Surendran Anudeep, Plank Michael J, Simpson Matthew J
School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.
School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand.
Proc Math Phys Eng Sci. 2020 Oct;476(2242):20200501. doi: 10.1098/rspa.2020.0501. Epub 2020 Oct 21.
Population dynamics including a strong Allee effect describe the situation where long-term population survival or extinction depends on the initial population density. A simple mathematical model of an Allee effect is one where initial densities below the threshold lead to extinction, whereas initial densities above the threshold lead to survival. Mean-field models of population dynamics neglect spatial structure that can arise through short-range interactions, such as competition and dispersal. The influence of non-mean-field effects has not been studied in the presence of an Allee effect. To address this, we develop an individual-based model that incorporates both short-range interactions and an Allee effect. To explore the role of spatial structure we derive a mathematically tractable continuum approximation of the IBM in terms of the dynamics of spatial moments. In the limit of long-range interactions where the mean-field approximation holds, our modelling framework recovers the mean-field Allee threshold. We show that the Allee threshold is sensitive to spatial structure neglected by mean-field models. For example, there are cases where the mean-field model predicts extinction but the population actually survives. Through simulations we show that our new spatial moment dynamics model accurately captures the modified Allee threshold in the presence of spatial structure.
包括强阿利效应在内的种群动态描述了长期种群生存或灭绝取决于初始种群密度的情况。阿利效应的一个简单数学模型是，低于阈值的初始密度会导致灭绝，而高于阈值的初始密度会导致生存。种群动态的平均场模型忽略了可能通过竞争和扩散等短程相互作用产生的空间结构。在存在阿利效应的情况下，非平均场效应的影响尚未得到研究。为了解决这个问题，我们开发了一个基于个体的模型，该模型既包含短程相互作用，又包含阿利效应。为了探索空间结构的作用，我们根据空间矩的动态推导了IBM的一个数学上易于处理的连续近似。在平均场近似成立的长程相互作用极限下，我们的建模框架恢复了平均场阿利阈值。我们表明，阿利阈值对平均场模型忽略的空间结构很敏感。例如，在某些情况下，平均场模型预测灭绝，但种群实际上存活了下来。通过模拟我们表明，我们的新空间矩动态模型在存在空间结构的情况下准确地捕捉到了修正后的阿利阈值。
