两种群随机动力系统的信息熵和动力学研究

Information entropies and dynamics in the stochastic ecosystem of two competing species

随机种群动力学模型是研究种群间以及种群与不确定性环境间相互作用的动力学行为的数学模型.本文从概率密度以及信息熵流、熵产生的演化角度探讨了两种群生态系统的Ito(或Statonovich)意义下随机模型的动力学行为.利用Fokker-Planck方程及其边界条件和信息熵定义导出信息熵流(平均散度)和熵产生的关系式,并通过数值路径积分法捕捉到熵流的非线性变化趋势以及信息熵的极值点位置与概率密度的快速迁移和分岔的联系.应用数值路径积分法计算结果表明Ito(或Statonovich)意义下两种随机模型的概率密度和信息熵的极值点位置不同但演化趋势一致.

Using the models of stochastic population dynamics, the competitions and interactions of interspecies and between species and the stochastic environment are studied. In this paper, the stochastic ecosystems （in Ito or Statonovich model） of two competing species are investigated through evaluating probability densities and information entropy fluxes and productions of two species. The formulas of entropy flux （i.e. expectation of divergence） and entropy production are educed for numerical calculations, through the corresponding Fokker-Planck equation with its condition and the definition of Shannon entropy. The nonlinear characteristics of entropy fluxes are captured and the relationships are found between the extremal points of entropy productions and the rapid transitions or bifurcations. The numerical results obtained with path integration method show that the probability densities and Shannon entropies of these two stochastic models （in Ito or Statonovich meaning） have the same evolutional tendency but with different points of extrema.