Introducing a Population into a Steady Community: The Critical Case, the Center Manifold, and the Direction of Bifurcation

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, R0. When R0 exceeds 1, the invading population meets with success, and when R0 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 R0 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.