Classical Competition Theory

The development of competition theory in population bioecology was in- fluenced strongly by Gause’s (1934) experiments on the coexistence of closely related species of beetles in controlled environments. He found that mixing two populations in the laboratory invariably caused one population to disappear. Gause summarized his findings by proposing a general ecological law. His principle of competitive exclusion holds that two species that occupy essentially the same fundamental niche cannot coexist in equilibrium.

Subsequent research has ruled out competitive exclusion as a general principle. For example, it turns out to be simple to produce coexistence of closely related species in the laboratory: one makes the environment more complex, creating subenvironments in which the inferior competitor can find refuge or may possess an adaptive advantage. Nonetheless, this “principle” has proved useful in directing attention to the crucial role of niche overlap in competition processes. It has also stimulated the application of general models of population dynamics to concrete ecological processes.

In order to make the foregoing discussion concrete, we consider the simple case of one environmental dimension, where niche intersection takes the form of simple overlap. Following Levins (1968), we use a graphic representation in which the adaptive capacity of each population is summarized by a fitness function which relates growth rates (fitness) to levels of the environmental condition. A fitness function tells the speed of growth of populations of various phenotypes bearing levels of a trait, y, in a particular environment. The fitness function tells which level of y is optimal in the sense of providing maximum population growth rates as well as how sharply departures from the optimum are penalized in terms of reduced growth.

In an organizational example, y might be the proportion of revenues that firms spend on research and development, and the environmental dimension might be the speed of technical change in the industry. During an era in which an industry’s technology is developing rapidly, high expenditures on research and development produce high growth rates for the firms, as well as high profits. Such firms attract imitators, and high research and development firms become more numerous. When technological change slows and new discoveries come farther apart, heavy expenditures on research and development produce lower payoffs, and imitation of technical pioneers is less likely. So the fitness of firms displaying high values of y is lower at low rates of technical change.

Figure 5.1 shows hypothetical fitness functions for three populations. In this example population B competes with both A and C, but A and C do not compete with each other. However, the overlap of the niches of A and B is considerable, while the overlap of B and C is slight. What happens in the regions of overlap? Do populations A and B coexist, sharing the resources? Does one exclude the other? Although these questions have not been raised in the context of organization theory, they are not new questions. Since these issues have been investigated for some time in the con-text of other disciplines, notably population biology, it makes sense to try to build on existing theory and models.

Figure 5.1 Hypothetical fitness functions

Alfred Lotka (1925) and Vito Volterra (1927) independently proposed models of population dynamics that incorporate effects of competition between populations. They began with models that imply that population growth of isolated populations has an S-shaped growth path. They assumed that the growth rate of an isolated population is given by the product of a growth rate and the current size of the population:

where N denotes the size of the population. The growth rate, pN, is defined as the difference between the birth and the death rates of the population:

Suppose that birth and death rates are constant, that is, that they do not vary with the size of the population. Then the model in (5.1) implies exponential growth. But such growth processes are not realistic in finite environments, as Malthus insisted. The growth model can be made more realistic by assuming that birth and death rates vary with density (the size of the population). Lotka and Volterra assumed that the birth rate falls approxi- mately linearly with the size of the population:

and that the death rate increases approximately linearly with population size (assuming that the resources available are finite):

Substituting the last two equations into the growth model (5.1) gives (5.2)

with π1a0 b0 > 0 and π2 = a1 + b1 > 0. This is the model of logistic population growth. It holds that the population grows almost exponentially at low values of N, but that competition for fixed resources eventually drives the growth rate toward zero.

Note that (5.2) and the restrictions on the signs of coefficients imply a steady state for the population. Setting equation (5.2) equal to zero shows steady states at N = 0 and N = π12. The non-zero steady state of the population is usually called the carrying capacity of the environment for the population in question. It has traditionally been denoted by K in bioecology.

The logistic model of population growth can be expressed in terms of the carrying capa city:

The coefficient r = a0 – b0 is called the intrinsic growth rate. It tells the speed with which the population grows in the absence of resource constraints. That is, when the population size is small compared to the carrying capacity, the growth rate essentially equals r. When the population size equals the carrying capacity, the growth rate is zero. If population size exceeds the carrying capacity (perhaps because some shock has reduced the carrying capacity), the growth rate is negative.

The parameterization in equation (5.3) provides a substantively appealing way to introduce competition. Two populations compete if the size of each population lowers the carrying capacity for the other. The Lotka- Volterra (LV) model of competitive interactions, which plays a prominent role in much contemporary population bioecology, assumes that the effect of the density of the competitor on the realized carrying capacity is linear. In the case of two comp eting populations, the model is

Comparing (5.4a) and (5.4b) with (5.3) shows that the presence of the competitor reduces the carrying capacity for the first population from K1 to K1 – a12N2. The so-called competition coefficients, a12 and a21, tell how the carrying capacity for each population declines with the density of the competitor. This model decomposes the growth rate for each population into the effects of three components: (1) ri , the intrinsic properties of the form that affect its speed of growth in the absence of resource limitations and competition; (2) Ki limits on growth that reflect generalized conditions of resource availability; and (3) aij, competition with specific populations (where the coefficient indicating the effect of intrapopulation competition, aii, has been scaled to unity).

Even though the LV model builds on simple notions of density-dependence in birth and death rates and of the effects of competitive interactions, the system of equations in (5.4) does not have a known solution. Therefore, analysis of questions of coexistence of competing populations, that is, whether models like (5.3) have stable steady states with non-zero sizes of both populations, relies on study of the qualitative behavior of the system of differential equations (see, for example, Wilson and Bossert 1971). The general result is that coexistence requires that the effects of density on mortality rates within populations must be stronger than the competitive effects between populations.

This is easy to see in the case of two competing populations. Setting equations (5.4a) and (5.4b) equal to zero and solving for the non-zero equilibrium values of N1 and N2 shows that stable coexistence requires that

Therefore, very similar populations (that is, populations whose competition coefficients are very near unity) can coexist only under a precise K2!K1 ratio. Any shock to the system that alters either carrying capacity is likely to drive the system away from the special conditions that support coexistence. Since this is an unstable equilibrium, the system will not tend to restore itself to the condition of coexistence.

Analysis of LV systems provides a way to assess Gause’s principle. If two populations occupy essentially the same niche, both competition coefficients are close to unity, because addition of a member of the competing population has almost the same dampening effect on population growth as addition of a member of the population in question. Coexistence of competing species is extremely unstable when competition coefficients are close to unity. This does not mean that the principle of competitive exclusion follows logically from the LV model; but it suggests that his principle does describe accurately the instability of coexistence of populations that occupy the same niche.

The general (/-dimensional) LV system has the form:

The matrix of competition coefficients, sometimes called the community matrix, governs stability. Although this matrix is not generally symmetric because the competition coefficients for pairs of populations are not equal, it does equal the product of a symmetric matrix of overlaps and a diagonal matrix whose entries indicate the total resource use of a population (Roughgarden 1979). The equilibrium point of this dynamic system satisfies

If one or more of Nj are zero or negative, it follows that not all I populations can coexist in a stable equilibrium, and competitive exclusion occurs. If the vector of the sizes of the populations is strictly positive, the stability of the coexistence of all I populations depends on the properties of the community matrix.

Stable coexistence requires that the matrix of competition coefficients has a non-zero determinant. Levin (1970) proved that, in the context of populations whose growth depends on resources and constraints, this re- quirement means that at least I distinct resources and constraints are needed to support the stable coexistence of / populations. Thus the number of coexisting populations is constrained by the number of resources and constraints. Earlier (Hannan and Freeman 1977) we suggested that this qualitative result could be applied in organizational analysis. We proposed that the diversity of a community of organizations, that is, the number of coexisting organizational populations, increases when new resources and constraints are added to social systems and declines when resources and constraints are eliminated.

Even though LV models do not have explicit solutions, they can still be estimated with data. Interestingly, the first attempt we have seen to do so with nonexperimental data is Carroll’s (1981) analysis of growth and decline in populations of organizations (see also Tuma and Hannan 1984, chaps. 11 and 14). By estimating an exact-discrete approximation to the LV model, Carroll was able to estimate competition coefficients directly from data. Recently Brittain and Wholey (1988) have used Carroll’s approach to estimate competition coefficients among subdivisions of the American semiconductor manufacturing industry.

Source: Hannan Michael T., Freeman John (1993), Organizational Ecology, Harvard University Press; Reprint edition.

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