Global stability of spatially homogeneous equilibria in migration-selection models (Q2794982)

The paper looks at a migration-selection model, namely a system of ordinary differential equations (ODE) describing the genetic evolution of a locus with \(n\) alleles in a diploid population with constant size, under selection and geographically subdivided into a finite number \(d\) of panmitic colonies with non-assortative migration among such colonies. Uniform selection (a genotype has the same fitness in all colonies) is assumed.NEWLINENEWLINEIn the absence of migration, the subsystems of ODE corresponding to the different colonies are identical and have the same equilibria. Let \(\hat {p}=(p_1, p_2, \dots, p_n)\) (vector of the different allele frequencies) be such a subsystem equilibrium and assume it is globally asymptotically stable (g.a.s.). The full system of ODE has as a g.a.s. equilibrium the \(d \times n\) vector \(\tilde {p}=(\hat {p} | \hat {p} | \dots | \hat {p})\) obtained by concatenating \(d\) copies of the vector \(\hat {p}\). \textit{T. Nagylaki} and \textit{Y. Lou} conjectured that \(\tilde {p}\) is a g.a.s. equilibrium of the full system even when there is migration. Under mild assumptions, they proved the conjecture [Theor. Popul. Biol. 72, No. 1, 21--40 (2007; Zbl 1125.92045)] when \(\hat {p}\) is an interior equilibrium, i.e., \(p_i >0\) for all \(i=1,2,\dots ,n\). The conjecture remained open when \(\hat {p}\) is not an interior point. Assuming all equilibria are regular (have Jacobian matrix with eigenvalues all different from zero), this paper proves the conjecture for boundary points in the particular case \(n=3\) by proving and using some results on quasiconcave Lyapunov functions for each of different possible cases (this case-by-case approach cannot be used for a general value of \(n\)).NEWLINENEWLINEThe paper then extends the result to the case of geographically continuously spread populations described by a system of partial differential equations on the densities across space of the different allele frequencies. This system takes into account migration trends and spatial diffusion.