Maximum entropy production, carbon assimilation, andthe spatial organization of vegetation in river basinsManuel del Jesusa,b, Romano Fotia, Andrea Rinaldoc,d,1, and Ignacio Rodriguez-Iturbea,1

aDepartment of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544; bEnvironmental Hydraulics Institute “Instituto de Hidráulica(IH) Cantabria,” Universidad de Cantabria, Parque Científico y Tecnológico de Cantabria (PCTCAN), 39011 Santander, Spain; cLaboratory of Ecohydrology,School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland; and dDipartimentoIngegneria Civile, Edile ed Ambientale (ICEA), Universitá di Padova, 35131 Padua, Italy

Contributed by Andrea Rinaldo, November 5, 2012 (sent for review September 23, 2012)

The spatial organization of functional vegetation types in riverbasins is a major determinant of their runoff production, biodiver-sity, and ecosystem services. The optimization of different objectivefunctions has been suggested to control the adaptive behaviorof plants and ecosystems, often without a compelling justification.Maximum entropy production (MEP), rooted in thermodynamicsprinciples, provides a tool to justify the choice of the objectivefunction controlling vegetation organization. The application ofMEP at the ecosystem scale results in maximum productivity (i.e.,maximum canopy photosynthesis) as the thermodynamic limittoward which the organization of vegetation appears to evolve.Maximum productivity, which incorporates complex hydrologicfeedbacks, allows us to reproduce the spatial macroscopic organi-zation of functional types of vegetation in a thoroughly monitoredriver basin, without the need for a reductionist description of theunderlying microscopic dynamics. The methodology incorporatesthe stochastic characteristics of precipitation and the associated soilmoisture on a spatially disaggregated framework. Our results suggestthat the spatial organization of functional vegetation types in riverbasins naturally evolves toward configurations corresponding todynamically accessible local maxima of the maximum productivityof the ecosystem.

ecohydrology | ecology | vegetation patterns

Vegetation patterns in river basins shift in response to changesin precipitation and temperature, whose local impact is linked

to the soil and geomorphological structure of the basin (1, 2).Semiarid ecosystems are particularly responsive to these changesbecause of the sensitivity of the soil moisture dynamics to climate,soil, and vegetation conditions (3–5). Moreover, the drainagestructure of river basins determines the aspect (i.e., orientation)of the hillslopes and the location of the snow line, which in turnheavily impact the radiation balance and the dynamics of the soilmoisture (6–8).Given the number of factors affecting the distribution of vege-

tation in a river basin, the implementation of optimization frame-works based on plant-scale or canopy-scale dynamics is ratherproblematic (9). The optimization of different objective functionshas been suggested to control the adaptive behavior of plantsand ecosystems (10–13), often without a compelling justification.Furthermore, a detailed investigation of microscopic dynamicsis generally unnecessary for the analysis of the macroscopic stateof open systems (14). Maximum entropy production (MEP) (13,15, 16) provides a general thermodynamic framework directlyapplicable at the ecosystem scale. For stationary states of open,nonequilibrium systems, MEP states are preferably selected amongall stationary states because they can exist within a greater numberof environments (17). In a system including litter and soil organiccarbon, the steady-state chemical entropy export is associatedwith the conversion of photosynthates to CO2. MEP, therefore,is equivalent to the maximization of canopy photosynthesis (9).We adopt MEP, i.e., maximum productivity, as the objective

criterion for the selection of different functional vegetation typesand their spatial organization at the ecosystem scale. We use it to

describe and analyze the type and spatial distribution of vege-tation in the Upper Rio Salado (URS) basin in New Mexico,located near the Sevilleta Long-Term Ecological Research (LTER)site. The URS is a 466-km2 semiarid basin, which exhibits a pro-nounced heterogeneity along a strong topographical gradient (from1,985 m above sea level to 2,880 m above sea level) with highlynonuniform precipitation (18). The climate regime is typical of asemiarid environment with rainfall increasing with elevation from220 to 325 mm during the growing season, which goes from Maythrough September. These heterogeneous conditions induce theexistence of complex vegetation structures, presenting a notice-able global gradient of functional vegetation types with elevationbut also local structures and gradients related to site orientation,i.e., aspect. These characteristics make the URS basin an attractivelocation to test the organizational criterion (a detailed descriptionof the basin can be found in SI Materials and Methods).

MEPAs mentioned before, MEP serves as a general thermodynamiclimit toward which complex, far from equilibrium systems evolve(9, 13, 15, 19). MEP postulates that these systems are driven towardMEP states, which are preferably selected among all attainablestates because they can exist within a greater number of environ-ments (17). We use MEP to study the organization of vegetation atthe ecosystem scale on a river basin.In this paper MEP is applied to the carbon balance at the

ecosystem scale [following the derivation of Dewar (9)], to specifythe objective function controlling the organization of differentfunctional vegetation types in a river basin. Applying MEP to thecarbon balance implies considering the entropy produced byphotosynthesis, a process strongly related to plant transpiration,which is controlled by fluxes of water and energy. Thus, explicitlyaccounting for the entropy production on the carbon cycle of ariver basin implicitly includes important effects of other fluxessuch as radiation, rainfall, and latent heat in the production ofentropy, although these fluxes are not explicitly included in theexpression for entropy production.Our system, for the application of MEP, consists of the veg-

etation of an entire river basin and the portion of soil in whichcarbon is stored as organic matter. Fig. 1 shows a scheme of thecarbon balance in this system.Photosynthesis captures atmospheric carbon, in the form of

carbon dioxide, which is introduced into the system as carbohy-drates. These carbohydrates are mainly used with two purposes.

Author contributions: M.d.J., R.F., A.R., and I.R.-I. designed research; M.d.J., R.F., A.R., andI.R.-I. performed research; M.d.J., R.F., and I.R.-I. analyzed data; and M.d.J., A.R., and I.R.-I.wrote the paper.

The authors declare no conflict of interest.

Freely available online through the PNAS open access option.1To whom correspondence may be addressed. E-mail: andrea.rinaldo@epfl.ch or irodrigu@princeton.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1218636109/-/DCSupplemental.

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The first one is to maintain plant vital functions. Carbohydratesare thus used and degraded via autotrophic respiration that gen-erates carbon dioxide, which is emitted to the atmosphere. Thesecond purpose is to support plant growth and to repair damagedplant tissues. The damaged tissues fall down and become part ofthe soil organic matter, which is used by different organisms tosupport their vital functions. This heterotrophic respiration resultsin the emission of carbon dioxide back to the atmosphere (9).Entropy production is evaluated as the product of the mass flux

across the boundaries times the chemical potential of the sub-stance transported, divided by the temperature at which the in-terchange takes place (19). For the case of the carbon cycle (9),

σsystem =1TðP  μP −RA   μR −RB   μRÞ;

where σsystem is the entropy production rate of the system, P is thephotosynthesis rate, RA and RB are the autotrophic and hetero-trophic respiration rates, respectively, μP is the chemical poten-tial of the photosynthates, and μR is the chemical potential of theproducts of respiration (9). MEP states that the carbon cycle atthe ecosystem scale is controlled by the maximization of σsystem.However, this expression can be simplified.Assuming the chemical potentials of the autotrophic and het-

erotrophic respirations are equal,

σsystem =1TðP  μP − ðRA +RBÞ  μRÞ:

In the steady state (achieved on timescales of the order of tensof years) the carbon flux of photosynthesis is compensated bythe autotrophic and heterotrophic respiration fluxes P≈RA +RB(9). Introducing this relation in the expression for the entropyproduction rate:

σsystem =1TðPðμP − μRÞÞ:

Finally, assuming isothermal transformations (a justification ofthis assumption is provided in SI Materials and Methods),

σsystem ∝P:

Therefore, maximizing the entropy production rate of the carboncycle of a river basin is equivalent to maximizing canopy photo-synthesis. Thus, the most probable state for vegetation, i.e., the oneexplaining its actual spatial organization, corresponds to the onemaximizing canopy photosynthesis (9).

Results and DiscussionFig. 2 shows the observed vegetation pattern of the URS basin(20) as well as the patterns obtained for several values of theannealing parameter (θ) throughout the simulated annealingprocedure. Because the data do not allow for a spatial distinctionbetween Bouteloua gracilis and Bouteloua eriopoda, the two grassspecies taken as representatives of all of the grass species presentat the basin, both types are merged together as grasses whencomparing the simulated patterns with the observed ones. Thesimulated vegetation patterns correspond to steady states of thesystem for the prescribed annealing parameters.Vegetation profiles provide an objective and quantitative com-

parison between the observed spatial distribution of vegetationtypes and the distribution obtained at different values of theannealing parameter. Fig. 3 shows the profiles of the differentfunctional vegetation types as a function of the distance from theoutlet, measured along the drainage network. The patterns ofvegetation for θ= 4× 10−3 (Fig. 3, Right) show a very good agree-ment with the observed ones. Only the profile of the simulated

Fig. 1. Ecosystem carbon balance. Adapted from Dewar (9). Black dashed line represents the system boundary. The region with dotted background rep-resents the ecosystem, whereas the region with white background represents the surrounding environment of the system. Blue box is processes generating anincoming flux of carbon to the ecosystem. Green boxes are different forms in which carbon appears within the ecosystem. Orange boxes are processesexporting carbon outside the ecosystem.

A B C D

Fig. 2. Spatial vegetation distributions. (A) Observed vegetation distribution at the Upper Rio Salado basin. (B) Steady-state vegetation distribution for θ= 0in the simulated annealing algorithm. (C) Steady-state vegetation for θ= 4× 10−3. (D) Random vegetation distribution obtained for θ> 1:0.

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grasses in the upper part of the basin differs from the observedone. Fig. 3, Left also shows the vegetation profiles for θ= 0.Although the profiles for trees and shrubs are again very wellreproduced, there is now a complete absence of grasses. The re-duced accuracy in grass reproduction in one case and their com-plete absence for θ= 0 suggest that the presence of grasses in thereal basin results from local conditions on soil and topographythat preclude the vegetation pattern corresponding to the groundstate (θ= 0) to dominate and instead lead to a feasible optimalpattern (21) corresponding to a temperature slightly larger thanzero. Feasible optimality is also a concept found in the evolutionof channel networks in river basins where networks are con-strained to settle in suboptimal configurations correspondingto local (rather than global) minima of energy dissipation, anexact feature of steady-state landscape evolution equations in thesmall gradient approximation (22).The productivity of the basin vegetation patterns associated

with different temperatures of the simulated annealing procedureis shown in Fig. 4. The maximum productivity is achieved for θ= 0.The simulation best reproducing the existing vegetation structure,however, is obtained for θ= 4× 10−3, which is very close to theabsolute optimum.A synthesis of river basin metabolism is contained in the al-

lometric relationship between the total average transpiration in asubbasin and the total amount of runoff present on the averagein the network of the same subbasin (18). Fig. 5 shows such a

relationship for the observed vegetation organization of the URSbasin and for that corresponding to the vegetation pattern as-sociated with a maximum productivity (i.e., θ= 4× 10−3) in thesimulated annealing procedure.The slope of the allometric relationship for the real basin

(0:672± 2:1× 10−4) is almost identical to the slope of the optimalbasin obtained in the simulations (0:674± 1:6× 10−4), presentinga very small difference (0:4%) between them. This behaviorwas expected because slope is controlled by the structure of thedrainage network (23). The location of the line, however, dependson the proportionality constant between Ti (the total transpirationin a subbasin) and ΣAi (the total amount of runoff present onaverage in a subbasin). This proportionality constant is in turndetermined by the vegetation organization in the basin. Fig. 5shows that the location of both lines is also very similar, the realbasin presenting a proportionality constant of 12:734± 0:026 andthe simulated one that of 13:159± 0:020 (a difference of 3:3%).As expected, the relationship for the optimal case is locatedslightly above the one for the real basin. The accuracy valuespresented above are equal to 1 SD, estimated using the boot-strapping method (24) on every variable studied.The allometric relationships for the case of a basin covered

by a random vegetation and that of a basin fully covered byB. gracilis are also presented in Fig. 5. In both cases the linesstill exhibit a 2=3 slope (0:670± 1:5× 10−4 for the random caseand 0:664± 1:0× 10−4 for the B. gracilis one). However, the

Fig. 3. Vegetation profiles show the dominant presence of different functional vegetation types at distances from the basin outlet measured through thedrainage network. (Left) Comparison between the observed vegetation profiles and the simulated ones for θ= 0. (Right) The same comparison for θ= 4 × 10−3.

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location of the lines is lower than in the optimal basin obtainedin the simulations. The proportionality constants in these casesare 11:184± 0:016 for the former and 11:175± 0:010 for the latter.These values present a difference of approximately 12% comparedwith the real basin. This lower value of the proportionality con-stant indicates a lower efficiency of the basin in its use of the areait occupies to produce transpiration (or assimilation or entropy,which in this case are equivalent).The agreement between the modeled and observed vegetation

profiles, and also between the modeled and observed allometricrelationships, suggests that the vegetation of river basins tendsto organize around the maximum productivity thermodynamiclimit, equivalent to the state of maximum entropy productionin these far from equilibrium open systems.The maximum productivity thermodynamic limit, derived and

warranted by maximum entropy production, provides a frame-work to forecast changes in the organization of functional veg-

etation types in river basins arising from ongoing and futurechanges in precipitation and temperature patterns.

Materials and MethodsSoil moisture dynamics and water balance are modeled at every 28:5× 28:5‐m2

pixel of a grid covering the basin, by means of the steady-state probabilitydistribution function of soil moisture (25, 26). It makes use of meteorologicaldata (average rate of storm arrival, average storm depth, and average tem-perature) and of potential evapotranspiration values computed by means ofthe Penman–Monteith model to solve the stochastic water balance.

Meteorological data are obtained from six stations in the Rio Salado region(27). To distribute meteorological magnitudes over the basin, the variablesmeasured only at the stations are correlated with variables measured forthe whole basin. Rainfall characteristics (average rate of storm arrivals andaverage storm depth) are correlated with pixel elevation. Temperature andincoming shortwave solar radiation are correlated with pixel aspect and ele-vation (Fig. S1). This procedure serves to downscale the meteorological var-iables to the pixel scale. Details about the datasets and the methodologies ofspatial downscaling are described in SI Materials and Methods.

Plant transpiration is obtained from the stochastic water balance andused to estimate plant assimilation (28–30). Computing plant transpirationby means of a stochastic model allows us to explicitly incorporate the ran-domness of precipitation as well as the spatially dependent characteristicsof the climate, soil, and geomorphological structure of the watershed toinvestigate the impact of different patterns of the regional functional typesof vegetation on the productivity of the basin. Because almost all assimila-tion in URS takes place during the growing season (31), the characterizationof the hydrologic dynamics and its associated vegetation productivity arereferred to the period from May to September.

The average productivity is estimated at each site and over the wholebasin. The total ecosystem assimilation is obtained by summing up the in-dividual assimilations at every pixel of the basin. Assimilation is computedby multiplying the average growing season transpiration at every pixel bythe water use efficiency (WUE) of the vegetation type occupying the pixel.WUE values and the characteristics of the four predominant plant speciesin the URS basin that have been used for the numerical simulations (32) arepresented in Tables S1 and S2. The spatial distribution of soil textures isshown in Fig. S2. Soil parameters (26, 33) associated with each of the threesoil textures found within the basin are presented in Table S3.

The optimal vegetation distribution, i.e., the one maximizing ecosystemproductivity, was obtained by means of a simulated annealing procedure(34, 35). The simulated annealing procedure defines a dimensionless

Fig. 4. Assimilation vs. annealing parameter curve. The curve shows thesteady-state value of assimilation attained by the vegetation patterns result-ing from the simulated annealing procedure for every annealing parameter inthe experiment schedule. The dot represents the simulation that comparesbest with the observations (θ= 4× 10−3).

Fig. 5. Allometric synthesis of river basin metabolism. Shown is total transpiration per subbasin vs. total amount of runoff present on average in thesubbasin. The total amount of runoff present in the subbasin is the sum of the runoff of all of the links contained within the considered subbasin. The totalamount of runoff is surrogated by the nested sum of total contributing areas (22). The exponent is very close to 2=3 as expected for a directed planar networktransporting metabolites with constant velocity to every element of a body (23). The black line corresponds to the real basin, whereas the dashed red linecorresponds to the simulation that compares best with the observations (θ= 4×10−3). The green line represents the random vegetation case and the magentaline corresponds to the basin fully covered by Bouteloua gracilis.

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parameter, called the annealing parameter, which accounts for all of thedifferent factors not explicitly included in the modeling and controls thefeatures of the optimal configurations. For a given annealing parameter,the simulated annealing algorithm proceeds to tentatively update (i.e.,change) the vegetation at a randomly chosen pixel by a vegetation typerandomly chosen among those existing in the whole basin. If there is an in-crease of the total ecosystem assimilation with the updated vegetation, thechange is accepted. If not, the change may be accepted with a probability

p = e−jΔjφ θ [1]

that depends on the absolute value of the decrease in total ecosystem as-similation (Δ) and the annealing parameter (θ). The constant φ is assigneda value of 1 and is measured in the same units of assimilation [kg CO2=d]. Itserves to transform the temperature into a dimensionless parameter.

As a consequence of the formulation of the acceptance probability, whenθ=0 (cold system), changes are accepted only if they increase total systemproductivity (therefore fulfilling the maximum productivity thermodynamiclimit). This process leads to the vegetation distribution ground state forthe basin under the imposed constraints and climatic conditions. Whentemperatures of the simulated annealing process are larger than 0, changes

decreasing the overall productivity may be accepted with a prescribedprobability (Eq. 1), depending on the magnitude of the decrease. Forlarge values of the temperature parameter, its effect dominates over theeffect of the driving thermodynamic limit (maximum productivity). In thesesituations a random vegetation organization is obtained (Fig. 2D).

For the purpose of this study a large number of different vegetationconfigurations over the basin as well as their corresponding productivitiesare analyzed for a set of temperatures. This procedure allows us to considerdifferent degrees of disruption to the optimal solution corresponding to thedifferent strength of constraints affecting dynamic accessibility.

ACKNOWLEDGMENTS. We thank J. M. Limousin, J. Medeiros, M. Thomey,and S. Collins for generously providing field data related to water useefficiency and assimilation for the different plant species used in the analyses.We also thank K. K. Caylor of Princeton University for helping with datainterpretation. M.d.J. acknowledges support provided by the Spanish Ministryof Science and Innovation (Formación de Profesorado Universitario GrantAP2007-02225). The simulations presented in this article were performed oncomputational resources supported by the Princeton Institute for Computa-tional Science and Engineering and the Office of Information Technology’sHigh Performance Computing Center and Visualization Laboratory at Prince-ton University.

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