Bifurcation curves of positive steady-state solutions for a reaction-diffusion problem of lake eutrophication

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• 作者：Tzung-Shin Yeh ; ^{tsyeh@mail.nutn.edu.tw}
• 关键词：Bifurcation curve ; Exact multiplicity ; Holling type-III functional response ; Time map
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 May 2017
• 年：2017
• 卷：449
• 期：2
• 页码：1708-1724
• 全文大小：479 K
• 卷排序：449

文摘

We study exact multiplicity and bifurcation curves of positive solutions for a multiparameter Dirichlet problem{u″(x)+λ[a−bu+up1+up]=0,−1<x<1,u(−1)=u(1)=0, where p>1p>1, a,ba,b are positive dimensionless parameters, and λ>0λ>0 is a bifurcation parameter. For fixed p>1p>1, assume that either 0<a≤a1,p0<a≤a1,p, b>0b>0 and (a,b)(a,b) lies below the curveΓ1={(a,b):a(t)=tp[p−1−tp](tp+1)2,b(t)=ptp−1(tp+1)2,0<t<p−1p+1p},or 0<a≤a2,p0<a≤a2,p and (a,b)(a,b) lies on or above the curve Γ1Γ1 for some positive constants a1,pa1,p and a2,pa2,p. Then on the (λ,‖u‖_{∞)(λ,‖u‖∞)-plane, we give a classification of three qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, and a monotone increasing curve. Hence we are able to determine the exact multiplicity of positive solutions by the values of a, b and λ.}