Department of Mathematics, University of Pittsburgh, 4200 5th Avenue, Pittsburgh, 15260, PA, USA.
Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, ON, L8S 4K1, Canada.
Math Biosci. 2024 Oct;376:109279. doi: 10.1016/j.mbs.2024.109279. Epub 2024 Aug 13.
We introduce a class of discrete single species models with distributed delay in the reproductive process and a cohort dependent survival function that accounts for survival pressure during that delay period. These delay recurrences track the mature population for species in which individuals reach maturity after at least τ and at most τ+τ breeding cycles. Under realistic model assumptions, we prove the existence of a critical delay threshold, τ˜. For given delay kernel length τ, if each individual takes at least τ˜ time units to reach maturity, then the population is predicted to go extinct. We show that the positive equilibrium is decreasing in both τ and τ. In the case of a constant reproductive rate, we provide an equation to determine τ˜ for fixed τ, and similarly, provide a lower bound on the kernel length, τ˜ for fixed τ such that the population goes extinct if τ≥τ˜. We compare these critical thresholds for different maturation distributions and show that if all else is the same, to avoid extinction it is best if all individuals in the population have the shortest delay possible. We apply the model derivation to a Beverton-Holt model and discuss its global dynamics. For this model with kernels that share the same mean delay, we show that populations with the largest variance in the time required to reach maturity have higher population levels and lower chances of extinction.
我们介绍了一类具有分布滞后的离散单物种模型，该模型的繁殖过程中存在滞后，且具有与滞后期间的生存压力相关的群体依赖生存函数。这些延迟复发跟踪成熟种群，适用于个体在至少 τ 个和至多 τ+τ 个繁殖周期后达到成熟的物种。在现实的模型假设下，我们证明了存在一个临界延迟阈值 τ˜。对于给定的延迟核长度 τ，如果每个个体至少需要 τ˜时间单位才能达到成熟，那么预测种群将会灭绝。我们表明，正平衡点随着 τ 和 τ 的增加而减少。在恒定繁殖率的情况下，我们提供了一个方程来确定 τ˜对于固定 τ，并且类似地，对于固定 τ，提供了核长度 τ˜的下限，使得如果 τ≥τ˜，则种群将会灭绝。我们比较了不同成熟分布的这些临界阈值，并表明如果其他所有条件相同，为了避免灭绝，种群中所有个体最好具有最短的延迟。我们将模型推导应用于 Beverton-Holt 模型，并讨论了其全局动力学。对于具有相同平均延迟的核的这种模型，我们表明，在达到成熟所需时间上具有最大方差的种群具有更高的种群水平和更低的灭绝几率。
