Abstract
ABSTRACTAn important problem in spatial ecology is to understand how population-scale patterns emerge from individual-level birth, death, and movement processes. These processes, which depend on local landscape characteristics, vary spatially and may exhibit sharp transitions through behavioural responses to habitat edges, leading to discontinuous population densities. Such systems can be modelled using reaction–diffusion equations with interface conditions that capture local behaviour at patch boundaries. In this work we develop a novel homogenization technique to approximate the large-scale dynamics of the system. We illustrate our approach, which also generalizes to multiple species, with an example of logistic growth within a periodic environment. We find that population persistence and the large-scale population carrying capacity is influenced by patch residence times that depend on patch preference, as well as movement rates in adjacent patches. The forms of the homogenized coefficients yield key theoretical insights into how large-scale dynamics arise from the small-scale features.
Highlights
Spatial ecology aims to explain observed spatial distribution patterns of populations and to predict their response to landscape alterations
We present a homogenization approach to the multiscale problem of how individual behavioural responses to sharp transitions in landscape features, such as forest edges, affect population-dynamical outcomes
The dynamics of populations on large spatial and temporal scales are of great interest in theoretical ecology, for example in conservation and invasion biology
Summary
Spatial ecology aims to explain observed spatial distribution patterns of populations and to predict their response to landscape alterations. We present a homogenization approach to the multiscale problem of how individual behavioural responses to sharp transitions in landscape features, such as forest edges, affect population-dynamical outcomes. Othmer [15] applied the method of homogenization to a model of Fickian diffusion assuming continuous population density. Other authors incorporated the more general interface conditions (3)–(4), but proceeded to scale density and space in Equations (2)–(4) to obtain continuous densities so that the classical techniques apply [13]. This approach is unsatisfactory and does not generalize, for example to multispecies models. We discuss ecological insights from our approach as well as future directions and application in the final section
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