David Tilman (1976, 1987) and others pointed out that the Lotka-Volterra equations were "phenomenological" and not "mechanistic." That is, competition coefficients were merely measures of the effect of one species on the growth rate of another. They are estimated from experiments in which two species are grown together. Therefore they are not an independently derived value that allows one to predict coexistence or competitive exclusion, or, in the latter case, which of two species should win. Furthermore, a competition coefficient does not help determine the mechanism of competition; we have no information on what resource the species might be competing for. If competition is really concerned with a resource in short supply, we need to understand what the resource is and how each species is using it before we can understand the potential competitive interaction.

Tilman, a particularly strong advocate of a mechanistic approach to competition (1976, 1981, 1982), developed what is now known as resource-based competition theory. In so doing he brought together ideas from a variety of disciplines, including microbiology, enzyme kinetics, and agricultural chemistry. For example, the idea that population growth is constrained by the depletion of critical resources can be traced to the agricultural chemist Liebig (1840) and his law of the minimum. Liebig asserted that a population increases until the supply of a single critical resource becomes limiting. For example, plant growth may continue until the amount of phosphorus, nitrogen, light, or soil moisture becomes limiting. According to Liebig's law, if plant growth is constrained by phosphorus and a farmer adds phosphorus fertilizer, plant growth will continue until another resource, such as nitrogen, becomes limiting. If the farmer adds nitrogen, then soil moisture may become the limiting factor. Liebig's law is overly simple in that two or more resources may interact to limit a population, but it puts resource supply into the context of population regulation, and therefore competition.

In the resource-based approach to competition, we need to couple the availability of resources to population growth. We begin by considering a renewable resource. If the supply rate is not affected by the population of potential consumers, the rate of supply (for example, phosphorus arriving at a small lake via a local stream) could be considered a constant (Eqn. 7.11), where Rt is the quantity of the resource i and kRi is the supply rate:

If this one resource sets the limit of growth for a population (as in Liebig's law of the minimum), the growth rate depends on both the resource level and the density of the population itself. That is, dN/dt is a function of both the resource supply rate, kRi, and population size, N. Assume that each individual must consume the resource at rate q to maintain itself. We ignore the possibility of storing the resource for later use. The population of N individuals will consume the resource at the rate qN. The remaining resources may eventually be lost downstream or in the lake sediments. Alternatively, they may be taken up by the consumers and used for the production of new individuals in the population. The supply rate of the resource available for reproduction is therefore kRi - qN. Suppose each individual converts the resource into new individuals with efficiency b. Population growth can now be written as:

We can rearrange this equation to:

C kRi F

Equation 7.13 is a form of the logistic equation with r = bkRi and the carrying capacity equal to kRi/q. We have now coupled resource supply with population growth rate. The growth rate of the population is proportional to the supply of the critical resource and the carrying capacity is the resource supply divided by the amount needed for maintenance per individual.

Now envision the relationship between resource availability and population growth. As described earlier, in a density-dependent population, the per capita growth rate declines with population size, yielding a negative slope. Logistic-like equations assume that the resources become scarcer as populations grow. However, if we graph per capita growth versus an increasing supply of resources, the predicted slope is positive (Fig. 7.8). As the resource becomes more abundant, the population growth rate increases. Per capita growth, however, finally levels off and declines to zero when some other resource limits the population (as proposed by Liebig).

The next step is to introduce a mortality rate, m. Instead of assuming that resources are simply turned into births, we assume instead that a certain minimum level of the resource is needed to maintain the population. Therefore, as shown in Fig. 7.9, we introduce the concept of R*, the level of the resource needed to balance mortality. If the resource is provided at the rate R*, then dN/dt = 0, growth just balances mortality, and the population

Resource supply

Figure 7.9 Per capita growth as a function of resource availability, but with a constant mortality rate. R* = the amount of resource producing a per capita growth rate of zero.

Resource supply

Figure 7.9 Per capita growth as a function of resource availability, but with a constant mortality rate. R* = the amount of resource producing a per capita growth rate of zero.

Time

Figure 7.10 Population and resource dynamics.

Population size Resource

Time

maintains itself. If the resource is provided at a level less than R*, growth is less than mortality, and the population declines. Conversely, if the resource is provided at a level greater than R*, we have growth (dN/dt > 0).

Examine Fig. 7.10. As the population grows, the resource is increasingly depleted, and once the resource declines to the level R*, the population should stop growing. The population thereafter remains steady at a size determined by the resource quantity R*.

Tilman (1976) realized that an independently derived R* could be used to predict population dynamics and, ultimately, competitive interactions. He first used the

Michaelis-Menton enzyme kinetics equation, which is normally employed to describe the relationship between cellular metabolism and substrate concentrations. This equation had also been used by microbiologists to describe the growth rate of bacteria on organic substrates (Monod 1950).

In the Michaelis-Menton equation, the growth rate, ¡1, on a given substrate or resource, R, is set equal to the maximum growth rate modified by the concentration of the resource and by a value known as K¡, the half-saturation constant for the resource in question. Kp is the concentration of the resource that produces half the maximum growth rate. If ¡ max is the maximum growth rate of the population, and R is the resource or substrate, the resultant growth rate (¡) according to the Michaelis-Menton equation is:

In Monod's version of this equation, instead of ¡1 we substitute dN/dt, and we use b instead of We will also now define the half-saturation constant for any given resource as Kt, rather than Kp. The result is Equation 7.15:

Per capita growth is shown in Equation 7.16

This equation tells us that at very high resource levels, the expression R/(K + R) is very close to one and the per capita growth is simply b. This is the maximum growth rate, ignoring the mortality rate, m, and is equivalent to the unmodified intrinsic rate of increase rmax. We can add a constant mortality rate to Monod's equation as follows:

We can set dN/Ndt (per capita growth) equal to zero and solve for R*, which is the resource level at which growth stops. The solution, as shown in Equation 7.18, allows us to predict R* if we know three variables: (i) the half-saturation constant, K, (ii) the maximum growth rate, b, and (iii) the mortality rate, m. The advantage of this approach, as opposed to that of the Lotka-Volterra equations, is that a resource is identified and a variable, R*, can be derived from simple experiments, which can then be used to compare how populations of different species respond to different resource levels.

For example, Tilman (1976) identified critical resources for which two species of planktonic algae were likely to be competing. Two freshwater diatom species were grown under different levels of the important limiting nutrients, phosphate (PO-3) and silicate (SiO-4). Using a growth vessel known as a chemostat, he determined the growth rates of each species cultured singly and also determined the limiting concentrations of phosphate and silicate. From these experiments he determined the values of the half-saturation constants for each species for both phosphate and silicate. By examining the ratios of silicate to phosphate utilizations he was able to establish boundaries for competition. He found that when both species were limited by phosphate, the diatom Asterionella won in competition with Cyclotella. When both species were limited by silicate, Cyclotella won. However, when both nutrients were simultaneously in short supply, the two species coexisted. Tilman minimized mortality rates and was able to predict which species would win based on growth rates and the half-saturation constants. He found that the species with the lower half-saturation constant (and therefore lower R* in this case) would win if both species were limited by a single nutrient. However, if each species was limited by a different nutrient, the density of each species was held in check through intraspecific competition and the two species coexisted. The incorporation of the equations originally developed by MichaelislMenton and Monod, and the use of half-saturation constants, allowed Tilman to predict the results of competition involving two species and two resources.

Subsequent papers by Tilman (1981), Tilman et al. (1982), Hansen and Hubbell (1980), and a book by Tilman (1982), have become the foundations for "resource competition" as a distinct theory and an important area of inquiry. Tilman's work expanded from the laboratory work on diatoms and other algae to competition studies of terrestrial plant communities at the Cedar Creek Natural History Area in Minnesota. Hansen and Hubbell (1980) and Tilman (1982) elaborated on the Monod equation, but they still emphasized the critical parameter, R*. This parameter can be used to predict which one species will survive in a mixed-species culture when there is a single limiting nutrient. According to what is now known as the _R*-rule, for any given resource (R), if we determine the R*-value for each species when grown alone, the species with the lowest R* should competitively exclude all other species, given enough time and a constant environment.

In deriving their version of the R*-rule, Hansen and Hubbell (1980) assumed that two competitors are grown in a continuous culture with a continuous input of a nutrient (R) as well as an effluent rate, which is equivalent to a death rate, m. The growth rates for two competing species were defined as:

b RN

and b2RN2

where bi = maximum cell division rate (= rmax);

R = the concentration of the one limiting resource in the culture; K = half saturation constant for the limiting resource; m = death rate, here due to outflow;

Nj = concentration of cells in the culture (population size).

If we do an equilibrium analysis, and set dNi/dt = 0, the result is:

Thus one solution is that growth stops when the concentration R equals the halfsaturation constant. At that time the cell division rate is at half of its maximum level (ra = 0.5rm) and just equals the death rate. Thus, as explored above in a graphical analysis, the equilibrium resource availability, R*, occurs when the growth function intersects the line m representing mortality. This is the amount of resource needed to just sustain the population (growth just offsets mortality).

When Hansen and Hubbell (1980) analyzed Equations 7.19 and 7.20, they found that they are globally stable when either: (i) all competitors die out, or (ii) one species survives while the second species dies out - that is, when competitive exclusion occurs. Which species survives depends on the critical parameter, R*, which we already saw in Equation 7.18 as R* = mK,/(b - m).

R* must be less than R0, otherwise all species die out because of lack of resources. If all R*-values for all species are less than R0, then, according to the R*-rule, the species with the lowest R* wins. Again, three parameters, the half-saturation constant, the intrinsic rate of increase, and the death rate, combine to determine which species wins in competition for any given resource. This is not predicted from classical Lotka-Volterra competition theory. A species with a high affinity for the resource can still lose if it has a low growth rate (r-value) and a high death rate.

Hansen and Hubbell (1980) confirmed the expected results with several species of bacteria auxotrophic for tryptophan. In the example in Table 7.2, based on Hansen and Hubbell (1980), we expect population two to outcompete the other three species for this resource. Species three is expected to outcompete both populations one and four, and population one should win in competition with population four.

Tilman and others have tested the R*-rule on algae, other microorganisms, higher plants, and zooplankton. For example, Tilman (1982) grew two species of diatoms, Asterionella formosa and Synedra ulna, which require SiO2 for cell-wall structure, in a laboratory chemo-stat. The R*-values for A. formosa and S. ulna were 1.0 |M and 0.4 |M, respectively. When

Population, i |
Ki |
m |
b |
ra |
R* |

(g L"1 X 10-6) |
per hour |
per hour |
(= b - m) |
(g L-1 X 10-6) | |

per hour | |||||

One |
4.0 |
0.05 |
0.25 |
0.20 |
1.00 |

Two |
4.1 |
0.05 |
0.50 |
0.45 |
0.46 |

Three |
6.5 |
0.05 |
0.50 |
0.45 |
0.72 |

Four |
20.0 |
0.05 |
0.25 |
0.20 |
5.00 |

grown together, as the silicate levels eventually were reduced to 0.4 |M, Synedra hung on at its equilibrium number, whereas Asterionella went extinct.

See Grover (1997) for a relatively recent review covering both the status of the theory and relevant field and laboratory studies.

Resource competition: conclusion

Applying the R*-rule and resource models to higher animals has proved less useful and, as usual, theories developed for simple laboratory settings run into a number of problems and complications when applied to field situations. In addition, we should recognize that other modes of competition, such as interference competition, do not follow the simple rules of resource-based models. Moreover, competition on any given site does not necessarily involve all species in the community. The R*-rule would only apply to the species which have "shown up" in a given habitat patch. Therefore, as elaborated below (also see Tilman 1994, 1999, Hubbell 2001), a community is not limited to the species that are the superior competitors.

Was this article helpful?

## Post a comment