Structured population growth, pt.3

Discussion: Crouse et al. 1987

  1. What is the relationship between the Eigenvalue of a population transition matrix (\(\lambda\)) and the intrinsic growth parameter \(r\) we explored in the exponential growth model? What happens when \(\lambda = 1\)? What happens when \(r = 1\)?

  2. The authors choose to construct a stage-based matrix rather than an age-based population matrix. Why? How did the authors get data to construct the matrix?

  3. Table 4 of the paper provides the values for the loggerhead transition matrix. Draw this as a transition diagram.

  4. Consider Figure 1 of the paper, and address the following questions:

    • What is the name of the type of analysis that the authors conducted to create this figure? What does this analysis entail?
    • What do the X- and Y-axis of the two panels represent?
    • What is the takeaway from Panel B?

  5. The authors consider several management scenarios that would enable \(\lambda\) to reach a value of 1 or more. Biologically, why is this threshold relevant, and what do the authors find is the most likely way to achieve this goal? What is the “uncomfortable possibility” regarding sea turtle conservation that they suggest managers to confront?

Age vs. stage-based matrices

  • So far, we have conflated age and stage (e.g. “1-year-old” == “Juvenile”, “2-year-old” == “Subadult”, etc.)

  • But, there’s a potential issue – e.g. what if the juvenile stage extends for multiple year?

Another empirical example: Population dynamics of red cockaded woodpeckers

Red cockaded woodpeckers: an uncommon (near threatened) species

How to build a matrix population model for RCW?

Question 1: What are the stages?

RCWs are territorial and cooperatively breeding with a complex social structure.

Groups are composed of a monogamous breeding pair and non-breeding helpers.

Fledglings either remain on the natal territory as helpers or disperse

Almost all females disperse in either the fall or spring following fledging and ultimately obtain breeding positions if they survive, although they may act as floaters before acquiring a breeding opportunity.

Substantial numbers of male fledglings stay as helpers, and those that survive ultimately inherit the breeding position in their natal territory or fill breeding vacancies in neighboring territories. Other male fledglings disperse to either become breeders, solitary males, floaters, or (to a lesser extent) helpers in another territory

So… what are the stages? It’s complex!

So… what are the stages? It’s complex!

$arr
   row col Angle Value       rad    ArrowX       ArrowY TextX      TextY
1    1   1   NaN  0.04 0.1386076 0.2171772 8.158057e-01 0.215  0.6772152
2    2   1     0 0.042 0.1000000 0.1995288 3.000011e-01 0.200  0.2800000
3    3   1     0 0.373 0.2000000 0.2990575 2.000022e-01 0.300  0.1800000
4    4   1     0 0.016 0.3000000 0.3985863 1.000033e-01 0.400  0.0800000
5    5   1     0 0.047 0.4000000 0.4981151 4.441314e-06 0.500 -0.0200000
6    1   2     0  0.84 0.1000000 0.2001571 4.999999e-01 0.200  0.5200000
7    2   2   NaN 0.768 0.1386076 0.4171772 8.158057e-01 0.415  0.6772152
8    3   2     0 0.001 0.1000000 0.3995288 3.000011e-01 0.400  0.2800000
9    4   2     0 0.009 0.2000000 0.4990575 2.000022e-01 0.500  0.1800000
10   5   2     0 0.003 0.3000000 0.5985863 1.000033e-01 0.600  0.0800000
11   1   3     0  0.23 0.2000000 0.3003142 5.999998e-01 0.300  0.6200000
12   2   3     0  0.21 0.1000000 0.4001571 4.999999e-01 0.400  0.5200000
13   3   3   NaN 0.547 0.1386076 0.6171772 8.158057e-01 0.615  0.6772152
14   4   3     0 0.011 0.1000000 0.5995288 3.000011e-01 0.600  0.2800000
15   5   3     0 0.011 0.2000000 0.6990575 2.000022e-01 0.700  0.1800000
16   1   4     0   0.6 0.3000000 0.4004712 6.999996e-01 0.400  0.7200000
17   2   4     0  0.55 0.2000000 0.5003142 5.999998e-01 0.500  0.6200000
18   4   4   NaN 0.214 0.1386076 0.8171772 8.158057e-01 0.815  0.6772152
19   5   4     0 0.044 0.1000000 0.7995288 3.000011e-01 0.800  0.2800000
20   1   5     0  0.43 0.4000000 0.5006283 7.999995e-01 0.500  0.8200000
21   2   5     0 0.003 0.3000000 0.6004712 6.999996e-01 0.600  0.7200000
22   3   5     0 0.011 0.2000000 0.7003142 5.999998e-01 0.700  0.6200000
23   4   5     0 0.014 0.1000000 0.8001571 4.999999e-01 0.800  0.5200000
24   5   5   NaN 0.104 0.1386076 1.0171772 8.158057e-01 1.015  0.6772152

$comp
       x   y
[1,] 0.1 0.4
[2,] 0.3 0.4
[3,] 0.5 0.4
[4,] 0.7 0.4
[5,] 0.9 0.4

$radii
       x         y
[1,] 0.1 0.2772152
[2,] 0.1 0.2772152
[3,] 0.1 0.2772152
[4,] 0.1 0.2772152
[5,] 0.1 0.2772152

$rect
     xleft      ybot xright      ytop
[1,]   0.0 0.1227848    0.2 0.6772152
[2,]   0.2 0.1227848    0.4 0.6772152
[3,]   0.4 0.1227848    0.6 0.6772152
[4,]   0.6 0.1227848    0.8 0.6772152
[5,]   0.8 0.1227848    1.0 0.6772152