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Regularity estimates for continuous solutions of α-convex balance laws
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 |
${\left\{ {\begin{array}{*{20}{l}} {{u^{\prime \prime }}(x) + \lambda \left[ {ru(1 - \frac{u}{q}) - \frac{{{u^p}}}{{1 + {u^p}}}\% } \right] = 0{\text{,}} - {\text{1}} < x < 1{\text{,}}} \\ {u( - 1) = u(1) = 0{\text{, }}} \end{array}} \right.},$ |
$\begin{array}{l}{\Gamma _1} = \{ (q,r):q(a) = \frac{{a[2{a^p} - (p - 2)]}}{{{a^p} - (p - 1)}}{\rm{, }}\\\quad \quad \quad \quad \quad r(a) = \frac{{{a^{p - 1}}[2{a^p} - (p - 2)]}}{{{{({a^p} + 1)}^2}}}{\rm{, }}\sqrt[p]{{p - 1}}\% < a < C_p^*\} ;\end{array}$ |
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. |
[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. |
[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. |
[8] |
R.M.May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477. |
[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. |
[12] |
M.Scheffer, S.Carpenter, J.A.Foley, C.Folke and B.Walkerk, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596. |
[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, Proc.Amer.Math.Soc., 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. |
[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. |
[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. |
[8] |
R.M.May, Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477. |
[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. |
[12] |
M.Scheffer, S.Carpenter, J.A.Foley, C.Folke and B.Walkerk, Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596. |
[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, Proc.Amer.Math.Soc., 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. |






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