Our starting point was a classical resource competition model (Tilman 1982) in which Phalaris (Ph) biomass (indicated with BPh in Table 1 and 2) and Carex (C) biomass (BC) compete for two resources: soil nitrogen (N) and light (L). The model also includes accumulation of litter from both plant species (DPh and DC). The model equations are shown in Table 1, the interpretation and sources for parameter values are shown in Table 2. For further details about model parameterization see Appendix S1 in Supporting Information. Carex is a common species in native American sedge meadows, and may thus serve to mimic competition between Phalaris and native American wetland species more generally (Perry, Galatowitsch & Rosen 2004). For the sake of brevity, however, we will refer to the native competitor species as Carex from here.
Table 1. Model equations
Table 2. Model parameters and state variables
|gL,C||Maximum growth rate of Carex under light limitation||0.25||day−1||1|
|gL,Ph||Maximum growth rate of Phalaris under light limitation||0.25||day−1||1|
|kL,C||Light availability at which Carex reaches half its maximal growth rate (if light limited)||50||mol m−2||2|
|kL,Ph||Light availability at which Phalaris reaches half its maximal growth rate (if light limited)||21||mol m−2||2|
|gN,C||Maximum growth rate Carex under nitrogen limitation||0.25||day−1||1|
|gN,Ph||Maximum growth rate Phalaris under nitrogen limitation||0.25||day−1||1|
|kN,C||Nitrogen availability at which Carex reaches half its maximal growth rate (if N limited)||30||mg kg−1||3|
|kN,Ph||Nitrogen availability at which Phalaris reaches half its maximal growth rate (if N limited)||35||mg kg−1||3|
|mC||Mortality rate Carex||0.005||day−1||4,5|
|mPh||Mortality rate Phalaris||0.01||day−1||5|
|a||Turnover rate of nutrient supply||0.005||day−1||2,3|
|S||Nitrogen availability in absence of plants||8–100||mg kg−1||2,3|
|qN,C||Nitrogen content of tissue of Carex||15||mg g−1||1,6,7|
|qN,Ph||Nitrogen content of tissue of Phalaris||15||mg g−1||1,6,7|
|ρ||Soil bulk density||530||g.m−3||8|
|lRoot||Rooting depth of plant species||1||m||9|
|αN,C||Nutrient–litter feedback coefficient Carex||0.7||-||10|
|αN,Ph||Nutrient–litter feedback coefficient Phalaris||0.7||-||10|
|dC||Carex litter decomposition rate||0.003||day−1||10,11,12,13|
|dPh||Phalaris litter decomposition rate||0.003||day−1||10,11,12,13|
|L0||Light supply rate||0.86–50||mol m−2 day−1||2|
|γL,C||Light interception coefficient Carex||0.03||m2 g−1||14,15|
|γL,Ph||Light interception coefficient Phalaris||0.04||m2 g−1||14,15|
|αL,C||Light–litter feedback coefficient Carex||0.01||m2 g−1||14,15,16|
|αL,Ph||Light–litter feedback coefficient Phalaris||0.013||m2 g−1||14,15,16|
|QN,C||Nitrogen content of Carex litter at which it decomposes at rate dC||15||mg g−1||- (scaling parameter)|
|QN,Ph||Nitrogen content of Phalaris litter at which it decomposes at rate dP||15||mg g−1||- (scaling parameter)|
|cRate||Parameter determining the characteristic timescale of light dynamics||>> 1||day−1||- (scaling parameter)|
|BC||Aboveground living biomass of Carex||–||g m−2||–|
|BPh||Aboveground living biomass of Phalaris||–||g m−2||–|
|N||Nitrogen availability in soil||–||mg kg−1||–|
|DC||Aboveground litter mass of Carex||–||g m−2||–|
|DPh||Aboveground litter mass of Phalaris||–||g m−2||–|
|L||Light availability||–||mol m−2 day−1||–|
In the model, plant biomass dynamics comprise growth (see model term I in Table 1) and mortality (II). Growth is determined by the availability of only one of the two resources. Which resource is limiting depends on the maximum growth rates of the plant species (gN,i and gL,i, with i referring to either Ph or C) and on the saturation constants in the resource uptake function (kN,i and kL,i). However, which resource is limiting also depends on the relative availability of each resource (dynamically varying over time). Hence, which resource is limiting may vary over time. The per capita mortality rate (mi) is assumed to be constant. Note that mortality not necessarily involves the death of an individual plant, but also incorporates turnover of leaves, for example. We only consider above-ground biomass in the model, assuming that above-ground biomass determines a species’ competitiveness (i.e. its R* values, sensuTilman (1982)) for both light and nitrogen. This implies a constant root:shoot ratio and a similar allocation strategy, which seems a reasonable assumption when focusing on wetland plants with similar life-history traits.
The nitrogen dynamics follow Tilman (1982) and Daufresne & Hedin (2005). Nitrogen dynamics comprise supply (III), uptake by both plant species (IV) and release from decomposition of litter of both plant species (V). Nitrogen supply can be viewed as the sum of a constant input (S) and a loss term, which increases linearly with nitrogen availability in the system. The turnover rate of nitrogen in the system is determined by the parameter a. The amount of nitrogen that is taken up is determined by the amount of plant growth multiplied by the nitrogen content of plant tissue (qN,I, in mg g−1). Because plant density is expressed in g m−2, the nitrogen uptake by plants is also expressed on a per-area basis, namely mg m−2. Soil nitrogen availability, however, is expressed in mg kg−1 soil. Therefore, to convert the nitrogen uptake by plants (in mg m−2) into a decrease in soil nitrogen availability (in mg kg−1), the nitrogen uptake needs to be divided by the amount of soil per area (kg m−2). The amount of soil per area is the product of the soil bulk density (ρ, in kg m−3) and the rooting depth of the plant species (lRoot, in m). Here we assumed a constant rooting depth for the plant species, and that nitrogen within this rooting zone is well mixed. Also, we made the assumption that the nitrogen content of plant tissue is constant (following e.g. Eppinga et al. 2009). Finally, release occurs through decomposition and mineralization of litter, of which a constant fraction (determined by the parameters αN,Cand αN,Ph) becomes available again for plants (Daufresne & Hedin 2005). The decomposition rate of litter is assumed to increase linearly with litter quality (e.g. Enrìquez, Duarte & Sand-Jensen 1993), which is quantified in the model by the quotients qN,C/QN,C and qN,Ph/QN,Ph. Again we made the simplifying assumption that the nitrogen content of litter remains constant (following e.g. Eppinga et al. 2009).
The other limiting resource is light. Light dynamics comprise light input (VIII), interception by both plant species (IX) and both litter types (X) and a light loss term (XI). Light input (L0) is assumed to be constant. To model the rate of change in light availability (Table 1), all processes affecting light dynamics are governed by the rate parameter cRate. We note, however, that this rate parameter has no influence on the equilibrium light level of the system (Table 1). Light interception of both vegetation and litter is based on Lambert–Beer’s law for light absorption. In the case of vegetation and litter, absorption efficiency depends on two properties: the height:mass ratio of the vegetation or litter layer and its shoot density (see Appendix S2). To enable analytical treatment, the Lambert–Beer equation can be approximated by a hyperbolic function of biomass, summarizing the two properties into a single parameter (e.g. Reynolds & Pacala 1993). Because Lambert–Beer’s law is generally applicable to any light-absorbing material (e.g. Grace & Woolhouse 1973), we used the same approach to model light interception by plant litter. Since Phalaris, Carex and their litter have different properties in terms of their height:mass ratios and their shoot densities, values for γL,i for biomass and αL,I for litter were set accordingly (see Appendix S1, S2). It should be noted, however, that our approach does not consider vegetation height explicitly. This is a simplification, because the order of colonization may determine which species occupies the highest position in the canopy at a certain point in time, and thus intercepts most light at that point (Perry, Neuhauser & Galatowitsch 2003). We refer to Appendix S2 for a comparison between the approach adopted here and the approach taken in a previous model study (Perry, Neuhauser & Galatowitsch 2003). This previous model study included hierarchical competition explicitly for light (but not competition for nitrogen), for the same species considered here (Perry, Neuhauser & Galatowitsch 2003). Finally, a light loss term is included, because light cannot be stored. Light dynamics can be assumed to be relatively fast as compared to other processes in the system (i.e. cRate is relatively large). Hence, a quasi-steady-state approach can be used, meaning that light availability is assumed to equilibrate instantaneously (e.g. Reynolds & Pacala 1993; Table 1,2)
The model also includes a litter pool for each plant species, the dynamics following Daufresne & Hedin (2005). For each litter pool, the dynamics comprise input of dead biomass (VI) and litter losses through decomposition (VII), which have already been described above. Note that the litter pools are treated as separate state variables to enable different litter characteristics of the two species, such as litter quality and decomposition rate. This is a simplification, because empirical studies have shown that mass loss from litter mixtures occurs at a different rate than would be expected from the sum of the species-specific mass losses (e.g. Nilsson, Wardle & Dahlberg 1999).
As noted above, we parameterized the model to simulate competition between Phalaris and Carex for soil nitrogen and light. Parameter values were derived from various sources (Table 2; see Appendix S1). Parameters describing soil nitrogen uptake and light interception were calibrated to mimic the results of previous competition experiments between Phalaris and Carex (Perry & Galatowitsch 2004; Perry, Galatowitsch & Rosen 2004). In general, these previous experiments suggest that Phalaris is a stronger competitor for light (Perry & Galatowitsch 2004), whereas Carex is a stronger competitor for soil nitrogen (Perry, Galatowitsch & Rosen 2004). Here, we consider a plant species to be a stronger competitor for a resource if it has the lowest R* value for this resource. We calibrated the R* values of Phalaris and Carex for soil nitrogen and light by tuning the saturation constants in the resource uptake functions. If a saturation constant is small, it means that a species can attain a relatively high growth rate at relatively low resource availability. Hence, a small saturation constant reflects a strong competitor.
The saturation constants were set in a way that at low light availability (L0 = 0.86 mol m−2 day−1, all light availabilities follow the treatment levels in Perry & Galatowitsch 2004), both species went extinct. This agreed with the observation of Perry & Galatowitsch (2004) that biomass of both species at this light intensity was reduced by 99%. Under high light availability (L0 = 50 mol m−2 day−1) and intermediate nitrogen availability (forced in the model by setting the supply rate at S = 60 mg kg−1, all nitrogen supply rates are based on treatment levels in Perry, Galatowitsch & Rosen 2004), there was coexistence, which agreed with the observation that both species reached similar biomass at these levels of light and nitrogen (Perry & Galatowitsch 2004). At intermediate light availability (L0 = 17 mol m−2 day−1), Carex biomass was reduced more than that of Phalaris, as shown experimentally by Perry & Galatowitsch (2004). The exclusion of Carex that occurs in the model, however, was not observed within the time span of the experiment (Perry & Galatowitsch 2004). Saturation constants were set in a way that both species coexisted under intermediate nitrogen availability, which corresponded to the observation that both species reached similar biomass at intermediate soil nitrogen levels (Perry, Galatowitsch & Rosen 2004). At high nitrogen availability (S = 90 mg kg−1) Carex was excluded, which agreed with the observation that Carex biomass was low when grown with Phalaris at high nitrogen availability (Perry, Galatowitsch & Rosen 2004). At low nitrogen availability (S = 30 mg kg−1), however, Phalaris is excluded, which is in line with the observation that Phalaris biomass is low when grown with Carex at low nitrogen availability (Perry, Galatowitsch & Rosen 2004).
Model Analyses of Litter Feedbacks and Evolutionary Change
Including litter pool dynamics in the model creates the possibility of studying plant–litter feedbacks: changes in the litter pool may alter the availability of resources (nitrogen and light), which affects plant growth. In turn, these changes in plant growth feed back to changes in the litter pools. Because the model encompasses two resources, two types of feedbacks can be distinguished. First, decomposition of litter releases nutrients (in this case soil nitrogen), which become available again for plant uptake. This can change plant community composition in favour of weak competitors for nutrients and hence feed back to changes in litter pools. From here we refer to this feedback as nutrient–litter feedback. The strength of the nutrient–litter feedback is determined by the parameters αN,Ph and αN,C. Second, the presence of litter decreases availability for light, which may change the plant community composition in favour of strong competitors for light. Again, such a change in community composition may feed back to changes in litter pools. Henceforth we refer to this feedback as light–litter feedback. The strength of the light–litter feedback is determined by the parameters αL,Ph and αL,C. In the model litter feedbacks could be switched off by setting the feedback coefficients (αL,Ph, αL,C, αN,Ph and αN,C) to zero.
Our analyses focused on three Phalaris invasion scenarios. The first scenario considered how Phalaris invasion could be influenced by the nutrient–litter and light–litter feedbacks, through on-and-off switching of these feedbacks in a full-factorial design. We analysed graphically how these feedbacks affected the competitive outcome between Phalaris and Carex over a range of nitrogen and light supply parameter values. For the studied parameterization, we also calculated the extent to which litter feedbacks reduced the parameter space for coexistence.
In the second scenario, we considered the effect of two evolutionary changes that have been observed in invasive Phalaris genotypes: a higher growth rate and a higher C:N ratio as compared to native European genotypes. The evolutionary changes were documented in common-garden experiments in the greenhouse. For the growth rate experiment, 41 native European genotypes (from France and the Czech Republic) and 49 invasive genotypes (from Vermont and North Carolina, USA) were grown under the same nutrient and light conditions, for a period of 73 days. More details on this experiment can be found in Lavergne & Molofsky (2007). For the C:N ratio experiment, 46 native and 56 invasive genotypes were grown under the same nutrient and light conditions, for a period of 3 months (J. Molofsky et al. unpubl. data). During both experiments, dead biomass was removed from the pots, meaning that the results were not affected by above-ground litter. As noted above, Phalaris’ growth rate is determined in the model by the parameters gN,Ph and gL,Ph. The C:N ratio of Phalaris tissue is reflected by 1/qN,Ph, meaning that Phalaris’ C:N ratio increases with a decreasing value of the parameter qN,Ph. We analysed graphically the effect of higher values of gN,Ph and gL,Ph and a lower value of qN,Ph on the competitive outcome between Phalaris and Carex over a range of nitrogen and light supply parameter values. Taking the coexistence situation without feedbacks and evolutionary changes as a starting point, we also derived analytically the magnitude of evolutionary change that would be required to qualitatively alter the competitive outcome between Phalaris and Carex (all analytical analyses are presented in Appendix S3). We then compared these calculated critical points for the model parameterization with the observed changes in the experiments. The analytical results yield a general expression of critical points in terms of model parameters (see Appendix S3), but inserting the parameter values of a specific parameterization in this general expression yields a critical point for that parameterization. Apart from increasing invader competitiveness, evolutionary change may also enhance invader success by broadening the climatic niche of evolved genotypes (Gallagher et al. 2010). Thus, evolutionary change stimulating geographic expansion during the so-called spread phase of invasions (e.g. Theoharides & Dukes 2007) may also contribute to the success of Phalaris in North America (Lavergne & Molofsky 2007). To investigate this possibility, we also considered competition between Phalaris and native species in habitats that are outside the original climatic niche of Phalaris (i.e. warmer or colder). In order to focus the analysis on effects of evolutionary change in different climates, we assumed that the native species had the same characteristics as Carex considered in the other model scenarios. We thus considered competition between Phalaris and an ecological congener of Carex in a different climatic habitat. Habitat suitability was assumed to affect the maximum growth rate of Phalaris (gN,Ph and gL,Ph) following Tilman (2004), who assumed the maximum growth rate of plant species being dependent on temperature. We assumed a 20% reduction of Phalaris’ maximum growth rates in different climatic habitats. Under cultivated and relatively dry conditions within one site, a reduction in Phalaris’ photosynthesis rate of 20% was observed when the temperature was 5–6 °C away from the optimum temperature (Shurpali et al. 2009).
In the third scenario, we analysed effects of both litter feedbacks and evolutionary change (focusing on the effect of a higher C:N ratio in invasive genotypes) to explore whether these factors could interactively affect plant competition and invasion. In this scenario, we analysed how the stability of community and single-species equilibria changed with increasing C:N ratio for invasive Phalaris genotypes. We considered a system with higher nitrogen supply (S = 10.25 mg kg−1) and a system with lower nitrogen supply (S = 8 mg kg−1). Nitrogen supply rates were set in a way that in the absence of evolutionary change (but in the presence of litter feedbacks), Phalaris was excluded by Carex in the second system, but could coexist with Carex in the first system. Using analytical results (see Appendix S3), we also calculated for both systems the critical Phalaris C:N ratio at which the competitive outcome of the system changed. We examined how this critical Phalaris C:N ratio was affected by the absence or presence of litter feedbacks.
Finally, we performed two types of sensitivity analyses (see Appendix S4), to assess the model sensitivity to parameter values (using an elasticity analysis, e.g. Eppinga et al. 2009) and to assess the robustness of model results (by quantifying the sensitivity ranges of the model parameters, c.f. Eppinga et al. 2006).