Abstract
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 R_0 for time-independent and periodical models and establish the permanence dichotomy: if R_0le 1, extinction on all patches is imminent, and if R_0>1, 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.
Highlights
Population permanence in a patchy environment is a result of complex interactions between abiotic, biotic, and anthropogenic factors (Lewis et al 2017)
Mathematical models are often used for theoretical investigation of population dynamics and establishing conditions for population permanence
Intra-specific competition can act in different ways and have a whole spectrum of different forms, where the pure intraspecific competition and unstructured population-wide competition are on the opposite ends
Summary
Population permanence in a patchy environment is a result of complex interactions between abiotic, biotic, and anthropogenic factors (Lewis et al 2017). Monotonicity of certain positive operators and comparison principles have been used for studying global dynamics of ordinary, delay and partial differential equations, see Hirsch and Smith (2003), Smith (1995) and Zhao (2003) Application of these methods to age-structured population models and epidemic models can be found in Busenberg et al (1991), Diekmann et al (2001), Kuniya and Iannelli (2014), Kuniya et al (2018), Magal and Ruan (2018), Magal et al (2019) and Webb (1985). From a biological point of view, intra-cohort competition occurs as a result of ageing and growth Some species, such as certain insects, molluscs and fish, undergo metamorphosis, which is a complete change of physical appearance and structure and thereby a change of food preferences and habitats.
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