Bifurcation curves of positive steady-state solutions for a reaction-diffusion problem of lake eutrophication
The article contains the investigation of the exact multiplicity and bifurcation curves of positive solutions for a multiparameter Dirichlet problem \[ \begin{cases} u''(x)+\lambda \left [a-bu+\frac{u^{p}}{1+u^{p}} \right]=0, -1<x<1,\\ u(-1)=u(1)=0 ,\end{cases} \] where \(p\), \(a\), \(b\) are positive dimensionless parameters, \(\lambda>0\) is a bifurcation parameter, the function \(\frac{u^{p}}{1+u^{p}} \) with \(p>1\) is the Holling type III functional response function. For fixed \(p > 1\), it's assumed that either \(0 \leq a \leq a_{1,p}\), \(b > 0\) and \((a, b)\) lies below the curve \[ \Gamma_{1} =\left\{ (a,b): a(t)=\frac{t^{p}[p-1-t^{p}]}{(t^{p}+1)^{2}}, b(t)=\frac{pt^{p-1}}{(t^{p}+1)^{2}}, 0<t< \sqrt[p]{\frac{p-1}{p+1}}\right\} \] and the bifurcation curve is an \(S\)-shaped curve; at \(0 \leq a \leq a_{2,p}\) the point \((a, b)\) lies on or above the curve \(\Gamma_{1}\) for some positive constants \(a_{1,p}\) and \(a_{2,p}\). The main result of the article are two theorems, it includes a classification of three qualitatively different bifurcation curves on the \((\lambda, \| u\|_{\infty})\)-plane and corresponding exactly amount of positive solutions. The bifurcation curves have one of the shapes: an \(S\)-shaped curve, a broken \(S\)-shaped curve, and a monotone increasing curve. Hence, the article includes the method to determine the exact multiplicity of positive solutions by the values of \(a\), \(b\) and \(\lambda\).
scientific article; zbMATH DE number 6682118
scientific article