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FIGURE 4.15 I (a) From the top two graphs we see that no flyby of the saddle equilibrium (L, P, A) = (47.13, 37.71,11.92) is predicted in the experimental treatment cpa = 0.10. The second row of graphs shows a stochastic simulation, using the model (4.7) and parameters from Table 4.1, in which two saddle flybys occur. The bottom two graphs show evidence of two and possibly three saddle flybys in the data from one replicate of this treatment, (b) The two saddle flybys exhibited by the data from replicate 16 are displayed in state space.

FIGURE 4.15 I (a) From the top two graphs we see that no flyby of the saddle equilibrium (L, P, A) = (47.13, 37.71,11.92) is predicted in the experimental treatment cpa = 0.10. The second row of graphs shows a stochastic simulation, using the model (4.7) and parameters from Table 4.1, in which two saddle flybys occur. The bottom two graphs show evidence of two and possibly three saddle flybys in the data from one replicate of this treatment, (b) The two saddle flybys exhibited by the data from replicate 16 are displayed in state space.

FIGURE 4.16 I Data from the cpa = 0.10 treatment for weeks 40 through 80 are plotted in three-dimensional state space (open circles). Unlike in Fig. 4.8, only two replicates are shown. Replicate 16 has been eliminated because it does not reach the attractor because random perturbations caused repeated saddle flybys (Fig. 4.15). Also shown are the predicted deterministic LPA model attractor (a 26-cycle consisting of the solid circles).

FIGURE 4.16 I Data from the cpa = 0.10 treatment for weeks 40 through 80 are plotted in three-dimensional state space (open circles). Unlike in Fig. 4.8, only two replicates are shown. Replicate 16 has been eliminated because it does not reach the attractor because random perturbations caused repeated saddle flybys (Fig. 4.15). Also shown are the predicted deterministic LPA model attractor (a 26-cycle consisting of the solid circles).

ultimately oscillate out of phase. For example, in treatment cpa = 1.0 the deterministic attractor is a 6-cycle. In Fig. 4.17 we see that the L-stage components from all three replicates in this treatment resemble the periodic 6-cycle but are all, by the end of the experiment, out of phase with one another. Notice, as we have pointed out before, how averaging over replicates would destroy the deterministic signal! (Phase differences among replicates are also present in treatment cpa = 0.50.)

4.4 | CONCLUDING REMARKS

I am in a state of chaos.

The principal aim of the route-to-chaos experiment was to document the occurrence of a (deterministic) model predicted sequence of dynamic transitions — a route-to-chaos experiment—in a real biological

LPA model

LPA model

FIGURE 4.17 I The attractor in experimental treatment cpa = 1.00 is a 6-cycle whose L-stage component appears in the top graph. The remaining three graphs show that all three replicates from this treatment resemble the predicted transients and the 6-cycle attractor through week 14. At week 10 each begins a pattern in phase with the model 6-cycle. However, by the end of the experiment all replicates are out of phase with one another. Replicate 23 sustains an oscillatory pattern similar to the model 6-cycle to the end of the experiment. Replicate 3 suffers a random perturbation at week 16 that shifts its phase by one time unit. Replicate 18 experiences a "mild" saddle flyby during weeks 18 to 34 after which its phase is different from either of the other two replicates.

population. The experiment was designed to place cultures of flour beetles at selected locations along this predicted sequence, including locations where chaotic dynamics are predicted to occur. The experimental protocol required a straightforward manipulation of certain demographic parameters of the beetle populations, namely the death and recruitment rates of the adult life-cycle stage; other life-cycle stages were left un-manipulated. To document the responses of the beetle populations to these selected manipulations we collected census data from replicate and control cultures for 80 weeks.

Comparison of the data with the a priori predicted deterministic at-tractors indicates convincingly that the populations did undergo the predicted dynamic changes. Particularly striking are the two treatments at the ends of the bifurcation sequence where the attractors are simple, distinctively different, and hence readily apparent in the data. The close match between prediction and data is also strongly supported by statistical analysis. In particular, for the two deterministically predicted chaotic attractors at the treatments cpa — 0.05 and 0.35 the model statistically accounts for 92.5% of the variability in the transformed L-stage abundances and 98.8% of the P-stage abundances in the six cultures allotted to these treatments (according to the generalized R2 value calculated for these populations). Furthermore, bootstrap analyses show that chaos is highly prevalent for these two treatments within the estimated confidence regions for the estimated parameters.

Because the dynamics have a stochastic component we cannot, in a strict mathematical sense, label the populations in the chaotic treatments as chaotic (a property of deterministic dynamical systems). Nonetheless it is reasonable to conclude that their dynamics are "highly influenced" by deterministic chaos. By this we mean that their dynamic properties are closely connected to the model predicted attractors and chaos is prevalent throughout the confidence intervals for the parameter estimates. In the next chapter, we explore in more detail the influence that the chaotic attractor has on these populations.

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