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Non-local effects in an integro-PDE model from population genetics

2015; Cambridge University Press; Volume: 28; Issue: 1 Linguagem: Inglês

10.1017/s0956792515000601

1469-4425

Fu-Xiang Li, Kimie Nakashima, Wei Ni,

Evolution and Genetic Dynamics

In this paper, we study the following non-local problem: \begin{equation*} \begin{cases} \displaystyle u_t=d{1\over\rho}\nabla\cdot(\rho V\nabla u)+b(\bar{u}-u)+ g(x) u^2(1-u) &\displaystyle \quad \textrm{in} \; \Omega\times (0,\infty),\\[3pt] \displaystyle 0\leq u\leq 1 & \quad\displaystyle \textrm{in}\ \Omega\times (0,\infty),\\[3pt] \displaystyle \nu \cdot V\nabla u=0 &\displaystyle \quad \textrm{on} \; \partial\Omega\times (0,\infty).\vspace*{-2pt} \end{cases} \end{equation*} This model, proposed by T. Nagylaki, describes the evolution of two alleles under the joint action of selection, migration, and partial panmixia – a non-local term , for the complete dominance case, where g ( x ) is assumed to change sign at least once to reflect the diversity of the environment. First, properties for general non-local problems are studied. Then, existence of non-trivial steady states, in terms of the diffusion coefficient d and the partial panmixia rate b , is obtained under different signs of the integral ∫ Ω g ( x ) dx . Furthermore, stability and instability properties for non-trivial steady states, as well as the trivial steady states u ≡ 0 and u ≡ 1 are investigated. Our results illustrate how the non-local term – namely, the partial panmixia – helps the migration in this model.