Appendix B. Mortality rates.
The mortality rates of red deer females are affected by the density of adult hinds (females older than two years), and were previously described using a logistic functional form by (Milner-Gulland et al. 2004):
(B.1) |
where R is the vital rate for sex i and age j, a and b are constants (Table A1), and D is adult hind density. This is a density-dependent function which describes the average mortality rates for females belonging to the same sex and age class. While density in this function has a considerable effect, predicting approximately 95% increase in mortality between low and high densities for young (age 1–2) and old (age 9+) females, it has a relatively small effect, 40% increase in mortality, on young adults and prime age females (age 3–8).
In this paper the focus is on the effects of survival costs of reproduction on females. Evidence for costs of reproduction in red deer females comes from studies that have associated successful reproduction with a subsequent reduction in the mother’s survival, subsequent calf survival, and the calf’s future fertility (Clutton-Brock 1984; Clutton-Brock et al. 1983; Benton et al. 1995). Only the cost of reproduction in terms of the mother’s survival is considered here since the fertility cost for reproductive hinds is more complex to quantify. This is due to the additive effects of the hind’s reproductive history over potentially many years, and to the effect of inter-sibling competition within the matrilineal group (Coulson et al. 1997).
Eq. B.1 describes the average mortality rates for all females of a given age irrespective of their reproductive status and therefore needs to be separated into survival rates of reproductive (milk) and non-reproductive (yeld) hinds. Here, the mortality rates for yeld hinds are modeled as:
(B.2) |
where R_{k}_{,y,j }is the mortality rate for an individual k, that is of type y, yeld hind, and age j. a, b, and D are parameters as described for Eq B.1. The values of the constants are taken from Table 1. The 0.8 multiplier represents an assumed 20% decrease in mortality for yeld hinds in comparison to the average of milk and yeld rates given by (Milner-Gulland et al. 2004). It is based on estimated age-specific survival differences between yeld and milk hinds for the Rum population (Clutton-Brock 1984). The mortality rates for milk hinds are modelled as:
(B.3) |
where R_{k,m,},_{j} is the mortality rate for individual k that is a hind of type m, milk hind, and age j, and C_{j,l} is the mortality attributed to the survival cost of reproduction which is modelled as:
(B.4) |
where B_{j} is the age-specific cost-related mortality for a milk hind that is subject to the lowest cost level (Fig. B1), and l is the cost level. The pattern of the survival cost function is characterised by high mortality for young reproductive hinds that rapidly declines to zero for prime age females, and subsequently increases until senescence. A similar pattern has been described for Soay sheep (Ovies aries), (Tavecchia et al. 2005) and for humans (Caswell 1982).
FIG. B1. Age-specific rates of the survival cost of reproduction for red deer females shown for the lowest level (level 1), and for a high level (level 6), according to Eq. B.4. |
Stochastic variation in environmental conditions and demographic rates has a strong effect on mortality rates of ungulates including red deer (Benton et al. 1995). It is assumed to be normally distributed and is modeled as:
(B.5) |
where z_{k},_{j} is a random value drawn from a normal distribution with a mean 0 and a standard deviation varying with hind age and the current population density (Table B1). The effect of stochasticity on the population dynamics is examined by simulating two different rates of stochasticity: (i) Rum rate (Milner-Gulland et al. 2004) and (ii) twice the Rum rate.
TABLE B1. Constants for mortality and stochastic rates used in the red deer individual based model for hinds and calves following Milner-Gulland et al. (2004). Mortality rates are calculated according to Eq. 2 (yeld hinds) and Eq. 3 (milk hinds), with a stochastic variation, _{1} for densities below 14 hinds/km^{2}, and _{2} for densities above 14 hinds/km^{2}. The fits to the data from the red deer population on the Isle of Rum are shown in Milner-Gulland et al. (2004).
Age |
a |
b |
_{1} |
_{2} |
1 |
-3.76 |
0.1487 |
0.06794 |
0.1835 |
2 |
-4.24 |
0.1387 |
0.08703 |
0.08111 |
3-5 |
-4.468 |
0.0685 |
0.01235 |
0.03069 |
6-8 |
-4.45 |
0.1014 |
0.0339 |
0.05448 |
9-13 |
-4.853 |
0.1642 |
0.03083 |
0.05612 |
14+ |
-1.2 |
0.0184 |
0.1377 |
0.1832 |
Cost level 5 approximates the condition on Rum (see Model implementation in the main text). Hence, cost levels 1–4 simulate conditions for reproduction that are better than those on Rum, and cost levels 6–8 simulate conditions that are worse than on Rum.
LITERATURE CITED
Benton, T. G., A. Grant, and T. H. Cluttonbrock. 1995. Does environmental stochasticity matter - analysis of red deer life-histories on rum. Evolutionary Ecology 9:559–574.
Caswell, H. 1982. Optimal life histories and the age-specific costs of reproduction. Journal of Theoretical Biology 98:519–529.
Clutton-Brock, T. H. 1984. Reproductive effort and terminal investment in iteroparous animals. American Naturalist 123:212–229.
Clutton-Brock, T. H., F. E. Guinness, and S. D. Albon. 1983. The costs of reproduction to red deer hinds. Journal of Animal Ecology 52:367–383.
Coulson, T., S. Albon, F. Guinness, J. Pemberton, and T. CluttonBrock. 1997. Population substructure, local density, and calf winter survival in red deer (Cervus elaphus). Ecology 78:852–863.
Milner-Gulland, E. J., T. Coulson, and T. H. Clutton-Brock. 2004. Sex differences and data quality as determinants of income from hunting red deer Cervus elaphus. Wildlife Biology 10:187–201.
Tavecchia, G., T. Coulson, B. J. T. Morgan, J. M. Pemberton, J. C. Pilkington, F. M. D. Gulland, and T. H. Clutton-Brock. 2005. Predictors of reproductive cost in female Soay sheep. Journal of Animal Ecology 74:201–213.