Theory predicts plants grow roots to compete with only their closest neighbours

Abstract

The combination of individual-based selection with shared access to resources drives individuals to invest more than necessary in taking up their share of resources due to the threat of other individuals doing the same (competitive overinvestments). This evolutionary escalation of investment is common, from deer antlers and peacock feathers to tree height and plant roots. Because plant roots seem to be well intermingled belowground, the simplifying assumption that belowground resources are perfectly well mixed is often made in models—a condition that favours maximal fine-root overinvestments. Here, I develop simple models to investigate the role of space in determining the overlap among individuals belowground and resulting fine-root biomass. Without costs of growing roots through space, evolutionary optimization leads individuals to intermingle their fine roots perfectly and to invest just as much in these roots, whether there are two individuals competing or many. However, if there are any costs of sending roots through soil, investment in fine roots is constrained in amount and spatial extent. Dominant individuals are those that keep their roots in the soil closest to their own stem and the stems of their closest neighbours. These results highlight the importance of space in determining individual strategies as well as competitive networks.

1. Background

Few concepts in ecology are as classic and appealing as the ‘ghost of competition past’ (Rosenzweig as described in [1]). Individuals with strategies that free themselves from competition with others are favoured, and, over time, lead to the niche differentiation of species [2,3]. This avoidance of competition may not always be possible, however. Limitations of essential resources like light, water and nitrogen are challenges common to almost all plants leaving little possibility for differentiation. When individuals must share access to the same pool of a limiting resource, theory predicts the opposite of niche differentiation, that individual-level selection will lead to magnification of the negative effects of competition through ‘red queen’ and ‘tragedy of the commons’ dynamics both across and within species [4–7]. That is, successful individuals will be those that invest more in resource uptake than is necessary for taking up their share of the resources, because, if they do not, others will (an intensification of competition).

For plants, this escalation of the effects of competition is predicted for the allocation to fine-root biomass. Fine roots are often intermingled, with many individuals sharing access to the same resources [8–10]. If one individual has more fine roots than its neighbours do, it will take up more than its share of the resources, increasing its fitness. However, when all of the individuals take on this (individually) adaptive strategy, their fitness drops. They are now taking up approximately the same amount of resources but building more fine roots to do so. This is called fine-root overproliferation.

To test for fine-root overproliferation, experiments have been conducted in which the presence of a neighbour is manipulated while efforts are made to hold resource input per individual constant (usually by adding and removing a barrier between two plants in the same pot). The plants with access to one another’s space belowground often exhibit higher fine-root biomass and lower reproductive output, evidence that shared access to resources causes plants to grow ‘extra’ roots (overproliferation) [5,11–14].

Some studies cast doubt on the prevalence of fine-root overproliferation, however. Several species do not show an increase in fine roots in response to the experimental manipulation of neighbour presence [15]. And the experiments in which species do respond to neighbour presence are not completely free of confounding factors that may provide alternate explanations for the observed increase in fine-root biomass (such as total growing space [16,17]).

But still, with model/data comparisons, we have seen that considering the evolutionary effects of competition among individuals with shared access to belowground resources allows us to explain the observed dominant trade-off in carbon allocation of trees in closed-canopy forests [18,19] and complex plant responses to simple resource additions [20]. So why do some experiments appear to show a lack of fine-root overproliferation?

It may be the case that there are environments in which there is no benefit to an individual for growing its roots into the home soil of neighbours, thus individuals do not share access to resources belowground, and do not benefit from fine-root overproliferation. Alternatively, it could be the case that shared access to resources is so common that some species have not gained the adaptation of the ability to respond to changes in the presence of neighbours, causing them to exhibit fine-root overproliferation irrespective of the experimental treatment.

Most models of plant competition make the simplifying assumption that there is no spatial structure belowground, and that all individuals have access to the same resources in one common pool (a mean-field approximation [5,19–22]), unrealistic and ideal conditions for fine-root overproliferation. To investigate how spatial structure may diminish fine-root overproliferation, here I build spatially structured models of belowground competition and analyse them to find the evolutionarily stable strategies of plants, for both their fine-root biomass and spatial extent.

Other studies have also investigated the role of space in plant allocation to fine-root biomass. Assuming plants are completely spatially segregated belowground, but that there is an externally driven mixing of soil resources, Zea-Cabrera et al. [21] found that fine-root overproliferation increases with the degree of mixing. Investigating the evolution of fine-scale fine-root placement between two competing individuals, O’Brien et al. [23] found that individuals that balance the benefits of taking resources away from neighbours with the benefits of exploring the higher resource availability of root-free soil dominate. These studies leave a few unanswered questions, however. Plants can influence both the rate of mixing of belowground resources (through fine-root placement) and the availability of root-free soil (through growth and reproduction). Here, instead of imposing a rate of mixing, I find the evolutionarily stable overlap among individuals and instead of assuming there will be open space, I incorporate population dynamic feedbacks to determine the density of individuals.

Through these analyses, I find that evolutionary drivers for plants to seek out shared access to belowground resources are almost completely inescapable. Under almost all environmental conditions, the evolutionarily stable strategy (ESS) for individuals is to send a proportion of their roots to neighbours’ home soil, but only to their closest neighbours, thus increasing spatial structure of competitive networks, while also producing the greatest fine-root overproliferation possible. To derive these results, I will first present a model that isolates the influence of the number of individuals interacting from resource availability. This model will produce insights that facilitate understanding of the results of the second model, where individuals grow on a spatially explicit, two-dimensional grid. The dependence of results on environmental conditions will also be investigated.

2. Material and methods

Here, I describe two models designed to investigate the effects of competition among individuals for shared belowground resources structured through horizontal space. Model 1 is designed specifically to determine the role of the number of individuals interacting on belowground strategies, independent of density. Here, all individuals are equally close. This model is used to develop insights before moving to model 2. Model 2 is more realistic, including a spatially explicit two-dimensional grid and closed population dynamics. In both models, it is assumed that there are no differences among individuals in the depth distribution of their fine roots.

Table 1 lists the variables and parameters used throughout the paper.

Table 1.

Model symbols and definitions, in order of appearance.

symbol	description	units	default value
n	number of individuals competing	ind	3
k, subscript	individual identity	—
r	fine-root biomass	g
x	proportion of fine roots sent ‘away’	(unitless)
sk	resource availability at the home site of individual k	g
P	total resource uptake	g yr−1
us	resource uptake efficiency of fine roots	yr−1	1
R	resource input rate	g dm−2 yr−1	2
Δ	first-order resource loss rate	yr−1	0.02
a	area per individual	dm2	1
f	reproductive output	ind yr−1
cr	base fine root costs*	g g−1 yr−1	1.7
cf	cost of reproduction*	g ind−1	1
cx	cost of sending fine roots ‘away’*	g g−1 yr−1	1.5 (model 1); calculated (model 2)
nd	site-level individual density	ind dm−2	1 (model 1); calculated (model 2)
cx,0	soil and species specific cost of fine root growth through soil	g g−1 dm−1 yr−1	1.2 (equation (2.6), model 2)

Note: ind stands for individuals. dm stands for decimeters. *Costs include annualized building, maintenance and respiration costs. Cost of reproduction includes all costs associated with producing one new individual.

(a). Model 1: ‘n − competitor model’

Across environmental gradients, plant density and/or biomass often change in conjunction with the probability and strength of interactions among plants. To derive a clear understanding of belowground competition, we must disentangle the effects of the number of interacting individuals from resource inputs. In this first model, I assume the number of individuals with the potential to share resources (n individuals, abbreviated here ‘ind’) is independent of both resource inputs and individual plant strategies. Each individual (distinguished by subscript, k) takes up the same amount of space (1 dm²) and each unit of space has the same resource input rate.

For simplicity, assume the plants are herbaceous annuals. Productivity is determined by the uptake of a single belowground resource. Each individual has two types of space belowground: ‘home’ space and ‘away’ space (figure 1). Home space is the space that is closer to that individual than to any other individual. Away space is the home spaces of all of the n – 1 neighbours. Assume the home spaces of all neighbours are equally close. In model 2, this assumption will be relaxed.

Figure 1.

Illustrations of (a) model 1 and (b) model 2. Each hexagon represents one individual’s ‘home space’ and each individual’s roots are represented by lines matching the colour of their home space. (a) Each individual has the same amount of space and is equally connected to every other individual. Across the first three panels, the effect of changing n given all individuals have the same strategy of fine-root biomass (r) and placement (x, the proportion sent to neighbours’ home space) is presented. The final panel shows the generalized case of n individuals, each with their own strategy (r_k and x_k). (b) A portion of the hexagonal grid is depicted. Individual density (n_d) increases across the first three panels to the maximum (1 ind dm⁻²). As belowground home space decreases with density, so too does the cost of sending roots to neighbours’ home soil (equation (2.6)). The final panel shows the belowground strategy of one target individual (purple) for the case where the individual sends roots to the home space of: no neighbours (zero rings), six closest neighbours (one ring) and 18 closest neighbours (two rings).

Two traits of individual plants (subscript k) will be analysed with evolutionary analyses: total fine-root biomass (r_k), and the proportion of r_k sent away to the home space of neighbours (x_k). Note, in the model, these roots represent investment in resource uptake belowground generally and may be thought of as including other traits like investment in mycorrhizae.

In each site where an individual has roots, it takes up resources in proportion to its fine-root biomass in that site and the local resource availability (s_k, g; where k is the home individual of the site):

$$P_k=u_{\mathrm{s}}(1-x_k)r_k{s}_k+\sum _{\hspace{0.17em}j\ne k}\frac{u_{\mathrm{s}}{x}_k{r}_k}{n-1}{s}_{\hspace{0.17em}j},$$ 2.1

where P_k is the resource uptake (g yr⁻¹) of individual k from all sites and u_s is the resource uptake efficiency of fine roots (g g⁻¹ yr⁻¹). Site resource availability, s_k, is set by the resource input rate (R, g dm⁻² yr⁻¹), the rate of uptake by all individuals with roots at the site, and first-order leakage losses (Δ g g⁻¹ yr⁻¹),

$$\frac{\text{d}{s}_k}{\text{d}t}=aR-u_{\mathrm{s}}(1-x_k)r_k{s}_k-\sum _{\mathrm{\forall }j\ne k}{u}_{\mathrm{s}}\frac{x_{\hspace{0.17em}j}{r}_{\hspace{0.17em}j}}{n-1}{s}_k-\mathrm{\Delta }{s}_k,$$ 2.2

where a is the area of the home space (1 dm² in model 1, but will vary with population density in model 2). At equilibrium,

$$s_k=\frac{aR}{u_{\mathrm{s}}(1-x_k)r_k+\sum _{\mathrm{\forall }j\ne k}{u}_{\mathrm{s}}(x_{\hspace{0.17em}j}{r}_{\hspace{0.17em}j}/(n-1))+\mathrm{\Delta }}.$$ 2.3

Diffusion of the resource across sites is assumed to be negligible.

The resource uptake rate of each individual (P_k) is used to build and maintain fine roots (r_k), to pay for sending a portion away from home (x_k), and for all costs associated with reproduction (f_k).

$$P_k=(c_{\mathrm{r}}+c_{\mathrm{x}}{x}_k)r_k+c_{\mathrm{f}}{f}_k,$$ 2.4

where c_r (g g⁻¹ yr⁻¹), is the cost of building and maintenance for all fine-root biomass, c_x (g g⁻¹ yr⁻¹) is the additional cost incurred by sending roots away from home soil (x_k r_k), and c_f (g ind⁻¹) captures all costs of the plants which increase in proportion to the production of new individuals. Assuming there is no light limitation, this includes leaf biomass as well as flowers and other reproductive structures.

The cost of sending roots away from home soil (c_x) is a cost that may include the cost of additional structures needed to support the longer transport needed by roots operating outside of home soil, a cost of exploring a greater soil volume, or any other cost associated with growing roots through space. For comparison, I will also investigate the case where there are no costs to placing roots away from the main stem (c_x = 0).

Assuming mortality rate is independent of root biomass allocation strategies, f_k can be treated as a measure of fitness and can be easily found by rearranging equation (2.4).

(b). Model 2: ‘hex-grid competition model’

To consider more realistic spatial interactions among individuals in a population, in model 2, individuals are placed on a hexagonal two-dimensional grid (figure 1). The grid is wrapped on a torus to avoid edge effects. In this closed population, feedbacks between the costs of an individual’s fine-root strategy, allocation to reproductive biomass, and population density will now be incorporated.

For simplicity of implementation, if population density (n_d, ind dm⁻²) changes, the number of individuals on the grid is held constant at 96 individuals, while the spatial scale of the grid changes:

$$a=\frac{1}{n_{\mathrm{d}}},$$ 2.5

where a is the size of the home-space of one individual (figure 1). As the focus is on belowground competition, only conditions in which aboveground competition is negligible, where density is low enough for an open canopy, are considered (n_d < n_d,max = 1).

Because the density of individuals influences the distance between individuals, it must also influence the cost of sending roots to neighbours’ home soil. Here, I simply assume that the cost of sending roots to neighbours’ home soil is proportional to the farthest distance that the roots reach:

$$c_{\mathrm{x}}=\frac{1}{n_{\mathrm{d}}}\frac{2}{\sqrt{3}}\hspace{0.17em}{(\mathrm{no}.\hspace{0.17em}\mathrm{rings})}^2{c}_{\mathrm{x},0},$$ 2.6

where c_x,0 is an intrinsic property of the soil and/or the plant species characterizing the cost of growing roots through soil (dm⁻¹ yr⁻¹). The symbol ‘no. rings’ represents the number of rings of neighbours in the hexagonal grid reached by the plant’s roots (figure 1). A plant with a 0-ring strategy keeps all of its roots at home. A plant with a 1 or 2 ring strategy spreads its (x r) roots among the home spaces of its 6 or 18 (respectively) closest neighbours. I assume individuals spread the roots sent away evenly among all sites reached. Other functional forms, including a tapering of root density by distance were considered but did not change the qualitative results.

For these annual plants, mortality is annual (1 yr⁻¹) and thus the equilibrium individual density (${\hat{n}}_{\mathrm{d},k}$) of a monoculture of strategy k occurs when individuals produce one new individual per year:

$$f_k({\hat{n}}_{\mathrm{d},k})=1.$$ 2.7

(c). Analysis

I analyse both models to find the plant belowground strategies that are likely to dominate landscapes. Fine roots (r), the proportion of fine roots a plant sends away (x) and the number of rings of neighbours’ home soil a plant reaches with its roots (model 2 only) are the traits that define belowground strategy.

Following an adaptive dynamics approach [24], I assume exactly one individual has the mutant strategy and the rest of the individuals have the resident strategy in each model. Through a process of successive selection at the individual level, the strategy expected to dominate is the strategy (if it exists) that cannot be invaded by any other, the ESS.

$$f_{\mathrm{M}}<f_{\mathrm{R}=\mathrm{ESS}}\mathrm{\forall }\text{M}\ne \text{ESS}.$$ 2.8

I also check that these strategies may be reached by mutations of small effect, that they are convergence stable strategies (CSSs), and, in the case of model 2, that they produce stable population densities.

A generic method for finding a putative two-dimensional ESS/CSS sets numerically has been written for this project and can be found online along with the rest of the code of this study (code written in R [25]; https://github.com/cfarrior/compterrVP). Appendix A shows plots that verify the putative solutions are ESSs and CSSs.

Model 2 has three traits, but it was found that the ESS of the trait no. rings does not depend on the exact values of r and x. Thus, the ESS of no. rings was found first and then the two dimensional algorithms were used to find r_ESS and x_ESS.

3. Results

Costs of sending roots through space increase reproductive output

If there are no costs of sending roots away from home soil (c_x = 0), the competitively dominant strategies (ESSs) are the same as if I used a classic mean-field model (model 1; figure 2, grey lines versus dots):

$$x_{\mathrm{ESS}}\hspace{0.17em}(c_{\mathrm{x}}=0)=1-\frac{1}{n}\text{and}{r}_{\mathrm{ESS}}\hspace{0.17em}(c_{\mathrm{x}}=0,n>1)=\frac{aR}{c_{\mathrm{r}}}-\frac{\mathrm{\Delta }}{u_{\mathrm{s}}}.$$ 3.1

Figure 2.

The effect of the number of individuals with access to the same belowground resources (n, model 1) on (a) ESS fine-root biomass (r_ESS), (b) the proportion of roots sent to neighbours’ home soil (x_ESS), (c) reproductive output (f_ESS) and (d) the degree of fine-root overproliferation. The degree of fine-root overproliferation is measured as the amount of potential reproduction diverted to belowground competition ((f_max − f_ESS)/f_max, where f_max is f_ESS for n = 1). Results shown are for cases of zero (grey) and non-zero (black, 1.5) costs of extending roots to neighbours’ home space (c_x). Grey lines are the predictions for a model with mean-field resource availability (equation 3.1). If not otherwise indicated, parameter values are found in table 1.

If c_x is positive, however, fine-root biomass (r) and the proportion of fine roots sent away from home (x) are lower than the mean-field case. This results in higher reproductive biomass (f) at evolutionary equilibrium (figure 2). The effects are strengthened with greater c_x (electronic supplementary material, figure B1).

(a). Fine-root overproliferation is highest if the fewest competitors interact

As in other studies, this model predicts that by adding the possibility of competition (increasing n from 1 to 2), the ESS r increases to the net detriment of allocation to reproductive biomass at evolutionary equilibrium (figure 2c), a classic tragedy of the commons/overproliferation of fine roots results. But, by adding a third potential competitor, the overproliferation is weakened. ESS r decreases and equilibrium f increases. Fine-root overproliferation decreases with the number of individuals competing if more than two individuals compete (figure 2d).

(b). Competitively dominant plants grow roots close to (but not entirely at) home

Analyses of model 2 show the competitively dominant strategy is almost always to send roots to home spaces of the closest neighbours only (figure 3). Only when there are no costs of sending roots away (c_x,0 = 0) is sending roots farther than the closest neighbours a good strategy. And only when the environment requires so many fine roots for resource uptake without competitors that there is no benefit to having even more roots (low R, high c_x,0, and high Δ) is the best strategy to send no roots away (x_ESS = 0, figure 4; electronic supplementary material, figure B3).

Figure 3.

The results of pairwise competition among individuals with differing strategies of fine root placement (model 2, no. rings). Individuals keep roots at home (0 rings or neighbours), send them to their closest ring of neighbours’ home soil (one ring, six neighbours), or send them to the closest two rings of neighbours’ home soil (two rings, 18 neighbours). Plus (+) marks cases in which the mutant outperforms the average resident ($f_{\mathrm{M}}>\overline{\hspace{0.17em}{f}_{\mathrm{R}}}$), minus (−) marks cases in which the average resident outperforms the mutant ($f_{\mathrm{M}}<\overline{\hspace{0.17em}{f}_{\mathrm{R}}}$) and zero (0) indicates no difference ($f_{\mathrm{M}}=\overline{\hspace{0.17em}{f}_{\mathrm{R}}}$). Here, the strategy of sending roots to the first ring of neighbours (6) is the competitive dominant (ESS). This result was found for all cases where c_x > 0 and x_ESS ≠ 0.

Figure 4.

Evolutionarily stable strategies of belowground competition across environmental gradients: (a) R resource input rate, (b) c_x,0, the cost of extending roots through space (equation (2.6)) and (c) Δ the first-order resource-dependent leakage loss rate. Each panel shows results for fine-root biomass (r_ESS, g, right axis), the proportion of fine roots sent to compete with neighbours (x_ESS, left axis), and the equilibrium population density for individuals of that strategy (n_d, ind dm⁻², left axis). Parameters values not indicated here can be found in table 1. (Online version in colour.)

(c). Belowground competition across environmental gradients

Across a gradient of increasing resource input rate (R), the competitively dominant strategy (ESS) changes, increasing r, x and the resulting n_d (figure 4). If the cost of sending roots through space (c_x,0) increases, ESS r and x decrease. This decrease in fine-root overproliferation causes an increase in n_d. Across an increasing gradient in the rate of resource-dependent leakage losses (Δ), ESS r and x again decrease. However, these decreases do not represent a decrease in fine-root overproliferation and are accompanied by a decrease in individual density (n_d).

In the more resource-poor environments, low R or high Δ, the model predicts no overproliferation of fine roots (x_ESS = 0). However, if there are no resource-dependent leakage losses (Δ = 0), this is never the case (figure 4; electronic supplementary material, figure B2-4).

4. Discussion

I have presented here two simple models to investigate the belowground strategies of plants. Both models predict that only in the extreme case of no cost of extending roots closer to neighbours than oneself will plant communities satisfy the assumptions of a mean-field approximation (well-mixed resource availability belowground). With any cost of growing roots through soil, the ESS is to keep a higher proportion of fine roots at home (lower x_ESS), to invest less in fine-root biomass (lower r_ESS), and to grow the roots in only home soil and the home soil of closest neighbours (1 ‘ring’). The cost of growing roots through space weakens fine-root overproliferation, benefitting the reproduction of plants with the ESS.

Although I unrealistically assume that there is no diffusion of resources across sites, overlapping root placements effectively serve to ‘mix’ the resource availability. And unlike in Zea-Cabrera et al.’s model [21], this mixing is a result of biological feedbacks, not a model input. Individuals’ fine-root biomass (r) and the proportion sent to neighbours’ home soil (x) determine the spatial structure of resource availability. Finding the evolutionarily stable strategies of r and x determines the mixing of resource availability across space because of the resulting plant strategies themselves. Thus, even with a level of mixing of resource availability determined by the competitively dominant plants themselves, the addition of costs of growing roots through soil still has a net benefit on the reproductive output of the evolutionarily stable dominant plants.

From model 1, while controlling for resource input per individual, I find the highest levels of fine-root overproliferation occur when the fewest individuals have access to one another’s space belowground (two individuals, figure 2). This occurs because each individuals’ fine-root biomass influences both their own resource availability and that of their neighbours. When an individual has more roots than its neighbours do, it takes up more resources for itself and it decreases the amount of resources the neighbours can take up. With fewer individuals competing, the fine-root biomass sent away is more concentrated by neighbour and has a greater (negative) effect on neighbour fitness. Because relative fitness is what matters for invasion (equation (2.8)), the ESS occurs at higher levels of fine-root biomass when fewer individuals compete.

In model 2, with many individuals competing on a grid, the most successful (evolutionarily stable) plants are those that do not send their roots as far as possible, but grow roots only in their home soil and the home soil of their closest neighbours. As in model 1, this concentration of fine roots among competitors provides the greatest benefits to individuals with ‘extra’ fine roots. The analyses predict that sending roots to the closest neighbours’ home soil is the best strategy whether plants are close to one another and the cost of sending roots to neighbours’ soil is low or plants are far apart and the cost of sending roots to neighbours’ soil is high.

There are cases where the evolutionarily stable fine-root placement (no. rings) differs, however. In the (unrealistic) case that there are no costs of sending roots through soil (c_x,0 = 0), the most successful plants are those that spread roots evenly over all available space. In the opposite extreme, if the leakage loss rate (Δ) is high and environmental input rate (R) is low and/or the cost of sending roots through soil (c_x,0) are high, the most successful plants are those that keep all of their roots at home (x_ESS and no. rings = 0). In the latter case, x_ESS is zero because the fine-root biomass necessary to take up resources from home soil is so high that there are no potential gains for extra fine roots to a competitor (figure 4; electronic supplementary material, figure B5).

Note, if there are no leakage losses that increase with resource availability (Δ is zero), it is never the best strategy to keep all roots at home (electronic supplementary material, figures B2 and B3). And whether resource-dependent leakage losses of limiting resources are common (Δ > 0) is still the subject of some debate [26,27].

In total, these findings support a conclusion that there are only rarely circumstances where plants are truly free from the pressures of competitors. Without circumstances when there are no competitors, plants would never benefit from a strategy that enables them to detect the absence of neighbours. This may explain why some species do not change allocation strategies in response to the presence of neighbours [15,28]. These findings are supported by analyses of fine-root distributions in soil, which show that the roots of several species are commonly intermingled in forests [8,9] and grasslands [10]. And meta-analyses show that competition can be strong even in resource-poor and sparse environments [29–31].

Although this model does not deal explicitly with facilitation, it does predict that, even in resource-poor environments, unless resource-dependent leakage occurs, negative interactions between individuals arise from the biological feedbacks. This is not necessarily in contradiction with studies that find a net positive effect of neighbours in stressful environments [14,32]. Facilitation among plants may be driven by physiological interactions with the environment and may be independent of the competitive interactions considered here [33–35]. Tang et al. [33] show that the interconnection of competition for resources and tolerance of a non-resource stress complicate studies comparing competition and facilitation. Qi et al. [34] outline a clear stress gradient hypothesis that takes into account this complication. De Parseval et al. [35,36] have recently produced detailed spatial models of plant competition and facilitation. However, these models do not yet include the potentially important feedbacks of competitive (ecological/evolutionary) optimization.

Likewise, considerations of many of the fascinating aspects of belowground plant ecology—plant/microbial interactions [37,38], depth distributions [39], nutrient hotspots [40,41], hydraulic lift [42], facilitation among individuals [33,35] and even size variation [43]—are likely to interact in interesting ways with the predictions derived here, especially the result that plants should have constrained competitive networks.

5. Conclusion

Through building and analysing strategically simple models, new predictions of the effects of belowground competition are presented. It is predicted that individuals almost always benefit from restraining their fine-root extent to overlap with their closest, and only their closest, neighbours. This leads to the highest payoffs of ‘extra’ fine roots and thus maximizes the effects of competition while minimizing the number of individuals sharing access to resources at any one site.

Analyses predict that the cost of growing roots through space restrains overproliferation of fine roots, enhancing reproductive biomass for all individuals in the community. Only if plants are in environments where foraging for resources belowground is extremely tough are there benefits to an individual to keep all roots in home soil.

These insights help explain the ubiquity of the importance of competition across environmental gradients and give specific predictions for the extent of belowground fine-root extension. They also point to the potential importance of considerations of the cost of growing roots through space for quantitative predictions of allocation strategies including overproliferation of fine roots which, in particular through trade-offs with woody biomass, can have significant impacts on ecosystem carbon storage [44].

Data accessibility

All of the code written for this project can be found online at https://github.com/cfarrior/compterrVP.

Competing interests

I have no competing interests.

Funding

I thank the National Institute for Mathematical and Biological Synthesis, an Institute sponsored by the National Science Foundationthrough NSF Award no. DBI-1300426, with additional support from The University of Tennessee, Knoxville and the University of Texas at Austin for funding.