Tang Sanyi, Chen Lansun
Institute of Mathematics, Academy of Mathematics and System Sciences, Academia Sinica, Beijing 100080, People's Republic of China.
Bull Math Biol. 2003 May;65(3):479-95. doi: 10.1016/S0092-8240(03)00005-3.
In most models of population dynamics, increases in population due to birth are assumed to be time-independent, but many species reproduce only during a single period of the year. A single species stage-structured model with density-dependent maturation rate and birth pulse is formulated. Using the discrete dynamical system determined by its Poincaré map, we report a detailed study of the various dynamics, including (a) existence and stability of nonnegative equilibria, (b) nonunique dynamics, meaning that several attractors coexist, (c) basins of attraction (defined as the set of the initial conditions leading to a certain type of attractor), (d) supertransients, and (e) chaotic attractors. The occurrence of these complex dynamic behaviour is related to the fact that minor changes in parameter or initial values can strikingly change the dynamic behaviours of system. Further, it is shown that periodic birth pulse, in effect, provides a natural period or cyclicity that allows multiple oscillatory solutions in the continuous dynamical systems.
在大多数种群动态模型中，因出生导致的种群增长被假定为与时间无关，但许多物种仅在一年中的单个时期繁殖。构建了一个具有密度依赖成熟率和出生脉冲的单物种阶段结构模型。利用由其庞加莱映射确定的离散动力系统，我们报告了对各种动态的详细研究，包括（a）非负平衡点的存在性和稳定性，（b）非唯一动态，即几个吸引子共存，（c）吸引域（定义为导致某种类型吸引子的初始条件集），（d）超瞬态，以及（e）混沌吸引子。这些复杂动态行为的出现与以下事实有关：参数或初始值的微小变化会显著改变系统的动态行为。此外，结果表明周期性出生脉冲实际上提供了一个自然周期或循环性，使得连续动力系统中存在多个振荡解。
