The timing of interventions plays a central role in managing and exploiting biological populations. However, few studies in the literature have addressed its effect on population stability. The Seno equation is a discrete-time equation that describes the dynamics of single-species populations harvested according to the proportional feedback method at any moment between two consecutive censuses. Here we study a discrete-time equation that generalizes the Seno equation by considering the management and exploitation of populations through the target-oriented chaos control method. We investigate the combined effect of timing, targeting, and control on population stability, focusing on global stability. We prove that high enough control values create a positive equilibrium that attracts all positive solutions. We also prove that it is possible to determine parameter values to stabilize the controlled populations at any preset population size. Finally, we investigate the parameter combinations for which the management and exploitation are optimized in different scenarios.

中文翻译：

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面向目标的混沌控制中的干预时间

干预的时机在管理和利用生物种群方面起着核心作用。然而，文献中很少有研究涉及其对人口稳定性的影响。Seno 方程是一个离散时间方程，描述了在两次连续普查之间的任何时刻根据比例反馈方法收获的单一物种种群的动态。在这里，我们研究了一个离散时间方程，该方程通过考虑通过面向目标的混沌控制方法对种群的管理和利用来推广 Seno 方程。我们研究时间、目标和控制对人口稳定性的综合影响，重点关注全球稳定性。我们证明了足够高的控制值会产生一个吸引所有正解的正均衡。我们还证明可以确定参数值以将受控种群稳定在任何预设的种群规模。最后，我们研究了在不同场景下优化管理和开发的参数组合。