V

(lnfc

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0e+00

2e+05

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1e+06

Algal density (cells ml-1)

0e+00

2e+05

4e+05

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Algal density (cells ml-1)

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Figure 5 Gradient matching method. (a) The regression spline used to estimate the gradient at each sampling point (lines) fit to the algal dynamics (circles) from Figure 1. (b) The estimated gradients (circles) are plotted against algal density along with the fit gradient model (line) of eqn [9]. (c) Observed algal dynamics (circles) and simulated dynamics assuming continuous logistic growth and process error (lines). Each line type is a separate simulation.

Time (days)

Figure 5 Gradient matching method. (a) The regression spline used to estimate the gradient at each sampling point (lines) fit to the algal dynamics (circles) from Figure 1. (b) The estimated gradients (circles) are plotted against algal density along with the fit gradient model (line) of eqn [9]. (c) Observed algal dynamics (circles) and simulated dynamics assuming continuous logistic growth and process error (lines). Each line type is a separate simulation.

with initial conditions m1 = ln(F1) and v2 = y2. The likelihood function is similar to that of a normal distribution, but with a mean and variance that change through time. While the likelihood function is more involved, the most likely parameter estimates are still found by numerically searching the likelihood function. For the algal dynamics of Figure 1, the most likely parameter estimates are r = 4.06, c= 0.70, a2 = 7.47 x 10~2, and r2 = 4.32 x 10-26

The parameter estimates indicate that the magnitude of process error is a good deal larger than the observation error. As a result, the statistical model behaves much like the discrete model with process error shown in Figure 4.

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