An integro-PDE model from population genetics
This paper is concerned with an integro-partial differential equation that models the evolution of the frequencies for two alleles at a single locus under the joint action of migration, selection, and partial panmixia. Let \(\Omega\) be a bounded region in \(\mathbb{R}^N\), let \(P(x,t)\) and \(1-P(x,t)\) be the frequencies of the two alleles at location \(x\) and generation \(t\), let \(V(x)\) be one half of the \(N\times N\) symmetric, positive definite, single generation covariance matrix of local migration, and let the vector \(M(x)\) represent the single generation mean displacement by local migration. Define a vector \(b\) by \(b(x)=\rho^{-1}\nabla\cdot (\rho V)-M\). Normalize the population \(\rho(x)\) by \(\int_{\Omega}\rho(x)dx=1\) and define the average of a function \(u(x,t)\) by \(\bar u(t)=\int_{\Omega}\rho(x)u(x,t)dx\). Then the equation can be written as \[ \begin{cases} P_t=\sum_{i,j=1}^N V_{ij}P_{x_ix_j}+b\cdot \nabla P+B(\bar P-P)+\lambda g(x)f(P) & \text{in } \Omega\times (0,\infty) \\ \nu \cdot V\nabla P=0 & \text{on } \partial \Omega\times (0,\infty) \end{cases} \] under the initial condition \(0\leq P(x,0)\leq 0\) in \(\Omega\), \(0<\bar P(0)<1\). Here \(B\) is a positive constant that represents the scaled rate of global panmixia, \(\lambda\) denotes the selection intensity, the functions \(g(x)\) and \(f(P)\) denote the spatial and frequency dependence of selection respectively, and \(\nu\) is the out-normal vector along the boundary \(\partial \Omega\). One of the main results obtained by the authors is the existence, uniqueness and global asymptotic stability of the nontrivial equilibrium, which implies the persistence of both alleles. Parameter dependence for the maintenance of the alleles is also studied. This work extends results in \textit{T. Nagylaki} [``Clines with partial panmixia'', Theor. Popul. Biol. 81, 45--68 (2012)] on conservative migration to arbitrary migration.
scientific article; zbMATH DE number 6140822