The most likely parameter estimates are found by minimizing the negative log-likelihood function using the same methods as in the previous section. For the algal dynamics of Figure 1 , the most likely parameter estimates are r - 1.54, K- 2.79 x 106, and cr2 - 0.029. The influence of process error on the algal dynamics can be seen in the model predictions (Figure 4). Stochasticity from process error causes trajectories to be correlated through time. For example, a trajectory that increases from process error (relative to the average of many trajectories) will tend to stay high because stochas-ticity directly influences the population abundance.

Continuous models with process error are known as stochastic differential equations, and tend to be more difficult to study than discrete models. A recent approach to fitting continuous models with process error is the method of gradient matching. Gradient matching involves two steps. The first step fits a local regression polynomial (also known as a spline) to the population dynamics, and estimates the gradient at each sampling point. The local nature of the polynomials means that gradient estimates become more independent with increasing distance between successive points. The second step fits o o o o o o o o o o o

Figure 4 Observed algal dynamics (circles) and predicted dynamics for discrete logistic growth with process error (lines) from eqn [8]. Each line type is a separate simulation.

Time (days)

Figure 4 Observed algal dynamics (circles) and predicted dynamics for discrete logistic growth with process error (lines) from eqn [8]. Each line type is a separate simulation.

the gradient estimates to the statistical model. For our algal example, we may postulate the following continuous model with process error:

where Wt is a normally distributed random variable with mean zero and variance u2, and Nt is the observed algal density at time t. Denoting the estimated gradient from the spline regression at time t as Z, the gradient matching model becomes z, =1 - + ^

The likelihood function is f{Z — z,\g(N,, r, K),a2

exp[

\Z2na2

For continuous logistic growth with process error, the most likely parameter estimates given the observed data are r = 1.82, K = 1.0 x 106, and a2 = 1.37 x 1010. The fit regression spline used to estimate the gradient, the estimated gradients, and three example trajectories are shown in Figure 5.

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