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[1] In this paper we address the following question: how can efficient water use strategies evolve and persist when natural selection favors aggressive but inefficient individual water use? A tragedy of the commons, in which the competitive evolutionary outcome is lower than the ecosystem optimum (in this case defined as maximum productivity), arises because of (1) a trade-off between resource uptake rate and resource use efficiency and (2) the open access character of soil water as a resource. Competitive superiority is determined by the lowest value of the steady state soil moisture, which can be minimized by increasing water uptake or by increasing drought tolerance. When the competing types all have the same drought tolerance, the most aggressive water users exclude efficient ones, even though they produce a lower biomass when in monoculture. However, plants with low water uptake can exclude aggressive ones if they have enough drought tolerance to produce a lower steady state soil moisture. In that case the competitive superior is also the best monoculture, and there is no tragedy of the commons. Spatial segregation in soil moisture dynamics favors the persistence of conservative water use strategies and the evolution of lower maximum transpiration rates. Increasing genetic relatedness between competing plants favors the evolution of conservative water use strategies. Some combinations of soil moisture spatial segregation and intensity of kin selection may favor the evolution and maintenance of multiple types of plant water use. This occurs because a cyclical pattern of species replacement can arise where no single type can exclude all other types.

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[2] A common feature of many previous theoretical approaches to plant water use is that they are based in the idea of physiological optimization by plants [e.g., Eagleson and Segarra, 1985; Eagleson, 1994; Cowan and Farquhar, 1977; Cowan, 1982, 1986]. However, the behavior of plants in competition may significantly deviate from optimality expectations because of the effects of frequency dependence [Cohen, 1970]. More precisely, functional types with aggressive resource use may displace types with efficient (optimal) resource use. This situation has been compared to the “tragedy of the commons” problem in resource economics [Hardin, 1968], because the community's evolutionary steady state has a lower ecosystem performance (e.g., productivity), rather than an alternative state in which all plants adopt an efficient resource use strategy [Gersani et al., 2001]. Conceptually this problem is similar to the classical problem of the evolution of cooperation [Hamilton, 1972], and as such we expect that genetic relatedness should be an important factor in the evolution of plant water use strategy. Likewise, models have shown that spatial structure has the potential to foster the evolution and persistence of cooperative behavior [Nowak and May, 1992], which in this context is analogous to an efficient water use.

[3] We explore these issues in this paper by addressing the following questions: (1) how does coexistence of contrasting water use strategies depend on the strength of spatial coupling in soil hydrology? (2) What is the effect of this spatial coupling on the evolution of water use characteristics? (3) What is the effect of genetic relatedness between competing plants on the evolution of water use characteristics? Throughout the paper we consider spatial structure only in resource competition, and assume global dispersal for simplicity. Our analysis is restricted to water controlled ecosystems where competition for other resources is either much less restrictive or is controlled by the soil water content.

2. A Simple Model of Plant Water Use

[4] We begin with a simple model for the daily dynamics of plant growth and soil moisture. The model is purposely rather simple and attempts only to capture, at the daily level, the main qualitative features that generate a “tragedy of the commons.” The model's state variables are plant biomass per unit area (B) and relative soil moisture (s), which is the volume of water in the soil divided by the volume of pore space in the soil. Daily transpiration ED(s) per unit leaf area is given as a function of s by:

where [s − s_{w}]_{+} = s − s_{w} if s > s_{w} and [s − s_{w}]_{+} = 0 otherwise. E_{m} is the maximum daily transpiration per unit leaf area, and k_{e} is the value of s − s_{w} at which ED = 0.5E_{m}. At s ≤ s_{w} the stomata are fully closed during the entire day. For the purposes of this paper we neglect cuticular transpiration and direct evaporation from the soil, and assume that ED = 0 when s < s_{w}. Daily assimilation per unit leaf area is given by

where A_{m} is the maximum daily net assimilation per unit leaf area, and ka is the value of s − s_{w} at which A_{D} = 0.5A_{m}. Change in both A_{D}(s) and E_{D}(s) follows from the stomata closing in response to decreasing values of s, so the responses of A_{D} and E_{D} with respect to s are very similar. Because of this we take k_{a} = k_{e} and refer to it as the “half saturation” soil moisture k. This assumption is supported by more detailed physiological models [e.g., Cowan and Farquhar, 1977].

[5] Our analysis, although mathematically carried out with equations in continuous time, needs to be interpreted at the daily timescale. Detailed simulations at the hourly level incorporating stomata control and transpiration dynamics were carried out by Daly et al. [2004a, 2004b] coupling the equations of the soil-plant-atmosphere continuum to a parsimonious model of the atmospheric boundary layer. The results were integrated at the daily timescale and were observed to closely match those obtained with the simplified daily level approach described by equations (1) and (2). Moreover, this approach is also justified by empirical studies carried out at the daily timescale [Federer, 1979; Schulze, 1986].

[6] With these definitions we can write the rates of change in plant biomass B and relative soil moisture s as:

where ρ is the specific leaf area, f_{L} is the leaf mass ratio, Y_{g} is the “growth yield” (the fraction of assimilation that remains after paying growth respiration costs), R_{r} is the root respiration coefficient, and q is the senescence rate (rate per unit biomass at which leaves and fine roots are shed). I is a constant rainfall input into the soil, Ks is the saturated hydraulic conductance, and c is an empirical constant characteristic of the soil type [Rodriguez-Iturbe and Porporato, 2004]. The term K_{s}s^{c} represents leakage losses from the active soil depth into deep layers. The parameters I, E_{m}, and K_{s} are normalized by the effective soil depth nZ_{r}, where n is the soil porosity and Z_{r} is the root depth. The above assumes steady state water flow, so that water lost at the leaf surface is immediately removed from the soil (i.e., there is no water storage in the system).

[7] We are fully aware of the importance of the stochastic character of rainfall in the dynamics of carbon assimilation by plants and in the competition for resources between them [e.g., Rodriguez-Iturbe and Porporato, 2004]. In fact we actively studying the impact of the stochastic rainfall fluctuations in the system described by equations (5) and (6) which then become stochastic in character. Nevertheless, as a baseline configuration against which to evaluate the importance of the many aspects of the stochastic input, it is valuable to first explore in detail the results when rainfall is maintained constant and equal to the mean value over a the region. A following publication will compare this case with that of a stochastic rainfall input.

[8] By grouping parameters together in the above system, we express it more compactly as

where γ = ρf_{L}E_{m}, α = A_{m}αρf_{L}Y_{g} and β = R_{r}(1 − f_{L}) + q. γ is the maximum transpiration rate per unit biomass, α is the maximum assimilation rate per unit biomass, and β is the loss rate per unit biomass resulting from root respiration and senescence. α-β is the maximum instantaneous growth rate. The system has two steady states, given bySteady state 1

Steady state 2

For the plant to grow (i.e., for steady state 2 to be stable) the following conditions must be satisfied:

Note that biomass in steady state 2 is proportional to the ratio of α and γ, which can be thought of as an intrinsic water use efficiency. Because assimilation and transpiration are both controlled by stomatal conductance, A_{m} and E_{m} are related by common physiological constraints. Increased stomatal opening increases the rate of CO_{2} diffusion into the leaf, but it also increases transpiration rate. Since assimilation saturates with intraleaf CO_{2} concentration while transpiration increases linearly with stomatal conductance, the relationship between assimilation and transpiration is one of diminishing returns [Cowan and Farquhar, 1977]. Accordingly, we use a saturating equation to model the relationship between A_{m} and E_{m}:

where A_{max} is the maximum A_{m}, and H is a half saturation constant. Equation (12) represents the joint evolutionary constraints on A_{m} and E_{m} which arise from the biophysical and physiological processes controlling leaf gas exchange. Individual strategies will correspond to a single point in the curve described by equation (12), with their (A_{m}, E_{m}) combination determining their intrinsic water use efficiency.

[9] With this assumption the intrinsic water use efficiency (ω) becomes

Note that ω decreases with increasing E_{m}. On the other hand, decreases with E_{m} (through A_{m}), and thus the subtracted term K_{s}()^{C} in equation (10) decreases. All else being equal, the maximum biomass is achieved by a strategy with an intermediate value of E_{m} (Figure 1). Decreasing the value of H increases the water use efficiency and moves the optimum E_{m} to lower values.

[10] Although the relationship between biomass and reproductive success is a very complex one and depends on a large number of factors, it is commonly accepted that a plant that assimilates more carbon has more resources to be reproductively successful than a plant with smaller assimilation. Thus in the context of this simple model, we use biomass as a measure of fitness. Conservative strategies (i.e., low E_{m}) achieve higher steady state biomass, and thus could potentially have a higher fitness than more aggressive water users. However, invasibility analysis shows that conservative strategies can always be invaded by a strategy with a higher E_{m}, but cannot invade an established aggressive strategy. For this analysis we compute the relative growth rate (RGR) of an invading strategy given the soil moisture conditions established by a resident strategy at its steady state biomass (the model can be interpreted as describing growth of a single plant or growth of a monoculture of plants with exactly the same strategy. In this case is the soil moisture established by the resident population at its carrying capacity). RGR is given by:

The condition for successful invasion is RGRI (_{R}) > 0, where RGRI is the RGR of the invader and _{R} is the steady state soil moisture established by the resident [Tilman, 1982]. We assume that invader has no effect on soil moisture because it occurs at a negligible density compared to the resident strategy. The results of the analysis are shown in Figure 2. Parameters for the analysis are given in Table 1. The main result is that a plant population can always be invaded by a mutant with a more aggressive strategy than the resident.

Table 1. Parameter Values Used for Invasibility Analysis

Parameter

Value

ρ, m^{2} g^{−1}

0.005

Y_{g}

0.85

q, d^{−1}

1/100

f_{l}

0.5

A_{max}, g m^{−2} d^{−1}

44

H, mm d^{−1}

3

K

0.2

R_{r}, d^{−1}

0.01

s_{w}

0.1

[11] The competitive winner is the strategy that has a lower value of steady state soil moisture , in agreement with the R* principle [Tilman, 1982]. From equation (9) we see that there are two ways in which a plant can become competitively successful: increasing maximum transpiration, and thus becoming the aggressive winner in a tragedy of the commons scenario, or decreasing its wilting point and avoiding the tragedy of the commons all together.

[12] An evolutionary escalation of water uptake, as depicted in Figure 2 is one way in which strategies may evolve to be more competitive. An alternative is to increase their drought tolerance by lowering their wilting point s_{w}. Figure 3 shows the effect of drought tolerance on competition between aggressive and efficient strategies. The left column of graphs shows the case where both strategies have equal drought tolerance (s_{w}). In this case, the aggressive water user is superior at all levels of soil moisture and thus excludes the efficient strategy. However, it is an inferior monoculture. The right column of graphs shows the outcome when there is a slight advantage in drought tolerance for the “conservative,” efficient strategy. In this case the efficient strategy has higher assimilation at soil moisture s < 0.21. Because it is superior at lower resource levels it excludes the aggressive strategy, as we expect from the R* principle. In this example the best monoculture is also the competitive superior, and in that sense the “tragedy of the commons” has been avoided. However, the winning strategy is still the one that is most aggressive in water uptake within its drought resistance class.

3. A Simple Spatial Model

[13] The model is given by the following four differential equations:

where the B_{1} and B_{2} are biomass of each plant, s_{1} and s_{2} are the respective relative soil moistures, α_{1} and α_{2} are the maximum assimilation rates per unit biomass, γ_{1} and γ_{2} are the maximum transpiration rates per unit biomass, β_{1} and β_{2} are the respiration costs per unit biomass, k_{1} and k_{2} are the half saturations of the assimilation and transpiration responses to soil moisture, and s_{w1} and s_{w2} are the wilting points. The α, γ, and β parameters as well as I, K_{s}, and c are defined as in equations (5) and (6). μ is the mixing rate, a measure of the strength of coupling between the two soil moisture compartments. Spatial structure in this model is represented only implicitly through the mixing rate between neighboring sites and it is equated to the heterogeneity in soil moisture within the coupled pair.

[14] We realize that the spatial linking involved in equations (17) and (18) is an extremely simplistic representation of a very complex reality. It is important to clarify that we are not assuming the coupling between the sites is due solely to underground lateral flow between. In fact the linkage is also affected by the presence of the vegetation and it seems reasonable to assume that in the absence of topographical controls, if moisture transfer takes place, it will happen as a function of the relative values of s_{1} and s_{2}. Much more complex relationships could be used for spatial interaction rather than the simple linear one used in equations (17) and (18), but this is an important one to use as a baseline for comparison with others that are being presently explored and will be reported in a future paper.

4. Coexistence and the Evolutionary Dynamics of Pair Competition

[15] In the context of this model coexistence can be considered at two levels: a local level consisting of a specific pair of sites coupled by horizontal soil water flux, and a regional level consisting of a population of such pairs. Local coexistence occurs when the steady state biomass of both species in a specific pair is positive. Figure 4 shows the effect of spatial coupling on the local dynamics of a species pair. In this example an aggressive species shares the site pair with a conservative species. They differ only in their maximum transpiration rate E_{m}, with all other parameters the same for both. When μ = 0.0 the sites in a pair are independent and there is trivial local coexistence. As μ increases beyond a threshold the aggressive species excludes the conservative species. Thus a minimum level of soil spatial segregation in hydrological dynamics is necessary for the conservative species to persist in a constant environment.

[16] A pair of sites with local coexistence will eventually be disturbed by some external force (e.g., fire, wind throw, etc.). Long-term coexistence at the regional level will depend on the reproductive success of the competing strategies, since that determines their capacity to recolonize disturbed sites. We assume steady state biomass is a proxy for reproductive success, and use it as the fitness measure. In this case fitness is frequency-dependent, since the steady state biomass achieved by a plant in one site depends on the type of plant that occurs in the paired site.

[17] Let p be the frequency of strategy x and 1 − p the frequency of strategy y over the set of all patches. Let E(x, y) refer to the steady state biomass achieved by strategy x when paired with strategy y according to the above model. Define average fitness F as:

F is the average steady state biomass of strategy x over the set of all patches, given that the competitor is strategy y and that the frequencies of x and y are given by p and (1 − p). For compactness in what follows we write only F(x) to refer to the fitness of strategy x. The competitive dynamics of x and y over the landscape is given by the change of p over time:

To study strategy coexistence and trait evolution we use invasibility analysis.

[18] We ask the question: can a strategy y invade an established population of strategy x? In this case p is close to 1 and 1 − p close to 0. Strategy x cannot be invaded if it has higher fitness than the invader, or:

Rearranging and using the fact that (1 − p) is very close to zero, gives the familiar condition defining an evolutionary stable strategy (ESS) [Maynard Smith, 1982]:

for all strategies y. If E(x, x) = E(y, x), an ESS must satisfy E(x, y) > E(y, y). For all practical purposes in our model condition (22) is enough to establish an ESS. We construct a “payoff” matrix E(x, y) for the plausible range of maximum transpiration rates E_{m} (1–10 mm d^{−1}). From this payoff matrix we identify ESS strategies using criterion 22. This is depicted in Figure 5, where the ESS is identified by the value of E_{m} at which the “best reply” curve (which gives the strategy in site 2 that yields the maximum steady state biomass given the strategy in site 1) intersects the 1:1 line. The ESS maximum transpiration rate increases with the mixing parameter μ, meaning that more spatial mixing in soil moisture favors an aggressive water use strategy ESS (Figure 6).

5. Effect of Genetic Relatedness on Water Use Strategy Evolution

[19] Many plants disperse their seeds near the parent plant, or in groups of seeds that germinate in close vicinity, so that neighboring plants are likely to be genetically related. Moreover, for clonally reproducing plants ramets may become physiologically independent constituting a clustered population of genetically identical individuals. This may have an effect on the evolution of water use strategies: a conservative, more “cooperative” strategy of water use may be selected if it increases the inclusive fitness of a genotype [Hamilton, 1972].

[20] We assume that of all the interactions an individual plant can establish, a fixed fraction η occurs with its genetic clones (as may be the case if for example a fixed percentage is produced by clonal rhizomes versus sexual reproduction), while the remaining fraction 1 − η occur with a random sample of the population. Then the fitness of strategy x becomes

Here fitness is an average over the possible relatedness classes [Grafen, 1979]. The individuals in the 1 − η fraction are not kin, but may still have the same strategy according to the relative abundances of the strategies in the landscape p and p − 1. The ESS condition in this case is

The idea behind this model is that as the probability of interacting with kin (given by η) increases, the plant should tend to adopt a cooperative water use strategy. If we define W(x, y) as the right hand side of inequality (24) then we see that W(x,x) = E(x, x). Thus the ESS criterion can be expressed compactly as W(x, x) > W(y, x) for all strategies y different from x. For each combination of the parameters μ and η, we generated a payoff matrix W(i, j) with i and j ranging from 1 to 10 mm d^{−1} of maximum transpiration E_{m}, and used the above criteria to identify ESS strategies. The result is very similar to the earlier simpler analysis, with one interesting difference: for certain parameter combinations there is no ESS (Figure 7).

[21] To clarify this point, Figure 8 compares the pairwise invasibility plots of a combination of μ and η that has an ESS (μ = 0.5, η = 0.5) and one that doesn't (μ = 0.9, η = 0.5). The nonexistence of an ESS can be verified in the plot by noticing that all the strategies are invadable by at least one other strategy (all rows in the plot have at least one filled marker, indicating the strategies that can invade it) (Figure 8b). In these cases the competitive and evolutionary outcomes have multiple strategies in coexistence.

[22] The example in Figure 8b has an interesting structure of species replacement. Strategies with increasing E_{m} replace each other, until the strategy with E_{m} = 5.5 mm d^{−1} becomes established. This strategy cannot be invaded by strategies in the close vicinity of its trait value (e.g., E_{m} = 7 or E_{m} = 5 mm d^{−1}), but it can be invaded by a much more conservative strategy (E_{m} = 1 mm d^{−1}). It is this fact that keeps strategies in the range 1–5 mm d^{−1} in the system, in a form of cyclical succession (or evolution). This is more clearly seen when we actually simulate the changes in strategy frequency over time. To do this we write the multiple species version of equations (23) and (20) as follows:

where N is the total number of species. Equation (25) specifies how the fitness of each type depends on the frequencies of all other types in the landscape, while equation (26) specifies how the frequency of the different types changes over time as a function of their fitness. Using the above equations we generate the dynamics of a community that starts with 10 strategies of maximum transpiration rate ranging from E_{m} = 1 to E_{m} = 10 mm d^{−1}, all initially with the same abundance. Figure 9 shows the dynamics for a case where there exists an ESS. Figure 10 shows a case where there is no ESS, and species are allowed to go extinct (after decreasing below an arbitrarily small value).

[23]Figure 11 has no ESS, but species abundances are maintained at a minimum, very small value (as would be the case with a constant immigration rate). Thus when p(t) becomes smaller than 0.0001 it is arbitrarily fixed at this value, so that all strategies are present at all times. When species are allowed to go extinct, the community exhibits one turn of the species cycle but then settles into the complete dominance of the most aggressive type. The conservative types that could invade it and restart the cycle have become extinct. This would be the case when a species has become regionally extinct. If all the species remain in the regional pool, then long distance dispersal events will maintain a small but significant supply of all strategies, allowing the most conservative types to reinvade the most aggressive types, restarting the cycle. An analogous dynamics was found by Maynard Smith and Brown [1986] in their study of body size and competition, where cyclical appearance of competitors occurred instead of a stable species mixture. It is interesting that this diversity-enhancing effect of kin selection does not occur at low values of soil moisture spatial coupling, and that the degree of genetic relatedness needed for multiple coexisting types becomes smaller as the soil moisture spatial coupling increases (Figure 7).

6. Discussion

[24] The model developed here serves to emphasize the two fundamental features that cause the “tragedy of the commons” effect. First, there is a trade-off between the rate of resource use (via E_{m}) and the efficiency of resource use (ω). The second feature is the open access character of the resource: all plants draw water from the same pool. Because of the trade-off, efficient strategies maintain soil water availability favorable for growth during a larger period of time, making them achieve higher monoculture steady state biomass. However, because of the open access character of the resource, aggressive water users can impose the consequences of their strategy on conservative competitors, eliminating any benefit of efficiency and excluding them by virtue of their higher resource uptake rate.

[25] Evolving drought tolerance can be a successful alternative to evolving higher maximum transpiration rates. However, drought tolerance requires adaptations that are costly to the plant and limit its capacity to have a high growth rate. Drought-adapted plants tend to have leaves with higher bulk elastic modulus by virtue of increased structural tissues, which requires a larger carbon investment per unit leaf area [Niinemets, 2001]. At the stem level, higher wood density conveys greater cavitation resistance at the cost of reduced growth rate and decreased stem storage capacity [Hacke et al., 2001]. Moreover, it is likely that these leaf and stem level adaptations need to occur consistently. The wilting point in leaves is often more negative than the pressures that induce cavitation in the root-stem xylem system [Sperry, 2000]. Thus a more resistant leaf would only be advantageous if the plant has the hydraulic investment to support it. These considerations suggest that drought tolerance is inevitably tied to slow growth and conservative water use.

[26] A recent experiment by Gersani et al. [2001] demonstrated a strong “tragedy of the commons” effect in soybean (indeed, it was this paper that introduced the term “tragedy of the commons” in analogy with Hardin's classic paper). Individual soybean plants had higher seed output when growing alone than when growing with a neighboring soybean, even though the amount of soil space per individual plant was kept the same in both experiments.

[27] This occurred because the presence of the competitor induced the soybean plants to allocate more resources to the roots, thereby reducing the final seed yield. In other words, soybeans became more aggressive in their uptake of soil resources in response to competition, with the consequence of reduced efficiency of resource use. Because soybean is a crop plant that has most likely been selected for stand yield, rather than individual fitness, it is surprising that it exhibits a strong tragedy of the commons effect. How important this effect is (or has been) in natural ecosystems is an open question worthy of further study. It is possible that some natural communities experienced the “tragedy of the commons” in the past, favoring the evolution of mechanisms and traits to avoid this effect.

[28] We have argued by means of a simple model that increasing the spatial mixing of soil water favors the establishment and evolution of aggressive water use strategies. Furthermore, our model suggests that genetic relatedness between competitors should favor the persistence and evolution of more conservative water use strategies. Is there any empirical evidence supporting these ideas? A study of desert shrubs in the Mojave desert revealed that Larrea tridentata (creosote bush) produces root allelochemicals that inhibit the root elongation of neighboring plants, both con specifics and of other species [Mahall and Callaway, 1992]. By virtue of this mechanism, Larrea individuals are able to have near exclusive access to the soil resources directly beneath their root system. In addition, Larrea propagates mostly by clonal reproduction, through rhizomes that produce a new stem at some distance from the parent stem, and eventually sever the connection. This mode of propagation results in regularly spaced pattern of physiologically independent clones.

[29] In the context of our model, both these features would be represented by a low mixing rate μ between neighboring sites, which the model showed was one of the conditions that would favor a conservative strategy. The fact that neighbors of Larrea are likely to be clones also suggests the possibility of kin selection. Thus our model would predict that Larrea should have a conservative strategy of water use. Studies of the water and carbon relations of Larrea do indicate that it is a species that has a strategy of very slow growth and remarkable drought resistance. The lifespan of a Larrea shrub can be longer than 100 years despite their very low stature, indicating extremely slow growth [Mahall and Callaway, 1992].

[30]Larrea can maintain positive assimilation in soils where the predawn water potential measures −10 MPa [Odening et al., 1974]. The cost of such a drought-resistant morphology is often a reduced growth rate resulting from increased construction cost of leaf and xylem tissues. The ensuing slow development of leaf area per unit ground area should result in low maximum rates of transpiration and a conservative water use. Furthermore, a study of stomatal conductance in Larrea shrubs in New Mexico [Ogle and Reynolds, 2002] found that conductance is confined to the range 0.0–0.1 mol m^{−2}s^{−1}, indicating a low maximum transpiration rate. Thus at least for this species it seems plausible that kin selection and spatial structure in water uptake have facilitated the evolution of an efficient strategy of water use.

Acknowledgments

[31] We are grateful to Steve Pacala and Henry Horn for their comments and suggestions throughout this research. We also thank Gabriel Katul and two other anonymous referees for their useful and thorough reviews which contributed to the improvement of the paper. E.Z. acknowledges the financial support of a grant from the Andrew W. Mellon Foundation to Simon Levin. I.R.-I. acknowledges the support of NSF through the Biocomplexity grant DEB0083566 and the National Center for Earth Surface Dynamics grant EAR-0120914.