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Effects of Noise and Time Delay on the Death and Propagation Processes Described by the Malthus-Verhulst Model

Author: CaiJianChun
Tutor: MeiDongCheng
School: Yunnan University
Course: Theoretical Physics
Keywords: ecosystem noise time delay statistical properties theoretical analysis stochastic simulation
CLC: Q14
Type: PhD thesis
Year: 2010
Downloads: 60
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Abstract


In this thesis, the stability, the death and propagation processes of thespecies described by the Malthus-Verhulst model with noises and time delay areinvestigated by the theory of stochastic dynamics and the method of stochasticsimulation. The whole thesis includes two parts. The ?rst part introduces thebackground and actuality to the study of the ecosystems, and the method oftheoretical analysis and stochastic simulation about the stochastic dynamicalsystems subjected to noise and time delay. In the second part, the Malthus-Verhulst model under the in?uence of weak Gaussian white noises, strong Gaus-sian white noises and Gaussian colored noises respectively are studied system-atically, the time delays in the death and propagation processes are considered,the e?ects of the delay time and the noises on the system are discussed.In Chapter 1, the background to the study of the ecosystems is introduced,mainly includes: The basic characters of the organisms, the discovery of theself-organization phenomena, the initiation of the theory about the dissipativestructure which solved the inconsistency between the biology and the physicsand appeared in Darwin’s theory of evolution, then the ecology was advancedfrom the phase of qualitative description to the phase of quantitative investi-gation. Furthermore, the actuality to the study of the ecosystems about thetemporal and spatial evolutive processes is introduced.In Chapter 2, the Brownian motion, the explanations given by Einsteinand Langevin severally, the Langevin equation are introduced ?rstly. Secondly, the method of theoretical analysis under the condition of the small parametersis listed: Based on the Liouville theorem, making use of the Novikov’s theoremand the functional analysis, the corresponding Fokker-Planck equation is ob-tained. The stationary probability density function is derived from the Fokker-Planck equation with the unvarying condition. The mean first passage time,the theory about the two-state transition and the steepest-descent approxima-tion are presented. Finally, the theory of decoupling cross-correlated noises, thestochastic equivalent method of using one noise instead of two noises, rewritingthe Langevin equation as the difference equation by the ?nite di?erence methodand the Monte Carlo method are elucidated.In Chapter 3, the e?ects of the delay time, the noise intensities, the cross-correlated intensity of the noises on the stationary probability density functionand the mean ?rst passage time of the Malthus-Verhulst model which under thein?uence of cross-correlated white noises and time delay are investigated. Theresults show that: The increasing delay time does not a?ect roughly the stabilityof population with short delay but strengthens it with long delay, and thepopulation of species is restricted in the lower level by the larger delay time. Thestability of population is weakened and the replacement of old individuals withyoung ones is accelerated by the increasing cross-correlation intensity betweentwo noises. When the noises intensities are strong, the stability of populationis enhanced by the decreasing multiplicative noise intensity and the increasingdelay time. The extinction phenomenon happens more easily in the specieswhich have the larger generation number. The replacement of old individualswith young ones is accelerated by the increasing multiplicative noise intensity,the increasing additive noise intensity and the decreasing delay time.In Chapter 4, the e?ects of the correlation times of the noises on the sta-tionary probability density function and the mean ?rst passage time of theMalthus-Verhulst model which under the in?uence of cross-correlated colorednoises and time delay are investigated. The results show that: The increasingcorrelation time of multiplicative noise strengthens the stability of populationand the correlation time of additive noise does not a?ect it. The increasingcorrelation time of multiplicative noise slows down the replacement of old indi- viduals with young ones, while the increasing correlation time of additive noisequicken it with the short delay time or the short correlation time of multi-plic‘the optimal correlation time of additive noise still quicken it with the longdelay time and the long correlation time of multiplicative noise.

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