Deterministic

DETERMINISTIC LĀ£

FIGURE 4.10 I Histograms of the 2000 bootstrap estimates of the deterministic Lyapunov exponents for each experimental treatment. Not visible on the Cpa ā€” 0.25 histogram are six positive Lyapunov exponents. (From B. Dennis, R. A. Desharnais, J. M. Cushing, Shandelle M. Henson, and R. F. Costantino, Estimating chaos and complex dynamics in an insect population, Ecological Monographs 71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.)

The LE histograms for the other treatments are highly skewed and peculiar (Fig. 4.10). For the cpa = 0.00 treatment, most of the bootstrapped LE values are negative, but there is a spike of frequency at 0. The sampling distribution of the LE estimates here is apparently a mixed discrete and continuous distribution, with the positive probability at 0 giving the proportion of times the dynamic behavior is estimated to be aperiodic cycling on an invariant loop. The cpa ā€” 0.10 treatment has a left-skewed, ramp-shaped histogram of LE values in the negative region, with a small but visible tail extending into the positive region (38 out of 2000 bootstrapped

LEs were positive). The cpa ā€” 1.00 treatment has a left-skewed, J-shaped histogram of LE values. The LE values for cpa = 1.00 were all strictly negative, although the histogram peak is near 0; the estimated cycles are weakly stable.

Though the LE histograms for the cpa = 0.05 and cpa = 0.35 treatments have negative portions, they extend considerably into the positive region (Fig. 4.10). For the cpa = 0.05 treatment, the skeleton attractor is highly variable in this region of parameter space (recall Fig. 4.9). A variety of stable cycles (1218 negative estimated LE values out of the 2000), as well as chaos (782 positive estimated LE values out of the 2000), are plausible behaviors estimated for the attractor, according to the bootstrapped LE values for cpa = 0.05. The bulk (1670 out of the 2000) of the bootstrapped LE values for the cpa = 0.35 treatment are positive. The remaining negative portion of the LE values for that treatment correspond to small windows of high-period stable cycles in an otherwise complex bifurcation diagram (recall Fig. 4.9).

The histograms for the bootstrapped SLE values are bell-shaped for all the treatments (Fig. 4.11).16 The control and cpa = 1.00 treatments have SLE histograms that are unambiguously within the negative region. The cpa = 0.05, 0.10, 0.25, 0.35, and 0.5 treatments have SLE histograms that are unambiguously positive. The cpa = 0.00 treatment has an SLE histogram that straddled 0. The normal-like shape of the histograms suggests that the SLE is a stable function of the model parameters for which the conventional asymptotic normality theory of maximum likelihood estimates might apply [110].

For each treatment the 2000 bootstrapped parameter set provides 2000 estimated attractors. We summarize the frequencies of different dynamic behaviors of those attractors in a series of pie diagrams in Fig. 4.12. Recall that we have no formal, rigorous proof for the existence of chaotic attractors for any parameter values. An attractor is termed "chaotic" for the construction of these pie diagrams if its estimated LE is positive.

The control and the cpa = 0.50 treatment are both robust in the sense that each shows entirely one type of attractor in Fig. 4.12, namely, a stable point equilibrium for the control and stable 3-cycles for the cpa = 0.50 treatment. The remaining treatments display more than one attractor type, although the cpa = 0.25 treatment is also reasonably robust in that it displays stable 8-cycles almost entirely, with only a tiny portion (0.3%) of the attractors being chaotic. These three treatments have ordinary bell-shaped histograms of bootstrapped LE values (Fig. 4.10).

16 The calculation of the bootstrapped SLEs for these histograms with appropriate numerical precision and safeguards required weeks of running time on a contemporary Pentium computer.

Chaotic Attractor

FIGURE 4.11 I Histograms of the 2000 bootstrap estimates of the stochastic Lyapunov exponents for each experimental treatment. (From B. Dennis, R. A. Desharnais, J. M. Cushing, Shandelle M. Henson, and R. F. Costantino, Estimating chaos and complex dynamics in an insect population, Ecological Monographs 71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.)

FIGURE 4.11 I Histograms of the 2000 bootstrap estimates of the stochastic Lyapunov exponents for each experimental treatment. (From B. Dennis, R. A. Desharnais, J. M. Cushing, Shandelle M. Henson, and R. F. Costantino, Estimating chaos and complex dynamics in an insect population, Ecological Monographs 71, No. 2 (2001), 277-303. Reprinted with permission from The Ecological Society of America.)

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