Plant competition experiments: testing hypotheses and estimating the probability of coexistence
The purpose of making plant competition studies is ultimately to be able to predict the dynamics of natural plant communities. Theoretically, this can be done by analyzing competition experiments using a competition model where the dynamic properties are known, i.e., for certain combinations of parameter values in the model we expect a certain behavior of a community (Pascual and Kareiva 1996).
However, it is still commonly found in the plant competition literature that competition or growth parameters in various competition models are estimated without discussing whether the observed effects are statistically significant (Cousens and O’Neill 1993). Additionally, when parameters have been estimated it has been difficult to discuss the importance of the findings, because the properties of the competition model have been unknown. Important questions like: “Can a plant invade a specific plant community?” or “Can two plant species coexist?” are often not answered on the basis of the estimated parameters, but solely with the use of biological intuition.
In this paper the analysis of plant competition experiments with variable plant densities is discussed, based on three published data sets where plant competition is measured at several proportions and combined densities. I will show how conclusions regarding plant community dynamics from typical plant competition experiments can be extended and based on a statistical framework.
Plant competition has usually been studied by measuring fitness-related traits of different plant species that are grown in experimentally manipulated mixtures (e.g., de Wit 1960, Marshall and Jain 1969, Wright 1981, Firbank and Watkinson 1985). Most of these experimentally based studies of plant competition have examined the effect on yield (plant biomass, number of fruits, or number of seeds) per area in a two-species design with variable plant densities or environments.
The combined density is almost always described as a linear function of the two densities, whereas effects of interaction and threshold densities have been less discussed [although see Law and Watkinson (1987), Francis and Pyke (1996), Cousens and O’Neill (1993)]. If a plant is a strong competitor only above a certain threshold density (e.g., competition by allelopathic inhibition), or when it is present at high density and the other species is present at low density, then the dynamic behavior of the plant community may change. In this paper, more complex competition models, as when the outcome of competition depends on the interaction between plant densities, will be integrated mathematically with simple competition models, thus allowing the complex models to be tested by a standard hierarchical statistical procedure.
Traditionally, the results of competition experiments have been reported using a model where different indices describing the effect of competition on yield in mixture plot is defined as functions of the yield in pure plots and the mixture proportion. The model was introduced by de Wit (1960), and much analytical work has been done using that model (Pakes and Maller 1990, Allen et al. 1996). However, the absolute fitness concept is not easily incorporated into the de Wit competition model (Inouye and Schaffer 1981) and the model is not readily comparable with the classical Lotka-Volterra competition model. Furthermore, the indices in the de Wit competition model lead to statistical difficulties (Connolly 1986, Skovgaard 1986). For these reason, I will take an approach similar to the plant competition model proposed by Firbank and Watkinson (1985), which is a hyperbolic competition model that is analogous to a discrete Lotka-Volterra competition model. The reason for choosing a hyperbolic function instead of a discrete Lotka-Volterra competition model is that the hyperbolic response function has a high structural flexibility and has empirically been shown to fit plant competition data well (Law and Watkinson 1987, Cousens 1991).
As for the Lotka-Volterra model and the de Wit model the condition for species coexistence is known and hypotheses on coexistence can be tested (e.g., Pascual and Kareiva 1996, Pakes and Maller 1990, Allen et al. 1996). However, the condition for coexistence in the hyperbolic plant competition model has until now not been examined. Here, I will examine the condition for the four possible outcomes of two-species competition (coexistence, one of the two species will always out-compete the other, or either species may out-compete the other depending on the initial conditions). The found conditions will be used to reparameterize the hyperbolic competition model in order to estimate the probabilities of each of the possible outcomes using Bayesian statistics. This will be exemplified for competition between Avena fatua and A. barbara (Marshall and Jain 1969) where the Bayesian posterior probabilities of the four possible outcomes will be estimated.
The Bayesian posterior probabilities of different ecological scenarios allow the construction of testable hypothesis from plant competition experiments, but the probabilities can also be used for ecological risk assessment of introduced species or genotypes into plant communities: (1) plant communities with few species (for example, the invasion of grasses, e.g., Deschampsia flexuosa into populations of Calluna vulgaris in North European heathland); (2) plant communities with high biodiversity, if the introduced species can be assumed to outcompete a specific plant that shares the same guild or multidimensional niche, or (3) invasion of selfing genotypes into a plant population.
Data sets. – Data on the number of total spikelets as a function of seed densities in Avena fatua and A. barbara were obtained from Fig. 13 in Marshall and Jain (1969). The data were given as means of four replicates and as an example of a data set from a plant competition experiment the means are shown in Fig. 1. For Trifolium pratense and Lolium multiflorum, data on the total biomass 7 wk after planting as a function of plant densities were obtained from Fig. 1 in Wright (1981). The data were given as means of two replicates. For Triticum aestivum and Agrostemma githago, data on the total biomass 3 mo after sowing as a function of seed densities were obtained from Fig. 6 in Firbank and Watkinson (1985). The experiments were done in three replicates, and all data were presented in the figure. Since only the mean values for the different treatments were reported in the first two experiments, the estimated probabilities will only be suggestive.
Hyperbolic response function. – The three data sets were fitted to a hyperbolic response function, first used by Bleasdale and Nelder (1960) and further analyzed by Mead (1970), Gillis and Ratkowsky (1978), and Seber and Wild (1989):
[Mathematical Expression Omitted] (1)
where y, is the yield, e.g., biomass or number of seeds [TABULAR DATA FOR TABLE 1 OMITTED] per unit area, [D.sub.i] is the density, [x.sub.i] is the combined plant density, [a.sub.i], [b.sub.i], [d.sub.i], and [f.sub.i] are shape parameters, and i is the species index.
The hyperbolic response function has some properties that are biologically significant. For species i we have that, if [d.sub.i] = [f.sub.i], then the yield per unit area ([y.sub.i]) has an asymptotic value for high densities ([x.sub.i]) (Mead 1970). Furthermore, if [d.sub.i]/[f.sub.i] = 3/2, then the hyperbolic response function corresponds to “the 3/2th power law of self-thinning” (Yoda et al. 1963, Seber and Wild 1989). Note that self-thinning is defined as a process that is occurring during a time interval and therefore cannot be described from only one sample in time. However, empirically it has been shown that “the 3/2th power law of self-thinning” can be observed in yield-density plots at a single point in time (Yoda et al. 1963), which can be explained by a simple allometric relationship between size and density (Seber and Wild 1989).
In the simple case when the shape parameters [d.sub.i] and [f.sub.i] are one, then the yield per plant at low density (intrinsic yield) is, 1/[a.sub.i], and the asymptotic value of yield per area is 1/[b.sub.i] at high density.
The combined plant density, [x.sub.i], is defined analogously to the Lotka-Volterra competition model in order to describe the competitive interaction between species. Although the combined plant density easily can be defined for several species, I will only consider the two species case (i = 1, 2):
[x.sub.1] = [D.sub.1] +[c.sub.2][D.sub.2] + [e.sub.1][D.sub.1][D.sub.2]
[x.sub.2] = [c.sub.1][D.sub.1] + [D.sub.2] + [e.sub.2][D.sub.1][D.sub.2] (2)
where [D.sub.i] is the density of species i, c,. is the competition coefficient, which measures the competitive effect of species i on the other species. If [c.sub.i] [greater than] 1, then species i has a larger negative effect on yield of the other species than the other species itself. If [c.sub.i] [less than] 1, then species i has a smaller negative effect on yield of the other species than itself, and if [c.sub.i] [less than] 0, then species i has a positive effect on yield of the other species. [e.sub.i] is a parameter that measures any effect of interaction between the two plant densities on the yield of species i. The effect of the two plant densities on the yield is assumed independent when [e.sub.i] = 0.
When the effect of the two plant densities on yield can be assumed to be independent, it is examined whether there is any significant “threshold effects” by an additional quadratic term in the expression of the combined densities:
[x.sub.1] = [D.sub.1] + [c.sub.2][D.sub.2] + [g.sub.11][[D.sub.1].sup.2] + [g.sub.12][[D.sub.2].sup.2]
[x.sub.2] = [c.sub.1][D.sub.1] + [D.sub.2] + [g.sub.21][[D.sub.1].sup.2] + [g.sub.22][[D.sub.2].sup.2] (3)
If [g.sub.ij] cannot be assumed to be zero, the biological significance of the quadratic term can be interpreted from a graphical representation of the model.
Hypotheses. – The first two mutually exclusive hypotheses that were tested in the three data sets were whether [d.sub.i] = [f.sub.i] (H2) and [d.sub.i]/[f.sub.i] = 312 (H3). After these initial tests, it was tested whether the number of shape parameters in the hyperbolic response function could be reduced (Mead 1970). Then it was tested whether [f.sub.i] = 1 (H4), and whether [d.sub.i] = 1 (H5). After these shape parameter reduction tests, it was tested whether the effect on yield of the two densities were independent, i.e., whether [e.sub.i] = 0 (H6). If this hypothesis was accepted, it was tested whether any of the additional quadratic terms introduced in Eq. 3 improved the fit significantly (H7, H8). Finally, it was tested whether the competition parameter, [c.sub.i], was equal to one (H9) or zero (H10). If [c.sub.1] = 1, then the densities of species 1 and species 2 had a similar effect on the yield of species 2. If [c.sub.1] = 0, then the density of species 1 has no effect on the yield of species 2.
A Box-Cox transformation of the data (Seber and Wild 1989) and a visual inspection of the residuals suggested that the data should be log transformed. After the data had been log transformed the residuals could be assumed to be approximately normally distributed. Both in the Avena fatua/A. barbata and Triticum aestivum/Agrostemma githago competition experiments there was one outlier, but in both cases the conclusions of the analyses were unaltered if the outlier was omitted. The results will be presented without the outliers. All tests were maximum log-likelihood ratio tests, where the log-likelihood functions (Seber and Wild 1989) were maximized by Newton-Raphson iterations.
Conditions for coexistence. – In the following, it is assumed that yield is measured as absolute fitness for each competing species as a function of the densities of seedlings for both competing plant species in the beginning of the growing season. Specifically, for an annual species, the absolute fitness is estimated as the number of seeds produced multiplied by a constant probability ([p.sub.i]) that the seed will germinate the following year. Thus, for an annual species, the number of seedlings the following year is a function of the densities of seedlings for both competing plant species and the seed germination probability:
[[z.sub.1].sup.t+1] = [p.sub.1][h.sub.1]([[z.sub.1].sup.t], [[z.sub.2].sup.t])
[[z.sub.2].sup.t+1] = [p.sub.2][h.sub.2]([[z.sub.1].sup.t], [[z.sub.2].sup.t]) (4)
where [[z.sub.i].sup.t] is the number of seedlings of species i in the beginning of the season in year t and [h.sub.i] is the hyperbolic response function for species i. The equilibrium values for [Mathematical Expression Omitted] could be found in the general case, but the expressions were too large to solve for any positive solutions. However in the case when [e.sub.i] = 0, and [g.sub.ij] = 0 the equilibrium values were
[Mathematical Expression Omitted] (5)
[Mathematical Expression Omitted] (6)
[Mathematical Expression Omitted]
[Mathematical Expression Omitted] (7)
[Mathematical Expression Omitted]. (8)
The nontrivial equilibrium Eqs. 7, for [c.sub.1][c.sub.2] [not equal to] 1, were set to zero and solved for [c.sub.i]:
[Mathematical Expression Omitted]
[Mathematical Expression Omitted]. (9)
A local stability analysis using the first term of the Taylor polynomial of recurrence equations (Eq. 4) evaluated at the nontrivial equilibrium (Eqs. 7) (the Jacobian matrix) could only be analyzed analytically in the case when 4 = 1, [f.sub.i] = 1, [e.sub.i] = 0, and [g.sub.ij] = 0. In this simple case the roots to the nontrivial equilibrium equations (Eqs. 7) are
[[c.sub.1].sup.*] = [b.sub.1] ([p.sub.2] – [a.sub.2]) / [b.sub.2]([p.sub.1] – [a.sub.1]) and [[c.sub.2].sup.*] = [b.sub.2]([p.sub.1] – [a.sub.1]) / [b.sub.1] ([p.sub.2] – [a.sub.2]). (10)
If ([p.sub.i] – [a.sub.i])/[b.sub.i] [less than] 0, then species i cannot exist on its own and it will always go extinct. If it is assumed that ([p.sub.i] – [a.sub.i])/[b.sub.i], [greater than] 0, then, as for the Lotka-Volterra competition equations, there are four different cases (Table 1). If [c.sub.1] [less than] [[c.sub.1].sup.*] and [c.sub.2] [less than] [[c.sub.2].sup.*], both [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are positive, and the absolute value of the leading eigenvalue of the Jacobian matrix is less than one. Therefore, the nontrivial equilibrium (Eqs. 7) is positive and stable when [c.sub.1] [less than] [[c.sub.1].sup.*] and [c.sub.2] [less than] [[c.sub.2].sup.*], and the two species will coexist and approach the equilibrium (Eqs. 7). If [c.sub.1] [less than] [[c.sub.1].sup.*], [c.sub.2] [greater than] [[c.sub.2].sup.*] then [Mathematical Expression Omitted] is negative, and species 2 will outcompete species 1 and approach the equilibrium (Eq. 6). If [c.sub.1] [greater than] [[c.sub.1].sup.*], [c.sub.2] [less than] [[c.sub.2].sup.*] then [Mathematical Expression Omitted] is negative, and species I will outcompete species 2 and approach the equilibrium (Eq. 8). If [c.sub.1] [greater than] [[c.sub.2].sup.*], [c.sub.2] [greater than] [[c.sub.2].sup.*], both [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are positive, but the absolute value of the leading eigenvalue of the Jacobian matrix is larger than one. Therefore, the equilibrium (Eqs. 7) is unstable and either one of the species might outcompete the other species depending on the initial conditions, and the population will either approach the equilibrium value (Eq. 6) or (Eq. 8). The initial condition that determines which species will outcompete the other species depends on the parameter values, and can be found by numerical simulations. Generally, a high initial density increased the chance for out-competing the other species. Specifically, if initially there was only one plant species close to its equilibrium density, then this species would outcompete other species that were introduced at low densities.
Numerical examinations of the recursion equations, as well as the eigenvalues of the Jacobian matrix in the more general model when [d.sub.i] [not equal to] 1 and [f.sub.i] [not equal to] 1, suggested that the above conclusions also were valid in that model. However, in the case when [c.sub.1] [greater than] [[c.sub.1].sup.*] and [c.sub.2] [greater than] [[c.sub.2].sup.*], the chance of outcompeting the other species did not increase monotonously with the initial density. For some parameter values, one of the species had an optimum initial density. At the optimal density of species 1, species 2 was outcompeted at a higher initial density, than for any other initial density of species 1. This effect is due to the nonasymptotic value of fitness at high densities.
Many plant species have a long-term seed bank strategy, i.e., not all surviving seeds germinate in the following growing season, but may instead germinate a number of years after they are produced. As a first approximation, the time to germination can be viewed as a geometric distribution in time with one shape parameter, q (Rees and Long 1993). If the recurrence equations (Eqs. 4) are expanded in time, so that the seed densities are functions of the seed production in the previous generations times, the probability that they will germinate as defined by the geometric distribution is:
[[z.sub.1].sup.t+1] = [p.sub.1]h([[z.sub.1].sup.t][[Gamma].sub.0,q] + [[z.sub.1].sup.t-1][[Gamma].sub.1,q] + . . ., [[z.sub.2].sup.t][[Gamma].sub.0,q] + [[z.sub.2].sup.t-1] [[Gamma].sub.1,q] + . . .)
[[z.sub.2].sup.t+1] = [p.sub.2]h([[z.sub.1].sup.t][[Gamma].sub.0,q] + [[z.sub.1].sup.t-1][[Gamma].sub.1,q] + . . ., [[z.sub.2].sup.t][[Gamma].sub.0,q] + [[z.sub.2].sup.t-1] [[Gamma].sub.1,q] + . . .) (11)
where [[Gamma].sub.u,q] = (1 – q)[q.sup.u] for (0 [less than] q [less than] 1) is the probability of germinating u years after they are produced. Using that the probabilities of the geometric distribution sums to one, it can be shown that the equilibrium values (Eqs. 5-8) remain unaltered, and a numerical simulation of the condition for coexistence showed the same conclusions as above. The only effect of adding a seed [TABULAR DATA FOR TABLE 2 OMITTED] bank to the model was that the rate of approach to an equilibrium point was a decreasing function of q.
Estimating the probability of coexistence. – In order to make testable ecological predictions, it will be useful to estimate the probability of coexistence, or the probability that one of the species outcompetes the other species, based on plant competition experiments. If the model is reparameterized so that [c.sub.i] = [[c.sub.i].sup.*] + [[Delta].sub.i], where [[c.sub.i].sup.*] is defined by Eqs. 9 or 10, then the signs of ([[Delta].sub.1], [[Delta].sub.2]) will discriminate between the four ecological scenarios. Thus, if the probability of [[Delta].sub.1] [less than] 0 and [[Delta].sub.2] [less than] 0 can be measured, then this is equal to the probability of coexistence. The probability of ([[Delta].sub.1], [[Delta].sub.2]) can be measured by estimating the Bayesian posterior probability distribution of the parameters in the reparameterized model, where the posterior probability distribution is the probability distribution of the parameters given the data. Although the posterior probability distribution is calculated for all the free parameters, only ([[Delta].sub.1], [[Delta].sub.2]) is interesting and the rest are considered to be nuisance parameters. The posterior probability distribution is found using Bayes’ theorem and an assumed prior distribution of the parameters (Edwards 1992, Pascual and Kareiva 1996). In the case where there is no previous competition experiment it is usual to assume that all values of the parameters are equally possible, i.e., that the prior distribution is assumed uniform (Edwards 1992). Under this simplifying assumption we have that the posterior probability distribution of ([[Delta].sub.1], [[Delta].sub.2]) is found as the value of the maximized likelihood function at ([[Delta].sub.1], [[Delta].sub.2]) divided by a normalizing constant that is the sum of the maximum likelihood functions evaluated for all ([[Delta].sub.1], [[Delta].sub.2]) (Edwards 1992).
After the posterior probability distribution is estimated, the probability of coexistence can be calculated as the volume under the posterior probability density curve where [[Delta].sub.1] [less than] 0 and [[Delta].sub.2], [less than] 0, since this is the parameter space where the condition for coexistence is met. Likewise, the volume where [[Delta].sub.1] [greater than] 0 and [[Delta].sub.2] [less than] 0 is equal to the probability that species I always will outcompete species 2. The volume where [[Delta].sub.1] [less than] 0 and [[Delta].sub.2] [greater than] 0 is equal to the probability that species 2 always will outcompete species 1. Finally, the volume where [[Delta].sub.1] [greater than] 0 and [[Delta].sub.2] [greater than] 0 is equal to the probability that any of the species may outcompete the other depending on the initial conditions.
Software. – A Mathematica notebook version 3.0 (Wolfram 1996), that can be used for estimating parameters and test the different hypotheses as well as estimating the Bayesian probabilities of the different ecological scenarios from a given data set, can be downloaded from the World Wide Web.(1)
In the three data sets, neither the hypothesis of an asymptotic yield for high combined densities (d = f), nor “the 3/2th power law of self-thinning” (d/f = 3/2) could be rejected (H2 and H3, Table 2). Since the hypotheses are mutually exclusive, this conclusion was due to insufficient power in the analysis.
The parameters d and f did not make the fit significantly better for any of the three data sets (H4 and H5, Table 2) and they could both be set to 1. There was no significant effect of interaction between the species densities on yield in any of the data sets (H6, Table 2), and each plant density could be assumed to influence yield independently. Furthermore, there was not a significant effect of the quadratic terms in the combined plant densities (H7 and H8, Table 2). The competition experiments could therefore be adequately described with only three parameters for each species (Table 3).
In the three experiments, the maximum likelihood estimates of the competition parameter for one of the species were higher than one and less than one for the other species (Table 3), suggesting that in all experiments there were a “strong” and a “weak competitor.” However, only three of the competition parameters were significantly different from one (H9, Table 2). Avena fatua had a significantly larger negative effect on yield of A. barbata than A. barbata itself. Both competition parameters in the competition experiment between Trifolium pratense and Lolium multiflorum were significantly different from one (H9, Table 2), and Trifolium pratense had no significant effect on the growth of Lolium multiflorum (H10, Table 2). This was probably due to the fact that Trifolium pratense is a smaller plant than Lolium multiflorum (1/a = 0.47 g compared to 5.42 g). Furthermore, Lolium multiflorum produced almost twice as much biomass at high densities compared to Trifolium pratense (1/b = 110.6 g/ box compared to 58.0 g/box). The statistical power in the experiment with Triticum aestivum and Agrostemma githago was apparently low and only the hypothesis that Agrostemma githago had no effect on the growth of Triticum aestivum could be rejected (H10, Table 2).
The probability of coexistence between Avena fatua and A. barbata was estimated based on the assumption that the number of seeds was twice the number of spikelets (Marshall and Jain 1969). There was no information of the probability of the seeds surviving to the next year and the probability of coexistence was estimated for different values of [p.sub.i]. The posterior probability distribution of ([[Delta].sub.fatua], [[Delta].sub.barbata]) is shown in Fig. 2, [TABULAR DATA FOR TABLE 3 OMITTED] where both species were assumed to have a probability of seed survival and germination of 0.5. The estimated posterior probability distribution had the maximum value in the maximum likelihood estimate of ([[Delta].sub.fatua], [[Delta].sub.barbata]), which was (0,345, -0.264). The distribution was not symmetric around the maximum likelihood estimate [ILLUSTRATION FOR FIGURE 2 OMITTED], but elongated diagonally, indicating that the two competition coefficients were positively correlated. The estimated probabilities of the four possible outcomes of the two-species competition, calculated by the volumes under the posterior probability density curve, are shown in Table 4.
If the seed germination probability was [less than]0.5, then the probability of A. fatua outcompeting A. barbata increased slightly (Table 4). Furthermore, if ([[Delta].sub.fatua], [[Delta].sub.barbata]) was set equal to (0, 0) and the likelihood function was maximized with respect to [p.sub.i], the maximum likelihood estimate of ([p.sub.fatua], [p.sub.barbata]) was (0.20, 0.30), suggesting that A. barbata most likely would outcompete A. fatua if ([p.sub.fatua] [less than or equal to] 0.20 and [p.sub.barbata] [greater than or equal to] 0.30). HOWever, this conclusion was only indicative since the posterior probability distribution was not symmetric around (0, 0).
The ultimate goal of plant competition studies is to predict the species composition in plant communities – [TABULAR DATA FOR TABLE 4 OMITTED] specifically, when two plant species compete, to discriminate between the four possible ecological scenarios (Table 1), apart from the trivial case where both species go extinct. Using a Bayesian approach it is possible to estimate the probabilities of the four outcomes based on data from a plant competition experiment. The estimated Bayesian probabilities should not be regarded as probabilities of different ecological hypothesis being true or not true (Edwards 1992), but rather as an alternative description of the data that focus on the dynamics of plant communities.
Based on the de Wit model, Marshall and Jain (1969) concluded that: “The present experiment clearly demonstrated that populations of Arena fatua and A. barbara have the properties of self-regulating systems in which frequency-dependent selection would allow stable cohabitation of the two species.” This is in contrast to the estimated Bayesian probability of coexistence of [less than]0.2 when the germination rate of the two species is assumed equal. Instead, the reanalyses of the data suggest that Avena fatua will outcompete A. barbata. This finding agrees with the observation that the two species most commonly are found in separate populations (Marshall and Jain 1969).
There was no support for complex competition models. This is somewhat surprising, since most plant species in nature are aggregated (e.g., Rees et al. 1996), one might expect that plant densities interact in determining the effect on yield. In experiments with animals significant deviations from the simple Lotka-Volterra competition model have been observed (Neill 1974, Smith-Gill and Gill 1978). The acceptance of the simple competition model may be due to the quality of the data and the fact that I did not have access to the original data for two of the competition experiments, which made the tests conservative.
The hyperbolic response function has been shown to give biased estimates of a and b when d is variable (Gillis and Ratkowsky 1978). The aim in this paper is, however, the testing of hypotheses on the competition coefficients, and since in all cases d and f could be assumed to be 1 I followed the suggestion by Mead (1979) and used the form of the hyperbolic equation originally suggested by Bleasdale and Nelder (1960). If d or f cannot be assumed to be 1 and unbiased estimates of a and b are desirable, then it is probably necessary to use one of the several reparameterizations of the hyperbolic response function (e.g., Gillis and Ratkowski 1978, Seber and Wild 1989, Fredshavn 1993). It can be argued that estimating the competition coefficients in a model with many shape parameters makes the competition coefficients estimates more “biologically realistic” because the fit of the model is better. For example, Firbank and Watkinson (1985) estimated the competition coefficients in a model with variable f, without examining whether or not this extra parameter was necessary. However, any tests on the parameter values will be more conservative when a model is “overparameterized.”
In the reanalyzed competition experiments, at least one of the competition coefficients was significantly different from 0, implying that there was significant interspecific competition in the three experiments. In two of the experiments the “weak competitor” had no significant effect on the growth on the “strong competitor.” For three species the competition coefficients were not significantly different from 1, i.e., the species had the same effect on the growth of the other species as the other species had on themselves. None of these finding disagree with the trends found in the original analysis of the data (Marshall and Jain 1969, Wright 1981, Firbank and Watkinson 1985).
Since the Bayesian posterior probabilities of the different ecological scenarios are estimated from plant competition experiments, the estimated probabilities can get no better than the experiments. If experiments are performed in the greenhouse, the probability estimates will in many cases not reflect the situation in nature. In natural environments growth conditions change from year to year. Hence, probabilities estimated from an experiment, which is conducted in only one environment, may not give adequate information on the dynamics of the community (Kareiva et al. 1996). Specifically, if the composition of the species in an ecosystem is characterized by plants that can sustain rare catastrophic events, such as hard frost or drought, then probabilities estimated from an experiment performed under commonly occurring conditions, will not adequately predict the dynamics of the plant community.
Important biological information on, for example, plant survival under rare conditions, which is not originally included in the model, might be incorporated by numerical simulation of an expanded model. Thus, even if there are specific ecological circumstances that are not accounted for by the simple model, the simple model may provide the “backbone” of a more realistic model.
In the case of stable coexistence and when one species consistently outcompetes another species, the equilibrium is independent of initial densities, for this reason I do not expect that spatial aggregation of plant species will influence the predicted outcome of competition. This expectation is justified by the empirical finding of Pacala and Silander (1990) that there was no effect of the spatial structure in a system where Abutilon theophrasti was predicted to outcompete Amaranthus retroflexus. However, in the case when either species might outcompete the other depending on the initial conditions, i.e., [c.sub.1] [less than] [[c.sub.i].sup.*] and [c.sub.2] [less than] [[c.sub.2].sup.*], both the spatial aggregation of species within a population and the geographic structure of populations into metapopulations will influence the dynamics of the plant community. Therefore, if either species might outcompete the other species it is necessary to model and numerically simulate specific plant communities with attention to the special characteristics of the investigated community, in order to predict the dynamics of the plant community. This can be done by using either neighborhood models (Pacala and Silander 1990) or metapopulation models.
It has been argued that theoretical models in ecology like the Lotka-Volterra models are mainly deductive and that they produce few testable predictions (e.g., Weiner 1995). Here I have suggested a possible route of constructing testable ecological predictions from a competition model in connection with manipulated experiments. As soon as the connection between empirical and theoretical work is made, it is my opinion that we will get better questions to answer by empirical methods and more relevant theoretical models.
Thanks to Jorgen J. Axelsen, Klaus S. Jensen, Gosta Kjellson, Hans Lokke, Ingrid Parker, Miguel Pascula, Vibeke Simonsen, Beate Strandberg, and Jacob Weiner who commented on a previous version of the manuscript.
1 URL = http://www.dmu.dk/TerrestrialEcology
Allen, A. J., L. J. S., Allen, and X. Gilliam, 1996. Dispersal and competition models for plants. Journal of Mathematical Biology 34:455-481.
Bleasdale, J. K. A., and J. A. Nelder, 1960. Plant populations and crop yield. Nature 188:342.
Connolly, J. 1986. On difficulties with replacement-series methodology in mixture experiments. Journal of Applied Ecology 23:125-137.
Cousens, R. 1991. Aspects of the design and interpretation of competition experiments. Weed Technology 5:664-673.
Cousens, R., and M. O’Neill, 1993. Density dependence of replacement series experiments. Oikos 66:347-352.
de Wit, C. T. 1960. On competition. Verslagen van Landbouwkundige Onderzoekingen 66.8:1-82.
Edwards, A. W. F. 1992. Likelihood. John Hopkins University Press, Baltimore, Maryland, USA.
Firbank, L. G., and A. R. Watkinson, 1985. On the analysis of competition within two-species mixtures of plants. Journal of Applied Ecology 22:503-517.
Francis, M. G., and D. A. Pyke, 1996. Crested wheatgrasscheatgrass seedling competition in a mixed-density design. Journal of Range Management 49:432-438.
Fredshavn, J. 1993. Modeling plant competition in monocultures of barley and oilseed rape. Acta Agriculturae Scandinavica, Section B, Soil and Plant Science 43:38-44.
Gillis, P. R., and D. A. Ratkowsky. 1978. The behavior of estimators of the parameters of various yield-density relationships. Biometrics 34:191-198.
Inouye, R. S., and W. M. Schaffer, 1981. On the ecological meaning of ratio (de wit) diagrams in plant ecology. Ecology 62:1679-1681.
Kareiva, P., I. M., Parker, and M. Pascual, 1996. Can we use experiments and models in predicting the invasiveness of genetically engineered organisms? Ecology 77:1670-1675.
Law, R., and A. R. Watkinson, 1987. Response-surface analysis of two-species competition: an experiment on Phleum arenarium and Vulpia fasciculata. Journal of Ecology 75: 871-886.
Marshall, D. R., and S. K. Jain, 1969. Interference in pure and mixed populations of Avena fatua and A. barbata. Journal of Ecology 57:251-270.
Mead, R. 1970. Plant density and crop yield. Applied Statistics 19:64-81.
—–. 1979. Competition experiments. Biometrics 35:4154.
Neill, W. E. 1974. The community matrix and interdependence of the competition coefficients. American Naturalist 108: 399-408.
Pacala, S. W., and J. A. Silander, Jr. 1990. Field tests of neighborhood population dynamic models of two annual weed species. Ecological Monographs 60:113-134.
Pakes, A. G., and R. A. Maller, 1990. Mathematical ecology of plant species competition: a class of deterministic models for binary mixtures of plant genotypes. Cambridge University Press, Cambridge, UK.
Pascual, M. G., and P. Kareiva, 1996. Predicting the outcome of competition using experimental data: Maximum likelihood and bayesian approaches. Ecology 77:337-349.
Rees, M., P. J., Grubb, and D. Kelly, 1996. Quantifying the impact of competition and spatial heterogeneity on the structure and dynamics of a four-species guild of winter annuals. American Naturalist 147:1-32.
Rees, M., and M. J. Long, 1993. The analysis and interpretation of seedling recruitments curves. American Naturalist 141:233-262.
Seber, G. A. F., and C. J. Wild, 1989. Nonlinear regression. Wiley, New York, New York, USA.
Skovgaard, I. 1986. A statistical model for competition experiments. Scandinavian Journal of Statistics 13:29-38.
Smith-Gill, S. J., and D. E. Gill, 1978. Curvilinearities in the competition equations: an experiment with ranid tadpoles. American Naturalist 112:557-570.
Weiner, J. 1995. On the practice of ecology. Journal of Ecology 83:153-158.
Wolfram, S. 1996. The mathematica book. Third edition. Wolfram Media/Cambridge University Press, Cambridge, UK.
Wright, A. J. 1981. The analysis of yield-density relationships in binary mixtures using inverse polynomials. Journal of Agricultural Science (Cambridge) 96:561-567.
Yoda, K., T., Kira, H., Ogawa, and K. Hozumi, 1963. Self-thinning in overcrowded pure stands under cultivated and natural conditions. Journal of Biology Osaka City University 14:107-129.
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