Ecologists have developed a traditional classification of population interactions, closely tied to a particular form of equation system , in which dN
Here, the parameter is the capacity of population i to increase in the absence of interactions, and the interaction coefficients a j quantify the impact of interactions among populations on the dynamics of population i. In most cases the intraspecific coefficient, ai, is negative due to competition among individuals of the same species. But the interaction coefficients among different species take various signs, leading to a canonical classification of pairwise interactions (Table 1). Though well entrenched in ecological thinking, this classification has at least two potential drawbacks. First, the nature of an interaction can depend on densities of the interacting species or other species, or upon other environmental factors. For example, an interaction might sometimes be competitive, and at other times
Competition among populations often arises from exploitation of common resources. In the simplest situations where a single nutrient resource is involved, a well-known result is that only one population persists at steady state while all others are excluded from the habitat. The winner is the population that reduces the nutrient to the lowest concentration at steady state when growing in the absence of other populations. The steady-state nutrient concentration when population i grows alone is denoted R*, and the prediction that it wins is called the R* rule. This prediction, in principle, is easily tested: grow all populations on their own and record R*, then grow them all together and see whether the population with the lowest value excludes others. Such tests have been conducted many times, mostly with bacteria or microalgae growing in laboratory cultures. The R* rule has been verified in virtually all cases where other complications can safely be ruled out.
Although stoichiometry can play a role in determining the R* value expressed by a population, stoichiometric constraints are especially evident in competition for multiple nutrient resources. Consider adapting equation system  to represent two populations that compete for two nutrient resources. With no other intraspecific or ii interspecific interactions, the reproductive rates of the two populations depend only on the concentrations of the two nutrient resources. The total concentration of nutrient j in the habitat consists of its dissolved concentration plus the amounts bound in individuals of both populations, leading to the mass-conservation constraints
i where Sj denotes the total concentration.
This mass-conservation constraint reduces the competition system to two dynamical degrees of freedom, permitting graphical analysis on the resource plane of nutrient concentration R1 versus concentration R2. On this plane, the 'zero net growth isocline' (ZNGI) for population i is a graph showing the combinations of nutrient concentrations supporting reproduction that exactly balances mortality. Steady states can only occur at these nutrient concentrations. The shape of a ZNGI graph depends on the physiological roles of the nutrient resources. Out of many possibilities, two suffice for illustration. For essential resources, each nutrient plays a unique physiological role that cannot be substituted by the other. Different nutrient elements used by autotrophs, such as nitrogen and phosphorus, are a good example. The absolute requirement for each nutrient to support reproduction leads to a ZNGI graph that is rectilinear, with a limb parallel to each nutrient concentration axis (Figure 1a). For substitutable resources, each nutrient can substitute for the other's physiological role. Different chemical forms of the same element, such as organic carbon substrates used by bacteria, are a good example. The availability of one nutrient at a given concentration reduces proportionally the amount of the other required to support reproduction, leading to a linear ZNGI graph with negative slope (Figure 1b).
If the ZNGI graphs for the two competing populations intersect, coexistence is possible. Biologically, such an intersection implies that one population is a superior competitor for the first resource and an inferior competitor for the second, while the other population is superior for the first resource and inferior for the second. Figure 2a shows an example of competition for essential resources, where species 1 is a superior competitor for resource 1 and an inferior competitor for resource 2, while species 2 is a superior competitor for resource 2 and an inferior competitor for resource 1.
Nutrient 1 concentration, R1
Nutrient 1 concentration, R
Nutrient 1 concentration, R
Figure 2 Competition between two populations for two essential nutrient resources. (a) ZNGI graphs (indicated by Zi) intersect at the circled point, making steady-state coexistence possible. (b) The shaded feasible region for steady states is superimposed, bounded by mass-conservation constraints for populations 1 and 2 (indicated by Mi), as are impact vectors implying stable coexistence (indicated by /i).
When coexistence of competitors at steady state is possible, it will not be achieved if inadequate nutrient supplies render coexistence infeasible, or if it is dynamically unstable. These questions are strongly affected by the competitors' stoichiometry for the nutrient resources in question. The mass-conservation constraints (eqns ) imply that on the plane of resource concentrations (R1 vs. R2), feasible steady states lie between two lines of positive slope passing through the supply point (S1, S2). If the intersection of the competitors' ZNGI graphs lies within these boundaries, then steady-state coexistence is feasible (Figure 2b). Biologically, the supplies of both nutrients are high enough to support both populations at steady state. The slopes of these mass-conservation constraints are the whole-organism stoichiometric ratios of the two nutrients for the two competitors. For example, if two algal populations compete for dissolved nitrogen and phosphorus, the slopes are their cellular N:P ratios.
The dynamical stability of coexistence depends on the same stoichiometric ratios, and can be analyzed graphically by plotting impact or consumption vectors for each population. These portray the relative impacts of the competitors on the two resources through consumption, at steady state. Only the slopes of these vectors affect stability of the steady state, and these slopes are the same as those of the mass-conservation constraints. Biologically, these vectors represent consumption of the two resources in the proportion required to maintain organismal stoichiometry. The arrangement of the impact vectors in Figure 2b shows an example of stable coexistence. Stability arises because each of the competitors consumes proportionally more of the resource for which it is the inferior competitor, enhancing the negative intraspecific effect of a population on its own growth rate, and ameliorating the negative impact on its competitor's growth rate. This result illustrates a general feature of competition theory that is easily derived from analyzing eqn  for two competing populations (the famous Lotka-Volterra equations of competition). Stable coexistence of competitors at steady state requires that intraspecific competitive effects be stronger than interspecific effects. Explicitly including the stoichiometry of resource consumption brings some biology into this rather abstract statement.
Explicitly including the stoichiometry of resource consumption also leads to an important testable prediction. When stable coexistence is possible (e.g., Figure 2b), manipulating the resource supply stoichiometry by moving the position of the supply point (S\,S2) around the resource plane alters the outcome of competition. When the supply point is moved down and to the right, increasing the supply ratio S\:S2, the shaded feasible region is moved beyond the intersections of the competitors' ZNGI graphs. Coexistence is then no longer feasible, and the relatively low supply of resource 2 leads species 2, the superior competitor for this resource, to exclude species 1. Moving the supply point in the opposite direction, up and to the left, has the opposite effect, producing exclusion of species 2 by species 1.
The predicted dependence of competitive outcomes on resource supply stoichiometry is called the resource ratio hypothesis, and it has been tested many times, primarily with diatoms or other microalgae competing for dissolved nutrient elements in laboratory cultures (e.g., Figure 3). Nearly all of these tests have verified the prediction, and some of the patterns of competitive dominance observed in the laboratory appear in the natural
Figure 3 Competition between two diatoms, Synedra filiformis (species 1) and Asterionella formosa (species 2), for the essential nutrients phosphorus (resource 1) and silicon (resource 2) in laboratory cultures. (a) The ZNGI graphs (Z) and impact vectors (Ii) predict that stable coexistence is possible. (b) The supply point (S1,S2) and mass-conservation constraints (Mi for one of the nutrient supply conditions was tested, for which stable coexistence was predicted and observed. Other supply conditions were tested for which exclusion of an inferior competitor was predicted, and trends toward this outcome were observed. Four other pairs of competing diatom species were tested in similar experiments, and observed results matched predictions based on parametrized, stoichiometric models. Data from Tilman D (1981) Tests of resource competition theory using four species of Lake Michigan algae. Ecology 62: 802-815.
distributions of algae. For example, some cyanobacteria are better competitors for nitrogen than are certain eukar-yotic algae, which are better competitors for phosphorus. Distributional data show an association of cyanobacterial dominance with a low N:P concentration ratio in lakes.
What happens when coexistence of two populations is feasible but unstable? For competitive interactions, instability usually implies that one competitor will persist while the other is excluded. The identity of the winner will depend on initial conditions, usually going to the advantage of the population with greater initial abundance, leading the term priority effect to be applied to such situations. However, instability and priority effects are unlikely in competitive interactions, if competitive abilities depend on organismal stoichiometry as explained below (see the secion titled 'Physiological variability in stoichiometry').
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