Pushing the Boundaries: Models for the Spatial Spread of Ecosystem Engineers
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Pushing the Boundaries: Models for the Spatial Spread of Ecosystem Engineers Frithjof Lutscher1
· Justus Fink2,3 · Yingjie Zhu2,4
Received: 17 July 2020 / Accepted: 3 October 2020 / Published online: 15 October 2020 © Society for Mathematical Biology 2020
Abstract Ecosystems engineers are species that can substantially alter their abiotic environment and thereby enhance their population growth. The net growth rate of obligate engineers is even negative unless they modify the environment. We derive and analyze a model for the spread and invasion of such species. Prior to engineering, the landscape consists of unsuitable habitat; after engineering, the habitat is suitable. The boundary between the two types of habitat is moved by the species through their engineering activity. Our model is a novel type of a reaction–diffusion free boundary problem. We prove the existence of traveling waves and give upper and lower bounds for their speeds. We illustrate how the speed depends on individual movement and engineering behavior near the boundary. Keywords Free boundary problem · Biological invasion · Ecosystem engineer · Reaction–diffusion equation · Traveling wave Mathematics Subject Classification 35K57 · 35R35 · 92D40
1 Introduction Many, if not most, mathematical models for the spatial spread of organisms and biological invasions are based on reaction–diffusion equations or, more recently, on integrodifference equations, and, ultimately, on the famous Fisher–KPP model for the spread of an advantageous gene (Fisher 1937; Kot et al. 1996; Lewis et al. 2016).
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Frithjof Lutscher [email protected]
1
Department of Mathematics and Statistics, Department of Biology, University of Ottawa, Ottawa, ON K1N 6N5, Canada
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Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, Canada
3
Present Address: Institute of Integrative Biology, ETH Zurich, 8092 Zurich, Switzerland
4
Present Address: College of Science, Changchun University, Changchun 130022, China
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Fisher’s model for the population density u(t, x) is given by u t = Du x x + f (u),
(1)
where D is the diffusion coefficient and f (u) = u(1 − u) the growth function. We use indices x, t to denote partial derivatives. Fisher’s famous result states that this equation has traveling waves for all speeds greater than or equal to the minimal speed c∗ = 2 D f (0). This is also the speed at which a locally introduced population will eventually spread (Weinberger 1982). Several empirical studies show that c∗ is a good estimator for the speed of spread of biological invasions (Andow et al. 1990; Lewis et al. 2016). One assumption that this model and most of its extensions make is that the environment in which the species spreads is unaffected by the presence of the species. Strictly speaking, this assumption is unlikely to apply to any species since almost all organisms affect their environment somehow. In many cases, the effect that the organism-induced changes to the environment have on the org
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