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[J]. Mathematical Biosciences and Engineering, 2018, 15(5): 1155-1164. doi: 10.3934/mbe.2018052
[1] | [ A. Anyamba,C. J. Tucker, Analysis of Sahelian vegetation dynamics using NOAA-AVHRR NDVI data from 1981-2003, Journal of Arid Environments, 63 (2005): 596-614. |
[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. |
[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,P. E. Greenwood, Sustained oscillations for density dependent Markov processes, Journal of Mathematical Biology, 63 (2011): 433-457. |
[5] | [ F. Borgogno,P. D'Odorico,F. Laio,L. Ridolfi, Mathematical models of vegetation pattern formation in ecohydrology, Rev. Geophys., 47 (2009): RG1005. |
[6] | [ P. D'Odorico,F. Laio,L. Ridolfi, Noise-induced stability in dryland plant ecosystems, PNAS, 102 (2005): 10819-10822. |
[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,R. W. Malone, Evaporation changes over the contiguous United States and the former USSR: A reassessment, Geophysical Research Letters, 28 (2001): 2665-2668. |
[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. |
[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. |
[13] | [ P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992. |
[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. |
[16] | [ C. S. Mantyka-Pringle,T. G. Martin,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. |
[18] | [ A. J. McKane,T. Biancalini,T. Rogers, Stochastic pattern formation and spontaneous polarization: The linear noise approximation and beyond, Bulletin of Mathematical Biology, 76 (2014): 895-921. |
[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. |
[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. |
[22] | [ M. Rietkerk,M. C. Boerlijst,F. van Langevelde,R. HilleRisLambers,J. van de Koppel,L. Kumar,H. H. T. Prins,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,J. van de Koppel, Site-specific properties and irreversible vegetation changes in semi-arid grazing systems, Oikos, 80 (1997): 241-252. |
[24] | [ J. Sheffield,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. |
[25] | [ J. A. Sherratt, An analysis of vegetation stripe formation in semi-arid landscapes, Journal of Mathematical Biology, 51 (2005): 183-197. |
[26] | [ J. A. Sherratt,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. |
[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. |
[28] | [ K. Siteur,M. B. Eppinga,D. Karssenberg,M. Baudena,M. F. P. Bierkens,M. Rietkerk, How will increases in rainfall intensity affect semiarid ecosystems?, Water Resources Research, 50 (2014): 5980-6001. |