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Introducing a population into a steady community: the critical case, the center manifold, and the direction of bifurcation
Boldin, Barbara, 1976-
In this paper we study deterministic, finite dimensional, continuous, as well as discrete time population invasion models. The ability of a newly introduced population, either a new species or a ...reproductively isolated subpopulation of one of the already present species, to settle in the community relies upon the basic reproduction ratio of the invader, ▫${\mathcal{R}}_0$▫. When ▫${\mathcal{R}}_0$▫ exceeds 1, the invading population meets with success, and when ▫${\mathcal{R}}_0$▫ is below 1, the invasion fails. The aim of this paper is to investigate the possible effects of an invasion when the parameters of a model are varied so that ▫${\mathcal{R}}_0$▫ of the invading population passes the value 1. We argue that population invasion models, regardless of the biology that underlies them, take a specific form that significantly simplifies the center manifold analysis. We make a uniform study of ecological, adaptive dynamics and disease transmission models and derive a simple formula for the direction of bifurcation from a steady state in which only the resident populations are present. Furthermore, we observe that among those bifurcation parameters that satisfy a certain condition, we acquire the same direction of bifurcation. The obtained mathematical results are used to gain insight into the biology of invasions. The theory is illustrated by several examples.
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