Introduction
 Top of page
 Summary
 Introduction
 Description
 Results
 Discussion
 Acknowledgements
 References
Stoichiometric relations in and differences between various components of ecosystems lead to powerful constraints on ecosystem development (e.g. Sterner & Elser, 2002; Sardans et al. 2012). Nitrogen (N) and phosphorus (P) are the two elements considered as limiting autotroph (plant) growth in most ecosystems. However, the increasing use of N and P fertilisers, as well as the formation of reactive N in various combustion processes, are increasing their availability in the biosphere. We should expect this to modify the stoichiometric constraints on plants in many ecosystems; possibly shifting ecosystems from N to P limitation (Peñuelas et al., 2012). In a review on responses of plants to N and P additions across marine, aquatic, and terrestrial ecosystems, Elser et al. (2007) showed that simultaneously adding both nutrients gave a much stronger response than either of them alone. Harpole et al. (2011) reached similar conclusions in a study of factorially designed experiments with N and P. In both studies, the authors concluded that ecosystems are frequently both N and P limited (colimited), which challenges the conventional view that plants are generally limited by one nutrient at a time (Liebig’s law of the minimum). Harpole et al. (2011) also suggest that we distinguish between simultaneous colimitation, when both N and P have to be added simultaneously to get a growth response, and independent colimitation, where a response is obtained when either N or P is added but the simultaneous addition may or may not add up to the two individual responses.
Davidson & Howarth (2007) criticised Elser et al.’s (2007) conclusions about colimitation on the basis of how nutrient doses were applied and what time scales the different experiments covered. They further noted that there currently does not exist a mechanistic understanding of how colimitation would function in the studied systems. Saito et al. (2008) suggested that colimitation could be classified in three categories:
Saito et al. (2008) suggest possible biochemical mechanisms for each of the three cases. We propose a fourth mechanism (described in detail later): (4) serially linked nutrients, where the rate control on growth by one nutrient depends on the rate control by another nutrient of another process. Note that this is not a serial colimitation in the terminology of Harpole et al. where the response to two nutrients depends on the order in which they are added.
In its simplest form, colimitation is expressed in Liebig’s law of the minimum (LM) (Liebig, 1840, 1855) stating that the nutrient in least supply relative to a plant’s requirement will limit its growth. Colimitation can, then, only occur at very strict nutrient ratios (optimum ratios) where two nutrients both limit growth (Knecht & Göransson, 2004). The existence of optimum ratios is supported by extensive laboratory experiments with thoroughly controlled nutrient supply (Ericsson, 1995), also indicating that the optimum ratio depends on the relative growth rate (Kerkhoff et al., 2006; Niklas, 2006; Niklas & Cobb, 2006; Ågren, 2008). Further support for optimum ratios is provided by the mechanistic approaches to ecological stoichiometry presented by Sterner & Elser (2002) and Ågren (2004). The idea of optimum ratios is also consistent with results from field experiments (Paris, 1992; Linder, 1995) as well as observations under natural conditions (Knecht & Göransson, 2004). Güsewell (2004) presented an extensive review of the current knowledge of N : P ratios that also support the existence of optimum ratios. However, in all cases it is difficult to know whether the optimum is actually a single value or rather a range of ratios.
A quite different approach to nutrient interactions is the ‘multiple limitation hypothesis’ (MLH) (Bloom et al., 1985) which suggests that organisms adjust their growth patterns such that they are limited by several resources simultaneously. Therefore, according to the MLH, colimitation will always exist and plants should always be driven towards the point of optimum nutrient ratios; this is the extension of category (3) in Saito et al.’s classifications of colimitation to include carbon (C). As Rubio et al. (2003) point out, MLH can easily be understood when it is a question of balancing resource acquisition in a plant such that C (a resource acquired by leaves) and nutrients (resources acquired by roots) provide equal limitations. A consequence of such balancing of resources is a variable root : shoot allocation (e.g. Ågren & Franklin, 2003) where a limitation in either N or P increases C allocation to roots (Ericsson, 1995). However, it is less obvious how MLH would apply to limitations in different nutrients acquired by the same roots. Also, an optimum nutrient ratio cannot be defined within the MLH framework itself but requires inclusion of additional plant properties.
The existence of the several mechanisms of interactions between nutrients does not really speak in favour of either LM or MLH, but they do provide evidence that plants do not need to obey strict LM behaviour. The existence of several mechanisms also complicates the understanding of what colimitation is. Then, under which conditions can we expect LM or MLH to best describe plant responses to nutrient limitations and what does it mean in terms of colimitation? In this paper we will explore these questions by formulating a model where plants invest assimilated C differentially in N and P uptake depending on their availability in the environment. We will do this in a terrestrial plant framework, but most parts should be applicable to other environments as well. Three questions will be in focus:
 1
Under which conditions can a plant increase its growth rate when N (or P) is the main limiting element if the supply of the other element (P or N) is increased?
 2
How are the plant nutrient concentrations affected by shifts in allocation of resources between uptake of N and P?
 3
How much can growth rates increase as a result of changes in allocation towards uptake of different elements?
Description
 Top of page
 Summary
 Introduction
 Description
 Results
 Discussion
 Acknowledgements
 References
Ågren (2004) showed that growth of plants could be described as a chain of reactions where P in ribosomes drives the production of enzymes, which, in turn, drive the fixation of C. In this analysis P and N were taken up at fixed relative rates. We will here extend his description to situations where the nutrient uptake is controlled by an external availability as well as by the plant’s allocation of resources to uptake (e.g. root growth, exudates, etc.). In the model we partition the plant’s contents of C, N, and P into three pools of labile (L) and three pools of structural (S) substrates (C_{L}, C_{S}, N_{L}, N_{S}, P_{L}, P_{S}). The structural pools drive processes and consist of tissues from which C, N and P cannot be remobilized. The labile pools consist of mobile forms of C, N and P and supply the substrates for processes. Uptake of C, N, and P also goes to the labile pools (Fig. 1) (cf. Thornley, 1972, 1995; Johnson & Thornley, 1987; see also Dynamic Energy Budget Theory, Kooijman, 2010). A summary of parameters used in the model is given in Table 1.
Table 1. List of symbols Symbol  Definition  Value  Units 


α_{N}  Halfvalue in production of structural N, Eqn 3  0.01  g N (g C)^{−1} 
α_{P}  Halfvalue in production of structural P, Eqn 5  0.0005  g P (g C)^{−1} 
ε  Relative allocation of resources to N and P uptake, Eqns 4, 6  ½ or variable  Dimensionless 
Φ_{CN}  Production rate of structural carbon, Eqn 1  3.18  g C (g N)^{−1} d^{−1} 
Φ_{CNL}  Production rate of labile carbon, Eqn 2  9.6  g C (g N)^{−1} d^{−1} 
Φ_{CL}  Rate of nutrient uptake, Eqns 2, 4, 6  2.0  (g C)^{−1} d^{−1} 
Φ_{NP}  Production rate of structural N, Eqn 3  5.0  g N (g P)^{−1} d^{−1} 
Φ_{PL}  Production rate of structural P, Eqn 5  0.08  g P (g N)^{−1} d^{−1} 
A_{N}  Amount of available N, Eqn 4  Variable  g N 
A_{P}  Amount of available P, Eqn 6  Variable  g P 
Y  Yield factor, Eqn 2  0.75  Dimensionless 
The formation of structural C (C_{S}) is proportional to the amount of structural N (N_{S}) (Ågren, 2004)
 (Eqn 1)
The production of labile C (C_{L}) (photosynthesis) is also determined by the amount of N_{S}. Labile C is drained when C_{S} is formed and C_{L} is also used for uptake of N and P at a rate Φ_{CL}C_{L}. We include a conversion cost (growth respiration) for forming structural C from labile C (Y) (McCree, 1974)
 (Eqn 2)
The formation of N_{S} is proportional to the amount of structural P (P_{S}) (Ågren, 2004). A Michaelis–Menten dependence on the availability of labile N (N_{L}) has been included to prevent the formation of N_{S} even in the absence of N_{L}. Here, and in the rest of the paper, we express all concentrations as fractions of structural C
 (Eqn 3)
The uptake of N to the labile pool is determined by allocation of resources to nutrient uptake (Φ_{CL}C_{L}), the fraction of the resources allocated to the N uptake (ε), and the availability of N in the environment (A_{N}). Labile N decreases when converted to N_{S}
 (Eqn 4)
The formation of P_{S} is similar to the formation of C_{S} and also driven by N_{S}
 (Eqn 5)
The uptake of labile P (P_{L}) is similar to the uptake of N_{L}
 (Eqn 6)
The availabilities should be understood in a general sense and include factors such as differences in nutrient mobilities. Although not stated explicitly, differential costs in uptake of N and P could also be included in the availabilities; a higher cost is mathematically equivalent to lower availability. We will, on the one hand, choose ε = ½ to simulate conditions where equal resources are allocated to N and P uptake (fixed allocation) and, on the other, find the ε that for given A_{N} and A_{P} maximises the relative growth rate (flexible allocation). The model could have been supplemented by maximum N and P concentrations in the plant; we have left this out for simplicity and restrict the ranges for A_{N} and A_{P} to those that keep the N and P concentrations within observed limits. Plant properties obtained at the extremes of A_{N} and A_{P} should, however, be treated cautiously and in particular for the fixed allocation.
In order to illustrate the properties of the model, we have chosen parameters suitable for approximately describing the growth of birch (Betula pendula Roth) seedlings (Table 1). The parameters Φ_{CN} and Φ_{NP} are taken directly from Ågren (2004), although Φ_{CN} has been adjusted to account for the introduction of the Michaelis–Menten factor. The other Φs have been set to values that produce realistic values of N and P concentrations in the plant. The two halfvalues (α_{N}, α_{P}) have been given values judged to correspond to levels of N_{L} and P_{L}, respectively. The yield factor accounting for growth respiration has been taken from McCree (1974).
Equations 1–6 can be combined to give an equation for steady state growth, for example, μ = 1/X dX/dt for all of the six state variables (X ). The resulting equation is quite complex but can be solved numerically, although care has to be taken to get the correct root. The model is quite insensitive to most parameters describing plant properties, and in particular to the two halfvalues (α_{N}, α_{P}). We will therefore only report the responses to changes in nutrient availabilities, A_{N} and A_{P}, and effects of the relative allocation to N vs P uptake, ε.
Results
 Top of page
 Summary
 Introduction
 Description
 Results
 Discussion
 Acknowledgements
 References
The model reproduces the standard relationship between plant nutrient concentrations and relative growth rate (e.g. Ingestad & Lund, 1979) (Fig. 2). The growth response is linear for N and slightly curvilinear for P. Increasing A_{N}/A_{P} decreases the relative growth rate slightly for a given plant N concentration, but increases the relative growth rate at fixed plant P concentrations.
The plant’s ability to allocate uptake effort towards the most limiting nutrient has a large influence on how nutrient concentrations in the plant vary with nutrient availability (Fig. 3).Without flexible allocation, increasing external N increases the plant N concentration and then also the growth rate, but the P uptake remains unchanged and the P concentration decreases. Analogously, increasing external P availability decreases the N concentration; a change in one element creates an opposite change in the other. By contrast, with flexible allocation an increase in the external availability of one element increases the concentrations of both elements in the plant; changes in the concentrations of the two elements go in the same direction. Also, the plant nutrient concentrations increase less rapidly with external availabilities and nutrient concentrations never reach the same levels as with the fixed allocation.
The effect of the flexible allocation on the plant’s relative growth rate is shown in Fig. 4. For all values of A_{N} and A_{P}, a plant that can allocate its uptake resources freely has a higher relative growth rate. It is only at A_{N} ≈ 20 A_{P} that equal allocation corresponds to the optimum allocation. This figure also illustrates the synergistic effects of N and P. Adding N and P simultaneously results in a larger growth response than the sum of the responses of the individual additions. With an increasing growth rate, leading to a higher relative demand for P than N, there is also a corresponding shift in allocation towards more P uptake; for example, ε is larger at the base of the diagonal arrow in Fig. 4 than at its point.
Fig. 5 illustrates the difference in relative growth rate between plants with flexible (μ_{flex}) and fixed (μ_{fix}) allocation at different external availabilities A_{N} and A_{P}. The effects are largest at the boundaries of availabilities. The response is not monotonic and has a maximum when A_{P} increases at fixed A_{N} in the lower right part (the region dominated by P limitation) and similarly when A_{N} increases at fixed A_{P} in the upper left part. The explanation for such maxima is that when the P availability is low, increased allocation to P uptake above equal allocation will not result in increased P uptake, because there is none to be had. At higher levels of P availability the allocation approaches the fixed one and changes in allocation do not increase growth over that obtained with ε = ½. In general, the possibilities for increases in the relative growth rate seem rather small (μ_{flex} is never > 10% larger than μ_{fix}); compare the absolute values in Fig. 4 with the possible changes in Fig. 5. It is only at low availabilities of one of the elements that the flexible allocation makes a significant difference. However, because of the potential for large excess uptakes at imbalanced supply, this observation should be taken as more qualitative than quantitative.
The flexible allocation mechanism also affects how the supply ratio is reflected in the plant nutrient ratio (Fig. 6). Without the flexible allocation, the external ratio is mirrored by the plant, whereas with the flexible allocation, the response is constrained. With a flexible allocation a given A_{N}/A_{P} does not correspond to a single value of c_{N}/c_{P}, because the latter responds not only to the ratio of N and P in the supply, but also to the absolute values of the supply. However, it should be observed that the model exaggerates the response to the external supply because real plants would not take up unlimited amounts of a nutrient and the curve would flatten out at the ends.
Discussion
 Top of page
 Summary
 Introduction
 Description
 Results
 Discussion
 Acknowledgements
 References
As Fig. 4 shows, there are at least two independent and different mechanisms causing interactions between nutrients and making plant behaviour deviate from LM; in a strict LM world the isoclines would be piecewise linear with line segments parallel to the axes. First, the use of one nutrient depends on the availability of another nutrient; the amounts of enzymes depend on the amounts of ribosomes. Second, adding one nutrient will reduce resources used for its uptake, increasing resources that can be invested in uptake of other nutrients. When both mechanisms are operating simultaneously, it seems that the effect of nutrients operating in a chain is more important than the flexibility in allocation (Fig. 4). Moreover, the deviation from strict LM behaviour depends on growth rate and increases as the growth rate increases.
A consequence of the gradual shift between limitation by N and P in the model is, on the one hand, that the concept of optimum ratios becomes less sharp; there is no longer a precisely defined ratio where the limitation shifts from one element to the other. This agrees with the observations of Koerselman & Meuleman (1996) and Güsewell (2004) that there is a range of N : P values where limitation shifts from one element to the other. On the other hand, Paris (1992) analysed an agricultural experiment with N and P application to corn and found that a strict LM model fitted observations better than response curves with smoother transitions between N and P limitations. However, it might be difficult experimentally to distinguish the curved response curves in Fig. 4 from two straight lines meeting at a sharp corner. Another aspect is that optimum concentrations, according to our model, are no longer single values but vary with nutrient supply ratios (Fig. 2) and that the supply of P relative to N has to increase with the relative growth rate (Fig. 4). At the same time, our predictions do not conform with MLH because there will not be a significant response to all levels of addition of a single element; such responses will be restricted to a narrow range around the optimum but can also depend on the availability of the other element (Fig. 4). Outside the narrow range around the optimum we predict responses according to LM. Our analysis, therefore, positions itself somewhere inbetween LM and MLH.
In a strict LM world, the N and P limitation would be of the serial type in the terminology of Harpole et al. (2011) but a response to, for example, N for a Plimited plant would also require a sufficiently large addition of P to push the plant into N limitation; under too small P additions the plant would remain Plimited and it is only in some region in the A_{N}–A_{P} space around the optimum N : P ratios where in practice a single element is not growth limiting. Coupling the use of nutrients and a flexible allocation of a plant’s resources towards uptake of the least available nutrient, amplifies the smoothness in the transition between limiting elements; the curves for flexible allocation change less sharply than those for fixed allocation in Fig. 4. This analysis could be extended to include optimization with respect to uptake allocation.
However, because of the way the two elements interact in the growth process the colimitation would rather be of the superadditive independent type (Harpole et al., 2011). It should be noted that the possibility of detecting the character of the limitation will depend on the initial fertility of the system, because plants in infertile systems behave closely to LM whereas plants in fertile systems more clearly express the independent colimitation (Fig. 4). The same change in external availability, for example, brought about by fertilization, will also cause quite different responses depending upon the initial status of the system.
A critical question is: have plants have evolved schemes for allocating between uptakes of different nutrients? On the one hand, Rubio et al. (2003) argue that C allocated below ground will increase uptake of all nutrients, not just the limiting ones. On the other, Treseder & Vitousek (2001) and Fujita et al. (2010) show that P fertilization can reduce the activity of several mechanisms for P acquisition, whereas N fertilization can increase extracellular phosphatase activity and hence stimulate P uptake. It remains, therefore, an open question of how important changes in allocation can be for nutrient interactions. However, as Fig. 3 shows, it should be possible to observe the extent to which plants can direct the uptake towards specific elements. If there is no flexibility in allocation, adding one element will increase its concentration in the plant at the expense of the concentration of the other element. But a flexible allocation leads to increases in both elements when the external availability of either increases. If there is no flexibility in uptake, the plant N : P ratio will follow the supply N : P, whereas if there is flexibility, the plant N : P will depend not only on the N : P ratio of the supply, but also on the absolute levels of the supplies (Fig. 6).
Another effect of the flexible allocation mechanism is that it moderates the effect of external nutrient variability such that the plant N : P ratio varies considerably less than the external N : P (Fig. 6). Our model exaggerates the response in plant N : P to the external N : P at the extremes because we have not included any restrictions on uptake when internal concentrations tend to be too high. Including such a mechanism should bend the response curves at the ends for both the flexible and (even more so) for the fixed allocation in Fig. 6 to more closely resemble those proposed by Güsewell (2004).
The model presented here has similarities to and differences from other models of multielement limitations. The curved isoclines we show in Fig. 4 seem to be a general feature when organisms can allocate between uptakes of essential resources independently, if they are derived from specific physiological considerations or more general functions (e.g. Abrams, 1987; Tilman, 1987). However, because of the particular physiological interactions between N and P we obtain curved isoclines even in the absence of a flexible allocation; the latter just emphasizes this aspect. The difference between curved and straightline isoclines depends therefore both on flexibility and the organism’s use of the resources.
The Droop model (Droop, 1974) and a simpler version of our model here (Ågren, 2004) both fit well to chemostat experiments with phytoplankton (Bi et al., 2012). It is, therefore, to be expected that the models of chemostat experiments by Klausmeier et al. (2004, 2007), which are based on strict Liebig behaviour with the Droop model, show several similarities to our models and in particular the way the supply ratio of N and P is reflected in the organisms N : P ratio. Other aspects are more difficult to compare because we use a static nutrient environment, which is a reasonable simplifying assumption in a terrestrial environment but not under the dynamic conditions in a chemostat. In chemostat experiments the relative growth is also fixed whereas we derive it from environmental nutrient availabilities. However, as Bi et al. remark, the model by Ågren (2004) may explain the underlying biochemical principle for the Droop model.
We see several possibilities of extending this model. One obvious extension is to consider shifts between allocation to aboveground and belowground resources, by varying Φ_{CL} in our terminology. A more difficult extension would be to follow Klausmeier et al. (2007) and consider the consequences of a response time for changes in allocation (ε). However, this would require the inclusion of a temporal variability in nutrient availability in the environment; this is a much more complex issue in soils than in chemostats. Understanding and being able to predict the dynamics of soil nutrient availability, including intra and interspecific competition between roots, is required before more extensive applications in agriculture and other environmental contexts would become meaningful. However, our analysis helps explain why an increasing N deposition can induce an increased access to P (Marklein & Houlton, 2012) and possibly other elements, thus delaying an onset of N saturation.
In conclusion, we have demonstrated how the interaction between N and P is modified from a strict application of Liebig’s law of the minimum by associating nutrient uptake with a cost in assimilated C. The chainlike operation of N and P causes further departure from LM. The two mechanisms, alone or in combination, should make ecosystems in general appear as colimited by the two nutrients; the law of the minimum may, however, be a good firstorder approximation.