Streipert Sabrina H, Wolkowicz Gail S K
McMaster University, Hamilton, Ontario, Canada.
J Math Biol. 2021 Aug 7;83(3):25. doi: 10.1007/s00285-021-01652-9.
We propose an alternative delayed population growth difference equation model based on a modification of the Beverton-Holt recurrence, assuming a delay only in the growth contribution that takes into account that those individuals that die during the delay, do not contribute to growth. The model introduced differs from a delayed logistic difference equation, known as the delayed Pielou or delayed Beverton-Holt model, that was formulated as a discretization of the Hutchinson model. The analysis of our delayed difference equation model identifies a critical delay threshold. If the time delay exceeds this threshold, the model predicts that the population will go extinct for all non-negative initial conditions. If the delay is below this threshold, the population survives and its size converges to a positive globally asymptotically stable equilibrium that is decreasing in size as the delay increases. We show global asymptotic stability of the positive equilibrium using two different techniques. For one set of parameter values, a contraction mapping result is applied, while the proof for the remaining set of parameter values, relies on showing that the map is eventually componentwise monotone.
我们基于对贝弗顿 - 霍尔特递推式的修改，提出了一种另类的延迟种群增长差分方程模型，假定仅在增长贡献方面存在延迟，同时考虑到在延迟期间死亡的个体对增长没有贡献。所引入的模型不同于一种延迟逻辑斯谛差分方程，即所谓的延迟皮洛或延迟贝弗顿 - 霍尔特模型，该模型是作为哈钦森模型的离散化形式而制定的。对我们的延迟差分方程模型的分析确定了一个临界延迟阈值。如果时间延迟超过此阈值，该模型预测对于所有非负初始条件，种群将灭绝。如果延迟低于此阈值，种群将存活，并且其规模会收敛到一个正的全局渐近稳定平衡点，该平衡点的规模会随着延迟增加而减小。我们使用两种不同的技术展示了正平衡点的全局渐近稳定性。对于一组参数值，应用了压缩映射结果，而对于其余参数值的证明，则依赖于表明该映射最终在分量上是单调的。
