Clines induced by variable selection and migration

Abstract

Clines (non-uniform spatial distributions in the genetic composition of a population in equilibrium) are often modelled by non-constant solutionsu(x)∈ [0,1] of [D(x) u']' +h(x, u)= 0, – ∞ <x< ∞, wherehsatisfiesh(x, 0) =h(x, 1) = 0, andDis often taken to be identically one. The functionsDandhhave interpretations in terms of mobility, carrying capacity and natural selection. We define clines as stable solutions satisfyingu( – ∞ ) = 0,u(∞) = 1. All past analyses of clines have considered the case when (say) 0 is the favoured state for large negativex, and 1 for large positivex(i. e. ∫01h(x, u)duchanges sign from negative to positive asxincreases from – ∞ to ∞). In this paper, however, we assume that the state 0 is favoured for allx, although both 0 and 1 are stable as uniform states. Conditions are given that ensure the existence of stable clines, or their analogues for bounded habitats. Conditions are also given that ensure the non-existence of clines. The concept of stability is to be understood with reference to the corresponding nonlinear diffusion equation, and is used in a special technical sense.