具有衰减和竞争相容时滞的种群增长和竞争模型
Population Growth and Competition Models with Decay and Competition Consistent Delay
摘要：
我们推导出一个延迟的Logistic方程的另一种表达，其中人口的变化率涉及的增长率，取决于人口密度在较早的时间段。在我们的公式中，增长项的延迟与模型给出的人口瞬时下降率一致。我们的公式是[Arino et al.，J.~理论~生物学~ 241（1）：109--119，2006]，并考虑了成虫和幼虫之间的种内竞争。我们提供了一个完整的全球分析表明，没有持续的振荡是可能的。给出灭绝和生存之间的接口的阈值确定在模型中的参数。利用链传递集理论和合作时滞微分方程的比较定理确定了模型的全局动力学性质。我们扩展我们的延迟逻辑方程的两个物种的竞争建模系统。对于竞争模型，我们提供了局部稳定性，分岔图和自适应动力学的结果。假设具有较短延迟的物种比具有较长延迟的物种一次产生较少的后代，我们证明了存在一个临界值$τ^*$，使得延迟的进化趋势是接近$τ^*$。
英文摘要：
We derive an alternative expression for a delayed logistic equation in which the rate of change in the population involves a growth rate that depends on the population density during an earlier time period. In our formulation, the delay in the growth term is consistent with the rate of instantaneous decline in the population given by the model. Our formulation is a modification of [Arino et al., J.~Theoret.~Biol.~241(1):109--119, 2006] by taking the intraspecific competition between the adults and juveniles into account. We provide a complete global analysis showing that no sustained oscillations are possible. A threshold giving the interface between extinction and survival is determined in terms of the parameters in the model. The theory of chain transitive sets and the comparison theorem for cooperative delay differential equations are used to determine the global dynamics of the model. We extend our delayed logistic equation to a system modeling the competition of two species. For the competition model, we provide results on local stability, bifurcation diagrams, and adaptive dynamics. Assuming that the species with shorter delay produces fewer offspring at a time than the species with longer delay, we show that there is a critical value, $τ^*$, such that the evolutionary trend is for the delay to approach $τ^*$.
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