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Figure 14.16 U.S. population data with logistic equation fit by Pearl and Reed, and updated logistic equation fitted to years 1950-1990.

The parameters of the logistic equation may be obtained by fitting the equation to laboratory or field data. For example, Pearl and Reed, who originally proposed the logistical equation in 1920, fitted it to U.S. population data from 1790 to 1910, when the population was about 92 million. They obtained r0 = 0.03134 per year and K = 197,273,000. Thus, they projected a doubling of the U.S. population, followed by a leveling off. Notice that their prediction held for four more decades (Figure 14.16).

In 1950, however, the population started increasing much faster than predicted. The model fell victim to a basic problem with any model: The conditions on which it is based may change, resulting in either different parameter values or an altogether different model. In this case, the growth rate stopped decreasing as the logistic equation would have predicted. An updated model fitted to the data from 1950 to 1990 predicts almost the same r0 = 0.03281 but a much higher K = 324,796,000. Thus, the carrying capacity of the United States seems to have increased, suggesting that the U.S. population will approach 325 million in the next century if things remain the same. (The population in 2005 is about 296 million.) The increased carrying capacity may have been due to the postwar prosperity and exploitation of new resources (raw materials, markets for our products). This result was a jump in the growth rate of the U.S. population called the "baby boom.''

The logistic equation has led to a terminology to describe two basic reproductive strategies of organisms. An organism is called K-selected if it prospers by making efficient use of its resources, maximizing its carrying capacity. These organisms are more likely to be associated with crowded, stable ecosystems. They produce relatively small numbers of young but may nurture them to ensure their survival. Other organisms, called r-selected, adopt a strategy of rapid reproduction. These organisms are described as opportunistic. They can quickly take advantage of stressed conditions. They tend to produce large numbers of offspring but have much higher mortality. Weed species often fall into this category, as does that urban nuisance, the rat. The r-selected organisms will also have an advantage in temperate or arctic climates, where the short growing season favors rapidly growing species.

Laboratory experiments with single species populations have revealed behaviors that cannot be described by the models described above. Some species of Daphnia overshoot their carrying capacity and then oscillate around it. They are able to store food in the form of fat deposits, which delays the effects of resource limitations. This behavior can be described by using models with time-delay elements. For example, the logistic equation may be modified by using populations taken from \tau time units in the past, N(t — t):

Each of the foregoing models is deterministic, meaning that for each set of initial conditions, there is a unique prediction. However, the outcome in reality will be affected by unknown factors which produce random variation. A model that treats the variables as a random variable is a stochastic model. Each of the preceding models could be made stochastic by adding an independently and randomly distributed "error" at fixed intervals of time, before continuing the prediction to the next time step. The prediction of a stochastic model is not just a single value, but a value that has a probability distribution.

None of these models reflect interactions with other species. Whatever is the source of food for the organism will certainly be subject to its own population dynamics, in response to the predation. In the next several sections we address some of these issues.

Competition The final term in equation (14.25) represents the effect of individuals from a single population competing among themselves. If another organism, species j, is using the same limiting resource as the population being modeled, species i, the logistic equation can be modified to take this into account:

dt Ki where Ni is the population of species i, Nj the population of species j, and bj the positive number coefficient of competition. The coefficient of competition expresses how much the competing population effectively reduces availability of the resource to population i. A similar equation would be written for species j by switching the subscripts. If more than two species are interacting, additional terms of the form of bikNk would be included. A differential equation similar to (14.28) must be written for each interacting species, and the equations solved simultaneously.

A stability analysis of equation (14.28) reveals the conditions under which each species can survive. For two species we can form the dimensionless groups a12 = b12(K2/K1) and a21 = b21(K1/K2). If both a12 < 1 and a21 < 1, the species can coexist, although at a lower population than each by itself. If both a12 > 1 and a21 > 1, only one species can survive; which one depends on the exact dynamics and the initial conditions. If a12 > 1 and a21 < 1, species 1 will always win out. These dimensionless numbers can be interpreted as a measure of the competitiveness of each species. If both numbers are less than 1, competition is weak and the species can coexist. If both are greater than 1, competition is strong and only one species can survive.

Study of this model has led to the development of the principle of competitive exclusion, which states that species with identical resource requirements cannot coexist. Strong competition tends to result in the elimination of one species from the ecosystem. Thus, strong competition does not often occur in nature. In cases of apparent strong competition, it has often been found that the two species may have some significant difference. For example, five similar species of warblers in New England forests all eat insects. However, they feed at different parts of the forest canopy and in different ways, and they nest at different times. Strong predation by another species can keep several species sufficiently far below their carrying capacity that competition does not seriously limit the availability of the mutual resource. Darwin noticed this when he pointed out that grazing increases the diversity of plants in meadows. When the predation is limited, a single species tends to dominate. This is what has happened in many ecosystems affected by human activities. People have tended to remove top carnivores, such as grizzly bears and wolves in western North America. This actually had a negative impact on most of their prey, by increasing the role of competition among them. Thus, conservationists have the goal of reintroducing many of these predators to increase the diversity of the ecosystems.

The principle of competitive exclusion is linked to the concept of niche. The niche can be described as the resources and other factors favored by a species. Then the principle of competitive exclusion is equivalent to the statement made above that no two organisms can occupy the same niche.

Mutualism and Symbiosis Mutualism and symbiosis are simply the opposite of competition. Instead of a species reducing the carrying capacity of another species, it increases it. This can be modeled simply by changing the sign of the interaction term in equation (14.28):

dt Ki

If b12b21 < 1, there is a stable steady-state population for both species, which for each is greater than their respective carrying capacities. However, the model is not realistic for b12b21 > 1, in which case the mutual benefit is so great that both populations grow without limit.

Predator-Prey: Lotka-Volterra Equations One of the most interesting types of interactions are those between consumers (predators, parasites, and herbivores) and their hosts. Lotka and Volterra derived independently a model which although somewhat simplistic for modeling real populations, is nevertheless capable of describing some of the important behaviors found in predator-prey relationships. One important behavior is cycles in predator and prey populations (Figure 14.17). Intensive predation can cause a collapse in prey population. This is followed by a drop in predator population, due to the drop in food supply. As a result, the prey population can boom, followed by the predator to complete one cycle.

The Lotka-Volterra predator-prey model has the following form:

where H is the host or prey population, P is the predator population, and a, b, c, and d are coefficients. Multiplying through, the host equation is seen to have two terms: The aH

Figure 14.17 Oscillations in predator-prey populations: the predatory wasp Heterospilus prosopidis and its host the weevil Callosobruchus chinensis. (Data from Utida, 1957.)

Generation

Figure 14.17 Oscillations in predator-prey populations: the predatory wasp Heterospilus prosopidis and its host the weevil Callosobruchus chinensis. (Data from Utida, 1957.)

term gives it exponential growth when the predator is absent. The —bHP term is a negative interaction due to predation. The predator equation has a positive interaction corresponding to the negative one for the host. The —dP term provides an exponential death rate without which the population could never decrease. There is no exponential growth term for the predator, and there is no carrying capacity for either equation.

The Lotka-Volterra model has a stable steady-state or equilibrium solution (other than H = P = 0) when H = d/c and P = a/b. If values other than these are used as an initial condition, the solution will be a stable oscillation. The plot of host and predator populations vs. time (Figure 14.18a) shows that predator population peaks just after the host population. Examination of the phase-plane plot (Figure 14.18b) suggests that predator population peaks when the host population is at its greatest rate of decline.

More realistic models have been developed. For example, adding a term of the form —e H2 to the host equation makes it equivalent to a logistic equation with a negative self-interaction. Other modifications place an upper limit on the amount of prey that each predator can consume. Other factors that affect predator-prey dynamics include the

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