## Introduction

The development of a quantitative theory of coupled atmosphere–soil–vegetation dynamics is a fundamental ecological and hydrological problem. Such a theory is essential for understanding the global distribution of biomes and their correlation with climate patterns (Holdridge 1947; Stephenson 1990), and the regular changes in vegetation structure and composition observed along topographic and other gradients in microclimate (Whittaker 1967). It is also a key to parameterizing the land–surface boundary conditions of general circulation models (Betts 1999; Desborough 1999) and predicting ecosystem responses to disturbance and climate change (Neilson 1995; Daly *et al*. 2000).

In water-limited ecosystems, atmosphere–soil–vegetation dynamics can be framed as a water-balance problem. Plant growth is limited by the availability of water, which is mediated by a combination of soil properties, precipitation and evaporative demand, and the presence of plants themselves. In this context a recent review (Hatton *et al*. 1997) called for a re-examination of the pioneering work of Peter Eagleson and colleagues (Eagleson 1978a, b, c, d, e, f, g, 1982; Eagleson & Tellers 1982; Eagleson & Segarra 1985) as the potential basis of an ecohydrological theory of atmosphere–soil–vegetation interactions. Here our goal is to examine critically the theoretical foundation and operational utility of Eagleson's framework in an ecological context. We ask whether the ideas represented in the theory make ecological sense, and whether the model produces fruitful predictions and fundamental insights.

### equilibrium water balance and the ecological optimality hypotheses

Eagleson's model uses a statistical–dynamical representation of soil-moisture dynamics, integrated over intermittent precipitation events, to derive analytically the equilibrium partitioning of precipitation into runoff and evapotranspiration. The two surface fluxes depend on 13 parameters (Table 1), all of which can be measured or estimated, as well as the soil moisture concentration, *s*. The distribution of storm depths and interstorm periods provides the boundary conditions for equations describing runoff and evapotranspiration. Because the surface fluxes depend on soil moisture concentration, the equilibrium soil moisture concentration *s*_{0}, which represents the spatially and temporally averaged state of the soil, acts as a state variable. The system state and the expected values of annual runoff and evapotranspiration are then found by averaging over the entire distribution of boundary conditions and solving the equilibrium:

Parameter | Units | Description |
---|---|---|

e_{P} | cm day^{−1} | Average bare soil potential evaporation rate |

mP_{A} | cm | Average annual precipitation |

m_{R} | days | Mean storm duration |

m_{τ} | days | Mean rainy season length |

α | day^{−1} | Reciprocal mean time between storms |

κ | Parameter, gamma distribution of storm depths | |

λ | cm^{−1} | Parameter, gamma distribution of storm depths |

K(1) | cm day^{−1} | Saturated soil hydraulic conductivity |

ψ(1) | cm | Saturated soil matrix potential |

c | Soil pore disconnectedness index | |

n | Soil porosity | |

h_{0} | cm | Surface retention capacity |

k_{V} | Vegetation transpiration coefficient | |

M | Fractional canopy density |

*E*[

*P*

*] =*

_{A}*E*[

*ET*

*(*

_{A}*s*

_{0}, climate, soil, vegetation)] +

*E*[

*R*

*(*

_{A}*s*

_{0}, climate, soil)]( eqn 1)

where *P** _{A}* is annual precipitation,

*ET*

*is annual evapotranspiration,*

_{A}*R*

_{A}is annual runoff (groundwater and surface water), and

*E*[] denotes the expected value. We provide a more detailed description of the model below, and a full published derivation is available (Eagleson 1978a, b, c, d, e, f, g, 1982; Eagleson & Tellers 1982; Eagleson & Segarra 1985).

The analytical form of Eagleson's model contrasts with both process-based simulation models (Running & Coughlan 1988; Running & Gower 1991; Neilson 1995; Haxeltine *et al*. 1996) and correlative methods (Stephenson 1998; Iverson & Prasad 2002) for examining vegetation–climate relationships. Like the former, it utilizes mechanistic representations of hydrological processes. However, instead of numerically intensive simulation, Eagleson's statistical–dynamical approach uses probability distributions of climatic parameters to derive equilibrium distributions of hydrologic fluxes, given the necessary vegetation and soil parameters. One advantage of this theoretical approach is that it can be inverted; given the distribution of hydrological responses to precipitation, researchers can derive properties of the soil–vegetation system. Additionally, the analytical and probabilistic nature of Eagleson's theory significantly reduces the system parameterization, relative to most numerical simulations, and thus eases both sensitivity analyses and the generation and testing of hypotheses.

However, several of the below-ground model parameters are still difficult to estimate. Eagleson (1982) eliminated the need to measure them directly by imposing three new constraints on the model, termed the ‘ecological optimality hypotheses’, which represent a hierarchy of ecological processes that affect water-balance dynamics on time-scales from years (plant growth and demography) to millennia (soil evolution). While these constraints simply ease parameter estimation for hydrologists (Eagleson 1982; Chavez *et al*. 1994), they allow ecologists to use model output to make explicit predictions concerning the expected state of vegetation in an undisturbed, water-limited system. The three hypotheses are not assumptions that underlie model calculations. Instead, they are *post hoc* constraints that limit the allowable parameter space of the model. However, because of their importance for any ecological application of the model, we examine each hypothesis in turn.

#### CANOPY STRESS MINIMIZATION

According to the first hypothesis, over short time-scales (a few plant generations) the vegetation canopy density (*M*) will equilibrate with the climate and soil parameters to minimize the water stress of the component plants, which Eagleson (1982) equated with a maximization of the equilibrium soil moisture, *s*_{0}. Within the model this constraint corresponds to a minimization of evapotranspiration, *ET*_{A}, with respect to canopy density, *M*, to yield the optimal canopy density, *M*_{0} (Eagleson 1982).

The assumption of an equilibrium between a vegetation canopy and water supply is ecologically reasonable, e.g. the hydrological equilibrium of leaf area index (Grier & Running 1977; Larcher 1995). However, it is unlikely that this condition corresponds to a minimization of evapotranspiration, as hypothesized by Eagleson. This assumption is difficult to justify in an ecological and evolutionary context. The major components of plant fitness –development, survival, and fecundity (Crawley 1997) – all require photosynthesis and thus transpiration. Maximizing equilibrium soil moisture in a water-limited system also leaves the limiting resource available to competitors (Tilman 1982), while minimizing evapotranspiration effectively minimizes photosynthetic productivity. Because soil moisture is both a limiting resource and a buffer against stress, it seems most realistic to consider the situation as a trade-off, with the competitive benefits of plant transpiration (e.g. growth and reproduction) balanced against the mortality costs of stress (Tilman 1988; Ehleringer 1993; Tyree *et al*. 1994; Richards *et al*. 1997; Schwinning & Ehleringer 2001).

The first optimality hypothesis is often interpreted as a growth–stress trade-off (MacKay 2001), and Eagleson acknowledges this point of view in later work (Eagleson 1994), but the trade-off is not reflected in any published form of the ecological optimality hypotheses, including Hatton *et al*.'s (1997) review. Part of the problem is the need to incorporate a quantitative measure of stress into the model. In this regard, some recent ecohydrological research utilizing an approach similar to Eagleson’s, but explicitly incorporating water stress, seems quite promising (Rodriguez-Iturbe *et al*. 1999).

#### SUCCESSIONAL STRESS MINIMIZATION

The second optimality hypothesis predicts that over successional time (many plant generations) species turnover driven by repeated drought will generate an optimal equilibrium community composition whose transpiration efficiency (*k*_{V0}) maximizes the equilibrium soil moisture, *s*_{0}, again under the assumption that soil moisture acts primarily as a buffer against drought stress. Mathematically, as above, this is accomplished by minimizing the total evapotranspiration, *ET*_{A}, this time with respect to the vegetation transpiration coefficient *k** _{V}* (Eagleson 1982).

The hypothesis that a plant community that utilized less of the limiting resource could replace a community that used more directly contradicts most theoretical and empirical work on the successional dynamics of plant communities (Bazzaz 1979; Tilman 1988). Instead, most ecologists assume that productivity converges on a rate that balances the supply rate of the limiting resource (Tilman 1988; Enquist & Niklas 2001). Because soil moisture is the limiting resource here, this assumption implies that the system would evolve to minimize the equilibrium soil moisture, the opposite of Eagleson's hypothesis. As in the case of the first optimality hypothesis described above, the community dynamic may be better considered a trade-off between production and drought stress (Breshears & Barnes 1999; Rodriguez-Iturbe *et al*. 1999; Wainwright *et al*. 1999).

However, there are also technical problems with the application of the second hypothesis in the equilibrium water-balance model which render moot further discussion of its conceptual basis. As pointed out by Salvucci (1992), for a given canopy density, *M*, the equilibrium soil moisture increases monotonically as the transpiration coefficient, *k*_{V}, decreases. Thus the maximum equilibrium soil moisture corresponds to a vanishingly small *k** _{V}*, i.e. no transpiration, no photosynthesis. Together with the ecological concerns, this technical problem suggests that the second optimality hypothesis should be discarded. This conclusion echoes that of Salvucci (1992), although it was not mentioned in Hatton

*et al*.'s (1997) review.

#### MAXIMUM PRODUCTIVITY SOIL

The third optimality hypothesis addresses the coevolution of vegetation and soils over quasi-geological time-scales. The hypothesis predicts that vegetation will alter soil properties (saturated hydraulic conductivity, *K*(1), and pore disconnectedness index, *c*, specifically) to maximize the optimal canopy density, *M*_{0}, derived from the first optimality hypothesis. The rationale for this hypothesis is that the maximum optimal canopy density maximizes productivity, given the minimum stress constraint of the first optimality hypothesis (Eagleson 1982).

Hatton *et al*. (1997) refer to the maximum optimal canopy density as the ‘climatic climax density’, thus drawing parallels with the Clementsian tradition in ecology. However, under Clements's theory the climatic climax is approached over successional time, not over geological time (Clements 1936). Such a long approach to the climatic climax assumes the stability of both vegetation composition (the transpiration coefficient *k _{V}*

_{0}does not change) and climatic conditions over geological time, which is not in accord with the paleorecord (Delcourt & Delcourt 1984; Webb & Bartlein 1992; Davis & Shaw 2001). The third optimality hypothesis is linked to successional changes through its dependence on the first two optimality hypotheses; the constant transpiration coefficient,

*k*

_{V}_{0}, is derived from the second optimality hypothesis, and the optimal canopy density,

*M*

_{0}, is derived from the first. Given the questionable basis of the first optimality hypothesis and the biophysical impossibility of the second, as well as a paucity of data capable of evaluating the third hypothesis, it is difficult to evaluate its validity or ecological relevance.

Thus the ecological optimality hypotheses seem untenable from an ecological perspective. However, we emphasize that while the optimality hypotheses might be invalid, Eagleson's model and methods may still provide valuable insights because the model calculations themselves do not depend on the optimality hypotheses. In that spirit, we next explore the equilibrium water-balance model in some detail. We begin with a review of the model. Next, we validate our application of the model using some of Eagleson's published test data. Finally, we explore the model in new contexts using data from a water-limited site in the Los Piños Mountains, New Mexico, USA. Our goal is to understand the range of behaviour exhibited by the model and its sensitivity to the range of conditions possible at our site. The more general rationale of this work, however, is to explore Eagleson's approach as a tool for understanding water-balance limitations on vegetation structure and function. Despite flaws in the ecological optimality hypotheses, Eagleson's approach might provide insights for developing a more general predictive equilibrium theory of atmosphere–soil–vegetation dynamics.