Ecological Archives A019-032-A2

Melissa L. Snover and Selina S. Heppell. 2009. Application of diffusion approximation for risk assessments of sea turtle populations. Ecological Applications 19:774–785.

Appendix B. Population simulations.

In their criteria for listing a species as either critically endangered, endangered, or "of concern", the IUCN calls for observed, estimated, inferred, or suspected population size reductions of 90, 70, or 50%, respectively, over the last 10 yr or three generations, whichever is longer (up to a maximum of 100 yr; IUCN 2001). As sea turtles do not reach sexual maturity for as much as 40 yr, depending on species and population (Balazs and Chaloupka 2004), three generations can be quite a long time period. Time periods of 35–100 yrs need to be considered and good historical records do not generally exist over such time periods. Diffusion approximation models for population viability fall nicely into the language of the IUCN listing criteria (Dennis et al. 1991, Holmes 2001, Sæther and Engen 2002, Staples et al. 2005). We suggest using population viability analysis based on diffusion approximation to estimate risk of populations declines of 90, 70, or 50% (or quasi-extinction) at a time point of three generations (up to a maximum of 100 yr) in the future using information on the current trends of individual populations.

We used a stochastic age-structured matrix model with variable remigration intervals for breeding to simulate female sea turtle populations. We established ranges of values typical for sea turtle demographic parameters based on empirical data where possible (Table B1; survival: Crouse et al. 1987; Chaloupka 2002; Heppell et al. 2005; fecundity: Miller 1997; Van Buskirk and Crowder 1994; age to maturity: Zug and Parham 1996; Zug et al. 2002; Snover 2002; Heppell et al. 2005; Zug et al. 2006). To establish a range of remigration intervals, a fraction of the adult female population was allowed to breed at each time step (centered on a mean; Table B1; excluding first-time nesters), where the fraction was drawn randomly from a distribution. This was accounted for in the fecundity function f:

f = ESH I M SJ ρ ,

where E is the number of eggs per nest, SH is the hatching success rate or survival rate of the nest, I is the number of nests laid in a season per female, ρ is the proportion of eggs that are female, and M is the fraction of adult females breeding in a given year. Survival probability to age 1, SJ, is also included because we used a pre-breeding census model, such that f is the number of female yearlings produced per adult female (Caswell 2001). A set of mean demographic parameters were randomly selected as input to an age-based matrix model. If the dominant eigenvalue, λ, was 0.9 < λ < 1.1, the parameter combination was kept, otherwise a new combination was selected. Parameter combinations with population growth rates outside of this range are likely unrealistic for natural populations, and extinction risk assessment for populations with such extreme growth or decline rates is somewhat trivial and not a good test of our methods. We repeated this process to establish 800 age-based matrices.

We used the matrix models to simulate time series of nesting beach census data with annual variance similar to that observed in nature. The number of adult females at t = 0 was randomly selected from the range [100, 2000]. The initial population was assumed to be at stable age distribution; hence, we used the dominant right eigenvector of the original matrix (mean demographic parameters for each population; Caswell 2001) to establish the initial number of individuals in each age class. The population was then projected forward stochastically for t = 15, 20, 25, or 30 yr with each parameter selected annually from a distribution (Table B1). Variances of the fecundity parameters were set to be consistent with empirical data (Van Buskirk and Crowder 1994, Miller 1997). Values of variance around the survival rates and remigration intervals were adjusted to result in time series of nesting females with variance similar to that observed in nature. For each simulated trajectory, annual counts of total population size (excluding hatchlings to year 1), total adult females, nesting adult females, and observed nesting females were recorded. Number of observed nesting females was a fraction (p) of the total number of nesting females for the year, where p ~ U (0.5, 0.8). This adds an observation error to the simulated nesting beach data, which better approximates available observed data.

TABLE B1. Ranges of values used to select parameter values for model simulations.


Range and distribution
for mean value

Variance and distribution for
interannual variability

Annual juvenile survival rate

0.65–0.90, uniform

0.002, beta

Annual adult survival rate

0.75–0.98, uniform

0.0002, beta

Proportion of adult females nesting

0.22–0.65, uniform

0.02, beta

Eggs per nest

80–130, uniform

15, gamma

Survival of nest (hatch success rate)

0.50–0.95, uniform

0.002, beta

Nests per season

2–7, uniform

1, gamma

Sex ratio

0.45–0.65, uniform

0.002, beta

Age to maturity

10–40, uniform

static for the purposes of age-
based population matrix models

   Notes: For each simulation, a set of mean values was randomly selected, a population matrix set up, and the dominant eigenvalue λ calculated. At each time step, actual values for the parameters were randomly selected from the stated distributions with the original mean and fixed variance. Where possible, ranges of values are based on empirical data known about marine turtle populations.


Balazs, G. H., and M. Chaloupka.. 2004. Spatial and temporal variability in somatic growth of green sea turtles (Chelonia mydas) resident in the Hawaiian Archipelago. Marine Biology 145:1043–1059.

Caswell, H. 2001. Matrix population models. Sinauer Associates, Sunderland, Massachusetts, USA, 722 pp.

Chaloupka, M. 2002. Stochastic simulation modelling of southern Great Barrier Reef green turtle population dynamics. Ecological Modelling 148:79–109.

Crouse, D. T., L. B. Crowder, and H. Caswell. 1987. A stage-based population model for loggerhead sea turtles and implications for conservation. Ecology 68:1412–1423.

Dennis, B., P. L. Munholland, and J. M. Scott. 1991. Estimation of growth and extinction parameters for endangered species. Ecological Monographs 61:115–143.

Heppell, S. S., D. T. Crouse, L. B. Crowder, S. P. Epperly, W. Gabriel, T. Henwood, R. Márquez, and N. B. Thompson. 2005. A population model to estimate recovery time, population size, and management impacts on Kemp's ridley sea turtles. Chelonian Conservation and Biology 4:767–773.

Holmes, E. E. 2001. Estimating risks in declining populations with poor data. Proceedings of the National Academy of Science 98:5072–5077.

IUCN (Species Survival Commission). 2001. International Union of the Conservation of Nature Red List Categories and Criteria. IUCN, Gland, Switzerland. [Online, URL:< <>.]

Miller, J. D. 1997. Reproduction in sea turtles. Pages 51–81. in P. L. Lutz and J. A. Musick, Editors, The Biology of Sea Turtles, CRC Press, Boca Raton, Florida, USA.

Sæther, B-E., and S. Engen S. 2002. Including uncertainties in population viability analysis using population prediction intervals. Pages 191–212 in S. R. Beissinger and D. R. McCullough, editors. Population Viability. University of Chicago Press, Chicago, Illinois, USA.

Snover, M. L. 2002. Growth and ontogeny of sea turtles using skeletochronology: methods, validation and application to conservation. Ph.D. dissertation, Duke University, Durham, North Carolina, USA.

Staples D. F., M. L.Taper, and B. B. Shepard. 2005. Risk-based viable population monitoring. Conservation Biology 19:1908–1916.

Van Buskirk, J., and L. B. Crowder. 1994. Life-history variation in marine turtles. Copeia 1994:66–81.

Zug, G. R., G. H. Balazs, J. A. Wetherall, D. M. Parker, and S. K. K. Murakawa. 2002. Age and growth of Hawaiian green sea turtles (Chelonia mydas): and analysis based on skeletochronology. Fishery Bulletin 100:117–127.

Zug, G. R., M. Chaloupka, and G .H. Balazs. 2006. Age and growth in olive ridley sea turtles (Lepidochelys olivacea) from the North-central Pacific: a skeletochronological analyses. Marine Ecology 27:263–270.

Zug, G. R, and J. F. Parham. 1996. Age and growth in leatherback sea turtles, Dermochelys coriacea (Testudines, Dermochelyidae): a skeletochronological analysis. Chelonian Conservation Biology 2:244–249.

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