Appendix A. Dynamics and time series analysis of replicated open population models with process and measurement error, and examples of empirical dynamics.
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| FIG. A1. Phase portrait of the relationship between abundance at time t and at t+1 for the deterministic skeleton of (A) first order (B) third order and (C) fourth order models of density dependence, after transient dynamics were removed. The second order model exhibited point stability, hence is not shown. |
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| FIG. A2. Typical trajectories for the final 100 time steps of stochastic first through fourth order models with both process error and measurement error. Coefficients of variation are similar to those of real data. The first 1000 iterations generated dynamics under the base parameter values, whereas the final 15 iterations (right of gray dashed line) show transient dynamics when the model parameters were switched to the other model (first- to second-order, third- to fourth-order and vice versa) to mimic an experimental manipulation. |
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FIG. A3. Example of dynamics of Corallina vancouveriensis, Semibalanus cariosus, and Pollicipes polymerus in the presence and chronic absence of Mytilus californianus. The data come from two immediately adjacent plots. Note the large variation in abundance in the presence of mussels relative to that exhibited in the absence of mussels. |
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| FIG. A4. Results of (A) spectral analysis without simulated experimental perturbations, (B) spectral analysis comparison of transient and non-transient dynamics for first and fourth order models, and (C) non-parametric order of density dependent analysis on simulations of stochastic first through fourth order models. |