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Loggerhead Sea Turtles
Initialization
<< LinearAlgebra`MatrixManipulation`<< Graphics`Graphics`
Leslie Matrix
birthRates = 84 ê 5, 3 ê 5, 2 ê 5, 1 ê 5<;survivalRates = 81, 3 ê 4, 1 ê 2, 1 ê 4<;DisplayTogether@ListPlot@birthRates,PlotRange Ø 80, 1<,PlotStyle Ø 8PointSize@0.02D, ForestGreen<,AxesLabel Ø 8yr, None<D,
ListPlot@survivalRates,PlotStyle Ø 8PointSize@0.02D, MarsOrange<DD;
1.5 2 2.5 3 3.5 4yr
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loggerheads.nb 1
p = ZeroMatrix@4D;p@@1DD = birthRates;Do@p@@k + 1, kDD = survivalRates@@kDD,8k, 1, 3<D;p@@4, 4DD = survivalRates@@4DD;p êê MatrixFormikjjjjjjjjjjjjjjjj
4ÅÅÅ53ÅÅÅ5
2ÅÅÅ51ÅÅÅ5
1 0 0 00 3ÅÅÅ4 0 0
0 0 1ÅÅÅ21ÅÅÅ4
y{zzzzzzzzzzzzzzzz
Calculate the eigendata for this matrix.
Relate your results to the following display of this population's dynamics.
loggerheads.nb 2
ClearAll@xD;n = 20;mult@x_D := p.x;x@0D = 81, 0, 0, 0<;pops = NestList@mult, x@0D, nD;popDynamics =Table@Prepend@pops@@kDD ê Apply@Plus, pops@@kDDD êê N, kD, 8k, n<D;
TableForm@popDynamics,TableHeadings Ø 88"popDynamics"<, 8year, 1, 2, 3, 4<<D
year 1 2 3 4popDynamics 1 1. 0. 0. 0.
2 0.444444 0.555556 0. 0.3 0.444444 0.286738 0.268817 0.4 0.444444 0.311011 0.150489 0.09405575 0.444444 0.317999 0.166895 0.07066146 0.444444 0.314918 0.168992 0.07164537 0.444444 0.315327 0.167572 0.07265658 0.444444 0.315387 0.167822 0.07234619 0.444444 0.31535 0.167834 0.072371110 0.444444 0.315356 0.167818 0.072381411 0.444444 0.315357 0.167821 0.072377412 0.444444 0.315356 0.167821 0.072377913 0.444444 0.315356 0.167821 0.07237814 0.444444 0.315356 0.167821 0.072377915 0.444444 0.315356 0.167821 0.072377916 0.444444 0.315356 0.167821 0.072377917 0.444444 0.315356 0.167821 0.072377918 0.444444 0.315356 0.167821 0.072377919 0.444444 0.315356 0.167821 0.072377920 0.444444 0.315356 0.167821 0.0723779
After a a bumpy start, the population quickly converges to a steady state..
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ListPlot3D@popDynamics@@All, 82, 3, 4, 5<DDD;
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00.250.5
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Population Wave
birthRates = 80, 2 ê 3, 0, 1 ê 3<;survivalRates = 81, 3 ê 4, 1 ê 2, 1 ê 4<;DisplayTogether@ListPlot@birthRates,PlotStyle Ø 8PointSize@0.02D, ForestGreen<,AxesLabel Ø 8yr, None<D,
ListPlot@survivalRates,PlotStyle Ø 8PointSize@0.02D, MarsOrange<DD;
1.5 2 2.5 3 3.5 4yr
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p = ZeroMatrix@4D;p@@1DD = birthRates;Do@p@@k + 1, kDD = survivalRates@@kDD,8k, 1, 3<D;p@@4, 4DD = survivalRates@@4DD;p êê MatrixFormikjjjjjjjjjjjjjjjj0 2ÅÅÅ3 0 1ÅÅÅ31 0 0 00 3ÅÅÅ4 0 0
0 0 1ÅÅÅ21ÅÅÅ4
y{zzzzzzzzzzzzzzzz
Calculate the eigendata for this matrix.
Relate your results to the following display of this population's dynamics.
loggerheads.nb 5
ClearAll@xD;n = 20;mult@x_D := p.x;x@0D = 81, 0, 0, 0<;pops = NestList@mult, x@0D, nD;popDynamics =Table@Prepend@pops@@kDD ê Apply@Plus, pops@@kDDD êê N, kD, 8k, n<D;
TableForm@popDynamics,TableHeadings Ø 88"popDynamics"<, 8year, 1, 2, 3, 4<<D
year 1 2 3 4popDynamics 1 1. 0. 0. 0.
2 0. 1. 0. 0.3 0.470588 0. 0.529412 0.4 0. 0.64 0. 0.365 0.489552 0. 0.429851 0.0805976 0.0357498 0.65144 0. 0.312817 0.47197 0.0313293 0.428167 0.06853298 0.0567623 0.612619 0.0304991 0.300129 0.456028 0.0509098 0.412091 0.080971310 0.0779729 0.583582 0.0488622 0.28958311 0.442215 0.0710091 0.398597 0.088179112 0.0966955 0.557263 0.0671124 0.27892913 0.429119 0.089333 0.386124 0.095423914 0.113579 0.533464 0.0832916 0.26966515 0.417058 0.106321 0.374529 0.10209216 0.128805 0.512044 0.0979016 0.2612517 0.405899 0.122027 0.363824 0.1082518 0.142547 0.492699 0.111091 0.25366319 0.395593 0.136533 0.353932 0.11394220 0.154968 0.475217 0.12301 0.246805
The "waves" resulting from the pulsed breeding regime are evident in this image.
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ListPlot3D@popDynamics@@All, 82, 3, 4, 5<DDD;
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00.250.5
0.751
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We can also make a movie to visualize the dynamics of this population."The limiting behaviour in this case is a population wave."
Do@ListPlot@popDynamics@@k, 82, 3, 4, 5<DD,Ticks Ø 8Range@4D, Automatic<,PlotRange Ø 880, 4.1<, 80, .4<<,AxesLabel Ø 8"age group", "%"<,ImageSize Ø 400,PlotStyle Ø 8Red, PointSize@0.02D<D,8k, n<D
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1 2 3 4age group
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1 2 3 4age group
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1 2 3 4age group
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1 2 3 4age group
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1 2 3 4age group
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1 2 3 4age group
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1 2 3 4age group
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1 2 3 4age group
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1 2 3 4age group
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1 2 3 4age group
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1 2 3 4age group
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Loggerhead Sea TurtlesOriginal Population ParametersNote that the duration of the oldest age class is taken to be 100.
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birthRates = b = 80, 0, 0, 0, 76.5<;survivalRates = s = 8.6747, .7500, .6758, .7425, .8091<;durations = d = 81, 7, 8, 6, 100<;DisplayTogether@ListPlot@birthRates,PlotStyle Ø 8PointSize@0.02D, ForestGreen<,AxesLabel Ø 8yr, None<D,
ListPlot@survivalRates,PlotStyle Ø 8PointSize@0.02D, MarsOrange<DD;
2 3 4 5yr
20
40
60
Calculate the entries of the population matrix.
loggerheads.nb 18
ClearAll@s, d, g, P, G, pD;p = ZeroMatrix@5D;p@@2, 1DD = G@1D = s@@1DD;DoA
s := s@@kDD;d := d@@kDD;g@k_D :=
sd-1 - sd
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 - sd
;
P@k_D := H1 - g@kDL s;G@k_D := g@kD s;If@k ã 4, p@@1, 4DD = G@4D birthRates@@5DDD,8k, 2, 5<Ep@@1, 5DD = F@5D = s@@5DD birthRates@@5DD;Do@p@@k, kDD = P@kD,8k, 2, 5<D;Do@p@@k + 1, kDD = G@kD,8k, 2, 4<D;p êê MatrixFormikjjjjjjjjjjjjjjjjj0 0 0 3.96521 61.89620.6747 0.711488 0 0 00 0.0385117 0.661054 0 00 0 0.0147461 0.690667 00 0 0 0.0518328 0.8091
y{zzzzzzzzzzzzzzzzz
Calculate the eigendata for this matrix.
Loggerhead Sea Turtles99.9% Survivability of HatchlingsWhat if the survivability of hatchlings were 99.9% Note that the duration of the oldest age class is taken to be 100.
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birthRates = b = 80, 0, 0, 0, 76.5<;survivalRates = s = 8.999, .7500, .6758, .7425, .8091<;durations = d = 81, 7, 8, 6, 100<;DisplayTogether@ListPlot@birthRates,PlotStyle Ø 8PointSize@0.02D, ForestGreen<,AxesLabel Ø 8yr, None<D,
ListPlot@survivalRates,PlotStyle Ø 8PointSize@0.02D, MarsOrange<DD;
2 3 4 5yr
20
40
60
Calculate the entries of the population matrix.
loggerheads.nb 20
ClearAll@s, d, g, P, G, pD;p = ZeroMatrix@5D;p@@2, 1DD = G@1D = s@@1DD;DoA
s := s@@kDD;d := d@@kDD;g@k_D :=
sd-1 - sd
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 - sd
;
P@k_D := H1 - g@kDL s;G@k_D := g@kD s;If@k ã 4, p@@1, 4DD = G@4D birthRates@@5DDD,8k, 2, 5<Ep@@1, 5DD = F@5D = s@@5DD birthRates@@5DD;Do@p@@k, kDD = P@kD,8k, 2, 5<D;Do@p@@k + 1, kDD = G@kD,8k, 2, 4<D;p êê MatrixFormikjjjjjjjjjjjjjjjjj0 0 0 3.96521 61.89620.999 0.711488 0 0 00 0.0385117 0.661054 0 00 0 0.0147461 0.690667 00 0 0 0.0518328 0.8091
y{zzzzzzzzzzzzzzzzz
Calculate the eigendata for this matrix.
Loggerhead Sea TurtlesUsing Turtle Exclusion DevicesCalculate the entries of the population matrices corresponding to reduction r.
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Clear@msD;ms = 8<;DoAClearAll@s, d, g, P, G, pD;birthRates = b = 80, 0, 0, 0, 76.5<;survivalRates = s = 8.6747, .7500, .6758, .7425, .8091<;Do@s@@kDD = s@@kDD + 0.1 r H1 - s@@kDDL,8k, 3, 5<D;durations = d = 81, 7, 8, 6, 100<;p = ZeroMatrix@5D;p@@2, 1DD = G@1D = s@@1DD;DoA
s := s@@kDD;d := d@@kDD;g@k_D :=
sd-1 - sd
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 - sd
;
P@k_D := H1 - g@kDL s;G@k_D := g@kD s;If@k ã 4, p@@1, 4DD = G@4D birthRates@@5DDD,8k, 2, 5<E;
p@@1, 5DD = s@@5DD birthRates@@5DD;Do@p@@k, kDD = P@kD,8k, 2, 5<D;Do@p@@k + 1, kDD = G@kD,8k, 2, 4<D;ms = Append@ms, pD,8r, 0, 5<E
Map@MatrixForm, msD
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9ikjjjjjjjjjjjjjjjjj0 0 0 3.96521 61.89620.6747 0.711488 0 0 00 0.0385117 0.661054 0 00 0 0.0147461 0.690667 00 0 0 0.0518328 0.8091
y{zzzzzzzzzzzzzzzzz,
ikjjjjjjjjjjjjjjjjj0 0 0 4.58834 63.35650.6747 0.711488 0 0 00 0.0385117 0.688505 0 00 0 0.019715 0.708272 00 0 0 0.0599783 0.82819
y{zzzzzzzzzzzzzzzzz,
ikjjjjjjjjjjjjjjjjj0 0 0 5.26888 64.81690.6747 0.711488 0 0 00 0.0385117 0.714819 0 00 0 0.0258213 0.725126 00 0 0 0.0688743 0.84728
y{zzzzzzzzzzzzzzzzz,
ikjjjjjjjjjjjjjjjjj0 0 0 6.00722 66.27730.6747 0.711488 0 0 00 0.0385117 0.73988 0 00 0 0.0331803 0.741224 00 0 0 0.0785258 0.86637
y{zzzzzzzzzzzzzzzzz,
ikjjjjjjjjjjjjjjjjj0 0 0 6.80333 67.73770.6747 0.711488 0 0 00 0.0385117 0.763591 0 00 0 0.0418892 0.756568 00 0 0 0.0889324 0.885459
y{zzzzzzzzzzzzzzzzz,
ikjjjjjjjjjjjjjjjjj0 0 0 7.65682 69.19810.6747 0.711488 0 0 00 0.0385117 0.785876 0 00 0 0.0520239 0.771161 00 0 0 0.100089 0.904546
y{zzzzzzzzzzzzzzzzz=
Compute the dominant eigenvalue for each of these matrices, plot the eigenvalues l against the reductions r, and interpret your results
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