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:
- (eqn 1)
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:
- (eqn 2)
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*:
- (eqn 3)
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).
Figure 1. Model output for how volume should effect the optimal investment into roots (r*) (a and c), and the resulting net profit available for reproduction (G(r)) (b and d). The x-axis represents volume per plant. The graphs compare the model's predictions for an isolated plant (dashed lines) with two plants experiencing competition (solid lines). On the left side (a and b), nutrient concentration is fixed (N = 10) even as volume increases. As volume increases so does the total amount of available nutrients. On the right side (c and d), the total amount of nutrients present is fixed (VN = 100) such that as volume increases, N decreases. Sharing space always results in greater root proliferation per plant and less nutrient profit per plant. When nutrient concentration is held fixed, root proliferation and nutrient profit increase linearly with volume. When the total amount of nutrients is held fixed (VN constant) then the relationship between root proliferation and volume is hump-shaped, and nutrient profit declines with volume. For these graphs, search volume per unit root was set to a = 1 and the cost of producing a unit of root to c = 1.
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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):
- (eqn 4)
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:
- (eqn 5)
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:
- (eqn 6)
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:
- (eqn 7)
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().