Bifurcação e estabilidade de equilíbrios de um problema de Neumann parabólico com peso indefinido em genética populacional

Abstract

This work is concerned with a semilinear parabolic partial differential equation under a homogeneous Neumann boundary condition occuring in population genetics. It describes the evolution of gene frequencies under selection and migration effects in a bounded domain. In the nonlinear term appearing in the equation, which is of logistic type, one has a positive parameter and an indefinite sign weight function. Considering a suitable phase space one obtains a nonlinear dynamical system, actually a gradient system, in a such way that the equilibrium solutions, or stationary solutions, play a fundamental role in the dynamics viewpoint. The problem has two constant equilibria, called the trivial ones, inducing two curves, called trivial branches, and containing the trivial equilibrium solutions. The main aim of this dissertation is to study the bifurcation and stability structures of equilibria, which are completely established and also expressed through diagrams. Furthermore, to establish the behaviour of the only nontrivial equilibrium the problem has for each value of the parameter. A key ingredient in the analysis is the average of the weight function. Indeed, if the weight has nonzero average a global curve consisting of nontrivial exponentially stable equilibria bifurcates from a trivial branch − which is determined according to the sign of the average. But if the parameter is sufficiently small the problem admits the two trivial equilibria as the only equilibrium solutions, one of them exponentially stable and the other unstable. When the weight function has zero average a new curve has now a central role: from such curve bifurcates a global curve defined for all values of the parameter and consisting of nontrivial exponentially stable equilibria. Further, there is no bifurcation from the trivial branches, that ones containing unstable equilibria. Finally, the behaviour of the global bifurcation branch is also established as the parameter is large. Actually, that is achieved as long as one proves the only nontrivial equilibrium concentrates in a region where the weight function has a definite sign, as the parameter is large.