Clément, Ph.; Peletier, L. A. On a nonlinear eigenvalue problem occurring in population genetics. (English) Zbl 0569.34021 Proc. R. Soc. Edinb., Sect. A 100, 85-101 (1985). 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. Cited in 7 Documents MSC: 34L99 Ordinary differential operators 92D25 Population dynamics (general) 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:nonlinear eigenvalue problem; bifurcation point; example PDFBibTeX XMLCite \textit{Ph. Clément} and \textit{L. A. Peletier}, Proc. R. Soc. Edinb., Sect. A, Math. 100, 85--101 (1985; Zbl 0569.34021) Full Text: DOI References: [1] DOI: 10.1007/BF00280092 · Zbl 0361.92020 · doi:10.1007/BF00280092 [2] DOI: 10.1137/0128005 · Zbl 0295.35044 · doi:10.1137/0128005 [3] DOI: 10.1016/0022-0396(79)90006-8 · Zbl 0387.35025 · doi:10.1016/0022-0396(79)90006-8 [4] DOI: 10.1016/0022-0396(75)90084-4 · Zbl 0304.35008 · doi:10.1016/0022-0396(75)90084-4 [5] DOI: 10.1137/1018114 · Zbl 0345.47044 · doi:10.1137/1018114 [6] DOI: 10.2977/prims/1195188180 · Zbl 0445.35063 · doi:10.2977/prims/1195188180 [7] DOI: 10.1080/03605307808820080 · Zbl 0421.35072 · doi:10.1080/03605307808820080 [8] Peletier, Lecture Notes in Mathematics 665 pp 170– (1978) [9] Peletier, Lecture Notes in Mathematics 564 pp 365– (1976) [10] Nagylaki, Genetics 80 pp 595– (1975) · Zbl 0839.92011 [11] DOI: 10.1216/RMJ-1973-3-2-161 · Zbl 0255.47069 · doi:10.1216/RMJ-1973-3-2-161 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.