We study the evolution of the probability density of an a sexual, one locus population under natural selection and random evolution. This evolution is governed by a Fokker-Planck equation with degenerate coefficients on the boundaries, supplemented by a pair of conservation laws. It is readily shown that no classical or standard weak solution definition yields solvability of the problem. We provide an appropriate definition of weak solution for the problem, for which we show existence and uniqueness. The solution displays a very distinctive structure and, for large time, we show convergence to a unique stationary solution that turns out to be a singular measure supported at the endpoints. An exponential rate of convergence to this steady state is also proved.
机构:
Zhejiang Univ, Dept Math, Hangzhou 310027, Zhejiang, Peoples R ChinaZhejiang Univ, Dept Math, Hangzhou 310027, Zhejiang, Peoples R China
Qian, Chenyin
Shen, Zifei
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Zhejiang Normal Univ, Coll Math Phys & Informat Engn, Jinhua 321004, Peoples R ChinaZhejiang Univ, Dept Math, Hangzhou 310027, Zhejiang, Peoples R China
Shen, Zifei
Zhu, Junqiao
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Zhejiang Normal Univ, Xingzhi Coll, Jinhua 321004, Peoples R ChinaZhejiang Univ, Dept Math, Hangzhou 310027, Zhejiang, Peoples R China
机构:
Sapienza Univ Roma, Dipartimento Sci Base & Appl Ingn, Via Antonio Scarpa 16, I-00161 Rome, Rm, ItalyUniv Lisbon, Dept Matemat, Fac Ciencias, Edificio C6,Piso 1, P-1749016 Lisbon, Portugal