• Previous Article
    Regularity estimates for continuous solutions of α-convex balance laws
  • CPAA Home
  • This Issue
  • Next Article
    A concentration phenomenon of the least energy solution to non-autonomous elliptic problems with a totally degenerate potential
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:
[1]

S.R. CarpenterD. 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.
[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. LeeS. 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. LudwigD.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. LudwigD.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. SchefferS. CarpenterJ.A. FoleyC. 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. SugieR. Kohno and R. Miyazaki, On a predator-prey system of Holling type, On a predator-prey system of Holling type, 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

show all references

References:
[1]

S.R. CarpenterD. 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.
[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. LeeS. 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. LudwigD.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. LudwigD.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. SchefferS. CarpenterJ.A. FoleyC. 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. SugieR. Kohno and R. Miyazaki, On a predator-prey system of Holling type, On a predator-prey system of Holling type, 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

Figure 1.  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 $g(u)$
Figure 2.  (a) S-shaped bifurcation curve $\bar{S}$ of (1). (b)-(c) Broken S-shaped bifurcation curves $\bar{S}$ of (1)
Figure 3.  Graphs of $\eta ={{m}_{p}}$, $\eta ={{\eta }_{1, p}}$ and $\eta ={{\eta }_{2, p}}$ for $p\in (1, 10]$
Figure 5.  Graphs of functions $\eta =I(u)$, $\eta =J(u)$, $\eta =K(u)$ on $(0, \infty)$
Figure 6.  Graphs of functions $\eta =I(u)$, $\eta =J(u)$, $\eta =M(u)$ on $(0, \infty)$
Figure 4.  (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)
[1]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839

[2]

Chih-Yuan Chen, Shin-Hwa Wang, Kuo-Chih Hung. S-shaped bifurcation curves for a combustion problem with general arrhenius reaction-rate laws. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2589-2608. doi: 10.3934/cpaa.2014.13.2589

[3]

Sabri Bensid, Jesús Ildefonso Díaz. Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1757-1778. doi: 10.3934/dcdsb.2017105

[4]

Xue Dong He, Roy Kouwenberg, Xun Yu Zhou. Inverse S-shaped probability weighting and its impact on investment. Mathematical Control & Related Fields, 2018, 8 (3&4) : 679-706. doi: 10.3934/mcrf.2018029

[5]

Shanshan Chen, Junping Shi, Junjie Wei. The effect of delay on a diffusive predator-prey system with Holling Type-II predator functional response. Communications on Pure & Applied Analysis, 2013, 12 (1) : 481-501. doi: 10.3934/cpaa.2013.12.481

[6]

Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244

[7]

Tetsuya Ishiwata, Takeshi Ohtsuka. Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-35. doi: 10.3934/dcdsb.2019058

[8]

Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875

[9]

Paweł Góra, Abraham Boyarsky. Stochastic perturbations and Ulam's method for W-shaped maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1937-1944. doi: 10.3934/dcds.2013.33.1937

[10]

Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182

[11]

Sébastien Biebler. Lattès maps and the interior of the bifurcation locus. Journal of Modern Dynamics, 2019, 15: 95-130. doi: 10.3934/jmd.2019014

[12]

Jean-François Couchouron, Mikhail Kamenskii, Paolo Nistri. An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1845-1859. doi: 10.3934/cpaa.2013.12.1845

[13]

Jian Zu, Wendi Wang, Bo Zu. Evolutionary dynamics of prey-predator systems with Holling type II functional response. Mathematical Biosciences & Engineering, 2007, 4 (2) : 221-237. doi: 10.3934/mbe.2007.4.221

[14]

Qiumei Zhang, Daqing Jiang, Li Zu. The stability of a perturbed eco-epidemiological model with Holling type II functional response by white noise. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 295-321. doi: 10.3934/dcdsb.2015.20.295

[15]

Shuping Li, Weinian Zhang. Bifurcations of a discrete prey-predator model with Holling type II functional response. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 159-176. doi: 10.3934/dcdsb.2010.14.159

[16]

Amadeu Delshams, Josep J. Masdemont, Pablo Roldán. Computing the scattering map in the spatial Hill's problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 455-483. doi: 10.3934/dcdsb.2008.10.455

[17]

Marie-Claude Arnaud. A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map. Journal of Modern Dynamics, 2011, 5 (3) : 583-591. doi: 10.3934/jmd.2011.5.583

[18]

Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597

[19]

Zengji Du, Xiao Chen, Zhaosheng Feng. Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203

[20]

Ana Paula S. Dias, Paul C. Matthews, Ana Rodrigues. Generating functions for Hopf bifurcation with $ S_n$-symmetry. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 823-842. doi: 10.3934/dcds.2009.25.823

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (5)
  • HTML views (2)
  • Cited by (1)

Other articles
by authors

[Back to Top]