic knowledge of stochastic differential equations, including some of the basicdefinitions and basic theories. Finally we introduce the research results of stochasticLogistic model in polluted environments, and give the evaluation of the researchresults.In the fifth section, we first introduce the basic knowledge of Markovianswitching. Then we introduce the establishment and the research results of stochasticLogistic model with Markovian switching in polluted environments. By using the Itoformula, the law of large numbers of martingale, the random comparison theorem etc.,the sufficient conditions for local extinction, stochastic weakly persistence in themean and stochastic persistence of the population are obtained. An acute thresholdbetween local extinction and stochastic weakly persistence in the mean is established.