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On a nonlinear eigenvalue problem occurring in population genetics. (English) Zbl 0569.34021

We discuss the nonlinear eigenvalue problem \[ (P)\quad u''+\lambda f(x,u)=0,\quad -1<x<1,\quad \lambda \geq 0,\quad u'(-1)=u'(+1)=0,\quad 0\leq u(x)\leq 1, \] where \(f(x,u)=u(1-u)[u-a(x)]\) and \(a(x)=(1/2)[1- \epsilon r(x)+h]\) \(\epsilon\geq 0\), \(h\in {\mathbb{R}}\) with \(r(-x)=-r(x)\) and r’\(\geq 0\). For \(\epsilon =h=0\), the solution to Problem P is well known, and every solution, except \(u=0\) and \(u=1\), is unstable with respect to the corresponding parabolic problem. We show how the branch of increasing solutions changes as \(\epsilon\) becomes positive, and acquires a bifurcation point \(({\bar \lambda},\bar u)\) beyond which this branch becomes stable. If h becomes non-zero as well, this bifurcation point is shown to break up. As an illustration, we consider an example in which the branch of increasing solutions can be computed. Here \(f(x,u)=-u\), \(0\leq u<a(x)\), \(f(x,u)=1-u\), \(a(x)<u\leq 1\), where a(x) is given above.

MSC:

34L99 Ordinary differential operators
92D25 Population dynamics (general)
34B15 Nonlinear boundary value problems for ordinary differential equations
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