Input parameters: Age-at-first-reproduction 9 y; longevity 15 y;
Mortality (M, y-1) = - ln(0.01)/15 =
0.3070, Survival(S) = exp(-M) = 0.7356;
Fertility (m, assuming 50% females) = 47 eggcases per year/2 = 23.5, F (= discounted
fertility) = m Sa = 23.5 x 0.7356 = 17.2876 (using a post-breeding census).
Note 1: As I use the same survival rate for all age classes, the projetion matrix A looks exactly the same for pre- and post-breeding census. I use colors to indicate how the construction of the projection matrix is completely different.
Note 2: The discounted fertility F depends on both fertility m and the survival rates of the aduts (in the case of a post-breeding census). This is most important when calculatig the E-patternbecause the discounted fertilities contribute to the elasticity of the adults (E3). In the E-matrix below I show this by using dual entries (when there is reallly only one) and color them differently to show how the additions for the E-pattern has to be done to get the correct E-pattern. If we use a Life History Table (LHT) instead of a Leslie matrix, the problem does not arise becasue the LHT only contains fertilities m, no discounted fertilities F = m Sa.
Age at-first-reproduction may have to increased and fertility may have to be
decreased in analysis because:
"Egg cases of barndoor skates are the largest among species in the Northwest
Atlantic and may require as much as two years before hatching. Because their
egg cases are large, are deposited on the sea floor and take so long to hatch,
they may also be vulnerable to impacts from fishing gear as well as mortality
when they are caught in gear and discarded." http://www.mcbi.org/bdoor/skatest.html.
Life Cycle Graph:
Projection Matrix (A) using pre-breeding census |
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Age class | J1 | J2 | J3 | J4 | J5 | J6 | J7 | J8 | A9 | A10 | A11 | A12 | A13 | A14 | A15 | |
J1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 17.2876 | 17.2876 | 17.2876 | 17.2876 | 17.2876 | 17.2876 | 17.2876 | |
J2 | 0.7356 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
J3 | 0 | 0.7356 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
J4 | 0 | 0 | 0.7356 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
J5 | 0 | 0 | 0 | 0.7356 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
J6 | 0 | 0 | 0 | 0 | 0.7356 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
J7 | 0 | 0 | 0 | 0 | 0 | 0.7356 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
J8 | 0 | 0 | 0 | 0 | 0 | 0 | 0.7356 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
A09 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.7356 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |
A10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.7356 | 0 | 0 | 0 | 0 | 0 | 0 | |
A11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.7356 | 0 | 0 | 0 | 0 | 0 | |
A12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.7356 | 0 | 0 | 0 | 0 | |
A13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.7356 | 0 | 0 | 0 | |
A14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.7356 | 0 | 0 | |
A15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.7356 | 0 | |
0.7356* | ||||||||||||||||
Eigenvalues | Eigenvectors (R&L) | * Not part of Projection matrix, need to calculate F15 | ||||||||||||||
Real | Imaginary | Age struct | Reprod val | Abs. value | rho | Add age structure and reproductive values setting value of first age class to 1.000? Note that age structure and reproductive values are the solutions which go with lambda 1. | ||||||||||
1.1624 | 0.0000 | 0.3675 | 0.0035 | 1.1624 | ||||||||||||
0.9058 | 0.6401 | 0.2326 | 0.0056 | 1.1092 |
1.048 |
Approximate recovery time is ln(10)/rho = 49 yr | ||||||||||
0.9058 | -0.6401 | 0.1472 | 0.0089 | 1.1092 | 1.048 | |||||||||||
0.4558 | 0.5728 | 0.0932 | 0.0140 | 0.7321 | ||||||||||||
0.4558 | -0.5728 | 0.0590 | 0.0221 | 0.7321 | ||||||||||||
0.2584 | -1.0137 | 0.0373 | 0.0350 | 1.0461 | ||||||||||||
0.2584 | 1.0137 | 0.0236 | 0.0552 | 1.0461 | ||||||||||||
-0.1574 | 0.7151 | 0.0149 | 0.0873 | 0.7323 | ||||||||||||
-0.1574 | -0.7151 | 0.0095 | 0.1379 | 0.7323 | ||||||||||||
-0.4681 | -0.8873 | 0.0060 | 0.1345 | 1.0032 | ||||||||||||
-0.4681 | 0.8873 | 0.0038 | 0.1291 | 1.0032 | ||||||||||||
-0.6660 | 0.3284 | 0.0024 | 0.1207 | 0.7426 | ||||||||||||
-0.6660 | -0.3284 | 0.0015 | 0.1073 | 0.7426 | ||||||||||||
-0.9098 | -0.3373 | 0.0010 | 0.0862 | 0.9703 | ||||||||||||
-0.9098 | 0.3373 | 0.0006 | 0.0528 | 0.9703 | ||||||||||||
r | 0.1505 | (rate of increase, y-1) | Adding F = 0.15 y-1 to M = 0.31 y-1 will produce stationary population (r = 0.0, lambda = 1.0). F = 0.15 y-1 can be related to catch if we know/assume a mean N for the virgin population in the mid-1960's. Then one could determine if observed or estimated catch was larger than F = 0.15 y-1, which would produce a declining population as implied by 96% decline in catch rate since the mid-1960's. | |||||||||||||
Ro | 4.9549 | (expected number of replacements) | ||||||||||||||
T | 10.63 | (generation time - time for increase of Ro, yr) | ||||||||||||||
mu1 | 10.86 | (mean age of parents of offspring of a cohort, yr) | ||||||||||||||
Abar | 10.43 | (mean age of parents of population at the stable age distribution, yr) | ||||||||||||||
N (fundamental matrix) | ||||||||||||||||
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.7356 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.5412 | 0.7356 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.3981 | 0.5412 | 0.7356 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.2929 | 0.3981 | 0.5412 | 0.7356 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.2154 | 0.2929 | 0.3981 | 0.5412 | 0.7356 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.1585 | 0.2154 | 0.2929 | 0.3981 | 0.5412 | 0.7356 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.1166 | 0.1585 | 0.2154 | 0.2929 | 0.3981 | 0.5412 | 0.7356 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.0858 | 0.1166 | 0.1585 | 0.2154 | 0.2929 | 0.3981 | 0.5412 | 0.7356 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0.0631 | 0.0858 | 0.1166 | 0.1585 | 0.2154 | 0.2929 | 0.3981 | 0.5412 | 0.7356 | 1 | 0 | 0 | 0 | 0 | 0 | ||
0.0464 | 0.0631 | 0.0858 | 0.1166 | 0.1585 | 0.2154 | 0.2929 | 0.3981 | 0.5412 | 0.7356 | 1 | 0 | 0 | 0 | 0 | ||
0.0341 | 0.0464 | 0.0631 | 0.0858 | 0.1166 | 0.1585 | 0.2154 | 0.2929 | 0.3981 | 0.5412 | 0.7356 | 1 | 0 | 0 | 0 | ||
0.0251 | 0.0341 | 0.0464 | 0.0631 | 0.0858 | 0.1166 | 0.1585 | 0.2154 | 0.2929 | 0.3981 | 0.5412 | 0.7356 | 1 | 0 | 0 | ||
0.0185 | 0.0251 | 0.0341 | 0.0464 | 0.0631 | 0.0858 | 0.1166 | 0.1585 | 0.2154 | 0.2929 | 0.3981 | 0.5412 | 0.7356 | 1 | 0 | ||
0.0136 | 0.0185 | 0.0251 | 0.0341 | 0.0464 | 0.0631 | 0.0858 | 0.1166 | 0.1585 | 0.2154 | 0.2929 | 0.3981 | 0.5412 | 0.7356 | 1 | ||
Sum | 3.7449 | 3.7313 | 3.7129 | 3.6877 | 3.6536 | 3.6072 | 3.5441 | 3.4583 | 3.3417 | 3.1832 | 2.9678 | 2.6749 | 2.2768 | 1.7356 | 1.0000 | |
R (expected lifetime production) | ||||||||||||||||
4.95 | 6.74 | 9.16 | 12.45 | 16.92 | 23.00 | 31.26 | 42.50 | 57.77 | 55.03 | 51.31 | 46.24 | 39.36 | 30.01 | 17.29 | ||
Deleted 14 rows with 0 entries | ||||||||||||||||
Elasticity Matrix | J8 | A9 | A15 | |||||||||||||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0367 | 0.0232 | 0.0147 | 0.0093 | 0.0059 | 0.0037 | 0.0024 | ||
0.0367 |
0.0232 | 0.0147 | 0.0093 | 0.0059 | 0.0037 | 0.0024 | ||||||||||
0.0959 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0.0959 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0.0959 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0.0959 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0.0959 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0.0959 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0.0959 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0959 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0592 | 0 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0360 | 0 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0213 | 0 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0120 | 0 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0061 | 0 | 0 | ||
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.0024 | 0 | ||
E(fertility) = E1 | 0.0959 | Same as elasticity of each juvenile age class | E1_norm | 0.0875 | ||||||||||||
E(juvenile survival) = E2 | 0.8631 | ER2 = E2/E1 = | 9.00 | (alpha) | E1_norm | 0.7876 | ||||||||||
E(adult survival) = E3 |
0.1369 |
ER3 = E3/E1 = | 1.43 | [(1/E1) - alpha] | E3_norm | 0.1249 | ||||||||||
Sum = E1 + E2 + E3 | 1.0959 |
In pre- and post-breeding cycle the sum is 1 + E1 = 1.0959 |
Sum | 1.000 |