Stochastic dynamics and deterministic skeletons: Population behavior of Dungeness crab

Science

Volumn 276, Issue 5317, 1997, Pages 1431-1434

  Higgins, Kevin ^a,c   Hastings, Alan ^a   Sarvela, Jacob N ^b   Botsford, Louis W ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^c UNIVERSITY OF HELSINKI   (Finland)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE; BEHAVIOR; CRAB; DENSITY; DYNAMICS; ENVIRONMENT; MALE; NONHUMAN; POPULATION; PRIORITY JOURNAL; SEA; STOCHASTIC MODEL;

DENSITY DEPENDENCE; DUNGENESS CRAB; ENVIRONMENTAL FACTOR; POPULATION REGULATION; STOCHASTIC MODEL;

CANCER MAGISTER; DECAPODA;

EID: 0030616677     PISSN: 00368075     EISSN: None     Source Type: Journal    
DOI: 10.1126/science.276.5317.1431     Document Type: Article

References (42)

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22. Spatial structure may have important dynamical consequences for populations like Dungeness crab [A. Hastings and K. Higgins, Science 263, 1133 (1994)].
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25. Potentially each age class is affected by environmental perturbations that do not affect other age classes and by environmental perturbations that affect all age classes. Because we have only univariate observations (catch records for legal size Dungeness crab males), we can estimate the perturbations on the observed (harvested) class of crabs and the relative perturbations on other age classes - perturbations to all age classes are exactly correlated, but of varying intensities. To estimate perturbations unique to other classes would require multivariate observations.
26. i ≤ 1, i = 1 to 4. Because of these constraints there are only three independent ρ's.
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29. Catch records for Dungeness crab do not vary smoothly and show extreme year-to-year variation. From north to south, the coefficients of variation are 0.61, 0.49, 0.47, 0.57, 0.52, 0.85, 0.66, and 0.77 for the eight time series. The most variable time series change over almost two orders of magnitude [see (10) for details].
30. 2/41 A very complex bifurcation structure for models of this type (10) leads to a difficult fitting (optimization) problem, which we have solved with a novel approach. Our method of negotiating the goodness of fit surface is based on a hybrid of simulated annealing [W. L. Goffe, G. D. Ferrier, J. Rogers, J. Econometrics 60, 65 (1994)], Powell's direction method [W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge Univ. Press, Cambridge, ed. 2, 1992)], and a sophisticated step-size adjuster that we designed.
31. 2/41 A very complex bifurcation structure for models of this type (10) leads to a difficult fitting (optimization) problem, which we have solved with a novel approach. Our method of negotiating the goodness of fit surface is based on a hybrid of simulated annealing [W. L. Goffe, G. D. Ferrier, J. Rogers, J. Econometrics 60, 65 (1994)], Powell's direction method [W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: the Art of Scientific Computing (Cambridge Univ. Press, Cambridge, ed. 2, 1992)], and a sophisticated step-size adjuster that we designed.
32. L. W. Botsford, Can. J. Fish. Aquat. Sci. 38, 1295 (1981).
33. We cannot test all alternative models, which is a drawback of nonlinear versus linear modeling. We do strongly suspect that the model framework is not critical to the main result.
34. We calculated largest Lyapunov exponents for the stochastic model dynamics on the basis of expansion and contraction of an arbitrary perturbation vector using a Jacobian method. At successive locations on the stochastic attractor, the Jacobian was applied to the same renormalized perturbation vector. Chaotic attractors have a direction along which this vector grows (that is, a positive largest Lyapunov exponent). The largest Lyapunov exponents for the best-fitting model at each location are as follows: -0.43, Grays Harbor, Willapa Bay, and Columbia River; -0.39, Astoria and Warrenton; -0.43, Tillamook and Garibaldi; -0.31, Newport and Depoe Bay; -0.36, Coos and Winchester Bays; -0.26, Brookings, Gold Beach, and Port Orford; -0.057, Eureka and Crescent City; -0.30, Fort Bragg.
35. Oceanographic variables in this system exhibit a pronounced correlation structure (that is, El Niño). Assuming that oceanographic variables (for example, sea temperature) that impact biological processes (for example, development) are correlated, a model based only on biology should not be able to remove all of the determinism from the data. In fact, if the model left behind pure white noise, this would imply that biological mechanisms were accounting for the correlation structure of physical variables.
36. a, all models had simulated dynamics with very high variability. True confidence intervals (by bootstrapping) for this 10-parameter highly nonlinear model would require thousands of days of computer time, which is clearly not feasible. So we confirmed the qualitative behavior over a reasonable range of a subset of the parameters.
37. S. Tuljapurkar, Population Dynamics in Variable Environments (Springer-Verlag, New York, 1990); P. L. Chesson, in Community Ecology, J. Diamond and T. J. Case, Eds. (Harper & Row, New York, 1986), pp. 240-256.
38. S. Tuljapurkar, Population Dynamics in Variable Environments (Springer-Verlag, New York, 1990); P. L. Chesson, in Community Ecology, J. Diamond and T. J. Case, Eds. (Harper & Row, New York, 1986), pp. 240-256.
39. W. E. Ricker, J. Fish. Res. Board Can. 11, 559 (1954).
40. l without consequence.
41. We assume harvest rate, h, is a simple proportion of the legally harvestable males. More realistic would be some functional form for h that depends on the density of harvestable males. Effort data were not used because there does not seem to be any reliable way to deal with long-term changes in effort.
42. We thank J. E. Keizer and the Institute of Theoretical Dynamics for providing computer support; D. G. Hankin for helpful discussions on Dungeness crab life-history characteristics; and R. F. Costantino, B. Dennis, S. Ellner, J. Quinn, and two anonymous reviewers for helpful discussions and comments on the manuscript.