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 $g(u)$
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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 |
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) r ≤ η1, 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) r ≤ η2, 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.
Keywords:
S-shaped bifurcation curve,
broken S-shaped bifurcation curve,
Holling type-Ⅲ functional response,
diffusive logistic problem,
time map.
Mathematics Subject Classification:
Primary: 34B15, 34B18.
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. 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. |
| [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. |
| [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. |
| [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. |
| [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 |
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J. D. Murray, Mathematical Biology. I. An introduction, 3rd edition, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. |
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J. D. Murray, Mathematical Biology. Ⅱ. Spatial Models and Biomedical Applications, 3rd edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003. |
| [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. |
| [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. |
| [15] |
J. Sugie, R. 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. |
| [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. |
show all references
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
J. D. Murray, Mathematical Biology. Ⅱ. Spatial Models and Biomedical Applications, 3rd edition, Interdisciplinary Applied Mathematics, 18, Springer-Verlag, New York, 2003. |
| [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. |
| [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. |
| [15] |
J. Sugie, R. 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. |
| [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. |
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)
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