Abstract
We propose a spatiotemporal generalized von Bertalanffy (vonB) growth model that also includes between-individual (BI) variation and male/female correlation. The generalized vonB model includes the effect of maturation on growth. The model and the methodology are applied to a long time-series of survey observations of age and length for American plaice on the Grand Bank off the northeast coast of Canada. The bias in age-length data due to size selectivity of the survey gear is accounted for. The survey design includes length-stratified age sampling which is a type of response selective sampling design for growth model estimation. We propose and implement a conditional empirical proportion likelihood approach for these data. Neglecting this sampling scheme can lead to seriously biased estimation results. We found that a 6-parameter growth model is necessary for capturing the biphasic growth patterns of the American plaice on the Grand Bank, and the survey gear selectivity and BI variation are important for a good model fit. We proposed an empirically optimal BI variation model for this data. Our estimation results indicate that there are substantial differences in size-at-age for male and female American plaice, and this changes over time and between regions.
Similar content being viewed by others
References
Brett J (1979) Environmental factors and growth. Fish physiology, vol VIII. Bioenergetics and growth. Academic Press, Orlando, pp 599–677
Brody S (1945) Bioenergetics and growth. Reinhold Publishing Corporation, New York
Cadigan NG, Campana SE (2017) Hierarchical model-based estimation of population growth curves for redfish (sebastes mentella and sebastes fasciatus) off the eastern coast of canada. ICES J Mar Sci 74(3):687–697
Candy SG (2005) Fitting a Von Bertalanffy growth model to length-at-age data accounting for length-dependent fishing selectivity and length-stratified sub-sampling of length frequency samples, Document WG-FSA-SAM-05/13. CCAMLR, Hobart, Australia
Candy SG, Constable AJ, Lamb T, Williams R (2007) A von bertalanffy growth model for toothfish at heard island fitted to length-at-age data and compared to observed growth from mark-recapture studies. CCAMLR Sci 14:43–66
Colbourne E, Holden J, Senciall D, Bailey W, Snook S (2016) Physical oceanographic environment on the newfoundland and labrador shelf in nafo subareas 2 and 3 during 2015. NAFO Scientific Council Research Document 16(07)
Day T, Taylor PD (1997) Von bertalanffy’s growth equation should not be used to model age and size at maturity. Am Nat 149(2):381–393
Enberg K, Jørgensen C, Dunlop ES, Varpe Ø, Boukal DS, Baulier L, Eliassen S, Heino M (2012) Fishing-induced evolution of growth: concepts, mechanisms and the empirical evidence. Mar Ecol 33(1):1–25
Filipe PA, Braumann CA, Roquete CJ (2012) Multiphasic individual growth models in random environments. Methodol Comput Appl Prob 14(1):49–56
Gavaris S, Brodie W (1984) Results of comparative fishing between the at cameron and the wilfred templeman during july-august 1983. CAFSAC Res Doc 84(41):16
Goodyear CP (1995) Mean size at age: an evaluation of sampling strategies with simulated red grouper data. Trans Am Fish Soc 124(5):746–755
Goodyear CP (2019) Modeling growth: consequences from selecting samples by size. Trans Am Fish Soc 148:528–551
Hájek J (1971) Discussion of ‘an essay on the logical foundations of survey sampling, part one’ by d. basu. Foundations of statistical inference Edited by VP Godambe and DA Sprott Holt, Rinehart, Winston, Toronto, Ont, pp 236
Kristensen K (2018) Rdocumentation: sdreport. https://www.rdocumentation.org/packages/TMB/versions/1.7.14/topics/sdreport
Kristensen K, Nielsen A, Berg CW, Skaug H, Bell B (2015) Tmb: automatic differentiation and laplace approximation. arXiv preprint arXiv:1509.00660
Laslett GM, Eveson JP, Polacheck T (2002) A flexible maximum likelihood approach for fitting growth curves to tag recapture data. Can J Fish Aquat Sci 59(6):976–986
Lester N, Shuter B, Abrams P (2004) Interpreting the von bertalanffy model of somatic growth in fishes: the cost of reproduction. Proc R Soc Lond B Biol Sci 271(1548):1625–1631
Lester NP, Shuter BJ, Venturelli P, Nadeau D (2014) Life-history plasticity and sustainable exploitation: a theory of growth compensation applied to walleye management. Ecol Appl 24(1):38–54
Manabe A, Yamakawa T, Ohnishi S, Akamine T, Narimatsu Y, Tanaka H, Funamoto T, Ueda Y, Yamamoto T (2018) A novel growth function incorporating the effects of reproductive energy allocation. PLoS ONE 13(6):e0199346
Mills TC, Mills TC (1991) Time series techniques for economists. Cambridge University Press, Cambridge
Minte-Vera CV, Maunder MN, Casselman JM, Campana SE (2016) Growth functions that incorporate the cost of reproduction. Fish Res 180:31–44
Morgan MJ, Colbourne E (1999) Variation in maturity-at-age and size in three populations of american plaice. ICES J Mar Sci 56(5):673–688
Morgan MJ, Hoenig JM (1997) Estimating maturity-at-age from length stratified sampling. J Northwest Atl Fish Sci 21:51–64
Morgan MJ, Brodie W, Bowering W, Parsons DM, Orr D, (1998) Results of data conversions for american plaice in div. 3 lno from comparative fishing trials between the engel otter trawl and the campelen, (1800) shrimp trawl. Sci Counc Res Doc NAFO 98:10
Morin R, LeBlanc SG, Campana SE (2013) Bomb radiocarbon validates age and long-term growth declines in american plaice in the southern gulf of st. lawrence. Trans Am Fish Soc 142(2):458–470
Perreault AM, Zheng N, Cadigan NG (2019) Estimation of growth parameters based on length-stratified age samples. Can J Fish Aquat Sci 999:1–12
Piner KR, Lee HH, Maunder MN (2016) Evaluation of using random-at-length observations and an equilibrium approximation of the population age structure in fitting the von bertalanffy growth function. Fish Res 180:128–137
Pitt T (1966) Sexual maturity and spawning of the american plaice, hippoglossoides platessoides (fabricius), from newfoundland and grand bank areas. J Fish Board Canada 23(5):651–672
Prajneshu Venugopalan R (1999) von bertalanffy growth model in a random environment. Can J Fish Aquat Sci 56(6):1026–1030
Quince C, Abrams PA, Shuter BJ, Lester NP (2008) Biphasic growth in fish i: theoretical foundations. J Theor Biol 254(2):197–206
Quinn TJ, Deriso RB (1999) Quantitative fish dynamics. Oxford University Press, Oxford
Quist MC, Pegg MA, DeVries DR (2012) Age and growth. Fisheries techniques, 3rd edn. American Fisheries Society, Bethesda, pp 677–731
R Core Team (2018) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, URL https://www.R-project.org/
Region N (1995) American plaice, hippoglossoides platessoides, life history and habitat characteristics. In: Proceedings of the symposium on the potential for development of aquaculture in Massachusetts, vol 15, pp 17
Richards F (1959) A flexible growth function for empirical use. J Exp Bot 10(2):290–301
Roff DA (1983) An allocation model of growth and reproduction in fish. Can J Fish Aquat Sci 40(9):1395–1404
Roff DA et al (2000) Trade-offs between growth and reproduction: an analysis of the quantitative genetic evidence. J Evol Biol 13(3):434–445
Russo T, Baldi P, Parisi A, Magnifico G, Mariani S, Cataudella S (2009) Lévy processes and stochastic von bertalanffy models of growth, with application to fish population analysis. J Theor Biol 258(4):521–529
Sainsbury K (1980) Effect of individual variability on the von bertalanffy growth equation. Can J Fish Aquat Sci 37(2):241–247
Schnute J (1981) A versatile growth model with statistically stable parameters. Can J Fish Aquat Sci 38(9):1128–1140
Schnute JT, Richards LJ (1990) A unified approach to the analysis of fish growth, maturity, and survivorship data. Can J Fish Aquat Sci 47(1):24–40
Schueller AM, Williams EH, Cheshire RT (2014) A proposed, tested, and applied adjustment to account for bias in growth parameter estimates due to selectivity. Fish Res 158:26–39
Shelton AO, Satterthwaite WH, Beakes MP, Munch SB, Sogard SM, Mangel M (2013) Separating intrinsic and environmental contributions to growth and their population consequences. Am Nat 181(6):799–814
Soriano M, Moreau J, Hoenig JM, Pauly D (1992) New functions for the analysis of two-phase growth of juvenile and adult fishes, with application to nile perch. Trans Am Fish Soc 121(4):486–493
Stansbury D (1997) Conversion factors for cod from comparative fishing trials for engel 145 otter trawl and the campelen 1800 shrimp trawl used on research vessels. NAFO SCR Doc 97:73
Swain DP, Sinclair AF, Mark Hanson J (2007) Evolutionary response to size-selective mortality in an exploited fish population. Proc R Soc B Biol Sci 274(1613):1015–1022
Von Bertalanffy L (1960) Principles and theory of growth. In: Nowinski W (ed) Fundamental aspects of normal and malignant growth. Elsevier, Amsterdam, pp 137–259
Walsh SJ (1997) Efficiency of bottom sampling trawls in deriving survey abundance indices. Oceanogr Lit Rev 44(7):748–748
Wang YG, Thomas MR, Somers IF (1995) A maximum likelihood approach for estimating growth from tag-recapture data. Can J Fish Aquat Sci 52(2):252–259
Warren W, Brodie W, Stansbury D, Walsh S, Morgan J, Orr D (1997) Analysis of the 1996 comparative fishing trial between the alfred needler with the engel 145 trawl and the wilfred templeman with the campelen 1800 trawl. NAFO SCR Doc 97:68
Webber DN, Thorson JT (2016) Variation in growth among individuals and over time: a case study and simulation experiment involving tagged antarctic toothfish. Fish Res 180:67–76
Wheeland L, Dwyer K, Morgan J, Rideout R, Rogers R (2018) Assessment of american plaice in div. 3lno. NAFO SCR Doc 18(039)
Zheng N, Cadigan N (2019) Likelihood methods for basic stratified sampling, with application to von bertalanffy growth model estimation. Open J Stat 9(6):623–642
Acknowledgements
We thank all of the people involved in the collection and processing of these data. Research funding to NZ and NC was provided by the Ocean Frontier Institute, through an award from the Canada First Research Excellence Fund. Research funding to NC was also provided by the Ocean Choice International Industry Research Chair program at the Fisheries and Marine Institute of Memorial University of Newfoundland. Many thanks for the comments from the two anonymous reviewers that greatly improved this manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Handling Editor: Bryan F. J. Manly
Electronic supplementary material
Below is the link to the electronic supplementary material.
Appendices
Appendices
1.1 Appendix A: Stationary results
If the AR(1) models in (6) are stationary, the marginal distributions of \(\log (L_{\infty ,c,d,g})\) and \(\log (K_{c,d,g})\) are
with \(\omega _{\infty ,d,g}^2 = \delta _{\infty ,d,g}^2/(1-\varphi _{\infty ,d,g}^2)\) and \(\omega _{k,d,g}^2 = \delta _{k,d,g}^2/(1-\varphi _{k,d,g}^2)\), and the distributions of \(\log (L_{\infty ,1,d,g})\) and \(\log (K_{1,d,g})\) are given by (29).
Based on the stationary assumption, the marginal sex and division correlations are
1.2 Appendix B: Conditional distribution of a captured fish
The conditional distribution of a captured fish (i.e. \(C=1\)) is
In the last step we applied \(\mathrm {Pr}\{ C=1 \,|\, Y=y, A=a;\pmb {\theta } \} = \mathrm {Pr}\{ C=1 \,|\, Y= y \}\); that is, the probability of capture does not depend on age given the size of a fish.
Rights and permissions
About this article
Cite this article
Zheng, N., Cadigan, N. & Morgan, M.J. A spatiotemporal Richards–Schnute growth model and its estimation when data are collected through length-stratified sampling. Environ Ecol Stat 27, 415–446 (2020). https://doi.org/10.1007/s10651-020-00450-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10651-020-00450-8