Blythe S P, Nisbet R M, Gurney W S
Theor Popul Biol. 1984 Jun;25(3):289-311. doi: 10.1016/0040-5809(84)90011-x.
An integro-differential equation for the dynamics of a subpopulation of adults in a closed system where only the adults compete and where there is a distribution of maturation periods is described. We show how the careful choice of a general weighting function based on the gamma distribution with a shift in origin enables us to characterize adequately some observed maturation-period distributions, and also makes local stability and numerical analyses straightforward. Using these results we examine the progression in the behavior of the distributed-delay model as the distribution is narrowed toward the limit of a discrete delay. We conclude that while local stability properties approach those of the limiting equation very rapidly, the persistent fluctuation behavior converges more slowly, with the dominant period and maximum amplitude being least affected by the details of the distribution, and the fine structure of solutions being most sensitive. Finally, we examine the consequences for population modeling, and using several examples of insect populations, conclude that although quite often a full maturation-period distribution should be incorporated in a given model, in many cases a discrete-delay approximation will suffice.
描述了一个封闭系统中成年亚群动态的积分-微分方程，在该系统中只有成年个体相互竞争，且存在成熟期分布。我们展示了如何基于具有原点偏移的伽马分布精心选择一个通用加权函数，使我们能够充分表征一些观察到的成熟期分布，并且还使局部稳定性和数值分析变得直接明了。利用这些结果，我们研究了分布式延迟模型的行为随着分布向离散延迟极限变窄时的进展情况。我们得出结论，虽然局部稳定性特性非常迅速地趋近于极限方程的特性，但持续波动行为收敛得更慢，主导周期和最大振幅受分布细节的影响最小，而解的精细结构最敏感。最后，我们研究了对种群建模的影响，并通过几个昆虫种群的例子得出结论，尽管在给定模型中通常应纳入完整的成熟期分布，但在许多情况下离散延迟近似就足够了。
