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1A growing body of literature suggests that some plants may engage in a game of Tragedy of the Commons when competing for soil resources. Two annual plants sharing a whole space will produce more roots per individual and less reproductive yield per individual than one plant with half the space to itself.
2Several papers have recently suggested that the increase in soil volume in going from a single plant to two competitor plants (as has been the case in empirical tests of the above prediction) may produce the increased root proliferation independent of the competitor.
3We extend the Tragedy of the Commons model of Gersani et al. (2001) to explicitly consider the separate effects of volume per plant (V) and nutrient concentration per unit volume (N) on root proliferation and yield of plants that ‘own’ their space relative to plants that share their space with a competitor. We then use simulations resulting from the model to evaluate existing data and the new perspectives provided by recent challenges to the conceptual basis for the Tragedy of the Commons.
4When N is held constant, increasing V should produce an almost linear increase in root proliferation and net nutrient profit for reproduction. The increase in roots should be more pronounced for two plants sharing their space, and the increase in yield should be more pronounced for plants with exclusive access to half the space. When V ¥ N is held constant, roots per plant should at first increase and then decline with increased volume. The difference between a plant that ‘owns’ its space vs. one that shares its space with a competitor should be less pronounced as V increases.
5Synthesis. We use this model to help clarify some confusion surrounding the Tragedy of the Commons theory of below-ground competition by showing how issues raised by others are not incompatible with the approach used to generate the Tragedy of the Commons theory and highlight future opportunities for research on root growth strategies.
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Root competition between plants might be a game of nutrient foraging. We use ‘game’ in the game theoretic sense of describing interactions among individuals where, in this case, the root proliferation strategy of an individual influences its own nutrient uptake as well as the nutrient uptake of neighbouring plants. When root competition is modelled as a game, a Tragedy of the Commons emerges (Hardin 1968; Gersani et al. 2001). It is perhaps easiest to think of this result as a multiplayer prisoner's dilemma (Flood 1952; Dresher 1961). Two annual plants sharing a common space should grow more roots per individual and produce less seed yield per plant than two plants separated from each other by a divider or physical barrier. These predictions emerge from the costs and benefits of producing more roots. All else being equal, a plant should benefit most from growing roots into virgin space compared to space already occupied by some roots. And, a plant benefits more from proliferating roots into space occupied by a neighbour's roots (increasing inter-plant competition) than from proliferating roots into space occupied by its own roots (increasing intra-plant competition).
Nutrient uptake by a unit of roots has two components. There are nutrients that would have gone unexploited in the absence of this unit of roots and also ‘stolen’ nutrients that otherwise would have been harvested by other roots. A plant stops root growth to reduce the latter when alone, but enjoys both components when growing roots into space occupied by a neighbour. It does not pay to steal from oneself, while stealing nutrients from a neighbour is a different matter altogether.
However, growing roots to steal nutrients from neighbours results in a public cost. At the evolutionarily stable strategy (ESS) each plant invests in roots to ‘steal’ and this depresses collective yield. All plants would experience an increase in seed yield if all were to collectively produce fewer roots than this ESS level. However, a plant that unilaterally produces less will do much worse than those maintaining their ESS level of root proliferation. And a plant that produces more roots than a restrained group of plants at a group optimum will do much better (this will be shown in Fig. 2). The ESS is the adaptive root strategy even though it does not maximize collective seed yield. The predictions are similar to the model of Schieving & Poorter (1999), who proposed that there is the potential for an above-ground Tragedy of the Commons resulting in increased specific leaf areas (SLA).
The alternative hypothesis suggested by Hess & de Kroon (2007) and Semchenko et al. (2007) is straightforward. Two plants sharing space experience more total available volume than two plants confined by a divider to half the original space (part of the original experimental designs to test the Tragedy of the Commons). According to Hess & de Kroon (2007) the sharers may produce more roots per individual as an unavoidable consequence of trying to ‘find’ or spread to the boundaries of the available soil volume. Semchenko et al. (2007), in a related idea, suggests that competing plants may forage in a somewhat spatially explicit fashion. A heterogeneous environment may result as neighbouring plants draw down resources. Both ideas imply that plants become less efficient at foraging in the presence of neighbours, but the question that remains to be resolved is why.
Rather than being an adaptive tactic on the part of the plant playing nutrient foraging games, the appearance of a Tragedy of the Commons may happen as an invariable response to space itself. Is it space or inter-plant nutrient competition that explains root proliferation?
Here we extend the original model of Gersani et al. (2001) to explicitly include volume and nutrient availability to address the issues raised since the model's original publication. In the discussion, we further clarify some issues surrounding the Tragedy of the Commons theory of below-ground competition. Specifically, we (i) revisit the model in its simplest form and extend it to examine the role of volume; (ii) evaluate existing data and the new perspectives provided by recent challenges to the conceptual basis for the Tragedy of the Commons; (iii) discuss the importance of a priori hypothesis testing vs. the role of ad hoc hypotheses such as those advanced by Hess & de Kroon (2007); and (iv) highlight future opportunities for research on root growth strategies.
The Tragedy of the Commons model of below ground competition
roots and the single plant
We extend the model of Gersani et al. (2001) to explicitly consider the effects of nutrients and space on root proliferation of a single plant with its ‘own’ space vs. two plants sharing twice the space. We consider an annual plant where fitness is maximized by maximizing the amount of nutrients devoted to seed production (= yield) by the end of growing season.
The important assumptions are: (i) total nutrient uptake is an increasing yet decelerating function of total root production by all individuals within the space (one individual if ‘owned’ space, two individuals if ‘shared’ space); (ii) a plant's share of this total nutrient harvest is in proportion to its contribution to the total roots present; and (iii) doubling the space (which doubles nutrient availability if concentration is held constant) doubles the number of roots required to harvest the same proportion of available nutrients. The first two assumptions are from the Gersani et al. (2001) model, while the third is only implicit in the original model.
We first consider a single plant under the simplifying assumption that nutrient uptake is a random encounter rate phenomenon (space is implicit and ‘searching’ by roots is random). This model is spatially implicit as the location of plants within the soil volume is not specified (see O’Brien et al. 2007 for a spatially explicit version of the original model). Nutrient uptake can be written as:
where H, a function of r, is nutrient uptake over the growing season (units of moles of nutrients, for instance), V is the volume of the space (units of cm3), N is the nutrient concentration within the space (moles per cm3), a is the effective volume of soil encountered by a unit of root biomass (cm3 g−1), a/V represents the encounter probability of a unit of root biomass for any given molecule of nutrients (units of g−1) and r is the individual's root proliferation (units of dry mass, for instance). This model is one of a much more general family of nutrient uptake models that can fit the three assumptions.
Equation (1) gives total nutrient harvest as a function of root proliferation over the course of the growing season. The term VN represents the total amount of available nutrients (moles of nutrients). The encounter probability, a/V, is a familiar term from models of animal foraging. When applied to roots encountering molecules of nutrients it represents the probability that any given unit biomass of roots will encounter a given molecule of nutrients. Under the assumption of random encounter, this encounter probability will be a function of volume but not of nutrient concentration.
The amount of nutrients available for reproduction is given by:
where c represents the per unit cost of roots (units of moles of nutrients per gram of roots). For simplicity, we let per unit cost of roots be a constant (this assumption is not necessary, but it renders the predictions more transparent). The term G(r) can be thought of as the plant's net profit in units of soil nutrients. Hence, the cost term c not only includes the direct cost of producing roots, but also includes the cost of producing the appropriate amount of shoots and leaves necessary for supporting the plant and fixing the needed carbon.
To maximize G(r), the plant should proliferate roots until its marginal uptake, ∂H/∂r, equals the marginal cost of additional roots, c. This can be shown by setting ∂G/∂r = 0. This can then be solved for r to find the profit maximizing level of roots, r*:
Note how all four parameters of encounter, nutrient concentration, volume and the cost of roots influence optimal root proliferation. For instance, doubling nutrient concentration, N, while holding all else constant (including volume), will cause root proliferation to increase by ln(2) = 0.69.
Doubling volume, V, while holding nutrient concentration constant, doubles the optimal amount of roots (Fig. 1a). The plant's total harvest, H, and net harvest, G, also double in value (Fig. 1b).
Doubling volume while keeping the total amount of nutrients constant can cause an increase, a decrease or no change at all in r* (this is the scenario particularly relevant to Hess & de Kroon, 2007 who propose that r* must increase). In such scenarios the doubling of V is correlated with a halving of N and a halving of a/V. If the value of nutrients (e.g. high NUE, nutrient use efficiency) is substantially greater than the cost of producing roots (aN >> c) then an increase in volume while holding total nutrient availability, VN, constant results in an increase in r* (Fig. 1c)! This happens because the extra space requires the plant to search more extensively for the same but now more widely dispersed nutrients (regardless of distribution pattern; i.e. homogeneous or heterogeneous). However, this doubling of space with VN constant results in a smaller nutrient harvest and an even larger proportional decline in net harvest (Fig. 1d).
roots with two plants
Consider two plants sharing twice the volume as the single plant above. Total nutrient uptake is now determined by r = r1 + rr, and an individual's share of this harvest is given by ri/(r1 + rr):
where i refers to either plant i = 1 or 2. Note that the volume has been doubled by multiplying V by 2. This means that V now has units of volume per individual plant.
A plant's net nutrient profit is now:
where i = 1, 2. With two plants, a plant's net profit G now results from a game of nutrient foraging because the profit to one plant is, in part, determined by the presence of the other plant. A plant's nutrient uptake depends not only on its own root production but also on the root production of its neighbour. Moreover, a plant's fitness maximizing rooting strategy depends upon the amount of roots produced by its neighbour. The solution to this game is an ESS. Such a strategy does not necessarily maximize collective fitness (it usually does not), but it maximizes the fitness of an individual given that the other individual is also at its ESS.
At the ESS, each plant cannot increase its fitness by unilaterally altering its root production. The ESS level of root production for plant i must satisfy:
Solving this condition, the ESS root proliferation occurs when the average of a plant's marginal uptake rate and its average uptake rate equals the marginal cost of root production (Gersani et al. 2001). The ESS root production for each plant must satisfy:
This expression cannot be solved analytically. But, using implicit differentiation it follows that: (i) plants in pairs (two plants with 2V) will produce more roots per individual and less net nutrient profit than a single plant (one plant with V; Fig. 1); (ii) doubling nutrient concentration while holding volume constant will cause each plant to increase its roots more than ln(2) and less than twice; (iii) doubling volume while holding nutrients per unit volume fixed will double each plant's root proliferation; and (iv) doubling volume while holding total nutrients constant causes a proportionally smaller increase in root production than when just a single plant experiences a doubling of volume (Fig. 1a and c).
The first prediction results in a Tragedy of the Commons when two plants share 2V rather than each ‘owning’V. Why would each plant over-proliferate roots to the detriment of collective yield? The group optimum would be for each plant to produce the r* given by eqn (3). Figure 2a shows the relationship between net nutrient profit G1 and root production, r1, for plant 2 when plant 1 uses the group optimum of r* vs. when plant 1 uses the ESS of r″. In the first case, plant 2 can do much better than plant 1 by producing roots in excess of plant 1's r*. In the second case, plant 2 maximizes G1 by producing r″, the same level of roots as plant 1. Despite r″ being the ESS, G() > G() – hence the Tragedy of the Commons (Fig. 2b).
The second prediction indicates that the magnitude of the Tragedy of the Commons increases with nutrient concentration, N. With plants that ‘own’ their space, r* increases at a rate of ln(2) with N. With two plants sharing their space, r″ increases at greater than ln(2), and the proportional gap between G() and G() grows. With increasing nutrient concentration the two plants at their ESS squander a greater and greater fraction of their nutrients on proliferating roots.
The third prediction remains the same whether for a single plant with its own space or for two plants sharing their space. Doubling V while holding N constant will cause both r* and r″ to double. This likely explains the absence of the tragedy of the common response found by Murphy & Dudley (2007) when trying to replicate the results of Gersani et al. (2001). The original soil volume used by Gersani et al. (2001) is approximately 4–8 times that used in the experiments by Murphy & Dudley (2007) and it is possible that their data represent one point on the continuum represented in Fig. 1 whereas the results from Gersani et al. (2001) are likely at another point on those same graphs.
The fourth prediction actually has similarities to the second. Increasing V while holding VN constant causes the relative gap between r″ and r* to narrow – the magnitude of the Tragedy of the Commons declines. As V increases N must decline to keep VN fixed. And, decreasing N reduces the proportional gap between G() and G().
tests of the model and rooting volume
Tests of the Tragedy of the Commons model of root competition (Gersani et al. 2001) have involved comparing the root and/or seed biomass production of pairs of plants that either share a rooting volume or that ‘own’ half the space by virtue of a divider preventing below-ground interactions between the plants (Gersani et al. 2001; Maina et al. 2002; O’Brien et al. 2005). The logic can be seen in the model above: for a single plant, doubling volume while holding nutrient concentration constant will double optimal root proliferation. So, when two plants sharing a space produce more roots per plant than one plant with half the space the prediction has been upheld.
Hess & de Kroon (2007) suggest that other reasons may explain this outcome. A single plant when exposed to twice the space and a competitor plant may grow more roots to fill the available space. This may be true, but stating this does not refute the original model under test. However, it does indicate the need for additional tests to distinguish between two alternative hypotheses that make similar predictions regarding root proliferation. Previous studies on root proliferation and soil volume (Gurevitch et al. 1990; McConnaughay & Bazzaz 1991; Xu et al. 2001; Wijesinghe et al. 2005; and others mentioned below) have been highly instructive, but they do not, nor were they intended to, test the model.
The results of McConnaughay & Bazzaz (1991) provide a springboard for discussing alternative hypotheses for the role of volume for reproductive output. One alternative sees plants as unable to control their response to volume. Like ‘overeating’ the plant may over-reach in a manner that actually compromises reproduction. This maladaptive response to volume forms the basis for Hess & de Kroon's (2007) alternative explanation to the Tragedy of the Commons. A test of these alternatives is needed.
A more interesting alternative for the role of volume concerns whether volume itself (independent of volume multiplied by nutrient concentration) is a resource. Somewhat independent of the presence of a competitor, space by itself may provide the plant with greater access, flexibility and/or predictability to water uptake, trace nutrients, or other beneficial soil–plant interactions. However, the opposite prediction emerges. If space is a resource (surrogate or otherwise) in addition to the intended nutrients, then plants sharing a combined space should produce higher reproductive yield (per plant) than those separated by a barrier. Other than the McConnaughay & Bazzaz (1991) study, this appears to be a largely neglected but exciting area of research.
Imagining that space itself may be a surrogate resource independent of the nutrient solution, we controlled for nutrients and space in several ways (as noted by Hess & de Kroon 2007). First, O’Brien et al. (2005) included an experimental protocol that varied pot size (V vs. 0.5 V) while maintaining the same amount of nutrients (N fixed); and varied nutrient concentration (N vs. 0.5 N) while holding volume constant (V fixed). Root biomass was halved when either the nutrient availability or the rooting volume was halved. A comparison between these single plant treatments and parallel two-plant three-pot experiments (one pot exclusive to a plant and a third pot shared by both plants) showed that plants in the shared pot produced, as predicted by the ESS model, more root than they do when they experience the 50% reduction in nutrients or space in isolated individual plant treatments (O’Brien et al. 2005). To date, this is one of the few tests of the effects of volume and nutrient availability on plants known to display a Tragedy of the Commons when competing and the response of these plants to such manipulations is very different from their response to a competitor. It should be noted that all nutrient treatments were homogeneous in distribution and so it does not constitute a good test of the predictions of Semchenko et al. (2007).
The heterogeneity of nutrients in the experiments by Semchenko et al. (2007) was established by creating regions that roots could not explore. This is a valid experimental design but does not make for a simple comparison to the experimental protocols previously used to test the predictions of the Tragedy of the Commons model.
Second, we examined the role of space by dividing the exclusive pot in half with each half occupied by a root system from the same plant. In support of Hess & de Kroon (2007) and Semchenko et al. (2007), the presence of a divider in those pots did result in small (~5%) but significant reduction in root biomass (roots per unit volume) and reproductive yield (this supports the idea of space as a resource) when compared to the same arrangement but without a divider. However, this difference paled in comparison to the 60% increase in root production when an undivided pot was shared with another plant (O’Brien et al. 2005). This shows that alternatives suggested by Hess & de Kroon (2007) and by space as an additional resource are not mutually exclusive of the game theory model. Future tests can profitably keep this in mind.
Laird & Aarssen (2005) amended the original model of Gersani et al. (2001) to demonstrate that size asymmetries between competing plants can also result in decreased biomass allocation, as the larger plant controls a greater proportion of available soil volume. However, this idea is not mutually exclusive of the original predictions of the Tragedy of the Commons model (Laird & Aarssen 2005; O’Brien et al. 2007). In a spatially explicit form of the model, this becomes clear as the areas exclusively occupied by each plant and the shared space (where the Tragedy of the Commons occurs) shifts with any form of asymmetry (O’Brien et al. 2007). These changes ultimately favour the superior competitor not only because of the increased area from which it is able to exclude inferior neighbours (Laird & Aarssen 2005; O’Brien et al. 2007), but also because the shared region represents a much smaller proportion of its occupied soil space compared to that experienced by the inferior competitor (O’Brien et al. 2007).
These predictions are further supported by the fact that the results presented in Gersani et al. (2001) and O’Brien et al. (2005) show that between pair variation is significantly greater than within pair variation regardless of competition treatment and the final results of analysis are the same regardless of whether an individual plant is used from a pair or the average plant data of the pair is used (unpublished analyses). In addition, when dramatic asymmetric interactions have been an intentional part of the experimental design in competition experiments, plant root allocation appears to more closely follow the predictions of resource matching (Gersani et al. 1998) and do indeed lack the root over-proliferation responses observed in more symmetric experiments. Thus, some of the discrepancies in thought regarding the role of space are not really mutually exclusive; however, others represent fundamental differences in connecting principles and concepts to generate hypotheses and some emerge from a lack of understanding of volume in the original game-theory model of Gersani et al. (2001).
Root communication and the Tragedy of the Commons
Semchenko et al. (2007) see self–nonself root recognition as an essential element of the Gersani et al. (2001) model. We also see recognition as playing a role in determining an individual plant's root growth strategy (see Gersani et al. 2001, p. 663; O’Brien et al. 2005, p. 411). But, plants do NOT need to be ‘aware’ of the presence of neighbours in order for the model's predictions to hold. To play the game of the model, the plant need only non-cognitively assess the net returns from investing in additional roots. There is some evidence that phenotypic plasticity in resource allocation as it relates to nutrient foraging may well be determined by a combination of short- and long-distance signalling pathways consisting of various plant hormones and sugars (Robinson 1994; Bates & Lynch 1996; Schmidt & Schikora 2001; Forde 2002 and reference therein).
Whether a plant is alone in a patch or not, it can achieve the ESS if it ceases to produce additional root when new roots do not increase the plant's net resource uptake. The plant can do this if it can account for the whole-plant consequence of local root proliferation. Local rooting densities being equal, the net benefit of adding a new root to the whole plant is always greater in the neighbourhood of competing roots than of owned roots because in the latter case the added root will simultaneously diminish the return to owned roots. Different proximate mechanisms, including root communication and an internal ability to assess costs and benefits of additional growth, may be responsible for the Tragedy of the Commons response and provide a wealth of research opportunities for the future.
Although root communication is unnecessary for plants to display a Tragedy of the Commons root investment strategy, it is relevant to the subject (Semchenko et al. 2007). The ability to detect the roots of others could enhance a plant's ability to play the game. Or, it may allow plants to play different and additional sorts of games.
The most obvious example of additional games is allelopathy, where plants exude phytotoxins from their roots (or other tissues) that inhibit the growth of neighbours’ roots. This may be very dramatic such as the cell and eventual plant death caused by the (–)-catechin produced by Centaurea maculosa (Bais et al. 2003), or much more subtle, including the production of an as yet unidentified root exudate by Larrea tridentata that results in a decrease in the root elongation rates of neighbours (Mahall & Callaway 1991, 1992). As perennial plants (not the annual plants of the Gersani et al. (2001) model) the potential exists for iterated games that would not be available to annual plants.
It is also possible that root communication between plants permits cooperative solutions. Direct evidence for kin selection in plants has recently been provided by Dudley & File (2007). Cakile edentula displayed the root over-proliferation predicted by the Tragedy of the Commons theory when grown with unrelated neighbours but not when grown with siblings, although it is important to note that the change in root growth strategy did not appear to alter reproductive output. There are other species that may show similar responses with further investigations. Non-toxic communication through physical contact of the roots of Ambrosia dumosa results in the segregation of root systems among plants of the same population (Mahall & Callaway 1991, 1992, 1996).
Semchenko et al. (2007) looked for evidence of root communication when testing for root growth strategy. Using experimental protocols very similar to those used in early tests of the Tragedy of the Commons model, their results showed no evidence of water diffusible root communication or for root proliferation patterns predicted by the Tragedy of the Commons. However, inter-plant inhibition may have still occurred as a result of direct contact between plants as has been observed in A. dumosa (Mahall & Callaway 1991, 1992, 1996). A semi-permeable divider in some treatments permitted the relatively free movement of nutrients while segregating the roots of two plants. It would be interesting to know whether some root over-proliferation occurred along this membrane as the only means for the plants to steal nutrients from each other, particularly as the membranes acted to prevent inter-plant contact.
The results of Semchenko et al. (2007) did show a decrease in root growth at smaller volumes that was eliminated by the presence of activated charcoal suggesting that a water diffusible form of root communication may alter root growth strategies for some plants even in isolation. As stated above, similar declines in root biomass have also been observed in studies where plants display a Tragedy of the Commons and may help to prevent intra-plant competition (O’Brien et al. 2005). However, the ultimate fitness consequences of this response have yet to be tested.
First principles, concepts and hypothesis formulation
Hess & de Kroon (2007) and Gersani et al. (2001) represent different ways of linking first principles, concepts and particulars to form hypotheses. Gersani et al. apply the first principles of foraging theory to imagine a plant attempting to maximize net nutrient uptake. The plant's roots are treated as a strategy for achieving this objective. Change the circumstances – nutrients, volume, and competition – and the strategy changes predictably. The three hypotheses of Hess & de Kroon (2007) do not emerge from a body of theory or principles. They are empirically based without recourse to root proliferation as adaptations.
Empirically based predictions may provide less grist for the conceptual and empirical mill than those based on principles. For instance, the Gersani et al. (2001) model can be easily reformulated to incorporate specific information states, more realistic physiological and morphological models of nutrient uptake and root architecture, and so on. Such refinements will provide novel and different predictions that can be used to test between alternative hypotheses.
Other ways to explore the model
The value of Gersani et al. (2001) has been the ability to generate and suggest new and interesting models and hypotheses, the predictions of a Tragedy of the Commons, and the recent challenges to this model. How common is the Tragedy of the Commons in crops and wild plants? Do soybeans and Kenya beans truly display a Tragedy of the Commons or are the changes in root growth the result of different rooting volumes? Under what conditions does root communication amplify, remove or diminish the Tragedy of the Commons?
The works of Hess & de Kroon (2007), Semchenko et al. (2007), and others point to the need to consider space and volume explicitly. We have extended the Gersani et al. (2001) model to include a distance cost for the roots produced by plants on a spatially explicit landscape of nutrient availabilities (O’Brien et al. 2007). The model makes specific predictions regarding the amount and spread of roots, the degree of root overlap between plants. The model predicts where a plant will (i) have exclusive, uncontested rooting space; (ii) have space ceded to it by a competitor; (iii) cede space to a competitor; and (iv) have space in which both competitors grow roots.
Our extension of the Gersani et al. (2001) model shows why the first experiments on the Tragedy of the Commons favoured treatments of two plants sharing space and two plants separated by a divider. The model also shows what predictions can be tested by keeping space constant but simultaneously varying the number of competitors and the nutrient concentration. The degree of the Tragedy of the Commons should increase with nutrient concentration (for a fixed volume), and decline with volume when the total amount of nutrients is fixed. Furthermore, the over-proliferation of roots and the decline in reproductive yield per plant should increase (at a diminishing rate) with the number of plants (holding space and nutrients per plant constant). A greater diversity of treatments with respect to space, nutrients and number of plants should permit testing a broader suite of predictions.
Another exciting direction for research on the model concerns more diverse and sophisticated measures of root proliferation and reproductive output. Root architecture, root kinetics and physiology may be additional strategies that a plant can use to modulate the costs and benefits associated with nutrient uptake. Using seed biomass as the metric of yield may miss important but more subtle measures of reproductive success. Other metrics might include seed size, flowering date and duration of flowering and seed production. The importance of these strategies to an annual plant's fitness may be quite context dependent, and invite consideration of the plant species’ evolutionary context.
Finally, the model's application to annual plants simplifies the game of root proliferation. Perennial plants invite novel model extensions and empirical tests. As perennials, measures of fitness become more interesting and time-dependent, and with perennial plants the game becomes iterative. Iterative games have a much broader range of possible ESSs and strategies. There is now the opportunity for neighbouring plants to punish or reward the prior rooting strategies of others. A plant might be able to adjust its rooting strategy this round of the game to the rooting strategy of a neighbour during a previous round of the game. Craine (2006) modelled optimal carbon and nitrogen allocation to roots in isolated and competing temperate grasses, which are generally perennials. His simulations suggest that, while isolated plants benefit from minimal allocation to roots, plants in competition must allocate a large proportion of their available resources to roots and that this amount increases when neighbours increase their own allocation to roots. While his model did not allow for iterations of the game, the resulting simulations suggest that the Tragedy of the Commons may help to explain the previously unexplained, high root length densities observed in these species.
While it may be stretching credulity to expect plants to be so sophisticated, it remains to be seen whether these limitations truly exist in the plants, or only in our imagination. Regardless, it seems clear that viewing root proliferation as a game of nutrient foraging suggests experiments, avenues for empirical research, and ample opportunities for falsification.
We would like to thank Mike Hutchings, David Gibson, and Bruce Mahall for discussion of this topic and three anonymous referees for their comments.