Mathematics – Analysis of PDEs
Scientific paper
2012-04-03
Mathematics
Analysis of PDEs
Scientific paper
A Hamilton-Jacobi formulation has been established previously for phenotypical structured population models where the solution concentrates as Dirac masses in the limit of small diffusion. Is it possible to extend this approach to spatial models? Are the limiting solutions still in the form of sum of Dirac masses? Does the presence of several habitats lead to polymorphic situations? We study the stationary solutions of a structured population model, while the population is structured by continuous phenotypical traits and discrete positions in space. Several habitable zones are possible and the growth term varies from one zone to another, for instance because of a change in the temperature. The individuals can migrate from one zone to another with a constant rate. The mathematical modeling of this problem, considering mutations between phenotypical traits and competitive interaction of individuals within each zone via a single resource, leads to a system of coupled parabolic integro-differential equations. We study the asymptotic behavior of the stationary solutions to this model in the limit of small mutations. The limit, which is a sum of Dirac masses, can be described with the help of an effective Hamiltonian. The presence of migration can modify the dominant traits and lead to polymorphic situations. Numerical simulations show that in the time-dependent problem an oscillatory behavior may occur.
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