Department of Mathematics, Linköping University, Linköping, Sweden.
Stockholm Resilience Centre, Stockholm University, Stockholm, Sweden.
J Math Biol. 2021 Dec 4;83(6-7):68. doi: 10.1007/s00285-021-01701-3.
We consider an age-structured density-dependent population model on several temporally variable patches. There are two key assumptions on which we base model setup and analysis. First, intraspecific competition is limited to competition between individuals of the same age (pure intra-cohort competition) and it affects density-dependent mortality. Second, dispersal between patches ensures that each patch can be reached from every other patch, directly or through several intermediary patches, within individual reproductive age. Using strong monotonicity we prove existence and uniqueness of solution and analyze its large-time behavior in cases of constant, periodically variable and irregularly variable environment. In analogy to the next generation operator, we introduce the net reproductive operator and the basic reproduction number [Formula: see text] for time-independent and periodical models and establish the permanence dichotomy: if [Formula: see text], extinction on all patches is imminent, and if [Formula: see text], permanence on all patches is guaranteed. We show that a solution for the general time-dependent problem can be bounded by above and below by solutions to the associated periodic problems. Using two-side estimates, we establish uniform boundedness and uniform persistence of a solution for the general time-dependent problem and describe its asymptotic behaviour.
我们考虑了一个在几个时变斑块上具有年龄结构和密度依赖性的种群模型。我们基于两个关键假设来建立和分析模型。首先，种内竞争仅限于同一年龄个体之间（纯同群竞争），并且它会影响密度依赖性死亡率。其次，斑块间的扩散确保了在个体生殖年龄内，每个斑块都可以从其他任何斑块直接或通过几个中间斑块到达。我们使用强单调性证明了解的存在唯一性，并分析了在恒态、周期变和非周期变环境下的大时间行为。与下一代算子类似，我们引入了净生殖算子和基本繁殖数 [Formula: see text]，用于时不变和周期模型，并建立了持久性二分法：如果 [Formula: see text]，则所有斑块上的灭绝迫在眉睫，如果 [Formula: see text]，则所有斑块上的持久性得到保证。我们表明，一般时变问题的解可以被关联的周期问题的解上下界。通过双边估计，我们建立了一般时变问题的解的一致有界性和一致性持久性，并描述了其渐近行为。
