Complete classification of bifurcation curves for a multiparameter diffusive logistic problem with generalized Holling type-IV functional response. (English) Zbl 1457.34041
Summary: We study exact multiplicity and bifurcation curves of positive solutions for the diffusive logistic problem with generalized Holling type-IV functional response \[ \begin{gathered} u''(x)+\lambda \bigg[ ru(1-\frac{u}{q})-\frac{u}{1+mu+u^2}\bigg] =0,\quad-1<x<1, \\ u(-1)=u(1)=0, \end{gathered} \] where the quantity in brackets is the growth rate function and \(\lambda>0\) is a bifurcation parameter. On the \((\lambda, \|u\|_\infty)\)-plane, we give a complete classification of two qualitatively different bifurcation curves: a \(\mathrm{C}\)-shaped curve and a monotone increasing curve.
MSC:
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
Keywords:
bifurcation curve; exact multiplicity; diffusive logistic problem; Holling type-IV functional response; time mapReferences:
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