Concluding Remarks

[T] he advantages of the laboratory can... be brought to bear. Whole series of experiments can be performed yielding sequences of dynamical states whereby one regime gives way to the next in response to varying a control parameter____The upshot is that while any one of the individual data sets might fail to stand up to the rigors of statistical analysis, the cumulative effect of all the data is compelling.

In this chapter we considered a bifurcation sequence predicted by the deterministic LPA model as the adult death rate fia is varied. This sequence included two bifurcations: a period-doubling bifurcation and a reverse period-doubling bifurcation. These transitions in dynamics motivated the implementation of experiments designed to document their occurrence in laboratory cultures of flour beeties. The statistical analyses of the resulting data showed the accuracy of the model's descrip -tion and prediction of the data. Graphs of the data, in both time series and in three-dimensional state space, also show quite convincingly that the beede population numbers did undergo the predicted bifurcations when their adult death rates are manipulated experimentally, with only one exception. In the fj,a = 0.73 treatment of the RR strain experiment the model prediction was an equilibrium while data exhibit an oscillation very much like a 2-cycle. We argued, however, that even in this exceptional case the data are well accounted for by the LPA model, in that the treatment value ¡ia = 0.73 lies near a period-doubling bifurcation value of Ha and a 2-cycle is predicted over a significant portion of the parameter estimate confidence intervals.

The fact that parameter estimates come with confidence intervals is often overlooked. If the same type of attractor (e.g., an equilibrium) is predicted throughout parameter confidence intervals, then one expects that type of attractor to be more easily observed in data. In that case, one can assign that robust attractor type to the biological population with a certain degree of confidence. However, in the case of more complicated dynamics — such as those often associated with bifurcations and chaos— more than one type of attractor can exist within the confidence intervals. Indeed, it can happen that many types of attractors are intricately interspersed throughout confidence intervals. In such cases an assignment of a specific attractor type to the dynamics of the population must be appropriately qualified.

Although the bifurcation experiment was designed around the attractors predicted by the deterministic LPA model, we also saw the effects of stochasticity in the results. These effects are not just a "fuzzing out" of the attractors; they induce phase shifting of oscillations and quiescent episodes caused by saddle flybys. Stochasticity also means that each replicate of a treatment has its own unique history, which is explained by a mix of deterministic entities (transients as well as stable and unstable invariant sets) and stochasticity. Notice that the common practice of averaging over replicates is not necessarily a helpful way to deal with stochasticity. Indeed, averaging over replicates might erase information contained in these unique histories, can obscure the effects of the deterministic model, and can lead to erroneous identification of dynamic patterns.

The success of the bifurcation experiment—documenting model-predicted dynamic changes by means of controlled and replicated laboratory experiments — gives us confidence to investigate other, more complicated model-predicted bifurcation sequences, including sequences involving chaotic attractors. This we do in the next chapter.

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