Bifurcation structure of indefinite nonlinear diffusion problem in population genetics.

We study positive stationary solutions for the following Neumann problem in one-dimension space arising from population genetics: { u t = D u x x + g (x) u 2 (1 − u) in (0 , 1) × (0 , ∞) , u x (0 , t) = u x (1 , t) = 0 in (0 , ∞) , where g changes sign once in (0 , 1) and D is a positive parameter. This equation has a stationary positive solution u , where u − 1 has n zeros in (0 , 1). We denote this solution by an M (n) -solution (n = 1 , 2 , ⋯). We show that the M (n) -solution branch bifurcates from the trivial solution u = 1 and the M (n) -solution branch does not meet other bifurcating points and is extended globally to D → 0. When D is sufficiently small, the M (n) -solution has a very characteristic shape. Especially when n = 1 , we will show all possible shapes of M (1) -solutions. [ABSTRACT FROM AUTHOR]