Model Fitting

Once the lag structure of the dominant -negative feedback mechanism acting on a population of organisms has been decided upon, P1a can be used to fit the following models to the data:

Linear Logistic Model

First, a logistic model with Q = 1 and a single (dominant) time delay decided by diagnostic analysis is fit to the data using a standard linear regression routine (see figure):

  1. A = the y-intercept or maximum per-capita rate of change.
  2. C = the slope of the regression line or the coefficient of density dependence.
  3. K = the x-intercept or the equilibrium density (carrying capacity of the environment); remember that K = A / C.
  4. s = the standard deviation of the data around the regression line. This is an estimate of density independent environmental variability plus measurement and other errors.
  5. r2 = the coefficient of determination or percent of the variation in the data explained by the regression line. This is an estimate of the relative importance of density dependent factors in determining the observed dynamics, compared to density independent factors; i.e., the proportion of the variability in the data explained by density independent forces is assumed to be 1 - r2.

Nonlinear Logistic Model

If the data do not appear linear, a Marquardt / Newton-Raphson convergence routine can be employed (by pressing [F7]) to obtain a nonlinear fit to the logistic model. Here the coefficient Q is sequentially adjusted to obtain a best fit of model to data (see figure).

In order to initiate convergence, an initial value for the maximum per-capita rate of change must be supplied. This is a chance to impose some biological realism on the model. In the case of the sandhill crane, for example, the linear logistic model estimated that the maximum per-capita rate of increase was A = 1.7, which implies that each bird can produce 4-5 offspring per year (an estimate of the birth rate can be obtained from B = exp[1.7] -1 = 4.5). A much more reasonable maximum birth rate is 1 offspring per bird, or A = ln(2) = 0.7. Imposing this condition on the convergence produces quite a different R-function and a better fit (see Tutorial #3.)

Gompertz Model

If the program is in logarithmic mode at the time of model fitting ([F3] was pressed), it will fit the discrete Gompertz model to the data; i.e., R is regressed against ln (Nt-d) using the same procedures as before (see figure). The user should be aware, however, that overflow problems may occasionally be encountered during nonlinear convergence on logarithmic data.

Two-lag Models

On occasion the PRCF may indicate more than one dominant lag. In such cases a multiple regression routine can be used to fit two-lag arithmetic or logarithmic models to the data (see figure).