Guckenheimer J, Oster G, Ipaktchi A
J Math Biol. 1977 May 23;4(2):8-147.
The dynamics of density-dependent population models can be extraordinarily complex as numerous authors have displayed in numerical simulations. Here we commence a theoretical analysis of the mathematical mechanisms underlying this complexity from the viewpoint of modern dynamical systems theory. After discussing the chaotic behavior of one-dimensional difference equations we proceed to illustrate the general theory on a density-dependent Leslie model with two age classes. The pattern of bofurcations away from the equilibrium point is investigated and the existence of a "strange attractor" is demonstrated--i.e. an attracting limit set which is neither an equilibrium nor a limit cycle. Near the strange attractor the system exhibits essentially random behavior. An approach to the statical analysis of the dynamics in the chaotic regime is suggested. We then generalize our conclusions to higher dimensions and continuous models (e.g. the nonlinear von Foerster equation).
正如众多作者在数值模拟中所展示的那样，密度依赖型种群模型的动态过程可能极其复杂。在此，我们从现代动力系统理论的视角出发，对这种复杂性背后的数学机制展开理论分析。在讨论了一维差分方程的混沌行为之后，我们接着以具有两个年龄组的密度依赖型莱斯利模型为例来说明一般理论。研究了从平衡点出发的分岔模式，并证明了“奇怪吸引子”的存在——即一个既非平衡点也非极限环的吸引极限集。在奇怪吸引子附近，系统呈现出本质上的随机行为。我们提出了一种在混沌区域对动力学进行静态分析的方法。然后，我们将所得结论推广到更高维度和连续模型（例如非线性冯·福斯特方程）。
