Time Series Data

Resource and pest management agencies often sample or census the abundance of resource and pest populations annually. As a result of this activity, long series of observations are frequently obtained that describe the annual fluctuations of resource or pest populations over a number of years. Such a string of observations at equal intervals of time is called a time series. The logistic food chain equation is ideal for modeling time series data because it can be written as the linear equation,

remembering that

R_i is the realized per-capita rate of population change in a given physical environment, and that this quantity can be estimated from a time series by the quantity

R_i = ln X_{i , t} - ln X_{i , t}-1, which becomes the dependent variable in the multiple regression.

a_i is the maximum per-capita rate of population change in a given environment when population density is very small, and is estimated from the intercept of the multiple regression of R_i on X_{i , t}-1 , X_{i , t}-1 / X_i-1 , t-1 , and X_{i+1 , t}-1 / X_{i , t}-1 .

b_i is the regression coefficient of R_i on X_{i , t}-1 as estimated by multiple regression.

c_i is the regression coefficient of R_i on X_{i , t}-1 / X_i-1 , t-1 as estimated by multiple regression.

d_i+1 is the regression coefficient of R_i on X_{i+1 , t}-1 / X_{i , t}-1 as estimated by regression.

w_i and w_i+1 are initially set to zero for purposes of multiple regression analysis, but can later be fit by convergence methods if there is evidence that the consumers are generalists.

Morris (1959) reported the number of black-headed budworms per 100 ft² of foliage in an eastern spruce-fir forest. The budworm larvae obtained during sampling were either dissected or reared in the laboratory to determine whether they were parasitized. The numbers of budworms and parasitoids, adjusted to 100 m², are given below:

Budworm: 236.81, 1205.55, 5737.14, 2421.87, 129.17, 33.37, 35.52, 333.68, 1614.58, 2551.04, 3229.16

Parasitoids: 16.57, 108.50, 2466.97, 2349.21, 113.67, 10.34, 7.10, 46.71, 161.46, 714.29, 1420.83

and are plotted in figure 1.

Figure 1. Numbers of black-headed budworm larvae (blue) and number parasitized (red) per 100 m² of foliage.

Given the logistic R-function for a prey in a two species food chain attacked by a specialist predator

we fit it to the data by linear multiple regression, with the following parameter estimates: a_h = 2.08, b_h = 0.000255, d_c = 4.19, and a coefficient of determination 0.893. The corresponding parasitoid R-function

is also fit to the data with parameter estimates: a_c = 2.42, b_c = 0.0000145, d_c = 5.715, and a coefficient of determination 0.904. Because the partial coefficient of determination for the intra-specific effect was so small (p_b = 0.00038), this parameter can be set to b_c = 0.

Deterministic simulation shows that the model is damped stable in a constant environment (Fig. 2, left), but that cyclic dynamics persist in a randomly varying environment (Fig. 2, right).

Figure 2. Simulation of black-headed budworm and parasitoid dynamics using the logistic 2-species model fit to the data shown in figure 1: Deterministic simulation (left); Stochastic simulation (right).

Of course, we can also calculate the zero growth isoclines for both budworm and parasitoids using the equations

  prey isocline

  predator isocline

and the parameters estimated from the data (Fig. 3). Notice that the community equilibrium occurs at very low prey densities, implying that the parasitoids are very effective at regulating budworm densities at low levels. As we will see later, isocline structures can be very useful for designing optimal pest management strategies.

Figure 3. Zero growth isoclines for the budworm-parasitoid model.

Experimental Data

When data describing the long-term (multi-generation) fluctuations of predator and prey populations are not available, or when generations are so mixed that it becomes impossible to distinguish between them, model parameters must be estimated in different ways. For example, we could attempt to estimate the parameters from field or laboratory experiments or from information in the literature. Biomass conversion models are often more amenable to this kind of modeling because the biological meaning of the parameters is generally more apparent, and this facilitates their estimation from independent data. For instance, the demand for resources in the consumer functional response (the parameter d in Lotka-Volterra or Ivlev-Watt type models) can be estimated by measuring the attack rate of starved consumers when resource density is extremely high; the apparency of the resources, v in the Ivlev-Watt model, can be obtained by estimating the proportion of the resource population invulnerable to predation (or the host refuge q with v = 1 - q); the relative abundance of alternative hosts, w, can be estimated from preference experiments; and finally c, the efficiency of conversion of attacked hosts into consumer offspring can be determined by rearing the organisms in the laboratory or field. Examples of the model parameterization in this manner can be found in Gutierrez (1992), Gutierrez et al. (1993) and Turchin and Hanski (1997).

References

Gutierrez, A. P. 1992. Physiological basis of ratio-dependent predator-prey theory: the metabolic pool model as a paradigm. Ecology 73: 1552-1563.

Gutierrez, A. P., P. Neuenschwander and J. J. M. van Alphen. 1993. Factors affecting the establishment of natural enemies: biological control of the cassava mealybug in West Africa by introduced parasitoids. Journal of Applied Ecology 30: 706-721.

Morris, R. F. 1959. Single-factor analysis in population dynamics. Ecology 40: 580-588.

Turchin, P. and I. Hanski. 1997. An empirically based model for latitudinal gradient in vole population dynamics. American Naturalist 149: 842-874.