Continuum Modeling of Discrete Plant Communities: Why Does It Work and Why Is It Advantageous?

Understanding ecosystem response to drier climates calls for modeling the dynamics of dryland plant populations, which are crucial determinants of ecosystem function, as they constitute the basal level of whole food webs. Two modeling approaches are widely used in population dynamics, individual (agent)-based models and continuum partial-differential-equation (PDE) models. The latter are advantageous in lending themselves to powerful methodologies of mathematical analysis, but the question of whether they are suitable to describe small discrete plant populations, as is often found in dryland ecosystems, has remained largely unaddressed. In this paper, we first draw attention to two aspects of plants that distinguish them from most other organisms—high phenotypic plasticity and dispersal of stress-tolerant seeds—and argue in favor of PDE modeling, where the state variables that describe population sizes are not discrete number densities, but rather continuous biomass densities. We then discuss a few examples that demonstrate the utility of PDE models in providing deep insights into landscape-scale behaviors, such as the onset of pattern forming instabilities, multiplicity of stable ecosystem states, regular and irregular, and the possible roles of front instabilities in reversing desertification. We briefly mention a few additional examples, and conclude by outlining the nature of the information we should and should not expect to gain from PDE model studies.

[1]  S. Clarke,et al.  Exceptional seed longevity and robust growth: ancient Sacred Lotus from China , 1995 .

[2]  Yuval R. Zelnik,et al.  Regime shifts in models of dryland vegetation , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[3]  R. Lefever,et al.  Vegetation spots and stripes: Dissipative structures in arid landscapes , 2004 .

[4]  Jean Poesen,et al.  Soil and water components of banded vegetation patterns , 1999 .

[5]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[6]  E. Meron From Patterns to Function in Living Systems: Dryland Ecosystems as a Case Study , 2018 .

[7]  J. Scott,et al.  The ecophysiology of seed persistence: a mechanistic view of the journey to germination or demise , 2015, Biological reviews of the Cambridge Philosophical Society.

[8]  Kenneth Showalter,et al.  Multistability and tipping: From mathematics and physics to climate and brain-Minireview and preface to the focus issue. , 2018, Chaos.

[9]  D. Lathrop Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering , 2015 .

[10]  M. Tlidi,et al.  Strong interaction between plants induces circular barren patches: fairy circles , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[11]  Jonathan A. Sherratt,et al.  Pattern Solutions of the Klausmeier Model for Banded Vegetation in Semiarid Environments V: The Transition from Patterns to Desert , 2013, SIAM J. Appl. Math..

[12]  Yuval R. Zelnik,et al.  Gradual regime shifts in fairy circles , 2015, Proceedings of the National Academy of Sciences.

[13]  E. Louvergneaux,et al.  Origin of the pinning of drifting monostable patterns. , 2012, Physical review letters.

[14]  Marten Scheffer,et al.  Local Facilitation May Cause Tipping Points on a Landscape Level Preceded by Early-Warning Indicators , 2015, The American Naturalist.

[15]  Thorsten Wiegand,et al.  Adopting a spatially explicit perspective to study the mysterious fairy circles of Namibia , 2015 .

[16]  M. Ooi Seed bank persistence and climate change , 2012, Seed Science Research.

[17]  C. Klausmeier,et al.  Regular and irregular patterns in semiarid vegetation , 1999, Science.

[18]  J. Dawes,et al.  Localised pattern formation in a model for dryland vegetation , 2016, Journal of mathematical biology.

[19]  Luca Ridolfi,et al.  Mathematical models of vegetation pattern formation in ecohydrology , 2009 .

[20]  E. Meron,et al.  A mathematical model of plants as ecosystem engineers. , 2007, Journal of theoretical biology.

[21]  N. Ursino The influence of soil properties on the formation of unstable vegetation patterns on hillsides of semiarid catchments , 2005 .

[22]  G. Grafi A “mille-feuilles” of stress tolerance in the desert plant Zygophyllum dumosum Boiss.: Highlighting epigenetics , 2019, Israel Journal of Plant Sciences.

[23]  Golan Bel,et al.  Localized states qualitatively change the response of ecosystems to varying conditions and local disturbances , 2016, 1703.00285.

[24]  Yuval R. Zelnik,et al.  Implications of tristability in pattern-forming ecosystems. , 2018, Chaos.

[25]  T. Laux,et al.  Plant stem cell niches. , 2012, Annual review of plant biology.

[26]  Ehud Meron,et al.  Front Instabilities Can Reverse Desertification. , 2019, Physical review letters.

[27]  Norbert Juergens,et al.  The Biological Underpinnings of Namib Desert Fairy Circles , 2013, Science.

[28]  S. Carpenter,et al.  Catastrophic shifts in ecosystems , 2001, Nature.

[29]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[30]  Thorsten Wiegand,et al.  Discovery of fairy circles in Australia supports self-organization theory , 2016, Proceedings of the National Academy of Sciences.

[31]  Hezi Yizhaq,et al.  Banded vegetation: biological productivity and resilience , 2005 .

[32]  Walter R. Tschinkel,et al.  The Life Cycle and Life Span of Namibian Fairy Circles , 2012, PloS one.

[33]  Jonathan A. Sherratt,et al.  An Analysis of Vegetation Stripe Formation in Semi-Arid Landscapes , 2005, Journal of mathematical biology.

[34]  P. Coullet,et al.  A simple global characterization for normal forms of singular vector fields , 1987 .

[35]  M. Zeroni,et al.  Gas exchange of a desert shrub (Zygophyllum dumosum Boiss.) under different soil moisture regimes during summer drought , 1994, Vegetatio.

[36]  E. Meron,et al.  Ecosystem engineers: from pattern formation to habitat creation. , 2004, Physical review letters.

[37]  Alan C. Newell,et al.  ORDER PARAMETER EQUATIONS FOR PATTERNS , 1993 .

[38]  Donald L. DeAngelis,et al.  Spatially Explicit Modeling in Ecology: A Review , 2016, Ecosystems.

[39]  E. Meron,et al.  Diversity of vegetation patterns and desertification. , 2001, Physical review letters.

[40]  E. Zaady,et al.  Infiltration through three contrasting biological soil crusts in patterned landscapes in the Negev, Israel , 2000 .

[41]  G. Lord,et al.  Nonlinear dynamics and pattern bifurcations in a model for vegetation stripes in semi-arid environments. , 2007, Theoretical population biology.

[42]  C. S. Holling,et al.  STABILITY OF SEMI-ARID SAVANNA GRAZING SYSTEMS , 1981 .

[43]  Y. Pomeau Front motion, metastability and subcritical bifurcations in hydrodynamics , 1986 .

[44]  R. Veit,et al.  Partial Differential Equations in Ecology: Spatial Interactions and Population Dynamics , 1994 .

[45]  Golan Bel,et al.  Interplay between Turing mechanisms can increase pattern diversity. , 2014, Physical review letters.

[46]  N. Yadav,et al.  Seasonal Growth of Zygophyllum dumosum Boiss.: Summer Dormancy Is Associated with Loss of the Permissive Epigenetic Marker Dimethyl H3K4 and Extensive Reduction in Proteins Involved in Basic Cell Functions , 2018, Plants.

[47]  Joel s. Brown,et al.  The Selective Interactions of Dispersal, Dormancy, and Seed Size as Adaptations for Reducing Risk in Variable Environments , 1988, The American Naturalist.

[48]  S. Mooney,et al.  Branching Out in Roots: Uncovering Form, Function, and Regulation1 , 2014, Plant Physiology.

[49]  Ken Thompson,et al.  Climate change and plant regeneration from seed , 2011 .

[50]  E. Meron Nonlinear Physics of Ecosystems , 2015 .

[51]  R. Bandurski,et al.  THE ONE HUNDRED‐YEAR PERIOD FOR DR. BEAL'S SEED VIABILITY EXPERIMENT , 1981 .

[52]  E. Meron Pattern formation--A missing link in the study of ecosystem response to environmental changes. , 2016, Mathematical biosciences.

[53]  J. A. Bonachela,et al.  A theoretical foundation for multi-scale regular vegetation patterns , 2017, Nature.

[54]  S. Solomon,et al.  Reactive glass and vegetation patterns. , 2002, Physical review letters.

[55]  J. Bogaert,et al.  Determinants and dynamics of banded vegetation pattern migration in arid climates , 2012 .

[56]  G. Cunningham,et al.  An Ecologial Significance of Seasonal Leaf Variability in a Desert Shrub , 1969 .

[57]  Jonathan H. P. Dawes,et al.  Localized Pattern Formation with a Large-Scale Mode: Slanted Snaking , 2008, SIAM J. Appl. Dyn. Syst..

[58]  K. Tielbörger,et al.  What does the stress‐gradient hypothesis predict? Resolving the discrepancies , 2010 .

[59]  Johan van de Koppel,et al.  Regular pattern formation in real ecosystems. , 2008, Trends in ecology & evolution.

[60]  C. Crawford The Negev. The Challenge of a Desert , 1983 .

[61]  A. Prokushkin,et al.  Critical analysis of root : shoot ratios in terrestrial biomes , 2006 .

[62]  Alexandre Bouvet,et al.  Multistability of model and real dryland ecosystems through spatial self-organization , 2018, Proceedings of the National Academy of Sciences.

[63]  G. Grafi,et al.  The Dead Can Nurture: Novel Insights into the Function of Dead Organs Enclosing Embryos , 2018, International journal of molecular sciences.

[64]  Nicolas Barbier,et al.  The global biogeography of semi‐arid periodic vegetation patterns , 2008 .

[65]  O. Ovaskainen,et al.  Stochastic models of population extinction. , 2010, Trends in ecology & evolution.

[66]  Daniel Wetzel,et al.  pde2path - A Matlab package for continuation and bifurcation in 2D elliptic systems , 2012, 1208.3112.

[67]  Alan R. Champneys,et al.  Localized Hexagon Patterns of the Planar Swift-Hohenberg Equation , 2008, SIAM J. Appl. Dyn. Syst..

[68]  M. A. Muñoz,et al.  Patchiness and Demographic Noise in Three Ecological Examples , 2012, 1205.3389.

[69]  Ehud Meron,et al.  Complex patterns in reaction-diffusion systems: A tale of two front instabilities. , 1994, Chaos.

[70]  Bai-lian Li,et al.  Exactly Solvable Models of Biological Invasion , 2005 .

[71]  Massimo Pigliucci,et al.  Evolution of phenotypic plasticity: where are we going now? , 2005, Trends in ecology & evolution.

[72]  Tao Zhang,et al.  On the Increased Frequency of Mediterranean Drought , 2012 .

[73]  E. Meron,et al.  Gradual regime shifts in spatially extended ecosystems , 2012, Theoretical Ecology.

[74]  E. Gilada,et al.  A mathematical model of plants as ecosystem engineers , 2007 .

[75]  Steven F. Railsback,et al.  Individual-based modeling and ecology , 2005 .

[76]  Maarten B. Eppinga,et al.  Beyond Turing: The response of patterned ecosystems to environmental change , 2014 .

[77]  Sergei Petrovskii,et al.  Allee effect makes possible patchy invasion in a predator-prey system. , 2002 .

[78]  Hezi Yizhaq,et al.  Linear and nonlinear front instabilities in bistable systems , 2006 .

[79]  Edgar Knobloch,et al.  Spatial Localization in Dissipative Systems , 2015 .

[80]  B. Schutte,et al.  Seed survival in soil: interacting effects of predation, dormancy and the soil microbial community , 2011 .

[81]  R. Lefever,et al.  On the origin of tiger bush , 1997 .

[82]  S. Chapman,et al.  Asymptotics of large bound states of localized structures. , 2006, Physical review letters.

[83]  M. Rietkerk,et al.  Self-Organized Patchiness and Catastrophic Shifts in Ecosystems , 2004, Science.

[84]  Edgar Knobloch,et al.  Snakes and ladders: Localized states in the Swift–Hohenberg equation , 2007 .

[85]  H. Meinhardt Pattern formation in biology: a comparison of models and experiments , 1992 .

[86]  Jonathan A. Sherratt,et al.  Using wavelength and slope to infer the historical origin of semiarid vegetation bands , 2015, Proceedings of the National Academy of Sciences.

[87]  E. Meron,et al.  Dynamics and spatial organization of plant communities in water-limited systems. , 2007, Theoretical population biology.

[88]  J. Hansen,et al.  Perception of climate change , 2012, Proceedings of the National Academy of Sciences.

[89]  E. Meron,et al.  Reversing desertification as a spatial resonance problem. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[90]  Henry S. Greenside,et al.  Pattern Formation and Dynamics in Nonequilibrium Systems , 2004 .

[91]  H. Schenk,et al.  Hydraulic integration and shrub growth form linked across continental aridity gradients , 2008, Proceedings of the National Academy of Sciences.

[92]  D. Cohen A General Model of Optimal Reproduction in a Randomly Varying Environment , 1968 .

[93]  D. DeAngelis,et al.  Individual-based models in ecology after four decades , 2014, F1000prime reports.

[94]  E. Meron,et al.  Species coexistence by front pinning , 2014 .

[95]  H. Prins,et al.  VEGETATION PATTERN FORMATION IN SEMI-ARID GRAZING SYSTEMS , 2001 .

[96]  Eric P. Verrecchia,et al.  Physical properties of the psammophile cryptogamic crust and their consequences to the water regime of sandy soils, north-western Negev Desert, Israel , 1995 .

[97]  M. Silber,et al.  Transitions between patterned states in vegetation models for semiarid ecosystems. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[98]  Yuval R. Zelnik,et al.  Regime shifts by front dynamics , 2018, Ecological Indicators.

[99]  B. Cook,et al.  Unprecedented 21st century drought risk in the American Southwest and Central Plains , 2015, Science Advances.

[100]  Peter Cox,et al.  Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system , 2011, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[101]  J. Anderies,et al.  Resilience in social-ecological systems: identifying stable and unstable equilibria with agent-based models , 2019, Ecology and Society.

[102]  Petrich,et al.  Interface proliferation and the growth of labyrinths in a reaction-diffusion system. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[103]  Edgar Knobloch,et al.  To Snake or Not to Snake in the Planar Swift-Hohenberg Equation , 2010, SIAM J. Appl. Dyn. Syst..