The Application Of Stochastic Lotka-volterra Equation In Biological Population

Stochastic differential equations have developed rapidly in recent decades, andthey are widely applied in automatic control, ecology, system science, medicine,and financial aspects. In terms of biology, by using stochastic differentialequations as tools to study the important issues in biology can further reveal therule of developing in biology. This paper mainly discusses the solutions ofstochastic Lotka-Volterra equations and related models under the background ofthe biological species.This paper can be divided into two main parts. In the first part, we mainlyoutline the existence and uniqueness of global solutions and the related stability ofthree different stochastic differential equations. They are stochastic Logisticequation, stochastic Lotka-Volterra equation and stochastic delay Lotka-Volterraequation. Through comparative analysis, applying Euler scheme to numericalsimulations then reach a conclusion that by adding white noise to stochasticLotka-Volterra equation and stochastic delay Lotka-Volterra equation can indeedinhibit blasting of their solutions. Further more, the solution of stochastic delayLotka-Volterra equation has the nature of stochastic ultimate boundedness.In the second part, we analyst the predator-prey models with theBeddington-DeAngelis response to discuss about its related natures. By employingthe Milstein method into the numerical simulation and we draw the conclusion thatwhen the intensity of random noise is not high, the equilibriums of stochasticsystem asymptotically stable in the large if the equilibriums of the deterministicsystem is asymptotically stable. When the intensity of random noises reaches acertain level, the random noise will affect the stability of the equilibriums state ofdeterministic system.