Appendix S3. Small rodent specific demographic model.
Small rodents typically have a simple life cycle made up of 2-stages; nonreproductive juveniles and reproductive adults. This resulted in a 3-class matrix
population model:
0
0
𝐹(𝑅)
𝑛𝑗
𝑛𝑗
0 ] [𝑛𝑚]
[𝑛𝑚]
= [𝐽(𝑆)/2 𝐴(𝑆)
𝑛𝑓 𝑡+1
𝑛𝑓 𝑡
𝐽(𝑆)/2
0
𝐴(𝑆)
(1)
We derived maximum survival and reproductive potential by finding the
maximum mean values for small rodents from a variety sources (Hoffmann 1958; Krebs,
Keller, & Tamarin 1969; Ims 1997). Given that we conducted a sensitivity analysis, we
were not interested in the exact value, but in deriving a biologically realistic range of
values. As with lynx it is clear that density dependence can have effects on both
productivity and survival rates (Clarke 1955; Krebs et al. 1969) and so we allowed
density dependence to act on both. Although the specific cause of population cycles for
small rodents varies we used successive density dependence. This led to 3 direct
𝛽1𝐽𝑆 = 𝑒 𝜙1 (𝑁𝑡−1 )
(2)
𝛽1𝐴𝑆 = 𝑒 𝜙2 (𝑁𝑡−1 )
(3)
𝛽1𝑅 = 𝑒 𝜙3 (𝑁𝑡−1 )
(4)
and 3 successive time delayed
−𝑇𝐷 𝑁 )
𝛽2𝐽𝑆 = 𝑒 𝜑1 (∑−1
−𝑇𝐷 𝑁 )
𝛽2𝐴𝑆 = 𝑒 𝜑2 (∑−1
−𝑇𝐷 𝑁 )
𝛽2𝑅 = 𝑒 𝜑3 (∑−1
(6)
(7)
(8)
density-dependent parameters, that were multiplied by demographic potential each time
step. Because small rodents generally breed 3-4 times a year, we divided the period by 3
to obtain a period length in years. The strength of density dependence was set to
minimize the number of simulations going to extinction or carrying capacity.
Parameterization
Both direct (β1) the delayed density-dependent (β2) coefficients of observed
values were well within the range of values estimated from our simulated time series
(Fig. 1). Thus, we retain simulations within these ranges to be used in our sensitivity
analysis (see results).
References
Bjørnstad, O.N., Falck, W. & Stenseth, N.C. (1995) A geographic gradient in small
rodent density fluctuations: a statistical modelling approach. Proceedings of the
Royal Society B: Biological Sciences, 262, 127–33.
Clarke, J.R. (1955) Influence of Numbers on Reproduction and Survival in Two
Experimental Vole Populations. Proceedings of the Royal Society B: Biological
Sciences, 144, 68–85.
Hoffmann, R. (1958) The role of reproduction and mortality in population fluctuations of
voles. Ecological Monographs, 28, 79–109.
Ims, R.A. (1997) Determinants of geographic variation in growth and reproductive traits
in the root vole. Ecology, 78, 461–470.
Krebs, C., Keller, B. & Tamarin, R. (1969) Microtus population biology: Demographic
changes in fluctuating populations of M. Ochrogaster and M. Pennsylvanicus in
southern Indiana. Ecology, 50, 587–607.
Table 1. Parameters and sampling distributions used in small rodent population time
series simulations.
Parameter (s)
Description
Min
Max
J(S)
Juvenile survival
0.64
0.96
A(S)
Adult male and female survival potential
0.70
0.99
F(R)
Adult female reproductive rate
3
13
Φ
Strength of direct dependence
0.0000001
0.002
ψ
Strength of time-delayed density
0.0000001
0.002
dependence
TD
Time delay increment (e.g. years)
2
6
STRT.ind
Initial population size
20
400
Figure 1. Probability density histograms (histogram has a total area of 1) for direct (β1)
and delayed density-dependent (β2) coefficients and period length for time series
simulated under a small rodent life cycle and compared with an parameters estimated for
0.5
1.0
observed
simulated
0.0
Density
1.5
small rodents across a geographic cline (Bjørnstad, Falck, & Stenseth 1995).
−1.5
−1.0
−0.5
0.0
1 + b1
0.5
4
0
2
Density
6
8
−2.0
−0.5
b2
0.0
0.5
0.6
0.4
0.2
0.0
Density
0.8
1.0
−1.0
2
3
4
Period
5
6