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A stochastic model for water-vegetation systems and the effect of decreasing precipitation on semi-arid environments
1. | Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA |
2. | Ecology Center, Utah State University, Logan, UT 84322, USA |
3. | Mathematics Department, University of British Columbia, Vancouver, BC, Canada |
Current climate change trends are affecting the magnitude and recurrence of extreme weather events. In particular, several semi-arid regions around the planet are confronting more intense and prolonged lack of precipitation, slowly transforming part of these regions into deserts in some cases. Although it is documented that a decreasing tendency in precipitation might induce earlier disappearance of vegetation, quantifying the relationship between decrease of precipitation and vegetation endurance remains a challenging task due to the inherent complexities involved in distinct scenarios. In this paper we present a model for precipitation-vegetation dynamics in semi-arid landscapes that can be used to explore numerically the impact of decreasing precipitation trends on appearance of desertification events. The model, a stochastic differential equation approximation derived from a Markov jump process, is used to generate extensive simulations that suggest a relationship between precipitation reduction and the desertification process, which might take several years in some instances.
References:
[1] |
A. Anyamba and C. J. Tucker,
Analysis of Sahelian vegetation dynamics using NOAA-AVHRR NDVI data from 1981-2003, Journal of Arid Environments, 63 (2005), 596-614.
doi: 10.1016/j.jaridenv.2005.03.007. |
[2] |
L. Arriola and J. M. Hyman, Sensitivity analysis for uncertainty quantification in mathematical models, in Mathematical and Statistical Estimation Approaches in Epidemiology (eds. G. Chowell, J. M. Hyman, L. M. A. Betancourt, D. Bies), Springer, Netherlands, (2009), 195–247.
doi: 10.1007/978-90-481-2313-1_10. |
[3] |
B. C. Bates, Z. W. Kundzewicz, S. Wu and J. P. Palutikof (Eds.),
Climate Change and Water, Technical Paper of the Intergovernmental Panel on Climate Change, IPCC Secretariat, Geneva, 2008. |
[4] |
P. H. Baxendale and P. E. Greenwood,
Sustained oscillations for density dependent Markov processes, Journal of Mathematical Biology, 63 (2011), 433-457.
doi: 10.1007/s00285-010-0376-2. |
[5] |
F. Borgogno, P. D'Odorico, F. Laio and L. Ridolfi,
Mathematical models of vegetation pattern formation in ecohydrology, Rev. Geophys., 47 (2009), RG1005.
doi: 10.1029/2007RG000256. |
[6] |
P. D'Odorico, F. Laio and L. Ridolfi,
Noise-induced stability in dryland plant ecosystems, PNAS, 102 (2005), 10819-10822.
doi: 10.1073/pnas.0502884102. |
[7] |
C. Gardiner,
Stochastic Methods, fourth edition, Springer, Berlin, 2009. |
[8] |
V. S. Golubev, J. H. Lawrimore, P. Y. Groisman, N. A. Speranskaya, S. A. Zhuravin, M. J. Menne, T. C. Peterson and R. W. Malone,
Evaporation changes over the contiguous United States and the former USSR: A reassessment, Geophysical Research Letters, 28 (2001), 2665-2668.
doi: 10.1029/2000GL012851. |
[9] |
U. Helldèn, Desertification: Time for assessment?, Ambio, Forestry and the Environment, 20 (1991), 372–383. |
[10] |
D. J. Higham,
An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.
doi: 10.1137/S0036144500378302. |
[11] |
W. Horsthemke and R. Lefever,
Noise-Induced Transitions: Theory and Applications in Physics, Chemistry and Biology, Springer, Berlin, 1984. |
[12] |
C. A. Klausmeier,
Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.
doi: 10.1126/science.284.5421.1826. |
[13] |
P. E. Kloeden and E. Platen,
Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-662-12616-5. |
[14] |
T. G. Kurtz,
Strong approximation theorems for density dependent Markov chains, Stochastic Processes and Applications, 6 (1978), 223-240.
|
[15] |
C. A. Lugo and A. J. McKane, Quasicycles in a spatial predator-prey model Physical Review E, 78 (2008), 051911, 15pp.
doi: 10.1103/PhysRevE.78.051911. |
[16] |
C. S. Mantyka-Pringle, T. G. Martin and J. R. Rhodes,
Interactions between climate and habitat loss effects on biodiversity: a systematic review and meta-analysis, Global Change Biology, 18 (2012), 1239-1252.
|
[17] |
A. J. McKane and T. J. Newman, Stochastic models in population biology and their deterministic analogs Physical Review E, 70 (2004), 041902, 19 pages.
doi: 10.1103/PhysRevE.70.041902. |
[18] |
A. J. McKane, T. Biancalini and T. Rogers,
Stochastic pattern formation and spontaneous polarization: The linear noise approximation and beyond, Bulletin of Mathematical Biology, 76 (2014), 895-921.
doi: 10.1007/s11538-013-9827-4. |
[19] |
E. Meron and E. Gilad, Dynamics of plant communities in drylands: a pattern formation approach, in Complex Population Dynamics: Nonlinear Modeling in Ecology, Epidemiology and Genetics, (eds. B. Blasius, J. Kurths, L. Stone), World Scientific, Singapore, (2007), 49–75.
doi: 10.1142/9789812771582_0003. |
[20] |
NOAA National Centers for Environmental Information, State of the Climate: Drought for August 2016, published online September 2016, retrieved on October 5, 2016. Available from http://www.ncdc.noaa.gov/sotc/drought/201608 |
[21] |
L. Ridolfi, P. D'Odorico and F. Laio,
Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, New York, 2011.
doi: 10.1017/CBO9780511984730. |
[22] |
M. Rietkerk, M. C. Boerlijst, F. van Langevelde, R. HilleRisLambers, J. van de Koppel, L. Kumar, H. H. T. Prins and A. M. de Roos,
Self-Organization of vegetation in arid ecosystems, The American Naturalist, 160 (2002), 524-530.
|
[23] |
M. Rietkerk, F. van den Bosch and J. van de Koppel,
Site-specific properties and irreversible vegetation changes in semi-arid grazing systems, Oikos, 80 (1997), 241-252.
doi: 10.2307/3546592. |
[24] |
J. Sheffield and E. F. Wood,
Global trends and variability in soil moisture and drought characteristics, 1950-2000, from observation-driven simulations of the terrestrial hydrologic cycle, Journal of Climate, 21 (2008), 432-458.
doi: 10.1175/2007JCLI1822.1. |
[25] |
J. A. Sherratt,
An analysis of vegetation stripe formation in semi-arid landscapes, Journal of Mathematical Biology, 51 (2005), 183-197.
doi: 10.1007/s00285-005-0319-5. |
[26] |
J. A. Sherratt and G. J. Lord,
Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments, Theoretical Population Biology, 71 (2007), 1-11.
doi: 10.1016/j.tpb.2006.07.009. |
[27] |
J. A. Sherratt,
Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments Ⅴ: The transition from patterns to desert, SIAM Journal of Applied Mathematics, 73 (2013), 1347-1367.
doi: 10.1137/120899510. |
[28] |
K. Siteur, M. B. Eppinga, D. Karssenberg, M. Baudena, M. F. P. Bierkens and M. Rietkerk,
How will increases in rainfall intensity affect semiarid ecosystems?, Water Resources Research, 50 (2014), 5980-6001.
doi: 10.1002/2013WR014955. |
show all references
References:
[1] |
A. Anyamba and C. J. Tucker,
Analysis of Sahelian vegetation dynamics using NOAA-AVHRR NDVI data from 1981-2003, Journal of Arid Environments, 63 (2005), 596-614.
doi: 10.1016/j.jaridenv.2005.03.007. |
[2] |
L. Arriola and J. M. Hyman, Sensitivity analysis for uncertainty quantification in mathematical models, in Mathematical and Statistical Estimation Approaches in Epidemiology (eds. G. Chowell, J. M. Hyman, L. M. A. Betancourt, D. Bies), Springer, Netherlands, (2009), 195–247.
doi: 10.1007/978-90-481-2313-1_10. |
[3] |
B. C. Bates, Z. W. Kundzewicz, S. Wu and J. P. Palutikof (Eds.),
Climate Change and Water, Technical Paper of the Intergovernmental Panel on Climate Change, IPCC Secretariat, Geneva, 2008. |
[4] |
P. H. Baxendale and P. E. Greenwood,
Sustained oscillations for density dependent Markov processes, Journal of Mathematical Biology, 63 (2011), 433-457.
doi: 10.1007/s00285-010-0376-2. |
[5] |
F. Borgogno, P. D'Odorico, F. Laio and L. Ridolfi,
Mathematical models of vegetation pattern formation in ecohydrology, Rev. Geophys., 47 (2009), RG1005.
doi: 10.1029/2007RG000256. |
[6] |
P. D'Odorico, F. Laio and L. Ridolfi,
Noise-induced stability in dryland plant ecosystems, PNAS, 102 (2005), 10819-10822.
doi: 10.1073/pnas.0502884102. |
[7] |
C. Gardiner,
Stochastic Methods, fourth edition, Springer, Berlin, 2009. |
[8] |
V. S. Golubev, J. H. Lawrimore, P. Y. Groisman, N. A. Speranskaya, S. A. Zhuravin, M. J. Menne, T. C. Peterson and R. W. Malone,
Evaporation changes over the contiguous United States and the former USSR: A reassessment, Geophysical Research Letters, 28 (2001), 2665-2668.
doi: 10.1029/2000GL012851. |
[9] |
U. Helldèn, Desertification: Time for assessment?, Ambio, Forestry and the Environment, 20 (1991), 372–383. |
[10] |
D. J. Higham,
An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.
doi: 10.1137/S0036144500378302. |
[11] |
W. Horsthemke and R. Lefever,
Noise-Induced Transitions: Theory and Applications in Physics, Chemistry and Biology, Springer, Berlin, 1984. |
[12] |
C. A. Klausmeier,
Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.
doi: 10.1126/science.284.5421.1826. |
[13] |
P. E. Kloeden and E. Platen,
Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-662-12616-5. |
[14] |
T. G. Kurtz,
Strong approximation theorems for density dependent Markov chains, Stochastic Processes and Applications, 6 (1978), 223-240.
|
[15] |
C. A. Lugo and A. J. McKane, Quasicycles in a spatial predator-prey model Physical Review E, 78 (2008), 051911, 15pp.
doi: 10.1103/PhysRevE.78.051911. |
[16] |
C. S. Mantyka-Pringle, T. G. Martin and J. R. Rhodes,
Interactions between climate and habitat loss effects on biodiversity: a systematic review and meta-analysis, Global Change Biology, 18 (2012), 1239-1252.
|
[17] |
A. J. McKane and T. J. Newman, Stochastic models in population biology and their deterministic analogs Physical Review E, 70 (2004), 041902, 19 pages.
doi: 10.1103/PhysRevE.70.041902. |
[18] |
A. J. McKane, T. Biancalini and T. Rogers,
Stochastic pattern formation and spontaneous polarization: The linear noise approximation and beyond, Bulletin of Mathematical Biology, 76 (2014), 895-921.
doi: 10.1007/s11538-013-9827-4. |
[19] |
E. Meron and E. Gilad, Dynamics of plant communities in drylands: a pattern formation approach, in Complex Population Dynamics: Nonlinear Modeling in Ecology, Epidemiology and Genetics, (eds. B. Blasius, J. Kurths, L. Stone), World Scientific, Singapore, (2007), 49–75.
doi: 10.1142/9789812771582_0003. |
[20] |
NOAA National Centers for Environmental Information, State of the Climate: Drought for August 2016, published online September 2016, retrieved on October 5, 2016. Available from http://www.ncdc.noaa.gov/sotc/drought/201608 |
[21] |
L. Ridolfi, P. D'Odorico and F. Laio,
Noise-Induced Phenomena in the Environmental Sciences, Cambridge University Press, New York, 2011.
doi: 10.1017/CBO9780511984730. |
[22] |
M. Rietkerk, M. C. Boerlijst, F. van Langevelde, R. HilleRisLambers, J. van de Koppel, L. Kumar, H. H. T. Prins and A. M. de Roos,
Self-Organization of vegetation in arid ecosystems, The American Naturalist, 160 (2002), 524-530.
|
[23] |
M. Rietkerk, F. van den Bosch and J. van de Koppel,
Site-specific properties and irreversible vegetation changes in semi-arid grazing systems, Oikos, 80 (1997), 241-252.
doi: 10.2307/3546592. |
[24] |
J. Sheffield and E. F. Wood,
Global trends and variability in soil moisture and drought characteristics, 1950-2000, from observation-driven simulations of the terrestrial hydrologic cycle, Journal of Climate, 21 (2008), 432-458.
doi: 10.1175/2007JCLI1822.1. |
[25] |
J. A. Sherratt,
An analysis of vegetation stripe formation in semi-arid landscapes, Journal of Mathematical Biology, 51 (2005), 183-197.
doi: 10.1007/s00285-005-0319-5. |
[26] |
J. A. Sherratt and G. J. Lord,
Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments, Theoretical Population Biology, 71 (2007), 1-11.
doi: 10.1016/j.tpb.2006.07.009. |
[27] |
J. A. Sherratt,
Pattern solutions of the Klausmeier model for banded vegetation in semi-arid environments Ⅴ: The transition from patterns to desert, SIAM Journal of Applied Mathematics, 73 (2013), 1347-1367.
doi: 10.1137/120899510. |
[28] |
K. Siteur, M. B. Eppinga, D. Karssenberg, M. Baudena, M. F. P. Bierkens and M. Rietkerk,
How will increases in rainfall intensity affect semiarid ecosystems?, Water Resources Research, 50 (2014), 5980-6001.
doi: 10.1002/2013WR014955. |




Event | Transition | Jump | Jump rate |
Vegetation biomass loss | |||
Incoming water | |||
Water evaporation | |||
Increase vegetation by | |||
water take up | |||
Event | Transition | Jump | Jump rate |
Vegetation biomass loss | |||
Incoming water | |||
Water evaporation | |||
Increase vegetation by | |||
water take up | |||
Parameter | Definition | Estimated values | Units |
uptake rate of water | 1.5(trees) - 100(grass) | mm year |
|
yield of plant biomass | 0.002(trees) - 0.003(grass) | kg dry mass (mm) |
|
mortality rate | 0.18(trees) - 1.8(grass) | year |
|
precipitation | 250 - 750 | mm year |
|
evaporation rate | 4 | year |
|
Parameter | Definition | Estimated values | Units |
uptake rate of water | 1.5(trees) - 100(grass) | mm year |
|
yield of plant biomass | 0.002(trees) - 0.003(grass) | kg dry mass (mm) |
|
mortality rate | 0.18(trees) - 1.8(grass) | year |
|
precipitation | 250 - 750 | mm year |
|
evaporation rate | 4 | year |
|
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