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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:
 [1] S.R.Carpenter, D.Ludwig and W.A.Brock, Management of eutrophication for lakes subject to potentially irreversible change, Ecol.Appl., 9 (1999), 751-771. Google Scholar [2] J. Jiang and J. Shi, Bistability dynamics in some structured ecological models, in Spatial Ecology, Chapman & Hall/CRC Press, Boca Raton, FL, 2009.   Google Scholar [3] P.Korman and J.Shi, New exact multiplicity results with an application to a population model, Proc.Royal.Soc.Edinburgh Sect.A, 131 (2001), 1167-1182. doi: 10.1017/S0308210500001323.  Google Scholar [4] T.Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ.Math.J., 20 (1970), 1-13. doi: 10.1512/iumj.1970.20.20001.  Google Scholar [5] E.Lee, S.Sasi and R.Shivaji, S-shaped bifurcation curves in ecosystems, J.Math.Anal.Appl., 381 (2011), 732-741. doi: 10.1016/j.jmaa.2011.03.048.  Google Scholar [6] D.Ludwig, D.G.Aronson and H.F.Weinberger, Spatial patterning of the spruce budworm, J.Math.Biol., 8 (1979), 217-258. doi: 10.1007/BF00276310.  Google Scholar [7] D.Ludwig, D.D.Jones and C.S.Holling, Qualitative analysis of insect outbreak systems: the spruce budworm and the forest, J.Anim.Ecol., 47 (1978), 315-332. Google Scholar [8] R.M.May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477. Google Scholar [9] J. D. Murray, Mathematical Biology. I. An introduction, 3rd edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002.  Google Scholar [10] J. D. Murray, Mathematical Biology. Ⅱ. Spatial Models and Biomedical Applications, 3rd edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003.  Google Scholar [11] I.Noy-Meir, Stability of grazing systems: An application of predator-prey graphs, J.Ecol., 63 (1975), 459-481. Google Scholar [12] M.Scheffer, S.Carpenter, J.A.Foley, C.Folke and B.Walkerk, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596. Google Scholar [13] J.Shi and R.Shivaji, Persistence in reaction diffusion models with weak Allee effect, J.Math.Biol., 52 (2006), 807-829. doi: 10.1007/s00285-006-0373-7.  Google Scholar [14] J.Sugie and M.Katagama, Global asymptotic stability of a predator-prey system of Holling type, Nonlinear Anal., 38 (1999), 105-121. doi: 10.1016/S0362-546X(99)00099-1.  Google Scholar [15] J.Sugie, R.Kohno and R.Miyazaki, On a predator-prey system of Holling type, Proc.Amer.Math.Soc., 125 (1997), 2041-2050. doi: 10.1090/S0002-9939-97-03901-4.  Google Scholar [16] S.-H.Wang and T.-S.Yeh, S-shaped and broken S-shaped bifurcation diagrams with hysteresis for a multiparameter spruce budworm population problem in one space dimension, J.Differential Equations, 255 (2013), 812-839. doi: 10.1016/j.jde.2013.05.004.  Google Scholar
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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