从Logistic增长到指数增长的种群动力学模型
From Logistic Growth to Exponential Growth in a Population Dynamical Model
摘要：
提供资源的中心源（集线器）和大量享受和贡献此资源的组件之间的动态描述了许多现实生活中的情况。建模、控制和平衡这种动态是许多科学学科中出现的一个普遍问题。我们分析了一个随机动力系统表现出这种动态与乘性噪声。我们表明，该模型可以通过传递到描述组件和轮毂之间的质量比的变量来精确求解。我们推导出平均质量比的确定性方程。这个方程描述了逻辑增长。通过计算系统的样本和矩李雅普诺夫指数，我们得到了该模型的完整相图，并识别了三个区域。第一种机制描述了非中心组件和中心之间的完全平衡，在第二种机制中，整个资源主要集中在中心，而在第三种机制中，资源集中在少数非中心组件和中心上。令人惊讶的是，在大量组件的限制下，转换值不依赖于集线器提供的资源量。该模型在使用磁共振（MR）技术分析多孔介质的背景下有有趣的应用。
英文摘要：
Dynamics among central sources (hubs) providing a resource and large number of components enjoying and contributing to this resource describes many real life situations. Modeling, controlling, and balancing this dynamics is a general problem that arises in many scientific disciplines. We analyze a stochastic dynamical system exhibiting this dynamics with a multiplicative noise. We show that this model can be solved exactly by passing to variables that describe the mass ratio between the components and the hub. We derive a deterministic equation for the average mass ratio. This equation describes logistic growth. We derive the full phase diagram of the model and identify three regimes by calculating the sample and moment Lyapunov exponent of the system. The first regime describes full balance between the non-hub components and the hub, in the second regime the entire resource is concentrated mainly in the hub, and in the third regime the resource is localized on a few non-hub components and the hub. Surprisingly, in the limit of large number of components the transition values do not depend on the amount of resource given by the hub. This model has interesting application in the context of analysis of porous media using Magnetic Resonance (MR) techniques.
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