October  2018, 15(5): 1155-1164. doi: 10.3934/mbe.2018052

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

* Corresponding author

Received  June 16, 2017 Accepted  March 16, 2018 Published  May 2018

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.

Citation: Shannon Dixon, Nancy Huntly, Priscilla E. Greenwood, Luis F. Gordillo. A stochastic model for water-vegetation systems and the effect of decreasing precipitation on semi-arid environments. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1155-1164. doi: 10.3934/mbe.2018052
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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[5]

F. BorgognoP. D'OdoricoF. Laio and L. Ridolfi, Mathematical models of vegetation pattern formation in ecohydrology, Rev. Geophys., 47 (2009), RG1005.  doi: 10.1029/2007RG000256.  Google Scholar

[6]

P. D'OdoricoF. Laio and L. Ridolfi, Noise-induced stability in dryland plant ecosystems, PNAS, 102 (2005), 10819-10822.  doi: 10.1073/pnas.0502884102.  Google Scholar

[7]

C. Gardiner, Stochastic Methods, fourth edition, Springer, Berlin, 2009.  Google Scholar

[8]

V. S. GolubevJ. H. LawrimoreP. Y. GroismanN. A. SperanskayaS. A. ZhuravinM. J. MenneT. 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.  Google Scholar

[9]

U. Helldèn, Desertification: Time for assessment?, Ambio, Forestry and the Environment, 20 (1991), 372–383. Google Scholar

[10]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar

[11]

W. Horsthemke and R. Lefever, Noise-Induced Transitions: Theory and Applications in Physics, Chemistry and Biology, Springer, Berlin, 1984.  Google Scholar

[12]

C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.  doi: 10.1126/science.284.5421.1826.  Google Scholar

[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.  Google Scholar

[14]

T. G. Kurtz, Strong approximation theorems for density dependent Markov chains, Stochastic Processes and Applications, 6 (1978), 223-240.   Google Scholar

[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.  Google Scholar

[16]

C. S. Mantyka-PringleT. 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.   Google Scholar

[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.  Google Scholar

[18]

A. J. McKaneT. 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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[22]

M. RietkerkM. C. BoerlijstF. van LangeveldeR. HilleRisLambersJ. van de KoppelL. KumarH. H. T. Prins and A. M. de Roos, Self-Organization of vegetation in arid ecosystems, The American Naturalist, 160 (2002), 524-530.   Google Scholar

[23]

M. RietkerkF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

K. SiteurM. B. EppingaD. KarssenbergM. BaudenaM. 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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[5]

F. BorgognoP. D'OdoricoF. Laio and L. Ridolfi, Mathematical models of vegetation pattern formation in ecohydrology, Rev. Geophys., 47 (2009), RG1005.  doi: 10.1029/2007RG000256.  Google Scholar

[6]

P. D'OdoricoF. Laio and L. Ridolfi, Noise-induced stability in dryland plant ecosystems, PNAS, 102 (2005), 10819-10822.  doi: 10.1073/pnas.0502884102.  Google Scholar

[7]

C. Gardiner, Stochastic Methods, fourth edition, Springer, Berlin, 2009.  Google Scholar

[8]

V. S. GolubevJ. H. LawrimoreP. Y. GroismanN. A. SperanskayaS. A. ZhuravinM. J. MenneT. 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.  Google Scholar

[9]

U. Helldèn, Desertification: Time for assessment?, Ambio, Forestry and the Environment, 20 (1991), 372–383. Google Scholar

[10]

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.  doi: 10.1137/S0036144500378302.  Google Scholar

[11]

W. Horsthemke and R. Lefever, Noise-Induced Transitions: Theory and Applications in Physics, Chemistry and Biology, Springer, Berlin, 1984.  Google Scholar

[12]

C. A. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science, 284 (1999), 1826-1828.  doi: 10.1126/science.284.5421.1826.  Google Scholar

[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.  Google Scholar

[14]

T. G. Kurtz, Strong approximation theorems for density dependent Markov chains, Stochastic Processes and Applications, 6 (1978), 223-240.   Google Scholar

[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.  Google Scholar

[16]

C. S. Mantyka-PringleT. 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.   Google Scholar

[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.  Google Scholar

[18]

A. J. McKaneT. 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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[22]

M. RietkerkM. C. BoerlijstF. van LangeveldeR. HilleRisLambersJ. van de KoppelL. KumarH. H. T. Prins and A. M. de Roos, Self-Organization of vegetation in arid ecosystems, The American Naturalist, 160 (2002), 524-530.   Google Scholar

[23]

M. RietkerkF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

K. SiteurM. B. EppingaD. KarssenbergM. BaudenaM. 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.  Google Scholar

Figure 1.  Simulations corresponding to the Markov jump process (left) and the diffusion approximation (right). For comparison purposes the paths in both panels were generated using the same parameters and the same scaled time.
Figure 2.  State averages of precipitation anomalies for 2000-2016 in California (inches year$^{-1}$). The averaged anomaly (difference from long term average) during that period is $\approx$ -2.07 (inches year$^{-1}$) (-52.58 mm year$^{-1}$). The precipitation increase expected from El Niño for the winter 2015-2016 was scarcely above the long term state average. Data/image provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their Web site at http://www.esrl.noaa.gov/psd/
Figure 3.  Examples of how average time to desertification might be affected by a reduction in average annual precipitation. Parameters for trees were used in panel (a) and for grass in panel (b). The average of negative anomalies similar to that observed for the last years in California is around $\approx$ 50 mm year$^{-1}$. The model suggests that the sensitivity index $S_0\approx 2$, i.e. relative changes in the mean time to desertification are roughly twice the relative changes in average annual precipitation. For the simulations, $N = 500$ and the initial conditions were $\rho(0) = (0.1,0.1)$ (squares), $\rho(0) = (0.5,0.5)$ (stars) and $\rho(0) = (0.9,0.1)$ (triangles). Each time average was obtained from 50000 simulations. Panel (c) shows the histograms corresponding to the simulated times to desertification with an average annual precipitation of 200 and 250 mm year$^{-1}$ (for grass) on the left and right, respectively. The simulations used the same initial conditions $\rho(0) = (0.1,0.1)$.
Figure 4.  Left: time to desertification for $A = 250$ (dashes) and $A = 200$ (dot-dashes) as function of the system capacity $N$. The sensitivity of the time to desertification from the annual precipitation was computed for $N = 10000$ showing to be the same as when $N = 500$, i.e. $\approx 2$. As $N$ increases both times to desertification also increase, but they get reduced dramatically as $N$ gets smaller. Right: Difference between the curves in the contiguous plot. Although the difference increases, the sensitivity of the time to desertification from the annual precipitation is apparently similar in relatively large systems.
Table 1.  Possible transition events with their associated jumps if the system is at state $(n,m)$, where $n$ and $m$ represent units of biomass and water, respectively.
Event Transition Jump Jump rate
Vegetation biomass loss $(n,m)\rightarrow(n-1,m)$ $(-1,0)$ $d$
Incoming water $(n,m)\rightarrow(n,m+1)$ $(0,1)$ $s$
Water evaporation $(n,m)\rightarrow(n,m-1)$ $(0,-1)$ $v$
Increase vegetation by $(n,m)\rightarrow(n+1,m-1)$ $(1,-1)$ $b$
water take up
Event Transition Jump Jump rate
Vegetation biomass loss $(n,m)\rightarrow(n-1,m)$ $(-1,0)$ $d$
Incoming water $(n,m)\rightarrow(n,m+1)$ $(0,1)$ $s$
Water evaporation $(n,m)\rightarrow(n,m-1)$ $(0,-1)$ $v$
Increase vegetation by $(n,m)\rightarrow(n+1,m-1)$ $(1,-1)$ $b$
water take up
Table 2.  Parameters for semi-arid landscapes, taken from [12].
Parameter Definition Estimated values Units
$R$ uptake rate of water 1.5(trees) - 100(grass) mm year$^{-1}$ (kg dry mass)$^{-2}$
$J$ yield of plant biomass 0.002(trees) - 0.003(grass) kg dry mass (mm)$^{-1}$
$M$ mortality rate 0.18(trees) - 1.8(grass) year$^{-1}$
$A$ precipitation 250 - 750 mm year$^{-1}$
$L$ evaporation rate 4 year$^{-1}$
Parameter Definition Estimated values Units
$R$ uptake rate of water 1.5(trees) - 100(grass) mm year$^{-1}$ (kg dry mass)$^{-2}$
$J$ yield of plant biomass 0.002(trees) - 0.003(grass) kg dry mass (mm)$^{-1}$
$M$ mortality rate 0.18(trees) - 1.8(grass) year$^{-1}$
$A$ precipitation 250 - 750 mm year$^{-1}$
$L$ evaporation rate 4 year$^{-1}$
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