Demography of the Sandtiger Shark Carcharias taurus (Rafinesque, 1810). Aka Gray Nurse Shark in Australia, Spotted Ragged-Tooth Shark in South Africa.
In progress, evaluation of age-at-maturity (6), natural mortality (0.1543 yr^-1) based on 25 yr longevity. One litter of 2 every other year.

Lambda = 1.0362 (r = 0.03556 yr^-1) Ro = 1.41, using 3x3 matrix with variable juvenile stage duration, pregnant and resting stages. This population growth rate is a purely analytical projections assuming that the environment is constant and that density effects are unimportant (Caswell 2001).
Because the age-at-maturity is low (6 yr), one should not use an effective annual fecundity. Doing so, underestimates Ro and lambda because one full litter at age 6 is "worth" more than two "half-litters" at ages 6 & 7 and so forth. In this particular case, using an annual effective fecundity yields lambda = 1.0238 (2.4 % annual population increase compared to 3.6%).

Following Caswell H. (2001). Matrix Population Models.
Sinauer Associates, Inc., MA 01375 USA. 722 p.

Three-stage matrix model comprising, juveniles. pregnant adults, and resting adults following Brewster and Miller (2000) and Caswell (2000). Matrix population models are "the future" for demographic analysis of pelagic elasmobranchs on a global scale. That is, a 15x15 matrix can handle a species in 5 oceans with different mortality and fertility in each region in an open system (i.e. immigration and emigration can be included).

Age at maturity 6 years (Branstetter and Musick 1994). Duration of first stage T(1) = 5yr. Duration of second stage (pregnant) and third third stage are both 1 year long. This is like an infinite loop but it will get terminated, so to speak, by the survivorship curve (mortality). Longevity 25 years (5ln2/k, with k = 0.14 yr-1 from Branstetter and Musick 1994). Mortality, M = 0.1543 yr-1 (S = 0.8570) based on longevity of 25 years and using Hoenig (1983) for cetaceans; The litter size is 2, and the reproductive cycle is 2 years (Branstetter and Musick 1994, G. Cliff 1995 personal communication)

Stage-duration distributions: Sigma (i) = P (survival of an indiviudal in stage i) Gamma (i) = P (growth from i to i+1 | survival) = growth probability for surviving individuals G(i) = sigma (i) gamma (i) P(i) = sigma (i) (1-gamma (i) ) I use a variable stage duration for the juveniles with variance V(1) = 2 yr2 and a fixed stage distribution for the adults. I can calculate gamma (1) from equation 6.l14 on p. 164 in Caswell (2000) and gamma (2) from equation 6.103 on p. 161 and then calculate G(i) and P(i). It's an iterative process. Assume lambda = 1, calculate G(i)'s and P(i)'s, solve 2x2 matrix for lambda, continue until assumed lambda agrees with calculated lamdda. The elements of the projection matrix A with breakdown into transition matrix T and fertility matrix F (A = T + F): (with fertility F(2) = m(2) x G(2) = 1 x 0.8570 = 0.8570). F(2) and G2 &G3 are accidentally identical because m(2) = 1.0. G(3) is the above-the- diagonal matrix elements for the transition of a resting female becoming pregnant againg. It makes this matrix a non-Lefkovitch matrix.

A = 0.7463 0.8570 0 T = 0.7463 0 0 F = 0 0.8570 0 0.1108 0 0.8570. 0.1108 0 0.8570. 0 0 0 0 0.8570 0 0 0.8570 0 0 0 0

Solution using PobTools (free by Greg Hood) or GNU-Octave version 2.1.31 (free version of MATLAB) Lambda 1 = 1.0362, growth rate of stable population (r = ln (lambda 1) = 0.03556 yr-1); (lambda 2 = -0.8866, lamda 3 = 0.5966) will determine how fast stable populaton is reached); Age structure (right eigenvector for lamda 1 of A): w1 (juveniles)= 61.8%, w2 (pregnant) = 20.9%, w3 (resting) = 17.3%; Reproductive values (left eigenvector for lamda 1 of A): v1 (juveniles, age 0) = 1.00, v2 = 2.6 (pregnant, age 6), v3 = 2.2 (resting, age 7). (Note that the % values of individuals in an age-structured Leslie matrix or life history table can be added to obtain a approximate % value of indiviuals in a stage, but that the reproductive values are not additive.)

Sensitivy matrix (first derivative of lamda with respect to matrix elements of A) and elasticity matrix (relative change of lamda with respect to relative change of matrix elements i.e. d ln( lamda)/d ln (aij):

S = 0.4014 0.1358 0 E= 0.2891 0.1123 0 1.0507 0 0.2940   0.1123 0 0.2432 0 0.2940 0 0 0.2432 0

Life cycle graph: To be added.

The life cycle as a Markov chain. This is more powerful than the z-transformed life cycle graphs. See Cochran and Ellner (1992) for additional information. The fundamental matrix N = (I - T)-1 gives the mean time spent in each stage. Using MATLAB (or GNU-Octave) notation: N = inv (eye (3) - T) where eye (3) is the identity matrix of dimension 3, in this particular case.

N = 3.941 0 0 1.644 3.766 3.228 1.409 3.228 3.766   The matrix elements give the mean time spent in each stage. Thus sandiger shark spends, on average, 3.9 year as a juvenile, and then, on average, 1.6 yr as pregant female and 1. 4 yr as resting femalet. A pregant mature adult, in contrast, spends an average 3.8 yr being pregnant and 3.2 yr resting.. The sum of the columns give the mean time to death (= life expectancy) and is the same for all stages (7.0 yr). This appears to be a property of any stage-based model with a resting phase.

The fundamemtal matrix (N) gives the expected number of time steps spent in each transient state and the fertility matrix F gives the expected number of offspring per time step. Therefore, the matrix R = F N has entries r (i,j) that give the expected lifetime production. R projects the population from one generation to the next. Ro is the dominant eigenvalue of R in the most general case with different types of offspring. When there is only one type of offspring, R has only a single nonzero row, and the dominant eigenvalue is just r (1,1) = Ro (= 1.41).

R = 1.409 3.228 2.766 0 0 0 0 0 0   A sandtiger juvenile can be expected to produce 1.41 female offspring during its lifetime. The generation time T is ln Ro/ln lambda = 9.64. Mu1 can be calculated from the matrix (F N2) and is 10.5 yr. Following Coale, A. J. (1972). The growth and structure of human populations. Princeton University Press, Princeton NJ, 227p: The generation time T is defined by e(rT) = Ro (net reproductive rate). It is the time T required for the population to increase by a factor of Ro. The concept is the same when one calculates a doubling time, the time T1/2 required for the population to increase by a factor of 2 or a half-life T1/2 in the case of a decreasing (=decaying) population. However, T is not, somewhat surprisingly, equal to any of the several mean ages of childbearing that can be defined.. There are three mean ages of fertility of interest in this context: 1) the mean age (m bar) of the fertility schedule (mean age of childbearing in a cohort subject to no mortality), 2) the mean age (mu1) of the net fertility schedule (mean age of childbearing in a cohort), 3) the mean age (A bar) of childbearing in the stable populaton. (See Coale (1972) or Caswell (2000) for equations).

What is missing? I should provide a confidence interval for lambda, the growth rate of the stable population. In this particular case and most other demographic analyses for sharks, only a Monte Carlo uncertainty analysis is possible because available demographic data is based on estimates of fertility and little data on survival is available. We cannot use a series approximation (sometime called the "delta method") because the variance of the parameters is not known. We cannot use a bootstrap method because the resampling procedure requires the history of individuals to be known. See Caswell (2000) and Coale and Demeny (1966) how survivorship can be estimated from other species using model life tables. The life tables of other species are re-scaled according to age-at-first reproduction. The lemon shark would be the first species to be included for the construction of a model life table. Perhaps life tables of marine mammals (e.g. whales) should be included. We cannot use a stage-based matrix model to do this, we need to use an age-based matrix model (= Leslie matrix).