Department of Biology, University of York, York, YO10 5DD, UK.
Bull Math Biol. 2010 Aug;72(6):1361-82. doi: 10.1007/s11538-009-9496-5. Epub 2010 Jan 8.
This paper investigates the dynamics of biomass in a marine ecosystem. A stochastic process is defined in which organisms undergo jumps in body size as they catch and eat smaller organisms. Using a systematic expansion of the master equation, we derive a deterministic equation for the macroscopic dynamics, which we call the deterministic jump-growth equation, and a linear Fokker-Planck equation for the stochastic fluctuations. The McKendrick-von Foerster equation, used in previous studies, is shown to be a first-order approximation, appropriate in equilibrium systems where predators are much larger than their prey. The model has a power-law steady state consistent with the approximate constancy of mass density in logarithmic intervals of body mass often observed in marine ecosystems. The behaviours of the stochastic process, the deterministic jump-growth equation, and the McKendrick-von Foerster equation are compared using numerical methods. The numerical analysis shows two classes of attractors: steady states and travelling waves.
本文研究了海洋生态系统中生物量的动态变化。定义了一个随机过程，其中生物体在捕食较小的生物体时，其体型会发生跳跃式变化。通过对主方程的系统展开，我们推导出了一个用于宏观动力学的确定性方程，即所谓的确定性跳跃生长方程，以及一个用于随机波动的线性福克-普朗克方程。之前研究中使用的 McKendrick-von Foerster 方程被证明是一种一阶近似，适用于处于平衡状态的系统，其中捕食者比猎物大得多。该模型具有幂律稳态，与海洋生态系统中经常观察到的对数间隔内质量密度近似恒定的情况一致。通过数值方法比较了随机过程、确定性跳跃生长方程和 McKendrick-von Foerster 方程的行为。数值分析显示了两类吸引子：稳态和传播波。
