Abstract
The spatial organization of functional vegetation types in river basins is a major determinant of their runoff production, biodiversity, and ecosystem services. The optimization of different objective functions has been suggested to control the adaptive behavior of plants and ecosystems, often without a compelling justification. Maximum entropy production (MEP), rooted in thermodynamics principles, provides a tool to justify the choice of the objective function controlling vegetation organization. The application of MEP at the ecosystem scale results in maximum productivity (i.e., maximum canopy photosynthesis) as the thermodynamic limit toward which the organization of vegetation appears to evolve. Maximum productivity, which incorporates complex hydrologic feedbacks, allows us to reproduce the spatial macroscopic organization of functional types of vegetation in a thoroughly monitored river basin, without the need for a reductionist description of the underlying microscopic dynamics. The methodology incorporates the stochastic characteristics of precipitation and the associated soil moisture on a spatially disaggregated framework. Our results suggest that the spatial organization of functional vegetation types in river basins naturally evolves toward configurations corresponding to dynamically accessible local maxima of the maximum productivity of the ecosystem.
Keywords: ecohydrology, ecology, vegetation patterns
Vegetation patterns in river basins shift in response to changes in 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 changes because of the sensitivity of the soil moisture dynamics to climate, soil, and vegetation conditions (3–5). Moreover, the drainage structure of river basins determines the aspect (i.e., orientation) of the hillslopes and the location of the snow line, which in turn heavily impact the radiation balance and the dynamics of the soil moisture (6–8).
Given the number of factors affecting the distribution of vegetation in a river basin, the implementation of optimization frameworks based on plant-scale or canopy-scale dynamics is rather problematic (9). The optimization of different objective functions has been suggested to control the adaptive behavior of plants and ecosystems (10–13), often without a compelling justification. Furthermore, a detailed investigation of microscopic dynamics is generally unnecessary for the analysis of the macroscopic state of open systems (14). Maximum entropy production (MEP) (13, 15, 16) provides a general thermodynamic framework directly applicable at the ecosystem scale. For stationary states of open, nonequilibrium systems, MEP states are preferably selected among all stationary states because they can exist within a greater number of environments (17). In a system including litter and soil organic carbon, the steady-state chemical entropy export is associated with 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 types and their spatial organization at the ecosystem scale. We use it to describe and analyze the type and spatial distribution of vegetation 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 pronounced heterogeneity along a strong topographical gradient (from 1,985 m above sea level to 2,880 m above sea level) with highly nonuniform precipitation (18). The climate regime is typical of a semiarid environment with rainfall increasing with elevation from 220 to 325 mm during the growing season, which goes from May through September. These heterogeneous conditions induce the existence of complex vegetation structures, presenting a noticeable global gradient of functional vegetation types with elevation but also local structures and gradients related to site orientation, i.e., aspect. These characteristics make the URS basin an attractive location to test the organizational criterion (a detailed description of the basin can be found in SI Materials and Methods).
MEP
As mentioned before, MEP serves as a general thermodynamic limit toward which complex, far from equilibrium systems evolve (9, 13, 15, 19). MEP postulates that these systems are driven toward MEP states, which are preferably selected among all attainable states because they can exist within a greater number of environments (17). We use MEP to study the organization of vegetation at the 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 specify the objective function controlling the organization of different functional vegetation types in a river basin. Applying MEP to the carbon balance implies considering the entropy produced by photosynthesis, a process strongly related to plant transpiration, which is controlled by fluxes of water and energy. Thus, explicitly accounting for the entropy production on the carbon cycle of a river basin implicitly includes important effects of other fluxes such as radiation, rainfall, and latent heat in the production of entropy, although these fluxes are not explicitly included in the expression for entropy production.
Our system, for the application of MEP, consists of the vegetation of an entire river basin and the portion of soil in which carbon is stored as organic matter. Fig. 1 shows a scheme of the carbon balance in this system.
Fig. 1.
Ecosystem carbon balance. Adapted from Dewar (9). Black dashed line represents the system boundary. The region with dotted background represents the ecosystem, whereas the region with white background represents the surrounding environment of the system. Blue box is processes generating an incoming flux of carbon to the ecosystem. Green boxes are different forms in which carbon appears within the ecosystem. Orange boxes are processes exporting carbon outside the ecosystem.
Photosynthesis captures atmospheric carbon, in the form of carbon dioxide, which is introduced into the system as carbohydrates. These carbohydrates are mainly used with two purposes. The first one is to maintain plant vital functions. Carbohydrates are thus used and degraded via autotrophic respiration that generates carbon dioxide, which is emitted to the atmosphere. The second purpose is to support plant growth and to repair damaged plant tissues. The damaged tissues fall down and become part of the soil organic matter, which is used by different organisms to support their vital functions. This heterotrophic respiration results in 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 substance transported, divided by the temperature at which the interchange takes place (19). For the case of the carbon cycle (9),
 |
where 
is the entropy production rate of the system, P is the photosynthesis rate, 
and 
are the autotrophic and heterotrophic respiration rates, respectively, 
is the chemical potential of the photosynthates, and 
is the chemical potential of the products of respiration (9). MEP states that the carbon cycle at the ecosystem scale is controlled by the maximization of 
. However, this expression can be simplified.
Assuming the chemical potentials of the autotrophic and heterotrophic respirations are equal,
 |
In the steady state (achieved on timescales of the order of tens of years) the carbon flux of photosynthesis is compensated by the autotrophic and heterotrophic respiration fluxes 
(9). Introducing this relation in the expression for the entropy production rate:
 |
Finally, assuming isothermal transformations (a justification of this assumption is provided in SI Materials and Methods),
 |
Therefore, maximizing the entropy production rate of the carbon cycle of a river basin is equivalent to maximizing canopy photosynthesis. Thus, the most probable state for vegetation, i.e., the one explaining its actual spatial organization, corresponds to the one maximizing canopy photosynthesis (9).
Results and Discussion
Fig. 2 shows the observed vegetation pattern of the URS basin (20) as well as the patterns obtained for several values of the annealing parameter (θ) throughout the simulated annealing procedure. Because the data do not allow for a spatial distinction between Bouteloua gracilis and Bouteloua eriopoda, the two grass species taken as representatives of all of the grass species present at the basin, both types are merged together as grasses when comparing the simulated patterns with the observed ones. The simulated vegetation patterns correspond to steady states of the system for the prescribed annealing parameters.
Fig. 2.
Spatial vegetation distributions. (A) Observed vegetation distribution at the Upper Rio Salado basin. (B) Steady-state vegetation distribution for 
in the simulated annealing algorithm. (C) Steady-state vegetation for 
. (D) Random vegetation distribution obtained for 
.
Vegetation profiles provide an objective and quantitative comparison between the observed spatial distribution of vegetation types and the distribution obtained at different values of the annealing parameter. Fig. 3 shows the profiles of the different functional vegetation types as a function of the distance from the outlet, measured along the drainage network. The patterns of vegetation for 
(Fig. 3, Right) show a very good agreement with the observed ones. Only the profile of the simulated grasses in the upper part of the basin differs from the observed one. Fig. 3, Left also shows the vegetation profiles for 
. Although the profiles for trees and shrubs are again very well reproduced, there is now a complete absence of grasses. The reduced accuracy in grass reproduction in one case and their complete absence for 
suggest that the presence of grasses in the real basin results from local conditions on soil and topography that preclude the vegetation pattern corresponding to the ground state (
) to dominate and instead lead to a feasible optimal pattern (21) corresponding to a temperature slightly larger than zero. Feasible optimality is also a concept found in the evolution of channel networks in river basins where networks are constrained to settle in suboptimal configurations corresponding to local (rather than global) minima of energy dissipation, an exact feature of steady-state landscape evolution equations in the small gradient approximation (22).
Fig. 3.
Vegetation profiles show the dominant presence of different functional vegetation types at distances from the basin outlet measured through the drainage network. (Left) Comparison between the observed vegetation profiles and the simulated ones for 
. (Right) The same comparison for 
.
The productivity of the basin vegetation patterns associated with different temperatures of the simulated annealing procedure is shown in Fig. 4. The maximum productivity is achieved for 
. The simulation best reproducing the existing vegetation structure, however, is obtained for 
, which is very close to the absolute optimum.
Fig. 4.
Assimilation vs. annealing parameter curve. The curve shows the steady-state value of assimilation attained by the vegetation patterns resulting from the simulated annealing procedure for every annealing parameter in the experiment schedule. The dot represents the simulation that compares best with the observations (
).
A synthesis of river basin metabolism is contained in the allometric relationship between the total average transpiration in a subbasin and the total amount of runoff present on the average in the network of the same subbasin (18). Fig. 5 shows such a relationship for the observed vegetation organization of the URS basin and for that corresponding to the vegetation pattern associated with a maximum productivity (i.e., 
) in the simulated annealing procedure.
Fig. 5.
Allometric synthesis of river basin metabolism. Shown is total transpiration per subbasin vs. total amount of runoff present on average in the subbasin. 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 total amount of runoff is surrogated by the nested sum of total contributing areas (22). The exponent is very close to 
as expected for a directed planar network transporting metabolites with constant velocity to every element of a body (23). The black line corresponds to the real basin, whereas the dashed red line corresponds to the simulation that compares best with the observations (
). The green line represents the random vegetation case and the magenta line corresponds to the basin fully covered by Bouteloua gracilis.
The slope of the allometric relationship for the real basin (
) is almost identical to the slope of the optimal basin obtained in the simulations (
), presenting a very small difference (
) between them. This behavior was expected because slope is controlled by the structure of the drainage network (23). The location of the line, however, depends on the proportionality constant between 
(the total transpiration in a subbasin) and 
(the total amount of runoff present on average in a subbasin). This proportionality constant is in turn determined by the vegetation organization in the basin. Fig. 5 shows that the location of both lines is also very similar, the real basin presenting a proportionality constant of 
and the simulated one that of 
(a difference of 
). As expected, the relationship for the optimal case is located slightly above the one for the real basin. The accuracy values presented above are equal to 1 SD, estimated using the bootstrapping 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 by B. gracilis are also presented in Fig. 5. In both cases the lines still exhibit a 
slope (
for the random case and 
for the B. gracilis one). However, the location of the lines is lower than in the optimal basin obtained in the simulations. The proportionality constants in these cases are 
for the former and 
for the latter. These values present a difference of approximately 
compared with the real basin. This lower value of the proportionality constant indicates a lower efficiency of the basin in its use of the area it 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 allometric relationships, suggests that the vegetation of river basins tends to organize around the maximum productivity thermodynamic limit, equivalent to the state of maximum entropy production in these far from equilibrium open systems.
The maximum productivity thermodynamic limit, derived and warranted by maximum entropy production, provides a framework to forecast changes in the organization of functional vegetation types in river basins arising from ongoing and future changes in precipitation and temperature patterns.
Materials and Methods
Soil moisture dynamics and water balance are modeled at every 
pixel of a grid covering the basin, by means of the steady-state probability distribution function of soil moisture (25, 26). It makes use of meteorological data (average rate of storm arrival, average storm depth, and average temperature) and of potential evapotranspiration values computed by means of the 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 variables measured only at the stations are correlated with variables measured for the whole basin. Rainfall characteristics (average rate of storm arrivals and average storm depth) are correlated with pixel elevation. Temperature and incoming shortwave solar radiation are correlated with pixel aspect and elevation (Fig. S1). This procedure serves to downscale the meteorological variables to the pixel scale. Details about the datasets and the methodologies of spatial downscaling are described in SI Materials and Methods.
Plant transpiration is obtained from the stochastic water balance and used to estimate plant assimilation (28–30). Computing plant transpiration by means of a stochastic model allows us to explicitly incorporate the randomness of precipitation as well as the spatially dependent characteristics of the climate, soil, and geomorphological structure of the watershed to investigate the impact of different patterns of the regional functional types of vegetation on the productivity of the basin. Because almost all assimilation in URS takes place during the growing season (31), the characterization of the hydrologic dynamics and its associated vegetation productivity are referred to the period from May to September.
The average productivity is estimated at each site and over the whole basin. The total ecosystem assimilation is obtained by summing up the individual assimilations at every pixel of the basin. Assimilation is computed by multiplying the average growing season transpiration at every pixel by the water use efficiency (WUE) of the vegetation type occupying the pixel. WUE values and the characteristics of the four predominant plant species in the URS basin that have been used for the numerical simulations (32) are presented in Tables S1 and S2. The spatial distribution of soil textures is shown in Fig. S2. Soil parameters (26, 33) associated with each of the three soil textures found within the basin are presented in Table S3.
The optimal vegetation distribution, i.e., the one maximizing ecosystem productivity, was obtained by means of a simulated annealing procedure (34, 35). The simulated annealing procedure defines a dimensionless parameter, called the annealing parameter, which accounts for all of the different factors not explicitly included in the modeling and controls the features 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 type randomly chosen among those existing in the whole basin. If there is an increase of the total ecosystem assimilation with the updated vegetation, the change is accepted. If not, the change may be accepted with a probability
 |
that depends on the absolute value of the decrease in total ecosystem assimilation (
) and the annealing parameter (θ). The constant φ is assigned a value of 1 and is measured in the same units of assimilation [
]. It serves to transform the temperature into a dimensionless parameter.
As a consequence of the formulation of the acceptance probability, when 
(cold system), changes are accepted only if they increase total system productivity (therefore fulfilling the maximum productivity thermodynamic limit). This process leads to the vegetation distribution ground state for the basin under the imposed constraints and climatic conditions. When temperatures of the simulated annealing process are larger than 0, changes decreasing the overall productivity may be accepted with a prescribed probability (Eq. 1), depending on the magnitude of the decrease. For large values of the temperature parameter, its effect dominates over the effect of the driving thermodynamic limit (maximum productivity). In these situations a random vegetation organization is obtained (Fig. 2D).
For the purpose of this study a large number of different vegetation configurations over the basin as well as their corresponding productivities are analyzed for a set of temperatures. This procedure allows us to consider different degrees of disruption to the optimal solution corresponding to the different strength of constraints affecting dynamic accessibility.
Supplementary Material
Acknowledgments
We thank J. M. Limousin, J. Medeiros, M. Thomey, and S. Collins for generously providing field data related to water use efficiency and assimilation for the different plant species used in the analyses. We also thank K. K. Caylor of Princeton University for helping with data interpretation. M.d.J. acknowledges support provided by the Spanish Ministry of Science and Innovation (Formación de Profesorado Universitario Grant AP2007-02225). The simulations presented in this article were performed on computational resources supported by the Princeton Institute for Computational Science and Engineering and the Office of Information Technology’s High Performance Computing Center and Visualization Laboratory at Princeton University.
Footnotes
The authors declare no conflict of interest.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1218636109/-/DCSupplemental.
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