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March  2017, 16(2): 645-670. doi: 10.3934/cpaa.2017032

S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response

 Department of Applied Mathematics, National University of Tainan, Tainan 700, Taiwan, ROC

Received  April 2016 Revised  October 2016 Published  January 2017

Fund Project: The author is supported by NSC grant 101-2115-M-024-003

We study exact multiplicity and bifurcation curves of positive solutions for a multiparameter diffusive logistic problem with Holling type-Ⅲ functional response
 ${\left\{ {\begin{array}{*{20}{l}} {{u^{\prime \prime }}(x) + \lambda \left[ {ru(1 - \frac{u}{q}) - \frac{{{u^p}}}{{1 + {u^p}}}\% } \right] = 0{\text{,}} - {\text{1}} < x < 1{\text{,}}} \\ {u( - 1) = u(1) = 0{\text{, }}} \end{array}} \right.},$
where u is the population density of the species, p > 1, q, r are two positive dimensionless parameters, and λ > 0 is a bifurcation parameter. For fixed p > 1, assume that q, r satisfy one of the following conditions: (ⅰ) rη1, p* q and (q, r) lies above the curve
 $\begin{array}{l}{\Gamma _1} = \{ (q,r):q(a) = \frac{{a[2{a^p} - (p - 2)]}}{{{a^p} - (p - 1)}}{\rm{, }}\\\quad \quad \quad \quad \quad r(a) = \frac{{{a^{p - 1}}[2{a^p} - (p - 2)]}}{{{{({a^p} + 1)}^2}}}{\rm{, }}\sqrt[p]{{p - 1}}\% < a < C_p^*\} ;\end{array}$
(ⅱ) rη2, p* q and (q, r) lies on or below the curve Γ1, where η1, p* and η2, p* are two positive constants, and $C_{p}^{*}={{\left(\frac{{{p}^{2}}+3p-4+p\sqrt{%{{p}^{2}}+6p-7}}{4} \right)}^{1/p}}$. Then on the (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 q, r and λ.
Citation: Tzung-shin Yeh. S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response. Communications on Pure & Applied Analysis, 2017, 16 (2) : 645-670. doi: 10.3934/cpaa.2017032
References:

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References:
Classified graphs of growth rate per capita $g(u)=r(1-\frac{u}{q})-\frac{u^{p-1}}{1+u^{p}}$ on $(0, \infty)$ with fixed $p > 1$, drawn on the first quadrant of $(q, r)$-parameter plane according to the monotonicity of $g(u)$
(a) S-shaped bifurcation curve $\bar{S}$ of (1). (b)-(c) Broken S-shaped bifurcation curves $\bar{S}$ of (1)
Graphs of $\eta ={{m}_{p}}$, $\eta ={{\eta }_{1, p}}$ and $\eta ={{\eta }_{2, p}}$ for $p\in (1, 10]$
Graphs of functions $\eta =I(u)$, $\eta =J(u)$, $\eta =K(u)$ on $(0, \infty)$
Graphs of functions $\eta =I(u)$, $\eta =J(u)$, $\eta =M(u)$ on $(0, \infty)$
(a) Graph of $N_{1}(B_{1, p}(\eta _{1, p}))+N_{2}(C_{2, p}(\eta _{1, p}))$ for $p\in \lbrack1.01, 10]$ (left). (b) Graph of $N_{3}(B_{1, p}(\eta _{2, p}))+N_{4}(C_{2, p}(\eta _{2, p}))$ for $p\in \lbrack 1.01, 10]$ (right)
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