Capitán José A, Delius Gustav W
Grupo Interdisciplinar de Sistemas Complejos, Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, E-28911 Leganés, Madrid, Spain.
Phys Rev E Stat Nonlin Soft Matter Phys. 2010 Jun;81(6 Pt 1):061901. doi: 10.1103/PhysRevE.81.061901. Epub 2010 Jun 1.
A striking feature of the marine ecosystem is the regularity in its size spectrum: the abundance of organisms as a function of their weight approximately follows a power law over almost ten orders of magnitude. We interpret this as evidence that the population dynamics in the ocean is approximately scale-invariant. We use this invariance in the construction and solution of a size-structured dynamical population model. Starting from a Markov model encoding the basic processes of predation, reproduction, maintenance respiration, and intrinsic mortality, we derive a partial integro-differential equation describing the dependence of abundance on weight and time. Our model represents an extension of the jump-growth model and hence also of earlier models based on the McKendrick-von Foerster equation. The model is scale-invariant provided the rate functions of the stochastic processes have certain scaling properties. We determine the steady-state power-law solution, whose exponent is determined by the relative scaling between the rates of the density-dependent processes (predation) and the rates of the density-independent processes (reproduction, maintenance, and mortality). We study the stability of the steady-state against small perturbations and find that inclusion of maintenance respiration and reproduction in the model has a strong stabilizing effect. Furthermore, the steady state is unstable against a change in the overall population density unless the reproduction rate exceeds a certain threshold.
生物数量作为其体重的函数,在近十个数量级上大致遵循幂律。我们将此解释为海洋中种群动态近似尺度不变性的证据。我们在构建和求解一个大小结构动态种群模型时利用了这种不变性。从一个编码捕食、繁殖、维持呼吸和内在死亡率等基本过程的马尔可夫模型出发,我们推导出一个描述数量对体重和时间依赖性的偏积分微分方程。我们的模型是跳跃增长模型的扩展,因此也是基于麦肯德里克 - 冯·福斯特方程的早期模型的扩展。只要随机过程的速率函数具有某些缩放特性,该模型就是尺度不变的。我们确定了稳态幂律解,其指数由密度依赖过程(捕食)的速率与密度独立过程(繁殖、维持和死亡)的速率之间的相对缩放决定。我们研究了稳态对小扰动的稳定性,发现模型中包含维持呼吸和繁殖具有很强的稳定作用。此外,除非繁殖率超过某个阈值,否则稳态对总体种群密度的变化是不稳定的。