Sea turtles (Crouse, Crowder, Caswell).sagews

The Loggerhead Sea Turtle

This model was first studied in a famous paper by Deborah Crouse, Larry Crowder, and Hal Caswell, published in 1987.

It uses seven life stages:

1. First-year (eggs and hatchlings)

2. Small juveniles (approx 1–7 years old)

3. Large juveniles (approx 8–15 years old)

4. Sub-adults (approx 16–21 years old)

5. Novice breeders (approx 22 years old)

6. First-year remigrants (approx 23 years old)

7. Mature breeders (approx 24 years old and older)

Here's the original matrix they set up:

C = matrix(RDF, [
    [0,      0,      0,      0,      127,      4,      80     ],
    [0.6747, 0.7370, 0,      0,        0,      0,       0     ],
    [0,      0.0486, 0.6610, 0,        0,      0,       0     ],
    [0,      0,      0.0147, 0.6907,   0,      0,       0     ],
    [0,      0,      0,      0.0518,   0,      0,       0     ],
    [0,      0,      0,      0,        0.8091, 0,       0     ],
    [0,      0,      0,      0,        0,      0.8091,  0.8089],
])
show("$C = $", C)

$C =$ $\displaystyle \left(\begin{array}{rrrrrrr} 0.0 & 0.0 & 0.0 & 0.0 & 127.0 & 4.0 & 80.0 \\ 0.6747 & 0.737 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0486 & 0.661 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0147 & 0.6907 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0518 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.8091 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.8091 & 0.8089 \end{array}\right)$

for eval in C.eigenvalues():
    show(eval)

$\displaystyle -0.0884454744686 + 0.119599712287i$

$\displaystyle -0.0884454744686 - 0.119599712287i$

$\displaystyle 0.265452810019$

$\displaystyle 0.371666767651$

$\displaystyle 0.746170195288 + 0.213056538519i$

$\displaystyle 0.746170195288 - 0.213056538519i$

$\displaystyle 0.945030980691$

T = transformation_matrix(C)
show("$T = $", T.round(4))

$T =$ $\displaystyle \left(\begin{array}{rrrrrrr} -0.7768 & 0.0 & -0.57 & -0.471 & -0.0127 & -0.2951 & 0.2908 \\ 0.6219 & 0.0901 & 0.8155 & 0.8699 & -0.9345 & 0.0 & 0.943 \\ -0.0384 & -0.012 & -0.1002 & -0.1461 & -0.0735 & 0.1838 & 0.1614 \\ 0.0007 & 0.0003 & 0.0035 & 0.0067 & 0.0106 & 0.0078 & 0.0093 \\ -0.0 & -0.0003 & 0.0007 & 0.0009 & 0.0008 & 0.0003 & 0.0005 \\ -0.001 & 0.001 & 0.0021 & 0.002 & 0.0009 & 0.0001 & 0.0004 \\ 0.001 & -0.0008 & -0.0031 & -0.0038 & -0.0007 & -0.0033 & 0.0026 \end{array}\right)$

Here's the diagonalization of C:

D = T.inverse()*C*T
show("$D = $", D.round(4))

$D =$ $\displaystyle \left(\begin{array}{rrrrrrr} -0.0884 & 0.1196 & -0.0 & 0.0 & 0.0 & -0.0 & -0.0 \\ -0.1196 & -0.0884 & -0.0 & -0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.2655 & -0.0 & -0.0 & 0.0 & 0.0 \\ -0.0 & -0.0 & 0.0 & 0.3717 & 0.0 & -0.0 & -0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.7462 & 0.2131 & -0.0 \\ 0.0 & -0.0 & 0.0 & -0.0 & -0.2131 & 0.7462 & -0.0 \\ -0.0 & 0.0 & -0.0 & 0.0 & -0.0 & -0.0 & 0.945 \end{array}\right)$

for eval in C.eigenvalues():
    show(abs(eval))

$\displaystyle 0.148750439102$

$\displaystyle 0.148750439102$

$\displaystyle 0.265452810019$

$\displaystyle 0.371666767651$

$\displaystyle 0.775991655201$

$\displaystyle 0.775991655201$

$\displaystyle 0.945030980691$

As you can see from the absolute values of the eigenvalues (above), the loggerhead sea turtle population (of the 1980s) was in decline, in fact at roughly 5.5% per year.

Below is a simulation of this system, as an interactive, which will show the long-term population decline. As usual, if you want to run this interactive, click the “Open in CoCalc” link at the top right of this worksheet to open this in your own CoCalc account. In this case, however, it's really not too necessary to see it, as the eigenvalues above don't lie!

@interact
def seaturtles(n=(0, 100, 1)):
    state = vector([100.0, 100.0, 100.0, 100.0, 100.0, 100.0, 100.0])
    for i in xrange(n+1):
        print("{:>8.2f}  {:>8.2f}  {:>8.2f}  {:>8.2f}  {:>8.2f}  {:>8.2f}  {:>8.2f}".format(
                *(float(s) for s in state)))
        state = C*state

Interact: please open in CoCalc

What happens if we protect nests and eggs?

C2 = matrix(RDF, [
    [0,      0,      0,      0,      150,      6,     100     ],
    [0.6747, 0.7370, 0,      0,        0,      0,       0     ],
    [0,      0.0486, 0.6610, 0,        0,      0,       0     ],
    [0,      0,      0.0147, 0.6907,   0,      0,       0     ],
    [0,      0,      0,      0.0518,   0,      0,       0     ],
    [0,      0,      0,      0,        0.8091, 0,       0     ],
    [0,      0,      0,      0,        0,      0.8091,  0.8089],
])
show("$C_2 = $", C2)

$C_2 =$ $\displaystyle \left(\begin{array}{rrrrrrr} 0.0 & 0.0 & 0.0 & 0.0 & 150.0 & 6.0 & 100.0 \\ 0.6747 & 0.737 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0486 & 0.661 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0147 & 0.6907 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0518 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.8091 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.8091 & 0.8089 \end{array}\right)$

The matrix below shows the changes between this new matrix $C_2$ and the original matrix $C$:

show("$C_2 - C = $", C2 - C)

$C_2 - C =$ $\displaystyle \left(\begin{array}{rrrrrrr} 0.0 & 0.0 & 0.0 & 0.0 & 23.0 & 2.0 & 20.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \end{array}\right)$

for eval in C2.eigenvalues():
    show(eval)

$\displaystyle -0.0949198087649 + 0.126208222902i$

$\displaystyle -0.0949198087649 - 0.126208222902i$

$\displaystyle 0.316495044674 + 0.0717306262597i$

$\displaystyle 0.316495044674 - 0.0717306262597i$

$\displaystyle 0.749499986831 + 0.22485415496i$

$\displaystyle 0.749499986831 - 0.22485415496i$

$\displaystyle 0.955449554519$

T2 = transformation_matrix(C2)
D2 = T2.inverse()*C2*T2
show("Diagonalization of $C_2$: ", D2.round(4))

Diagonalization of $C_2$: $\displaystyle \left(\begin{array}{rrrrrrr} -0.0949 & 0.1262 & 0.0 & 0.0 & -0.0 & -0.0 & 0.0 \\ -0.1262 & -0.0949 & -0.0 & -0.0 & -0.0 & -0.0 & 0.0 \\ -0.0 & 0.0 & 0.3165 & 0.0717 & -0.0 & 0.0 & 0.0 \\ 0.0 & -0.0 & -0.0717 & 0.3165 & 0.0 & -0.0 & -0.0 \\ -0.0 & -0.0 & 0.0 & 0.0 & 0.7495 & 0.2249 & 0.0 \\ -0.0 & -0.0 & 0.0 & 0.0 & -0.2249 & 0.7495 & -0.0 \\ 0.0 & 0.0 & -0.0 & 0.0 & -0.0 & 0.0 & 0.9554 \end{array}\right)$

for eval in C2.eigenvalues():
    show(abs(eval))

$\displaystyle 0.157918604426$

$\displaystyle 0.157918604426$

$\displaystyle 0.324521795951$

$\displaystyle 0.324521795951$

$\displaystyle 0.782502154159$

$\displaystyle 0.782502154159$

$\displaystyle 0.955449554519$

Once again, the absolute values of all the eigenvalues, displayed above, show that even with these measures to protect the nests and eggs, the loggerhead population would still decline (now by 4.5% per year, only a slight improvement).

Again, you can try the interactive below if you want to see the decline for yourself.

@interact
def seaturtles(n=(0, 150, 1)):
    state = vector([100.0, 100.0, 100.0, 100.0, 100.0, 100.0, 100.0])
    for i in xrange(n+1):
        print("{:>8.2f}  {:>8.2f}  {:>8.2f}  {:>8.2f}  {:>8.2f}  {:>8.2f}  {:>8.2f}".format(
                *(float(s) for s in state)))
        state = C2*state

Interact: please open in CoCalc

What happens if we instead work on protecting juveniles, and especially older juveniles?

C3 = matrix(RDF, [
    [0,      0,      0,      0,      127,      4,      80     ],
    [0.6747, 0.7470, 0,      0,        0,      0,       0     ],
    [0,      0.0531, 0.7597, 0,        0,      0,       0     ],
    [0,      0,      0.0403, 0.7289,   0,      0,       0     ],
    [0,      0,      0,      0.0710,   0,      0,       0     ],
    [0,      0,      0,      0,        0.8091, 0,       0     ],
    [0,      0,      0,      0,        0,      0.8091,  0.8089],
])
show("$C_3 = $", C3)

$C_3 =$ $\displaystyle \left(\begin{array}{rrrrrrr} 0.0 & 0.0 & 0.0 & 0.0 & 127.0 & 4.0 & 80.0 \\ 0.6747 & 0.747 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0531 & 0.7597 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0403 & 0.7289 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.071 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.8091 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.8091 & 0.8089 \end{array}\right)$

show("$C_3 - C = $", C3 - C)

$C_3 - C =$ $\displaystyle \left(\begin{array}{rrrrrrr} 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.01 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0045 & 0.0987 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0256 & 0.0382 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0192 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \end{array}\right)$

for eval in C3.eigenvalues():
    show(eval)

$\displaystyle -0.131007994203 + 0.169794165929i$

$\displaystyle -0.131007994203 - 0.169794165929i$

$\displaystyle 1.05704194752$

$\displaystyle 0.79314828337 + 0.299401266144i$

$\displaystyle 0.79314828337 - 0.299401266144i$

$\displaystyle 0.33158873707 + 0.189769049033i$

$\displaystyle 0.33158873707 - 0.189769049033i$

T3 = transformation_matrix(C3)
D3 = T3.inverse()*C3*T3
show("Diagonalization of $C_3$: ", D3.round(4))

Diagonalization of $C_3$: $\displaystyle \left(\begin{array}{rrrrrrr} -0.131 & 0.1698 & 0.0 & 0.0 & -0.0 & -0.0 & -0.0 \\ -0.1698 & -0.131 & -0.0 & -0.0 & -0.0 & 0.0 & 0.0 \\ 0.0 & -0.0 & 1.057 & -0.0 & -0.0 & 0.0 & -0.0 \\ -0.0 & 0.0 & 0.0 & 0.7931 & 0.2994 & 0.0 & 0.0 \\ 0.0 & -0.0 & -0.0 & -0.2994 & 0.7931 & 0.0 & -0.0 \\ -0.0 & 0.0 & -0.0 & 0.0 & 0.0 & 0.3316 & 0.1898 \\ -0.0 & 0.0 & 0.0 & 0.0 & 0.0 & -0.1898 & 0.3316 \end{array}\right)$

Now the dominant eigenvalue is larger than 1!!!

With these changes, the sea turtle population will grow healthily, at almost 6% per year!

for eval in C3.eigenvalues():
    show(abs(eval))

$\displaystyle 0.214460143917$

$\displaystyle 0.214460143917$

$\displaystyle 1.05704194752$

$\displaystyle 0.84777669087$

$\displaystyle 0.84777669087$

$\displaystyle 0.382051544327$

$\displaystyle 0.382051544327$

@interact
def seaturtles(n=(0, 100, 1)):
    state = vector([100.0, 100.0, 100.0, 100.0, 100.0, 100.0, 100.0])
    for i in xrange(n+1):
        print("{:>8.2f}  {:>8.2f}  {:>8.2f}  {:>8.2f}  {:>8.2f}  {:>8.2f}  {:>8.2f}".format(
                *(float(s) for s in state)))
        state = C3*state

Interact: please open in CoCalc

The happy ending to this story

The conclusion of the Crouse–Crowder–Caswell paper was that it was more effective to focus conservation efforts on protecting the juveniles, and especially the older juveniles. The greatest threat to those life stages of sea turtles is being caught in fishing nets and drowning*. Thus, as a direct result of this research, the U.S. passed a law requiring all fishing nets used by U.S. companies to be fitted with a “turtle excluder device”. Not long after, an international treaty was agreed upon whereby most other countries in the world also now require fishing trawlers to use such devices.

Here is a Youtube video showing how one of these turtle excluder devices works, and another video showing one in action, including a medium-sized juvenile sea turtle escaping from a net equipped with a T.E.D.

Today, decades later, this effort has been successful enough that the loggerhead sea turtle is now classified as a species of “least concern” by the IUCN.

* Remember that turtles don't have gills; they need to swim to the surface regularly to breath.