Figure 14.18 Simulation results of the Lotka-Volterra equations: (a) time-domain plot with H(0) = 100 and P(0) = 10; (b) phase-plane plot (P vs. H) for various initial conditions, and the equilibrium point.

availability of safe refuges, which limits predation when host population drops, and the presence of alternative food sources for the predator. Some of these factors make multiple steady-state values possible. This can lead to epidemics, when some environmental disturbance stimulates the system to jump to a different steady-state condition. Also, predator-prey populations can stabilize by some sort of accommodation to each other, reducing the amplitude of oscillations.

Empirical Models Another way to model single populations is to use autoregressive models of the form

N(i) = f1N(i - 1) + f2N(i - 2)+ f3N(i - 3) + ••• (14.31)

in which N(i) represents the population at equispaced time intervals, and N(i — k) is the population k time steps earlier (lagged by k). The coefficients can be obtained by multilinear regression techniques. Because this method is empirical, the coefficients cannot necessarily be identified with particular phenomena, such as growth, death, or predation rates. It is common to perform regressions of the form of equation (14.31) using logarithm-transformed variables instead of raw data. This can make the errors of the regression more normally distributed, and eliminates the possibility of negative predictions from the model. In addition, variables other than lagged population, such as the population of other species, can be used as dependent variables.

Multilinear regression software is capable of determining which coefficients, and therefore which independent variables, contribute significantly to the predictive ability of the model. Thus, it is possible to test which populations affect the dependent population and whether that effect is positive or negative.

Notice that the models discussed previously, including the exponential growth model (14.17), the logistic model (14.25), the competition model (14.28), the mutualism model (14.29), and the Lotka-Volterra predator prey model (14.30), can all be expressed as multivariable polynomials of the following form by combining terms and coefficients:

dt 1

Effects of even more species can be modeled by adding appropriate terms to the model, such as a4N1 N3. By examining the signs of the coefficients of a pair of interacting species and comparing to the signs in Table 14.5, the type of interaction can be determined from population dynamics. This information can even help analyze the trophic structure of an ecosystem.

Terms such as a4N1 N2N3 test for three-way interactions. Statistical tests can show whether such effects are significant. Otherwise, they can be dropped from the model. As Krebs has stated: "Species interactions are rarely one-on-one in natural communities, and the untangling of complex sets of species interactions is an important focus in ecology today.''

Stochastic Extinction A population that is in equilibrium (birth rate = death rate = b) can nevertheless experience population fluctuations due to random variations. Thus, there is a finite probability that a population could go extinct just by chance. The

TABLE 14.8 Extinction Probability for Birth Rate = 0.5 per Year


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