Stochastic Dynamical Characteristics For A Few Class Of Biological Population Models

In the present paper, the system stability and mean first-passage time subjected to different kinds of noises and time delays for several kinds of special biological population systems and the stochastic resonance phenomenon for several types of biological population systems including time delay or excluding time delay driven by different kinds of noises and periodic signals are investigated theoretically and numerically, and the main work and results are as following: 1) The stability and mean first-passage time for a class of biotic population systems excited by coloured correlated or white correlated noises are investigated. In accordance with Novikov’ theorem and Fox’s approach, the approximate Fokker-Planck equations for stochastic models are obtained. The expressions of steady probability distribution functions are put forward. The approximation expression for mean first-passage time is obtained by applying the way of steepest-descent approximation. In the light of the figures of SPDF and MFPT, the effects of the parameters of the systems, noises(In particular, the strength of coloured correlated noise and the correlation time) on stability and MFPT are discussed in detail. 2) The stability and mean first-passage time for a class of biological population systems including a single term of time delay or multiple terms of time delays are studied. By means of the approximation of probability density, the time-delayed biological population systems are tured into ones without terms of time delay. Moreover, the steady probability distribution function and mean first-passage time are acquired in the adiabatic limit. Moreover, in line with the expressions, the corresponding figures are drawn through the numerical computation. By analyzing the figures, we discuss the effects of different noise parameters and time delays on the stability and mean first-passage time of the systems. 3) The phenomena of stochastic resonance of the biological population systems driven by colored correlated noises, white correlated noises, additive or multiplicative periodic signals are investigated. According to unified colored noise approximation, the FPK equations corresponding to the Langvin’s equations are obtained. By applying the SR theory put forward by McNamara and Wiesenfeld, the expressions of transition rates and SNR are obtained. Moreover, on the basis of these formulas, the impacts of the parameters of the system, noises and noise correlation times on SNR and stochastic resonance are studied. 4) Stochastic properties of the asymmetric bistable cancer growth system subjected to the terms of time delay, the periodic signal and noises are inquired. In terms of small time-delay approximation and two-stated theory, the equations of the steady probability distribution function and output signal-to-noise ratio of the system in the adiabatic limit are gained. On the basis of the expressions, the influences of the parameters of noises, time delays upon the steady states and stochastic resonance of the system are investigated. 5) Stochastic resonance phenomenon in the forest logistic growth system induced by multiplicative non-Gaussian noise, additive Gaussian noise by using the path integral approach and unified colored noise approximation is studied. By means of translating the non-Gaussian noise into Gaussian noise, the expressions of SNR are obtained. Based on them, the effects of intensities of noises, noise correlation times and the departure from the Gaussian noise on the SNR and stochastic resonance are discovered. Meanwhile, by applying the adiabatic limit theory, under the influence of the periodic rectangular signal and the coloured noise, the effects of the intensities and correlation time of the noises, the asymmetric property and parameters of the system, the amplitude of the periodic rectangular signal on the stochastic resonance and SNR of the asymmetric monostable system are discussed.