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Minimization of carbon abatement cost: Modeling, analysis and simulation
Interaction between water and plants: Rich dynamics in a simple model
1. | School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China |
2. | Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA |
An ordinary differential equation model describing interaction of water and plants in ecosystem is proposed. Despite its simple looking, it is shown that the model possesses surprisingly rich dynamics including multiple stable equilibria, backward bifurcation of positive equilibria, supercritical or subcritical Hopf bifurcations, bubble loop of limit cycles, homoclinic bifurcation and Bogdanov-Takens bifurcation. We classify bifurcation diagrams of the system using the rain-fall rate as bifurcation parameter. In the transition from global stability of bare-soil state for low rain-fall to the global stability of high vegetation state for high rain-fall rate, oscillatory states or multiple equilibrium states can occur, which can be viewed as a new indicator of catastrophic environmental shift.
References:
[1] |
J. C. Alexander and J. A. Yorke,
Global bifurcations of periodic orbits, Amer. J. Math., 100 (1978), 263-292.
doi: 10.2307/2373851. |
[2] |
J. Arino, C.C. McCluskey and P. van den Driessche,
Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260-276.
doi: 10.1137/S0036139902413829. |
[3] |
F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology volume 40 of Texts in Applied Mathematics, Springer, New York, 2012.
doi: 10.1007/978-1-4614-1686-9. |
[4] |
S.R. Carpenter and W.A. Brock,
Rising variance: A leading indicator of ecological transition, Ecol. Lett., 9 (2006), 311-318.
doi: 10.1111/j.1461-0248.2005.00877.x. |
[5] |
S.-N. Chow and J. Mallet-Paret,
The Fuller index and global Hopf bifurcation, J. Differential Equations, 29 (1978), 66-85.
doi: 10.1016/0022-0396(78)90041-4. |
[6] |
E. Gilad, M. Shachak and E. Meron,
Dynamics and spatial organization of plant communities in water-limited systems, Theor. popul. biol., 72 (2007), 214-230.
doi: 10.1016/j.tpb.2007.05.002. |
[7] |
E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, Ecosystem engineers: From pattern formation to habitat creation Phys. Rev. Lett. 93 (2004), 098105.
doi: 10.1103/PhysRevLett. 93. 098105. |
[8] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields volume 42 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[9] |
K.P. Hadeler and P. van den Driessche,
Backward bifurcation in epidemic control, Math. Biosci., 146 (1997), 15-35.
doi: 10.1016/S0025-5564(97)00027-8. |
[10] |
S.-B. Hsu and T.-W. Huang,
Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.
doi: 10.1137/S0036139993253201. |
[11] |
S.-B. Hsu and T.-W. Huang,
Uniqueness of limit cycles for a predator-prey system of Holling and Leslie type, Canad. Appl. Math. Quart., 6 (1998), 91-117.
|
[12] |
C.G. Jones, J.H. Lawton and M. Shachak,
Organisms as ecosystem engineers, Ecosystem Management, (1996), 130-147.
doi: 10.1007/978-1-4612-4018-1_14. |
[13] |
C.G. Jones, J.H. Lawton and M. Shachak,
Positive and negative effects of organisms as physical ecosystem engineers, Ecology, 78 (1997), 1946-1957.
|
[14] |
C.A. Klausmeier,
Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.
doi: 10.1126/science.284.5421.1826. |
[15] |
A.J. Koch and H. Meinhardt,
Biological pattern formation: From basic mechanisms to complex structures, Rev. Mod. Phys., 66 (1994), 1481-1507.
doi: 10.1103/RevModPhys.66.1481. |
[16] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory volume 112 of Applied Mathematical Sciences, Springer-Verlag, New York, 1998. |
[17] |
R. Lefever and O. Lejeune,
On the origin of tiger bush, Bull. Math. Biol., 59 (1997), 263-294.
doi: 10.1016/S0092-8240(96)00072-9. |
[18] |
R.-S. Liu, Z.-L. Feng, H.-P. Zhu and D.L. DeAngelis,
Bifurcation analysis of a plant-herbivore model with toxin-determined functional response, J. Differential Equations, 245 (2008), 442-467.
doi: 10.1016/j.jde.2007.10.034. |
[19] |
M.-X. Liu, E. Liz and G. Röst,
Endemic bubbles generated by delayed behavioral response: Global stability and bifurcation switches in an SIS model, SIAM J. Appl. Math., 75 (2015), 75-91.
doi: 10.1137/140972652. |
[20] |
A. Manor and N.M. Shnerb,
Dynamical failure of Turing patterns, Europhys. Lett., 74 (2006), 837-843.
doi: 10.1209/epl/i2005-10580-5. |
[21] |
R.M. May,
Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.
doi: 10.1038/269471a0. |
[22] |
H. Meinhardt,
Pattern formation in biology: A comparison of models and experiments, Rep. Prog. Phys., 55 (1992), 797-849.
doi: 10.1088/0034-4885/55/6/003. |
[23] |
E. Meron, E. Gilad and J. von Hardenberg,
Vegetation patterns along a rainfall gradient, Chaos, Solitons Fract., 19 (2004), 367-376.
doi: 10.1016/S0960-0779(03)00049-3. |
[24] |
L. Perko, Differential Equations and Dynamical Systems volume 7 of Texts in Applied Mathematics, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4613-0003-8. |
[25] |
M. Rietkerk, S.C. Dekker, P.C. De Ruiter and J. van de Koppel,
Self-organized patchiness and catastrophic shifts in ecosystems, Science, 305 (2004), 1926-1929.
doi: 10.1126/science.1101867. |
[26] |
M. Scheffer, J. Bascompte and W.A. Brock,
Early-warning signals for critical transitions, Nature, 461 (2009), 53-59.
doi: 10.1038/nature08227. |
[27] |
M. Scheffer, S. Carpenter and J.A. Foley,
Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.
doi: 10.1038/35098000. |
[28] |
N. M. Shnerb, P. Sarah, H. Lavee and S. Solomon, Reactive Glass and Vegetation Patterns Phys. Rev. Lett. 90 (2003), 038101.
doi: 10.1103/PhysRevLett. 90. 038101. |
[29] |
H.-Y. Shu, L. Wang and J.-H. Wu,
Global dynamics of Nicholson's blowflies equation revisited: Onset and termination of nonlinear oscillations, J. Differential Equations, 255 (2013), 2565-2586.
doi: 10.1016/j.jde.2013.06.020. |
[30] |
A.M. Turing,
The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B Biol. Sci., 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[31] |
J. von Hardenberg, E. Meron, M. Shachak and Y. Zarmi, Diversity of vegetation patterns and desertification Phys. Rev. Lett. 87 (2001), 198101.
doi: 10.1103/PhysRevLett. 87. 198101. |
[32] |
J.-L. Wang, S.-Q. Liu, B.-W. Zheng and Y. Takeuchi,
Qualitative and bifurcation analysis using an SIR model with a saturated treatment function, Math. Comput. Modelling, 55 (2012), 710-722.
doi: 10.1016/j.mcm.2011.08.045. |
[33] |
J.-F. Wang, J.-P. Shi and J.-J. Wei,
Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331.
doi: 10.1007/s00285-010-0332-1. |
[34] |
W.-D. Wang,
Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.
doi: 10.1016/j.mbs.2005.12.022. |
[35] |
W.-D. Wang and S.-G. Ruan,
Bifurcation in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004), 775-793.
doi: 10.1016/j.jmaa.2003.11.043. |
[36] |
X.-L. Wang, W.-D. Wang and G.-H. Zhang,
Global analysis of predator-prey system with hawk and dove tactics, Stud. Appl. Math., 124 (2010), 151-178.
doi: 10.1111/j.1467-9590.2009.00466.x. |
[37] |
A.S. Watt,
Pattern and process in the plant community, J. Ecol., 35 (1947), 1-22.
doi: 10.2307/2256497. |
[38] |
X. Zhang and X.-N. Liu,
Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl., 348 (2008), 433-443.
doi: 10.1016/j.jmaa.2008.07.042. |
[39] |
L.-H. Zhou and M. Fan,
Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. Real World Appl., 13 (2012), 312-324.
doi: 10.1016/j.nonrwa.2011.07.036. |
show all references
References:
[1] |
J. C. Alexander and J. A. Yorke,
Global bifurcations of periodic orbits, Amer. J. Math., 100 (1978), 263-292.
doi: 10.2307/2373851. |
[2] |
J. Arino, C.C. McCluskey and P. van den Driessche,
Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260-276.
doi: 10.1137/S0036139902413829. |
[3] |
F. Brauer and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology volume 40 of Texts in Applied Mathematics, Springer, New York, 2012.
doi: 10.1007/978-1-4614-1686-9. |
[4] |
S.R. Carpenter and W.A. Brock,
Rising variance: A leading indicator of ecological transition, Ecol. Lett., 9 (2006), 311-318.
doi: 10.1111/j.1461-0248.2005.00877.x. |
[5] |
S.-N. Chow and J. Mallet-Paret,
The Fuller index and global Hopf bifurcation, J. Differential Equations, 29 (1978), 66-85.
doi: 10.1016/0022-0396(78)90041-4. |
[6] |
E. Gilad, M. Shachak and E. Meron,
Dynamics and spatial organization of plant communities in water-limited systems, Theor. popul. biol., 72 (2007), 214-230.
doi: 10.1016/j.tpb.2007.05.002. |
[7] |
E. Gilad, J. von Hardenberg, A. Provenzale, M. Shachak and E. Meron, Ecosystem engineers: From pattern formation to habitat creation Phys. Rev. Lett. 93 (2004), 098105.
doi: 10.1103/PhysRevLett. 93. 098105. |
[8] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields volume 42 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[9] |
K.P. Hadeler and P. van den Driessche,
Backward bifurcation in epidemic control, Math. Biosci., 146 (1997), 15-35.
doi: 10.1016/S0025-5564(97)00027-8. |
[10] |
S.-B. Hsu and T.-W. Huang,
Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.
doi: 10.1137/S0036139993253201. |
[11] |
S.-B. Hsu and T.-W. Huang,
Uniqueness of limit cycles for a predator-prey system of Holling and Leslie type, Canad. Appl. Math. Quart., 6 (1998), 91-117.
|
[12] |
C.G. Jones, J.H. Lawton and M. Shachak,
Organisms as ecosystem engineers, Ecosystem Management, (1996), 130-147.
doi: 10.1007/978-1-4612-4018-1_14. |
[13] |
C.G. Jones, J.H. Lawton and M. Shachak,
Positive and negative effects of organisms as physical ecosystem engineers, Ecology, 78 (1997), 1946-1957.
|
[14] |
C.A. Klausmeier,
Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.
doi: 10.1126/science.284.5421.1826. |
[15] |
A.J. Koch and H. Meinhardt,
Biological pattern formation: From basic mechanisms to complex structures, Rev. Mod. Phys., 66 (1994), 1481-1507.
doi: 10.1103/RevModPhys.66.1481. |
[16] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory volume 112 of Applied Mathematical Sciences, Springer-Verlag, New York, 1998. |
[17] |
R. Lefever and O. Lejeune,
On the origin of tiger bush, Bull. Math. Biol., 59 (1997), 263-294.
doi: 10.1016/S0092-8240(96)00072-9. |
[18] |
R.-S. Liu, Z.-L. Feng, H.-P. Zhu and D.L. DeAngelis,
Bifurcation analysis of a plant-herbivore model with toxin-determined functional response, J. Differential Equations, 245 (2008), 442-467.
doi: 10.1016/j.jde.2007.10.034. |
[19] |
M.-X. Liu, E. Liz and G. Röst,
Endemic bubbles generated by delayed behavioral response: Global stability and bifurcation switches in an SIS model, SIAM J. Appl. Math., 75 (2015), 75-91.
doi: 10.1137/140972652. |
[20] |
A. Manor and N.M. Shnerb,
Dynamical failure of Turing patterns, Europhys. Lett., 74 (2006), 837-843.
doi: 10.1209/epl/i2005-10580-5. |
[21] |
R.M. May,
Thresholds and breakpoints in ecosystems with a multiplicity of stable states, Nature, 269 (1977), 471-477.
doi: 10.1038/269471a0. |
[22] |
H. Meinhardt,
Pattern formation in biology: A comparison of models and experiments, Rep. Prog. Phys., 55 (1992), 797-849.
doi: 10.1088/0034-4885/55/6/003. |
[23] |
E. Meron, E. Gilad and J. von Hardenberg,
Vegetation patterns along a rainfall gradient, Chaos, Solitons Fract., 19 (2004), 367-376.
doi: 10.1016/S0960-0779(03)00049-3. |
[24] |
L. Perko, Differential Equations and Dynamical Systems volume 7 of Texts in Applied Mathematics, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4613-0003-8. |
[25] |
M. Rietkerk, S.C. Dekker, P.C. De Ruiter and J. van de Koppel,
Self-organized patchiness and catastrophic shifts in ecosystems, Science, 305 (2004), 1926-1929.
doi: 10.1126/science.1101867. |
[26] |
M. Scheffer, J. Bascompte and W.A. Brock,
Early-warning signals for critical transitions, Nature, 461 (2009), 53-59.
doi: 10.1038/nature08227. |
[27] |
M. Scheffer, S. Carpenter and J.A. Foley,
Catastrophic shifts in ecosystems, Nature, 413 (2001), 591-596.
doi: 10.1038/35098000. |
[28] |
N. M. Shnerb, P. Sarah, H. Lavee and S. Solomon, Reactive Glass and Vegetation Patterns Phys. Rev. Lett. 90 (2003), 038101.
doi: 10.1103/PhysRevLett. 90. 038101. |
[29] |
H.-Y. Shu, L. Wang and J.-H. Wu,
Global dynamics of Nicholson's blowflies equation revisited: Onset and termination of nonlinear oscillations, J. Differential Equations, 255 (2013), 2565-2586.
doi: 10.1016/j.jde.2013.06.020. |
[30] |
A.M. Turing,
The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B Biol. Sci., 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[31] |
J. von Hardenberg, E. Meron, M. Shachak and Y. Zarmi, Diversity of vegetation patterns and desertification Phys. Rev. Lett. 87 (2001), 198101.
doi: 10.1103/PhysRevLett. 87. 198101. |
[32] |
J.-L. Wang, S.-Q. Liu, B.-W. Zheng and Y. Takeuchi,
Qualitative and bifurcation analysis using an SIR model with a saturated treatment function, Math. Comput. Modelling, 55 (2012), 710-722.
doi: 10.1016/j.mcm.2011.08.045. |
[33] |
J.-F. Wang, J.-P. Shi and J.-J. Wei,
Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331.
doi: 10.1007/s00285-010-0332-1. |
[34] |
W.-D. Wang,
Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.
doi: 10.1016/j.mbs.2005.12.022. |
[35] |
W.-D. Wang and S.-G. Ruan,
Bifurcation in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl., 291 (2004), 775-793.
doi: 10.1016/j.jmaa.2003.11.043. |
[36] |
X.-L. Wang, W.-D. Wang and G.-H. Zhang,
Global analysis of predator-prey system with hawk and dove tactics, Stud. Appl. Math., 124 (2010), 151-178.
doi: 10.1111/j.1467-9590.2009.00466.x. |
[37] |
A.S. Watt,
Pattern and process in the plant community, J. Ecol., 35 (1947), 1-22.
doi: 10.2307/2256497. |
[38] |
X. Zhang and X.-N. Liu,
Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl., 348 (2008), 433-443.
doi: 10.1016/j.jmaa.2008.07.042. |
[39] |
L.-H. Zhou and M. Fan,
Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear Anal. Real World Appl., 13 (2012), 312-324.
doi: 10.1016/j.nonrwa.2011.07.036. |












λ | (µ0, µ1) | Transcritical at R = R0 | Saddle-node bifurcation | Hopf bifurcation points | Homoclinic bifurcation | B-T bifurcation |
0 < λ < 1 | Ⅰ | backward | R1 | none | none | R1 |
0 < λ < 1 | Ⅱ | backward | R1 | R3 | exist | R1 |
0 < λ < 1 | Ⅲ | backward | R1 | R2, R3 | none | R1 |
0 < λ < 1 | Ⅳ | forward | none | none | none | none |
0 < λ < 1 | Ⅴ | forward | none | R2, R3 | none | none |
λ ≥ 1 | Ⅵ | forward | none | none | none | none |
λ ≥ 1 | Ⅶ | forward | none | R2, R3 | none | none |
λ | (µ0, µ1) | Transcritical at R = R0 | Saddle-node bifurcation | Hopf bifurcation points | Homoclinic bifurcation | B-T bifurcation |
0 < λ < 1 | Ⅰ | backward | R1 | none | none | R1 |
0 < λ < 1 | Ⅱ | backward | R1 | R3 | exist | R1 |
0 < λ < 1 | Ⅲ | backward | R1 | R2, R3 | none | R1 |
0 < λ < 1 | Ⅳ | forward | none | none | none | none |
0 < λ < 1 | Ⅴ | forward | none | R2, R3 | none | none |
λ ≥ 1 | Ⅵ | forward | none | none | none | none |
λ ≥ 1 | Ⅶ | forward | none | R2, R3 | none | none |
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