Ecological Archives A021-131-A1

Daniel H. Monson, Daniel F. Doak, Brenda E. Ballachey, and James L. Bodkin. 2011. Could residual oil from the Exxon Valdez spill create a long-term population “sink” for sea otters in Alaska? Ecological Applications 21:2917–2932.

Appendix A. Form of the transition matrix model and modifier functions used to model source and sink population demographic rates, along with baseline survival and fecundity rates and final “best model” parameter estimates.

Model Construction

For each habitat type (source or sink), we model otter demography using both sex and age as state variables. The general form of the transition matrix model for one habitat area is thus:

– – – – Female – – – – – – Male – – – – (A.1)
– – 0 1 2 3 … 19 20 0 1 2 3 … 19 20
– 0 0 s1,fF1 s2,fF2 s3,fF3 – s19,fF19 0 0 0 0 0 – 0 0
– 1 s0,f 0 0 0 – 0 0 0 0 0 0 – 0 0
Female 2 0 s1,f 0 0 – 0 0 0 0 0 0 – 0 0
– 3 0 0 s2,f 0 – 0 0 0 0 0 0 – 0 0
– … – – – – – – – – – – – – – –
– 19 0 0 0 0 – 0 0 0 0 0 0 – 0 0
– 20 0 0 0 0 – s19,f 0 0 0 0 0 – 0 0
– 0 0 s1,fF1 s2,fF2 s3,fF3 – s19,fF19 0 0 0 0 0 – 0 0
– 1 0 0 0 0 – 0 0 s0,m 0 0 0 – 0 0
Male 2 0 0 0 0 – 0 0 0 s1,m 0 0 – 0 0
– 3 0 0 0 0 – 0 0 0 0 s2,m 0 – 0 0
– … – – – – – – – – – – – – – –
– 19 0 0 0 0 – 0 0 0 0 0 0 – 0 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 – s19,m 0

si,f is the survival rate of female otters from age i to i + 1, si,m is the equivalent rate for males, and Fi is the mean number of female or male newborns produced by a female at the end of her ith year of life (Table A1). Survival rates, but not reproduction, vary with year and habitat type, but we have suppressed these subscripts for clarity. Note that we set the maximum age of male of female otters at 20 years, that at 50:50 sex ratio at birth is assumed, and that there is no male limitation of female mating success. Functional forms for survival are described in the main text, and in Tables 1 and 2 of the parent manuscript.

Process

Call the matrix of the form shown in Eq. 1, with survival rates for the sink population in year t, Nsink, t Similarly, the source population matrix for year t will be Msource, t. If we let the vectors Nsink, t and Nsource, t contain the population numbers in each sex and age category, ordered as in Eq. A.1, for the sink and source populations, respectively, then the model can be written as:

Nsink, t + 1 = Msink, t Nsink, t + It(A.2)
Nsource, t + 1 = Msource, t Nsource, t - It(A.3)

Here, the vector It includes the size and sex dependent immigrants from the source to the sink population. As described in the main text, the total number of immigrants is adjusted to maintain the sink population size, and is allotted as 0.50 × 0.75 first year males, 0.50 × 0.75 second year males, 0.50 × 0.25 first year females, and 0.50 × 0.25 second year females. Itotal, t, the total immigrant number, is equal to sum(Nsink, t)-sum(Nsink, t Nsink, t), and It is a column vector equal to: {0, 0.125 Itotal, t, 0.125 Itotal, t, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0.375 Itotal, t,0.375 Itotal, t, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}

TABLE A1. Baseline fecundity and survival rates used in the Prince William Sound sea otter population matrix model. Fecundity is assumed to remain constant through time while survival rates are modified by the logistic or cloglog function (sink population) or the modified logistic function (source population).

Age Fecundity ♀ survival ♂ survival
0 0.00 0.56 0.56
1 0.00 0.59 0.47
2 0.38 0.96 0.95
3 0.60 0.96 0.94
4 0.73 0.95 0.93
5 0.81 0.94 0.91
6 0.85 0.93 0.88
7 0.87 0.92 0.85
8 0.89 0.9 0.81
9 0.90 0.88 0.75
10 0.90 0.86 0.69
11 0.90 0.83 0.60
12 0.91 0.8 0.51
13 0.91 0.77 0.40
14 0.91 0.73 0.29
15 0.60 0.69 0.19
16 0.00 0.64 0.10
17 0.00 0.58 0.05
18 0.00 0.52 0.01
19 0.00 0.46 0.00
20 0.00 0.40 0.00

TABLE A2. Parameter estimates and one-dimensional 95% likelihood profile confidence intervals for the single best-fit source-sink model (4a).  Numbers in parenthesis are actual numbers produced when using the corresponding parameter value.

  Sink population
survival (St, i-sink1) 		Source population
survival (St, i-source) 		Sink N Nt tshift WPWS N
1990 N0 		Sink population
survival (St, i-sink2)
Parameter θ1 θ2 θ3 θ5 θ6 θ7 θ9 θ13 θ14 θ15 θ16 θ17
Estimate -1.92 0.664 1.409 0.68 -0.002 -0.114 -0.26 (937) 15 (1994) 2156 2.11 -0.025 -0.094
Lower CI -2.18 -0.44 0.530 0.32 -0.018 -0.125 -0.94 (606) 16 (1993) 1900 1.51 -0.055 -0.107
Upper CI 0.68 0.860 1.670 0.82 0.036 -0.067 -0.22 (960) 12 (1997) 2430 2.68 0.003 -0.058

Modifier functions

Examples of the three functional forms used to modify baseline survival rates are presented in the figures (Fig. A1–A5) below. Each figure presents the baseline female survival rate in black and the colored lines present the resulting modified age-specific survival rates at t = 1 year, 5 years, 10 years, 15 years and 20 years post spill. Parameter values were chosen at random simply to illustrate the flexibility of the modifier functions as well as the similarities and differences in the three functional forms when the same parameter values are applied to all three.


(a)
(b)
(c)

FIG. A1. Intercept = 9.1710, t = -0.0560, i = 0.5986, t × i = -0.0889. (a) Logistic form, (b) Cloglog form, and (c) Modified Logistic form.



(a)
(b)
(c)

FIG. A2. Intercept = 0.2849, t = -0.0560, i = 0.5986, t × i = -0.0889. (a) Logistic form, (b) Cloglog form, and (c) Modified Logistic form.



(a)
(b)
(c)

FIG. A3. Intercept = 3.1714, t = 0.4259, i = 0.1330, t × i = -0.0726. (a) Logistic form, (b) Cloglog form, and (c) Modified Logistic form.



(a)
(b)
(c)

FIG. A4. Intercept = 2.7013, t = 0.9213, i = -0.4932, t × i = -0.0457. (a) Logistic form, (b) Cloglog form, and (c) Modified Logistic form.



(a)
(b)
(c)

FIG. A5. Intercept = -0.8400, t = 0.8997, i = 0.0744, t × i = -0.0436. (a) Logistic form, (b) Cloglog form, and (c) Modified Logistic form.

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