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Partial life-cycle models: how good are they?
Ecological Modelling 169 (2003) 313–325
Partial life-cycle models: how good are they? Madan K. Oli∗ Department of Wildlife Ecology and Conservation, 110 Newins-Ziegler Hall, University of Florida, Gainesville, FL 32611-0430, USA Received 17 September 2002; received in revised form 11 June 2003; accepted 16 July 2003
Abstract Partial life-cycle models are simplified, matrix-based population models that allow demographic analyses using incomplete demographic data. Number of parameters in most partial life-cycle models are fixed, and this permits direct interspecific comparison of population dynamics and life histories. Because of these desirable properties, partial life-cycle models have recently received substantial applications. However, adequacy of partial life-cycle models has not been investigated using empirical data. I tested the adequacy of a post-breeding census partial life-cycle model by examining whether and to what extent dynamical properties of the age-structured model are retained in the post-breeding census partial life-cycle model. I applied the age-structured Leslie matrix model and a post-breeding census partial life-cycle model to demographic data for 142 populations of mammals, and compared the projected population growth rate (λ), and elasticity of λ to changes in ages at first and last reproduction, juvenile survival, adult survival, and fertility estimated from the two models. Population growth rate estimated from the two models were practically indistinguishable, and elasticities estimated from the two models were very similar. These results suggest that dynamical properties of the age-structured model are generally captured in the partial life-cycle model, and that application of the post-breeding census partial life-cycle model to age-structured or partial demographic data will not substantially compromise precision of analyses nor conclusions of an investigation. These results have substantial implications in basic and applied population ecology, because age-specific demographic data are seldom available for many species of conservation concern, and also because partial life-cycle models permit perturbation analyses involving ages at first and last reproduction using standard analytical techniques. © 2003 Elsevier B.V. All rights reserved. Keywords: Age-structured models; Elasticity analysis; Matrix population models; Model comparison; Partial life-cycle models; Population dynamics; Sensitivity analysis
1. Introduction An important goal of population ecology is to understand, predict, and explain dynamics of biological populations (Akçakaya et al., 1997). Achieving this goal often requires application of a population model to demographic data. Of various demographic models currently available, matrix population models have become the most popular tools for investigating ∗ Tel.: +1-352-846-0561; fax: +1-352-392-6984. E-mail address: [email protected] (M.K. Oli).
the dynamics of age- or stage-structured populations (Caswell, 2001). In particular, variations of the age-structured Leslie matrix model (Leslie, 1945, 1948; Caswell, 2001) have become standard tools for demographic analyses of age-structured populations. However, the Leslie matrix model requires age-specific demographic data, which are seldom available for most species of conservation concern or those with long lifespans. Additionally, the Leslie matrix model or its variations do not allow perturbation analyses involving ages at first and last reproduction, because these variables do not appear as
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explicit model parameters in the characteristic equation. This is a major limitation of the Leslie matrix model, especially because age at first reproduction is an important life history variable with substantial influence on the dynamics of biological populations (Cole, 1954; Lewontin, 1965; Dobson and Oli, 2001; Oli and Dobson, 2001). Models that allow demographic analyses based on partial demographic data have been developed as an alternative to age-structured models. Such models are called “partial life-cycle models”, because they generally are based on life-cycle stages, and can be parameterized with partial demographic data (Oli and Zinner, 2001a). One of the earliest partial life-cycle models was derived by Cole (1954), which was further refined by Lewontin (1965), Eberhardt and O’Shea (1995), and Slade et al. (1998). Various versions of partial life-cycle models have been presented by Levin et al. (1996), Heppell et al. (2000), and Caswell (2001). Partial life-cycle models have recently been fully developed for post-breeding (Oli and Zinner, 2001a; Oli, 2003) and pre-breeding (Oli and Zinner, 2001b) census situations. Because partial life-cycle models can be parameterized with incomplete demographic data, and also allow perturbation analyses involving time variables (ages at first and last reproduction), these models have recently received substantial applications in basic and applied population ecology (Lande, 1988; Eberhardt and O’Shea, 1995; Slade et al., 1998; Heppell et al., 2000; Dobson and Oli, 2001; Oli et al., 2001). However, adequacy of most partial life-cycle models have not been tested. My objective was to empirically test the adequacy of the post-breeding census partial life-cycle model of Oli and Zinner (2001a). Specifically, I applied the Leslie matrix model and the post-breeding census partial life-cycle model to demographic data for 142 natural populations of mammals. I then compared population growth rate (λ), and elasticity of λ to changes in demographic variables estimated from the Leslie matrix model and the post-breeding census partial life-cycle model. Because most demographic studies of mammalian populations are conducted within post-breeding census framework (Caughley, 1977; Caswell, 2001), the post-breeding census partial life-cycle model of Oli and Zinner (2001a) was appropriate for the analyses of these data.
2. Methods 2.1. Demographic data and models I compiled demographic data for 142 populations of mammals from published sources (Appendix A). I used demographic data from a study if age-specific or otherwise detailed survival and at least average fecundity rates were available. Demographic data based only on the male segment of populations or those based on laboratory studies were not used. Demographic data were analyzed using the agestructured Leslie matrix model and the post-breeding census partial life-cycle model of Oli and Zinner (2001a). The life-cycle of an age-structured population can be graphically represented as an age-structured life-cycle graph, from which a corresponding population projection (Leslie) matrix can be derived (Fig. 1). The Leslie matrix summarizes age-specific demographic data; age-specific fertilities Fi appear on the first row of the matrix, and age-specific survival rates Pi appear on the lower subdiagonal of the matrix. Construction and analyses of the age-structured Leslie matrix models are described in detail by Caswell (2001). The partial life-cycle model is an approximation to the age-structured Leslie matrix model. In general, it is assumed that age-specific fertilities Fi are adequately approximated by an average fertility parameter F, age-specific survival Pi until reproduction is accomplished (i.e. juvenile survival) by a juvenile survival parameter Pj , and age-specific survival from the first reproductive event until age at last reproduction ω by an adult survival parameter Pa . Such a life-cycle, based on two life-history stages (a pre-reproductive juvenile stage, and a reproductive adult stage) can be represented by a partial life-cycle graph, from which a corresponding population projection matrix can be derived (Fig. 2). Assuming that reproduction begins at age α and terminates at age ω, and that demographic data are collected just after the birth-pulse (i.e. post-breeding census), the characteristic equation for this type of two-stage life-cycle is (Oli and Zinner, 2001a) 1 = FPj α−1 λ−α − FPj α−1 Pa λ−α−1 + FPj α λ−α−1 − FPj α Pa ω−α λ−ω−1 + Pa λ−1 .
(1)
The projected population growth rate λ is the largest real root of Eq. (1), and can be obtained numerically.
M.K. Oli / Ecological Modelling 169 (2003) 313–325
F2 1
2 P1
F3
F4
3
4
P2
P3
0 P1 A = 0 0 0
F2
F3
F4
F5
0 P2
0 0
0 0
0 0
0 0
P3 0
0 P4
0 0
315
F5
P4
5
Fig. 1. An example of an age-structured life-cycle graph for age at first reproduction (α) = 2 and age at last reproduction (ω) = 5. Reproduction begins at age α and terminates at age ω. In an age-structured life-cycle graph, both fertility (Fi ) and survival rates (Pi ) are age-specific. A corresponding age-structured population projection (Leslie) matrix also is given.
2.2. Parameter estimation and demographic analyses
Fi = Pi mi ,
For each population, I estimated age-specific survival (Pi ) and fertility (Fi ) using the post-breeding census formulation of Caswell (2001):
where li and mi are age-specific survivorship and fecundity, respectively. Using Pi and Fi thus estimated, a Leslie matrix A was constructed for each population. The population growth rate λ was then estimated as the dominant eigenvalue of the Leslie matrix. The sensitivity of λ to changes in a matrix entry aij was
Pi =
li li−1
,
(2)
1 Pj
(3)
F
F
F
2
3
4
Pj 0 Pj A = 0 0 0
Pa F 0 Pj
F 0 0
F 0 0
F 0 0
0 0
Pa 0
0 Pa
0 0
F
5 Pa
Fig. 2. A post-breeding census partial life cycle graph for age at first reproduction (α) = 2, and age at last reproduction (ω) = 5. In a partial life-cycle graph, age-specific fertilities (Fi ) are approximated by a parameter F, age-specific survival probabilities until the first reproductive event by a parameter Pj , and age-specific survival probabilities for age classes greater than ␣ by a parameter Pa . A population projection matrix corresponding the partial life-cycle graph also is given.
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estimated as (Caswell, 2001) v i wj ∂λ = , ∂aij v, w
(4)
where v and w are the left and right eigenvectors, respectively, corresponding to the dominant eigenvalue of the matrix A. Elasticity (i.e. proportional sensitivity) of λ to changes in aij was then estimated as eij = [∂λ/∂aij ][aij /λ]. For the post-breeding census partial life-cycle model, I estimated ages at first (α) and last (ω) reproduction as the first and last age class with non-zero fertility, respectively. When detailed demographic data were available, F, Pj , and Pa were estimated from the age-classified projection matrix as weighted averages, weighted according to the contribution of each age class to the stable age distribution (Oli and Zinner, 2001a): ω w i Fi F = i=α , (5) ω i=α w i ω−1 w i Pi Pa = i=α+1 , (6) ω−1 i=α+1 w i α w i Pi Pj = i=1 , (7) α i=1 w i
life-cycle model with those obtained from the Leslie matrix model. A close correspondence between results obtained from the two models would indicate that dynamical properties of the age-structured models are generally captured in the simplified partial life-cycle model. To test the adequacy of the post-breeding census partial life-cycle model, I compared λ and elasticity of λ to changes in model parameters (F, Pj , and Pa ) estimated from the two models. Because α and ω do not appear as model parameters in the Leslie matrix model, elasticity of λ to changes in these two parameters could not be compared between the two models. Also, F, Pj and Pa do not appear explicitly as parameters in the Leslie matrix model, and elasticity of λ to changes in F, Pj and Pa were estimated as (Oli and Zinner, 2001b) ω
∂λ xi, = ∂F
(8)
i=α
α
∂λ Pj yi, = ∂Pj λ
(9)
i=1
ω−1 ∂λ Pa yi, = ∂Pa λ
(10)
i=α+1
where wi is the ith entry of the right eigenvector corresponding to the dominant eigenvalue of the Leslie matrix A. The population growth rate was estimated as the largest real root of Eq. (1). The sensitivity of λ to changes in a model parameter p (where p is α, ω, F, Pj , or Pa ) was estimated as the partial derivative of λ with respect to p (Oli and Zinner, 2001a,b). Elasticity of λ to changes in p was then estimated as e(p) = [∂λ/λp][p/λ].
where x is a vector consisting of the first row of the elasticity matrix, and y is a vector consisting of the lower subdiagonal entries of the elasticity matrix. Elasticities of λ to changes in F, Pj and Pa calculated as above from the elasticity matrix of the age-structured model are directly comparable to those obtained from the partial life-cycle model.
2.3. Test of model adequacy
3. Results
The age-structured Leslie matrix model adequately considers the age-specific survival and fertility rates, whereas the partial life-cycle model assumes constant survival probabilities within each of the two life-history stages, as well as age-independent fertility rate. Consequently, the Leslie matrix model may be expected to yield accurate results, and an appropriate way to evaluate adequacy of the partial life-cycle model would be to compare results of the partial
I compiled demographic data for 142 populations of mammals, representing 110 species, 33 families, and 11 orders. Rodentia was the most represented order, with 42 populations of 27 species, followed by Artiodactyla (29 populations of 27 species), and Carnivora (27 populations of 20 species) (Appendix A). Ranges of values of demographic variables were α: 15 days to 15 years, ω: 84 days to 60 years; Pj : 0.111–0.969; Pa : 0.232–0.978; and F: 0.068–1.77; λ ranged from
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317
3.5 Y = -0.002 + 1.009X; R2 = 0.995
λ,partial LC model
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5
1.0
1.5
2.0
2.5
3.0
λ, Leslie matrix model Fig. 3. A plot of population growth rates (λ) estimated from the age-structured Leslie matrix model (X-axis) and the post-breeding census partial life-cycle model (Y-axis).
the Leslie matrix model (r = 0.99, P < 0.0001; Fig. 3). Likewise, elasticity of λ to changes in Pj (r = 0.93, P < 0.0001; Fig. 4), Pa (r = 0.91, P < 0.0001; Fig. 5) and F (r = 0.94, P < 0.0001; Fig. 6) were very similar. Overall, the difference between results
0.67 to 2.62. Thus, data used in this study were appropriate for testing the adequacy of the partial life-cycle model. The population growth rate λ estimated from the partial life-cycle model resembled those obtained from 0.8
Y = -0.014 + 0.974X; R2 = 0.870
e( ), partial LC model
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.2
0.4
0.6
0.8
e( ), Leslie matrix model Fig. 4. A plot of the elasticity of population growth rates (λ) to changes in juvenile survival e(Pj ) estimated from the age-structured Leslie matrix model (X-axis) and the post-breeding census partial life-cycle model (Y-axis).
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0.8 2
0.6
0.4
0.2
e(
), partial LC model
Y = -0.010 + 0.922X; R = 0.821
0.0
0.0
0.2
e(
0.4
0.6
0.8
), Leslie matrix model
Fig. 5. A plot of the elasticity of population growth rates (λ) to changes in adult survival e(Pa ) estimated from the age-structured Leslie matrix model (X-axis) and the post-breeding census partial life-cycle model (Y-axis).
1.0
2
e( ), partial LC model
Y = 0.011 + 1.095X; R = 0.882 0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
e( ), Leslie matrix model Fig. 6. A plot of the elasticity of population growth rates (λ) to changes in fertility e(F) estimated from the age-structured Leslie matrix model (X-axis) and the post-breeding census partial life-cycle model (Y-axis).
obtained from the two models appeared to be random around zero, suggesting that the partial life-cycle model produced relatively unbiased estimates of λ and elasticities.
4. Discussion Population growth rates estimated from the partial life-cycle model were practically indistinguishable
M.K. Oli / Ecological Modelling 169 (2003) 313–325
from those obtained from the Leslie matrix model. Elasticity of λ to changes in demographic variables estimated from the two models differed to some degree in some populations; however, these differences were fairly small and elasticities estimated from the two models were strongly correlated (r > 0.90). These results suggest that dynamical properties of age-structured models are generally retained in the partial life-cycle model of Oli and Zinner (2001a), and that results of the partial life-cycle model are robust. These findings should be comforting to researchers working on endangered or long-lived species for which age-structured demographic data are seldom available (Heppell et al., 2000; Sæther and Bakke, 2000). Matrix population models have received substantial applications in research areas as diverse as conservation biology, life-history theory, and population regulation (Heppell et al., 2000; Caswell, 2001; Dobson and Oli, 2001; Oli et al., 2001). When age-specific demographic data are available, the age-structured Leslie matrix model provides the best estimates of λ, and sensitivity or elasticity of λ to changes in age-specific demographic variables because it adequately incorporates age-specific demographic information. However, parameterization of the Leslie matrix model requires ≤2n − 1 age-specific parameters (where n = number of age classes). For example, one must estimate up to 10 age-specific fertility rates and 9 age-specific survival rates to fully parameterize an age-structured model for a population with 10 age classes. Collection of data for estimating age-specific demographic parameters is difficult, and such data are seldom available for rare or endangered species or for species with long lifespans, and ecologists frequently have to rely on incomplete demographic data for addressing basic and applied ecological problems. Theoretical as well as empirical studies have shown that age at first reproduction (the age at which reproduction begins) is an important life-history variable with substantial potential for influencing population dynamics and fitness (Cole, 1954; Lewontin, 1965; Stearns, 1992; Oli and Dobson, 1999; Oli et al., 2002). However, this variable does not appear as an explicit model parameter in the Leslie matrix model, and absolute or proportional sensitivity of λ to changes in age at first reproduction cannot be estimated using standard techniques. Although some age-specific information is lost, the post-breeding census partial life-cycle model of Oli
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and Zinner (2001a) has several desirable properties. First, variables such as α and ω explicitly appear as model parameters in the characteristic equation, and the sensitivity analyses as well as analyses of life-table response experiments (LTRE analyses; Caswell, 1989, 1996, 2001) involving these life-history variables can be directly estimated using standard analytical techniques (Levin et al., 1996; Heppell et al., 2000; Sæther and Bakke, 2000; Dobson and Oli, 2001; Oli et al., 2001). Second, the partial life-cycle model can be fully parameterized with five parameters (α, ω, Pj , Pa , and F) regardless of the number of age classes (whereas the Leslie matrix model requires ≤2n−1 parameters). Consequently, the partial life-cycle model allows demographic analyses based on incomplete demographic data, but it can also be parameterized with age-specific data when available. Third, because the number of parameters in the partial life-cycle model is fixed regardless of the number of age classes, this model allows direct comparison of the life-history or population dynamics among species with vastly different life histories or age structure (Silvertown et al., 1993; Heppell et al., 2000; Sæther and Bakke, 2000; Oli and Dobson, 2003). Finally, results presented in this paper suggest that the dynamical properties of the age-structured model are generally retained in the partial life-cycle model. When age-specific demographic data are available and when perturbation analyses involving ages at first and last reproduction are not desired, the age-structured model is the best option because it adequately considers age-specific demographic information. If perturbation analyses involving ages at first and last reproduction are desired, then the partial life-cycle model is preferable even when age-specific demographic data are available. If only partial demographic data are available, then the partial life-cycle model is the best option for demographic analyses. Finally, I note that timing of a census relative to the birth-pulse has important implications in the construction and analysis of matrix population models (Caswell, 2001; Oli and Zinner, 2001a,b). Consequently, estimation of parameters and choice of an appropriate model should be based on the timing of a census relative to the birth-pulse. If demographic data are collected immediately after the birth-pulse (post-breeding censuses), the post-breeding census partial life-cycle model of Oli and Zinner (2001a) should be used. If data are
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collected just before the birth-pulse (i.e. pre-breeding census), then the pre-breeding census partial life-cycle model of Oli and Zinner (2001b) is appropriate (see Slade et al., 1998 and Caswell, 2001 for similar models). Results presented here suggest that the partial life-cycle analysis will not generally compromise precision of the analyses nor conclusions of an investigation if parameters are estimated adequately and an appropriate model is used. Thus, the partial life-cycle model of Oli and Zinner (2001a) may be used with confidence for demographic analyses of populations with minimal demographic data.
Acknowledgements I thank Dr. Boris Schröder and an anonymous reviewer for insightful comments on the manuscript. This research was supported in part by a grant from the National Science Foundation (DEB-0224953) and the Florida Agricultural Experiment Station, and approved for publication as Journal Series No. R-09495.
Appendix A References for demographic data used in the analyses reported in this paper. Within each order, species are sorted in alphabetical order. When reproductive data were not reported in the original source, estimates of average fecundity compiled by Purvis and Harvey (1995) were used. Order Artiodactyla Aepyceros melampus: Spinage (1972); Alces alces: Boer (1988); Capreolus capreolus: Pielowski (1984); Cervus elaphus: Clutton-Brock et al. (1988), Lowe (1969); Connochaetes taurinus: Watson (1969), Attwell (1982); Damaliscus lunatus: Mertens (1985); Hemitragus jemlahicus: Caughley (1966, 1977); Kobus defassa ugandae: Spinage (1970); Kobus ellipsiprymnus: van Sickle et al. (1987); Kobus kob: Mertens (1985); Moschus berezovskii: Yang et al. (1990); Odocoileus hemionus: Medin and Anderson (1979); Odocoileus virginianus: Jensen (1995); Ovis aries: Clutton-Brock et al. (1997); Ovis canadensis: Woodgerd (1964); Ovis dalli: Hoefs and Bayer
(1983), Simmons et al. (1984); Phachochoerus aethiopicus: Rodgers (1984); Procapra gutturosa: Jiang et al. (1993); Pseudois nayaur: Wegge (1979); Rangifer t. tarandus: Messier et al. (1988); Rangifer tarandus: Leader-Williams (1988); Rupicapra rupicapra: Caughley (1970); Sus scrofa: Jezierski (1976), Ahmad et al. (1995); Syncerus caffer: Sinclair (1977); Tayassu pecari: Gottdenker and Bodmer (1998); Tayassu tajacu: Hellgren et al. (1995). Order Carnivora Acinonyx jubatus: Kelly et al. (1998); Ailuropoda melanoleuca: Wei et al. (1989); Alopex lagopus: Macpherson (1969); Canis lupus: Parker and Luttich (1986); Felis catus: Warner (1985); Helogale parvula: Waser et al. (1995); Lutra canadensis: Stephenson (1977); Lynx rufus: Crowe (1975), Rolley (1985); Martes pennanti: Kohn et al. (1993); Martes zibellina: Monakhov (1983); Meles meles: Rogers et al. (1997); Mephitis mephitis: Casey and Webster (1975); Mungos mungo: Waser et al. (1995); Mustela putorius: Weber (1989); Nyctereutes procyonoides: Helle and Kauhala (1993); Pathera leo: Packer et al. (1988, 1998); Urocyon cinereoargenteus: Michod and Anderson (1980), Fritzell et al. (1985); Ursus americanus: McLean and Pelton (1994, two populations), Yozdis and Kolenosky (1986); Ursus arctos: Knight and Eberhardt (1985); Vulpes vulpes: Nelson and Chapman (1982), Harris and Smith (1987, two populations). Order Chiroptera Carollia perspicillata: Flemming (1988); Eptesicus fuscus: Gaisler (1979); Myotis lucifugus: Humphrey and Cope (1976); Myotis myotis: Gaisler (1979); Pipistrellus pipistrellus: Thompson (1987); Pipistrellus subflavus: Davis (1966); Rhinolophus ferrumequinum: Corbet and Harris (1991). Order Insectivora Talpa europea: Lodal and Grue (1985). Order Lagomorpha Lepus europaeus: Kovacs (1983); Ochotona princeps: Smith (1974); Ochotona princeps: Millar and
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Zwickel (1972); Oryctolagus cuniculus: Cowan and Roman (1985); Sylvilagus floridanus: Lord (1961a,b). Order Perissodactyla Equus asinus: Perryman and Muchlinski (1987); Equus asinus: Choquenot (1991, two populations); Equus burchelli: Spinage (1972); Equus caballus: Garrott and Taylor (1990); Equus hemionus: Saltz and Rubenstein (1995). Order Pinnipedia Arctocephalus australis: Lima and Páez (1997); Callorhinus ursinus: Barlow and Boveng (1991); Callorhinus ursinus: Landers (1981); Eumetopias jubatus: Calkins and Pitcher (1982); Halichoerus grypus: Mansfield and Beck (1977); Leptonychotes weddelli: Croxall and Hiby (1983); Mirounga angustirostris: Le Boeuf and Reiter (1988); Mirounga angustirostris: Hindell (1991); Mirounga angustirostris: Bester and Wilkinson (1994); Phoca vitulina: Reijnders (1978). Order Primates Cebus olivaceus: Robinson and O’Brien (1991, two populations); Cercopithecus aethiops: Cheney et al. (1988); Macaca fascicularis: Crockett et al. (1996); Macaca fuscata: Richard (1985); Macaca mulatta: Richard (1985); Macaca mulatta: Jiang et al. (1989); Macaca sinica: Gage and Dyke (1988); Macaca thibetana: Li et al. (1995); Pan troglodytes: Courtenay and Santow (1989); Papio cynocephalus: Packer et al. (1998); Theropithecus gelada: Richard (1985); Theropithecus gelada: Dunbar (1980). Order Proboscida Loxodonta africana: Laws (1966, 1969). Order Rodentia Apodemus flavicollis: Bobek (1973); Castor canadensis: Payne (1984), Larson (1967); Clethrionomys glareolus: Bobek (1973); Cynomys gunnisoni: Cully (1997, two populations); Cynomys ludovicianus: Hoogland (1995); Dipodomys spectabilis: Waser and Jones (1991); Dipodomys stephensi: Price and Kelly
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(1994); Histrix africaeaustralis: van Aarde (1987); Liomys adspersus: Flemming (1971); Liomys salvini: Flemming (1974); Marmota flaviventris: Schwartz et al. (1998); Microtus oeconomus: Johannensen and Andreassen (1998); Myocastor coypus: Willner et al. (1983); Oryzomys capito: Flemming (1971); Proechimys semispinosus: Flemming (1971); Sciurus carolinensis: Barkalow et al. (1970); Spermophilus armatus: Slade and Balph (1974, three populations); Spermophilus beldingi: Sherman and Morton (1984); Spermophilus columbianus: Zammuto (1987, six populations); Spermophilus d. alaschani: Chen (1991); Spermophilus dauricus: Luo and Fox (1990); Spermophilus lateralis: Bronson (1979, four populations); Spermophilus townsendii: Smith and Johnson (1985); Tachyoryctes splendens: Jarvis (1973); Tamias striatus: Tryon and Snyder (1973); Tamiasciurus hudsonicus: Kemp and Keith (1970); Zapus hodsonius: Hoyle and Boonstra (1986); Zapus princeps: Falk and Millar (1987). Order Sirenia Trichechus manatus: Eberhardt and O’Shea (1995, Blue Spring). References Akçakaya, H.R., Burgman, M.A., Ginzburg, L.R., 1997. Applied Population Ecology. Sinauer, Sunderland. Ahmad, E., Brooks, J.E., Hussain, I., Khan, M.H., 1995. Reproduction in Eurasian wild boar in central Punjab. Pakistan Acta Theriol. 40, 163–173. Attwell, C.A.M., 1982. Population ecology of the blue wildebeest Connochaetes taurinus taurinus in Zululand, South Africa. Afr. J. Ecol. 20, 147–168. Barkalow, F.S., Hamilton, R.B., Soots, R.F., 1970. The vital statistics of an unexploited gray squirrel population. J. Wildl. Manage. 34, 489–500. Barlow, J., Boveng, P., 1991. Modeling age-specific mortality for marine mammal populations. Marine Mammal Sci. 7, 50–65. Bester, M.N., Wilkinson, I.S., 1994. Population ecology of southern elephant seals at Marion Island. In: Le Boeuf, B.J., Laws, R.M. (Eds.), Elephant Seals: Population Ecology, Behaviour, and Physiology. University of California Press, Berkeley, pp. 85–97. Bobek, B., 1973. Net production of small rodents in a deciduous forest. Acta Theriol. 18, 403–434. Boer, A.H., 1988. Mortality rates of moose in New Brunswick: a life table analysis. J. Wildl. Manage. 52, 21–25. Bronson, M.T., 1979. Altitudinal variation in the life history of golden-mantled ground squirrel (Spermophilus lateralis). Ecology 60, 272–279.
322
M.K. Oli / Ecological Modelling 169 (2003) 313–325
Calkins, D.G., Pitcher, K.W., 1982. Population assessment, ecology, and trophic relationships of Steller sea lions in the Gulf of Alaska. Alaska Department of Fish and Game. Final Report RU 243. Alaska Department of Fish and Game, Anchorage. Casey, G.A., Webster, W.A., 1975. Age and sex determination of stripped skunks (Mephitis mephitis) from Ontario, Manitoba, and Quebec. Can. J. Zool. 53, 223–226. Caswell, H., 1989. The analysis of life table response experiments. I. Decomposition of treatment effects on population growth rate. Ecol. Model. 46, 221–237. Caswell, H., 1996. Analysis of life table response experiments. 2. Alternative parameterizations for size- and stage-structured models. Ecol. Model. 88, 73–82. Caswell, H., 2001. Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer, Sunderland. Caughley, G., 1966. Mortality patterns in mammals. Ecology 47, 906–918. Caughley, G., 1970. Population statistics of Chamois. Mammalia 34, 194–199. Caughley, G., 1977. Analysis of Vertebrate Populations. Wiley, New York. Chen, J., 1991. Life table and reproduction of Alashan ground squirrel (Chinese). Acta Theriol. Sin. 11, 138–142. Cheney, D., Seyfarth, R.M., Andelman, S.J., Lee, P.C., 1988. Reproductive success in vervet monkeys. In: Clutton-Brock, T.H. (Ed.), Reproductive Success. Studies of Individual Variation in Contrasting Breeding Systems. University of Chicago Press, Chicago, pp. 384–402. Choquenot, D., 1991. Density-dependent growth, body condition, and demography in feral donkeys: testing the food hypothesis. Ecology 72, 805–813. Clutton-Brock, T.H., Albon, S.D., Guinness, F.E., 1988. Reproductive success in male and female red deer. In: Clutton-Brock, T.H. (Ed.), Reproductive Success. Studies of Individual Variation in Contrasting Breeding Systems. University of Chicago Press, Chicago, pp. 325–343. Clutton-Brock, T.H., Illius, A.W., Wilson, K., Grenfell, B.T., MacColl, A.D.C., Albon, S.D., 1997. Stability and instability in ungulate populations: an empirical analysis. Am. Nat. 149, 195–219. Cole, L., 1954. The population consequences of life-history phenomena. Q. Rev. Biol. 29, 103–137. Corbet, G.B., Harris, S., 1991. The Handbook of British Mammals. Blackwell, Oxford. Courtenay, J., Santow, G., 1989. Mortality of wild and captive chimpanzees. Folia Primat. 52, 167–177. Cowan, D.P., Roman, E.A., 1985. The construction of life-tables with special reference to the European wild rabbit (Oryctolagus cuniculus). J. Zool. (Lond.) 207, 607–609. Crockett, C.M., Kyes, R.C., Sajuthi, D., 1996. Modeling managed monkey populations: sustainable harvest of longtailed macaques on a natural habitat island. Am. J. Primat. 40, 343–360. Crowe, D.M., 1975. Aspects of ageing, growth, and reproduction in bobcats from Wyoming. J. Mammal. 56, 177–198. Croxall, J.P., Hiby, L., 1983. Fecundity, survival and site fidelity in Weddel seals, Leptonychotes weddelli. J. Appl. Ecol. 20, 19–32. Cully Jr., J.F., 1997. Growth and life history changes in Gunnison’s prairie dogs after a plague epizootic. J. Mammal. 78, 146–157.
Davis, W.H., 1966. Population dynamics of the bat Pipistrellus subflavus. J. Mammal. 47, 383–396. Dobson, F.S., Oli, M.K., 2001. The demographic basis of population regulation in Columbian ground squirrels. Am. Nat. 158, 236–247. Dunbar, R.I.M., 1980. Demographic and life history variables of a population of gelada baboons (Theropthecus gelada). J. Anim. Ecol. 49, 485–506. Eberhardt, L.L., O’Shea, T.J., 1995. Integration of Florida manatee life history data and population modeling. In: O’Shea, T.J., Ackerman, B.B., Percival, H.F. (Eds.), Population Biology of the Florida Manatee. US Department of the Interior, Washington, DC, pp. 269–279. Falk, J.W., Millar, J.S., 1987. Reproduction by female Zapus princeps in relation to age, size, and body fat. Can. J. Zool. 65, 568–571. Flemming, T.H., 1971. Population ecology of three species of Neotropical rodents. Miscellaneous publications of the museum of zoology. Univ. Michigan 143, 1–77. Flemming, T.H., 1974. The population ecology of two species of Costa Rican Heteromyid rodents. Ecology 55, 493–510. Flemming, T.H., 1988. The Short-Tailed Fruit Bat: A Study in Plant Animal Interactions. University of Chicago Press, Chicago. Fritzell, E.K., Hubert, G.F., Meyen, B.E., Sanderson, G.C., 1985. Age-specific reproduction in Illinois and Missouri raccoons. J. Wildl. Manage. 49, 901–905. Gage, T.B., Dyke, B., 1988. Model life tables for the larger Old World monkeys. Am. J. Primat. 16, 305–320. Gaisler, J., 1979. Ecology of bats. In: Stoddart, D.M. (Ed.), Ecology of Small Mammals. Chapman & Hall, London, pp. 281–342. Garrott, R.A., Taylor, L., 1990. Dynamics of feral horse population in Montana. J. Wildl. Manage. 54, 603–612. Gottdenker, N., Bodmer, R.E., 1998. Reproduction and productivity of white-lipped and collared peccaries in the Peruvian Amazon. J. Zool. (Lond.) 245, 423–430. Harris, S., Smith, G.C., 1987. Demography of two urban fox (Vulpes vulpes) populations. J. Appl. Ecol. 24, 75–86. Helle, E., Kauhala, K., 1993. Age structure, mortality, and sex ratio of the raccoon dog in Finland. J. Mammal. 74, 936–942. Hellgren, E.C., Synatzske, D.R., Oldenburg, P.W., Guthery, F.S., 1995. Demography of a collared peccary population in south Texas. J. Wildl. Manage. 59, 153–163. Heppell, S.S., Caswell, H., Crowder, L.B., 2000. Life histories and elasticity patterns: perturbation analysis for species with minimal demographic data. Ecology 81, 654–665. Hindell, M.A., 1991. Some life history parameters of a declining population of southern elephant seals, Mirounga leonina. J. Anim. Ecol. 60, 119–134. Hoefs, M., Bayer, M., 1983. Demographic characteristics of an unhunted Dall sheep (Ovis dalli dalli) population in southwest Yukon, Canada. Can. J. Zool. 61, 1346–1357. Hoogland, J., 1995. Black Tailed Prairie Dogs. University of Chicago Press, Chicago. Hoyle, J., Boonstra, R., 1986. Life history traits of the meadow jumping mouse, Zapus hudsonius, in Southern Ontario. Can. Field Nat. 100, 537–544. Humphrey, S.R., Cope, J.B., 1976. Population ecology of the little brown bat, Myotis lucifugus, in Indiana and North-central Kentucky. Am. Soc. Mammal. Special Publ. 4, 38–53.
M.K. Oli / Ecological Modelling 169 (2003) 313–325 Jarvis, J.U.M., 1973. The structure of a population of naked mole-rats, Tachyoryctes splendens (Rodentia: Rhyzomyidae). J. Zool. (Lond.) 171, 1–14. Jensen, A.L., 1995. Simple density dependent matrix model for population projection. Ecol. Model. 77, 43–48. Jezierski, W., 1976. Longevity and mortality rate in a population of wild boar. Acta Theriol. 22, 337–348. Jiang, H., Liu, Z., Yuan, X., Wang, H., 1989. Population structure of rhesus monkeys at Nanwan peninsula of Hainan Island (Chinese). Acta Theriol. Sin. 9, 254–261. Jiang, Z., Ma, Y., Gao, Z., 1993. A study on population structure and dynamic trend of Mongolian gazelle in China (Chinese). Acta Theriol. Sin. 13, 16–20. Johannensen, E.H.P., Andreaseen, H.P., 1998. Survival and reproduction of resident and immigrant female root voles (Microtus oceconomus). Can. J. Zool. 76, 763–766. Kelly, M.J., Laurenson, M.K., Fitzgibbon, C.D., Collins, D.A., Durant, S.M., Frame, W.F., Bertram, B.C.R., Caro, T.M., 1998. Demography of Serengeti cheetah (Acinonys jubatus) population: the first 25 years. J. Zool. (Lond.) 244, 473–488. Kemp, G.A., Keith, L.B., 1970. Dynamics and regulation of red squirrel (Tamiasciurus hudsonicus) populations. Ecology 51, 763–779. Knight, R.R., Eberhardt, L.L., 1985. Population dynamics of Yellowstone grizzly bears. Ecology 66, 323–334. Kohn, B.E., Payne, N.F., Ashbrenner, J.E., Creed, W.A., 1993. The fisher in Wisconsin. Wisconsin Dept. Nat. Resour. Tech. Bull. 183, 1–24. Kovacs, G., 1983. Survival patterns in adult European hares. Acta Zool. Fenn. 174, 69–70. Lande, R., 1988. Demographic models of the northern spotted owl (Strix occidentalis caurina). Oecologia 75, 601–607. Landers, R.H., 1981. A life table and biomass estimate for Alaskan fur seals. Fisheries Res. (Amsterdam) 1, 55–70. Larson, J.S., 1967. Age structure and sexual maturity within a western Maryland beaver (Castor canadensis) population. J. Mammal. 48, 408–413. Laws, R.M., 1966. Age criteria for the African elephant, Loxodanta a. africana. East Afr. Wildl. J. 4, 1–37. Laws, R.M., 1969. Aspects of reproduction in the African elephants, Loxodonta africana. J. Repr. Fert. (Suppl.) 6, 193– 218. Le Boeuf, B.J., Reiter, J., 1988. Life time reproductive success in northern elephant seals. In: Clutton-Brock, T.H. (Ed.), Reproductive Success. University of Chicago Press, Chicago, pp. 344–362. Leader-Williams, N., 1988. Reindeer on South Georgia. Cambridge University Press, New York. Leslie, P.H., 1945. On the use of matrices in certain population mathematics. Biometrica 33, 183–212. Leslie, P.H., 1948. Some further notes on the use of matrices in population mathematics. Biometrica 35, 213–245. Levin, L., Caswell, H., Bridges, T., DiBacco, C., Cabrera, D., Plaia, G., 1996. Demographic responses of estuarine polychaetes to pollutants: life table response experiments. Ecol. Appl. 6, 1295– 1313. Lewontin, R.C., 1965. Selection for colonizing ability. In: Baker, H.G., Stebins, G.L. (Eds.), The Genetics of Colonizing Species. Academic Press, New York, pp. 79–94.
323
Li, J., Wang, Q., Li, M., 1995. Studies on the population ecology of Tibetan monkeys (Macaca thibetana). III. Age structure and life table of Tibetan monkeys (Chinese). Acta Theriol. Sin. 15, 31–35. Lima, M., Páez, E., 1997. Demography and population dynamics of South American fur seal. J. Mammal. 78, 914–920. Lodal, J., Grue, H., 1985. Age determination and age distribution in populations of moles (Talpa europea) in Denmark. Acta Zool. Fenn. 173, 279–281. Lord, R.D.J., 1961a. Magnitudes of reproduction in cottontail rabbits. J. Wildl. Manage. 25, 28–33. Lord, R.D.J., 1961b. Mortality rates of cottontail rabbits. J. Wildl. Manage. 25, 33–40. Lowe, V.P.W., 1969. Population dynamics of the red deer (Cervus elaphus L.) on Rhum. J. Anim. Ecol. 38, 425–458. Luo, J., Fox, B.J., 1990. Life table comparison between two ground squirrels. J. Mammal. 71, 364–370. Macpherson, A.H., 1969. The dynamics of Canadian arctic fox populations. Can. Dept. Environ. Fisheries Marine Services Tech. Rep. 8, 1–48. Mansfield, A.W., Beck, B., 1977. The grey seal in eastern Canada. Can. Dept. Environ. Fisheries Marine Services Tech. Rep. 704, 1–81. McLean, P.K., Pelton, M.R., 1994. Estimates of population density and growth of black bears in the Smoky mountains. Int. Conf. Bear Res. Manage. 9, 254–261. Medin, D.E., Anderson, A.E., 1979. Modeling the dynamics of a Colorado mule deer population. Wildl. Monogr. 68, 1–77. Mertens, H., 1985. Structures de population et tables de survie de buffles, topis et cobs de buffon au parc national des Virunga. Zaire. Rev. Ecol. Terre Vie 40, 33–51. Messier, F., Huot, J., Le Henaff, D., Luttich, S., 1988. Demography of the George River Caribou herd: evidence of population regulation by forage exploitation and range expansion. Arctic 41, 279–287. Michod, R.E., Anderson, W.W., 1980. On calculating demographic parameters from age frequency data. Ecology 61, 265–269. Millar, J.S., Zwickel, F.C., 1972. Determination of age, age structure, and mortality of the pika, Ochotona princeps Richardson. Can. J. Zool. 50, 229–232. Monakhov, V.G., 1983. Age structure of the sable (Martes zibellina) populations (Russian). Z. Zhurnal 62, 1398–1406. Nelson, B.B., Chapman, J.A., 1982. Age determination and population characteristics of the wolf (Canis lupus labradorius Goldman) in northern Quebec and Labrador. Z. Säugetierkunde 47, 296–311. Oli, M.K., 2003. Partial life-cycle analysis: a simplified model for post-breeding census data. Ecol. Model., in press. Oli, M.K., Dobson, F.S., 1999. Population cycles in small mammals: the role of age at sexual maturity. Oikos 86, 557– 565. Oli, M.K., Dobson, F.S., 2001. Population cycles in small mammals: the ␣-hypothesis. J. Mammal. 82, 573–581. Oli, M.K., Dobson, F.S., 2003. The relative importance of life-history variables to population growth rate: Cole’s prediction revisited. Am. Nat. 161, 422–440. Oli, M.K., Slade, N.A., Dobson, F.S., 2001. Effect of density reduction on Uinta ground squirrels: an analysis of life table response experiments. Ecology 82, 1921–1929.
324
M.K. Oli / Ecological Modelling 169 (2003) 313–325
Oli, M.K., Hepp, G.R., Kennamer, R.A., 2002. Fitness consequences of delayed maturity in female wood ducks. Evol. Ecol. Res. 4, 563–576. Oli, M.K., Zinner, B., 2001a. Partial life cycle analysis: a model for birth-pulse populations. Ecology 82, 1180–1190. Oli, M.K., Zinner, B., 2001b. Partial life cycle analysis: a model for pre-breeding census data. Oikos 93, 376–387. Packer, C., Herbst, L., Pusey, A.E., Bygott, J.D., Cairns, S.J., Mulder, M.B., 1988. Lifetime reproductive success in lions. In: Clutton-Brock, T.H. (Ed.), Reproductive Success: Studies of Individual Variation in Contrasting Breeding Systems. University of Chicago Press, Chicago, pp. 363–383. Packer, C., Tatar, M., Collins, A., 1998. Reproductive cessation in female mammals. Nature 392, 807–811. Parker, G.R., Luttich, S., 1986. Characteristics of the wolf (Canis lupus labradorius) in northern Quebec and Labrador. Arctic 39, 145–149. Payne, N.F., 1984. Mortality rates of beaver in Newfoundland. J. Wildl. Manage. 48, 117–126. Perryman, P., Muchlinski, A., 1987. Population dynamics of feral burros at the Naval Weapons Center, China Lake, California. J. Mammal. 68, 435–438. Pielowski, Z., 1984. Some aspects of population structure and longevity of field roe deer. Acta Theriol. 29, 17–33. Price, M.V., Kelly, P.A., 1994. An age-structured demographic model for the endangered Stephen’s Kangaroo rat. Conserv. Biol. 8, 810–821. Purvis, A., Harvey, P.H., 1995. Mammal life history evolution: a comparative test of Charnov’s model. J. Zool. (Lond.) 237, 259–283. Reijnders, P.J.H., 1978. Recruitment in the harbour seal (Phoca vitulina) population in the Dutch Wadden Sea. Netherland J. Sea Res. 12, 164–179. Richard, A., 1985. Primates in Nature. W.H. Freeman, New York. Robinson, J.G., O’Brien, T.G., 1991. Adjustment in birth sex ratio in wedge-capped capuchin monkeys. Am. Nat. 138, 1173–1186. Rodgers, W.A., 1984. Warthog ecology in south east Tanzania. Mammalia 48, 327–350. Rogers, L.M., Cheeseman, C.L., Mallinson, P.J., Clifton-Hadley, R., 1997. The demography of a high density badger (Meles meles) population in the west of England. J. Zool. (Lond.) 242, 705–728. Rolley, R.E., 1985. Dynamics of a harvested bobcat population in Oklahoma. J. Wildl. Manage. 49, 283–292. Sæther, B.E., Bakke, Ø., 2000. Avian life history variation and contribution of demographic traits to the population growth rate. Ecology 81, 642–653. Saltz, D., Rubenstein, D.I., 1995. Population dynamics of a reintroduced asiatic wild ass (Equus hemionus) herd. Ecol. Appl. 5, 327–335. Schwartz, O.A., Armitage, K.B., Van Vuren, D., 1998. A 32-year demography of yellow-bellied marmots (Marmota flaviventris). J. Zool. (Lond.) 246, 337–346. Sherman, P.W., Morton, M.L., 1984. Demography of Belding’s ground squirrels. Ecology 65, 1617–1628. Silvertown, J., Franco, M., Pisanty, I., Mendonza, I., 1993. Comparative plant demography—relative importance of life-cycle components to finite rate of increase in woody and herbaceous perennials. J. Ecol. 81, 465–476.
Simmons, N.M., Bayer, M.B., Sinkey, L.O., 1984. Demography of Dall’s sheep in the Makenzie Mountains, Northwest Territories. J. Wildl. Manage. 48, 156–162. Sinclair, A.R.E., 1977. The African Buffalo. A Study in Resource Limitation of Populations. University of Chicago Press, Chicago. Slade, N.A., Balph, D.F., 1974. Population ecology of Uinta ground squirrels. Ecology 55, 989–1003. Slade, N.A., Gomulkiewicz, R., Alexander, H.M., 1998. Alternatives to Robinson and Redford’s method of assessing overharvest from incomplete demographic data. Conserv. Biol. 12, 1–8. Smith, A.T., 1974. The distribution and dispersal of pikas: consequences of insular population structure. Ecology 55, 1112– 1119. Smith, G.W., Johnson, D.R., 1985. Demography of a Townsend ground squirrel population in southwestern Idaho. Ecology 66, 171–178. Spinage, C.A., 1970. Population dynamics of the Uganda defassa waterbuck (Kobus defassa ugandae Neumann) in Queen Elizabeth Park. J. Anim. Ecol. 39, 51–78. Spinage, C.A., 1972. African ungulate life tables. Ecology 53, 645–652. Stearns, S.C., 1992. The Evolution of Life Histories. Oxford University Press, Oxford. Stephenson, A.B., 1977. Age determination and morphological variation of Ontario otters. Can. J. Zool. 55, 1577–1583. Thompson, M.J.A., 1987. Longevity and survival of female pipistrelle bats (Pipistrellus pipistrellus) on the Vale of York. Engl. J. Zool. (Lond.) 211, 209–214. Tryon, C.A., Snyder, D.P., 1973. Biology of the eastern chipmunk, Tamias striatus: life tables, age distributions, and trends in population numbers. J. Mammal. 54, 145–168. van Aarde, R.J., 1987. Demography of a Cape porcupine, Hystrix africaneaustralialis, population. J. Zool. (Lond.) 213, 205–212. van Sickle, J., Attwell, C.A.M., Craig, G.C., 1987. Estimating population growth rate from an age distribution of natural death. J. Wildl. Manage. 51, 941–948. Warner, R.E., 1985. Demography and movements of free ranging domestic cats in rural Illinois. J. Wildl. Manage. 49, 340–346. Waser, P.M., Jones, W.T., 1991. Survival and reproductive effort in banner-tailed kangaroo rats. Ecology 72, 771–777. Waser, P.M., Elliott, L.F., Creel, N.M., Creel, S.R., 1995. Habitat variation and mongoose demography. In: Sinclair, A.R.E., Arcese, P. (Eds.), Serengeti II. University of Chicago Press, Chicago, pp. 421–447. Watson, R.M., 1969. Reproduction in wildebeest, Connochaetes taurinus albojubatus Thomas, in the Serengeti region, and its significance to conservation. J. Reprod. Fert. Suppl. 6, 287–310. Weber, V.D., 1989. Zur Populationsbiologie schweizerischer Iltisse (Mustela putorius L.). Z. Jagdwissenschaften 35, 86–99. Wegge, P., 1979. Aspects of the population ecology of blue sheep in Nepal. J. Asian Ecol. 1, 10–20. Wei, F., Hu, J., Xu, G., Jiang, M., Deng, Q., Zhong, Z., 1989. A study on the life table of wild giant pandas (Chinese). Acta Theriol. Sin. 9, 81–86. Willner, G.R., Dixon, K.R., Chapman, J.A., 1983. Age determination and mortality of the nutria (Myocastor coypus) in Maryland. Z. Säugetierkunde 48, 19–34.
M.K. Oli / Ecological Modelling 169 (2003) 313–325 Woodgerd, W., 1964. Population dynamics of bighorn sheep on Wildhorse Island. J. Wildl. Manage. 28, 381–391. Yang, Q., Hu, J., Peng, J., 1990. A study on the population ecology of forest musk deer (Muschus berezovskii) at the north of Hengduan mountains (Chinese). Acta Theriol. Sin. 10, 255– 262.
325
Yozdis, P., Kolenosky, G.B., 1986. A population dynamics model of black bears in eastcentral Ontario. J. Wildl. Manage. 50, 602–612. Zammuto, R.M., 1987. Life histories of mammals: analyses among and within Spermophilus columbianus life tables. Ecology 68, 1351–1363.
Partial life-cycle models: how good are they? Madan K. Oli∗ Department of Wildlife Ecology and Conservation, 110 Newins-Ziegler Hall, University of Florida, Gainesville, FL 32611-0430, USA Received 17 September 2002; received in revised form 11 June 2003; accepted 16 July 2003
Abstract Partial life-cycle models are simplified, matrix-based population models that allow demographic analyses using incomplete demographic data. Number of parameters in most partial life-cycle models are fixed, and this permits direct interspecific comparison of population dynamics and life histories. Because of these desirable properties, partial life-cycle models have recently received substantial applications. However, adequacy of partial life-cycle models has not been investigated using empirical data. I tested the adequacy of a post-breeding census partial life-cycle model by examining whether and to what extent dynamical properties of the age-structured model are retained in the post-breeding census partial life-cycle model. I applied the age-structured Leslie matrix model and a post-breeding census partial life-cycle model to demographic data for 142 populations of mammals, and compared the projected population growth rate (λ), and elasticity of λ to changes in ages at first and last reproduction, juvenile survival, adult survival, and fertility estimated from the two models. Population growth rate estimated from the two models were practically indistinguishable, and elasticities estimated from the two models were very similar. These results suggest that dynamical properties of the age-structured model are generally captured in the partial life-cycle model, and that application of the post-breeding census partial life-cycle model to age-structured or partial demographic data will not substantially compromise precision of analyses nor conclusions of an investigation. These results have substantial implications in basic and applied population ecology, because age-specific demographic data are seldom available for many species of conservation concern, and also because partial life-cycle models permit perturbation analyses involving ages at first and last reproduction using standard analytical techniques. © 2003 Elsevier B.V. All rights reserved. Keywords: Age-structured models; Elasticity analysis; Matrix population models; Model comparison; Partial life-cycle models; Population dynamics; Sensitivity analysis
1. Introduction An important goal of population ecology is to understand, predict, and explain dynamics of biological populations (Akçakaya et al., 1997). Achieving this goal often requires application of a population model to demographic data. Of various demographic models currently available, matrix population models have become the most popular tools for investigating ∗ Tel.: +1-352-846-0561; fax: +1-352-392-6984. E-mail address: [email protected] (M.K. Oli).
the dynamics of age- or stage-structured populations (Caswell, 2001). In particular, variations of the age-structured Leslie matrix model (Leslie, 1945, 1948; Caswell, 2001) have become standard tools for demographic analyses of age-structured populations. However, the Leslie matrix model requires age-specific demographic data, which are seldom available for most species of conservation concern or those with long lifespans. Additionally, the Leslie matrix model or its variations do not allow perturbation analyses involving ages at first and last reproduction, because these variables do not appear as
0304-3800/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0304-3800(03)00277-1
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explicit model parameters in the characteristic equation. This is a major limitation of the Leslie matrix model, especially because age at first reproduction is an important life history variable with substantial influence on the dynamics of biological populations (Cole, 1954; Lewontin, 1965; Dobson and Oli, 2001; Oli and Dobson, 2001). Models that allow demographic analyses based on partial demographic data have been developed as an alternative to age-structured models. Such models are called “partial life-cycle models”, because they generally are based on life-cycle stages, and can be parameterized with partial demographic data (Oli and Zinner, 2001a). One of the earliest partial life-cycle models was derived by Cole (1954), which was further refined by Lewontin (1965), Eberhardt and O’Shea (1995), and Slade et al. (1998). Various versions of partial life-cycle models have been presented by Levin et al. (1996), Heppell et al. (2000), and Caswell (2001). Partial life-cycle models have recently been fully developed for post-breeding (Oli and Zinner, 2001a; Oli, 2003) and pre-breeding (Oli and Zinner, 2001b) census situations. Because partial life-cycle models can be parameterized with incomplete demographic data, and also allow perturbation analyses involving time variables (ages at first and last reproduction), these models have recently received substantial applications in basic and applied population ecology (Lande, 1988; Eberhardt and O’Shea, 1995; Slade et al., 1998; Heppell et al., 2000; Dobson and Oli, 2001; Oli et al., 2001). However, adequacy of most partial life-cycle models have not been tested. My objective was to empirically test the adequacy of the post-breeding census partial life-cycle model of Oli and Zinner (2001a). Specifically, I applied the Leslie matrix model and the post-breeding census partial life-cycle model to demographic data for 142 natural populations of mammals. I then compared population growth rate (λ), and elasticity of λ to changes in demographic variables estimated from the Leslie matrix model and the post-breeding census partial life-cycle model. Because most demographic studies of mammalian populations are conducted within post-breeding census framework (Caughley, 1977; Caswell, 2001), the post-breeding census partial life-cycle model of Oli and Zinner (2001a) was appropriate for the analyses of these data.
2. Methods 2.1. Demographic data and models I compiled demographic data for 142 populations of mammals from published sources (Appendix A). I used demographic data from a study if age-specific or otherwise detailed survival and at least average fecundity rates were available. Demographic data based only on the male segment of populations or those based on laboratory studies were not used. Demographic data were analyzed using the agestructured Leslie matrix model and the post-breeding census partial life-cycle model of Oli and Zinner (2001a). The life-cycle of an age-structured population can be graphically represented as an age-structured life-cycle graph, from which a corresponding population projection (Leslie) matrix can be derived (Fig. 1). The Leslie matrix summarizes age-specific demographic data; age-specific fertilities Fi appear on the first row of the matrix, and age-specific survival rates Pi appear on the lower subdiagonal of the matrix. Construction and analyses of the age-structured Leslie matrix models are described in detail by Caswell (2001). The partial life-cycle model is an approximation to the age-structured Leslie matrix model. In general, it is assumed that age-specific fertilities Fi are adequately approximated by an average fertility parameter F, age-specific survival Pi until reproduction is accomplished (i.e. juvenile survival) by a juvenile survival parameter Pj , and age-specific survival from the first reproductive event until age at last reproduction ω by an adult survival parameter Pa . Such a life-cycle, based on two life-history stages (a pre-reproductive juvenile stage, and a reproductive adult stage) can be represented by a partial life-cycle graph, from which a corresponding population projection matrix can be derived (Fig. 2). Assuming that reproduction begins at age α and terminates at age ω, and that demographic data are collected just after the birth-pulse (i.e. post-breeding census), the characteristic equation for this type of two-stage life-cycle is (Oli and Zinner, 2001a) 1 = FPj α−1 λ−α − FPj α−1 Pa λ−α−1 + FPj α λ−α−1 − FPj α Pa ω−α λ−ω−1 + Pa λ−1 .
(1)
The projected population growth rate λ is the largest real root of Eq. (1), and can be obtained numerically.
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F2 1
2 P1
F3
F4
3
4
P2
P3
0 P1 A = 0 0 0
F2
F3
F4
F5
0 P2
0 0
0 0
0 0
0 0
P3 0
0 P4
0 0
315
F5
P4
5
Fig. 1. An example of an age-structured life-cycle graph for age at first reproduction (α) = 2 and age at last reproduction (ω) = 5. Reproduction begins at age α and terminates at age ω. In an age-structured life-cycle graph, both fertility (Fi ) and survival rates (Pi ) are age-specific. A corresponding age-structured population projection (Leslie) matrix also is given.
2.2. Parameter estimation and demographic analyses
Fi = Pi mi ,
For each population, I estimated age-specific survival (Pi ) and fertility (Fi ) using the post-breeding census formulation of Caswell (2001):
where li and mi are age-specific survivorship and fecundity, respectively. Using Pi and Fi thus estimated, a Leslie matrix A was constructed for each population. The population growth rate λ was then estimated as the dominant eigenvalue of the Leslie matrix. The sensitivity of λ to changes in a matrix entry aij was
Pi =
li li−1
,
(2)
1 Pj
(3)
F
F
F
2
3
4
Pj 0 Pj A = 0 0 0
Pa F 0 Pj
F 0 0
F 0 0
F 0 0
0 0
Pa 0
0 Pa
0 0
F
5 Pa
Fig. 2. A post-breeding census partial life cycle graph for age at first reproduction (α) = 2, and age at last reproduction (ω) = 5. In a partial life-cycle graph, age-specific fertilities (Fi ) are approximated by a parameter F, age-specific survival probabilities until the first reproductive event by a parameter Pj , and age-specific survival probabilities for age classes greater than ␣ by a parameter Pa . A population projection matrix corresponding the partial life-cycle graph also is given.
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estimated as (Caswell, 2001) v i wj ∂λ = , ∂aij v, w
(4)
where v and w are the left and right eigenvectors, respectively, corresponding to the dominant eigenvalue of the matrix A. Elasticity (i.e. proportional sensitivity) of λ to changes in aij was then estimated as eij = [∂λ/∂aij ][aij /λ]. For the post-breeding census partial life-cycle model, I estimated ages at first (α) and last (ω) reproduction as the first and last age class with non-zero fertility, respectively. When detailed demographic data were available, F, Pj , and Pa were estimated from the age-classified projection matrix as weighted averages, weighted according to the contribution of each age class to the stable age distribution (Oli and Zinner, 2001a): ω w i Fi F = i=α , (5) ω i=α w i ω−1 w i Pi Pa = i=α+1 , (6) ω−1 i=α+1 w i α w i Pi Pj = i=1 , (7) α i=1 w i
life-cycle model with those obtained from the Leslie matrix model. A close correspondence between results obtained from the two models would indicate that dynamical properties of the age-structured models are generally captured in the simplified partial life-cycle model. To test the adequacy of the post-breeding census partial life-cycle model, I compared λ and elasticity of λ to changes in model parameters (F, Pj , and Pa ) estimated from the two models. Because α and ω do not appear as model parameters in the Leslie matrix model, elasticity of λ to changes in these two parameters could not be compared between the two models. Also, F, Pj and Pa do not appear explicitly as parameters in the Leslie matrix model, and elasticity of λ to changes in F, Pj and Pa were estimated as (Oli and Zinner, 2001b) ω
∂λ xi, = ∂F
(8)
i=α
α
∂λ Pj yi, = ∂Pj λ
(9)
i=1
ω−1 ∂λ Pa yi, = ∂Pa λ
(10)
i=α+1
where wi is the ith entry of the right eigenvector corresponding to the dominant eigenvalue of the Leslie matrix A. The population growth rate was estimated as the largest real root of Eq. (1). The sensitivity of λ to changes in a model parameter p (where p is α, ω, F, Pj , or Pa ) was estimated as the partial derivative of λ with respect to p (Oli and Zinner, 2001a,b). Elasticity of λ to changes in p was then estimated as e(p) = [∂λ/λp][p/λ].
where x is a vector consisting of the first row of the elasticity matrix, and y is a vector consisting of the lower subdiagonal entries of the elasticity matrix. Elasticities of λ to changes in F, Pj and Pa calculated as above from the elasticity matrix of the age-structured model are directly comparable to those obtained from the partial life-cycle model.
2.3. Test of model adequacy
3. Results
The age-structured Leslie matrix model adequately considers the age-specific survival and fertility rates, whereas the partial life-cycle model assumes constant survival probabilities within each of the two life-history stages, as well as age-independent fertility rate. Consequently, the Leslie matrix model may be expected to yield accurate results, and an appropriate way to evaluate adequacy of the partial life-cycle model would be to compare results of the partial
I compiled demographic data for 142 populations of mammals, representing 110 species, 33 families, and 11 orders. Rodentia was the most represented order, with 42 populations of 27 species, followed by Artiodactyla (29 populations of 27 species), and Carnivora (27 populations of 20 species) (Appendix A). Ranges of values of demographic variables were α: 15 days to 15 years, ω: 84 days to 60 years; Pj : 0.111–0.969; Pa : 0.232–0.978; and F: 0.068–1.77; λ ranged from
M.K. Oli / Ecological Modelling 169 (2003) 313–325
317
3.5 Y = -0.002 + 1.009X; R2 = 0.995
λ,partial LC model
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5
1.0
1.5
2.0
2.5
3.0
λ, Leslie matrix model Fig. 3. A plot of population growth rates (λ) estimated from the age-structured Leslie matrix model (X-axis) and the post-breeding census partial life-cycle model (Y-axis).
the Leslie matrix model (r = 0.99, P < 0.0001; Fig. 3). Likewise, elasticity of λ to changes in Pj (r = 0.93, P < 0.0001; Fig. 4), Pa (r = 0.91, P < 0.0001; Fig. 5) and F (r = 0.94, P < 0.0001; Fig. 6) were very similar. Overall, the difference between results
0.67 to 2.62. Thus, data used in this study were appropriate for testing the adequacy of the partial life-cycle model. The population growth rate λ estimated from the partial life-cycle model resembled those obtained from 0.8
Y = -0.014 + 0.974X; R2 = 0.870
e( ), partial LC model
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.2
0.4
0.6
0.8
e( ), Leslie matrix model Fig. 4. A plot of the elasticity of population growth rates (λ) to changes in juvenile survival e(Pj ) estimated from the age-structured Leslie matrix model (X-axis) and the post-breeding census partial life-cycle model (Y-axis).
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0.8 2
0.6
0.4
0.2
e(
), partial LC model
Y = -0.010 + 0.922X; R = 0.821
0.0
0.0
0.2
e(
0.4
0.6
0.8
), Leslie matrix model
Fig. 5. A plot of the elasticity of population growth rates (λ) to changes in adult survival e(Pa ) estimated from the age-structured Leslie matrix model (X-axis) and the post-breeding census partial life-cycle model (Y-axis).
1.0
2
e( ), partial LC model
Y = 0.011 + 1.095X; R = 0.882 0.8
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
e( ), Leslie matrix model Fig. 6. A plot of the elasticity of population growth rates (λ) to changes in fertility e(F) estimated from the age-structured Leslie matrix model (X-axis) and the post-breeding census partial life-cycle model (Y-axis).
obtained from the two models appeared to be random around zero, suggesting that the partial life-cycle model produced relatively unbiased estimates of λ and elasticities.
4. Discussion Population growth rates estimated from the partial life-cycle model were practically indistinguishable
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from those obtained from the Leslie matrix model. Elasticity of λ to changes in demographic variables estimated from the two models differed to some degree in some populations; however, these differences were fairly small and elasticities estimated from the two models were strongly correlated (r > 0.90). These results suggest that dynamical properties of age-structured models are generally retained in the partial life-cycle model of Oli and Zinner (2001a), and that results of the partial life-cycle model are robust. These findings should be comforting to researchers working on endangered or long-lived species for which age-structured demographic data are seldom available (Heppell et al., 2000; Sæther and Bakke, 2000). Matrix population models have received substantial applications in research areas as diverse as conservation biology, life-history theory, and population regulation (Heppell et al., 2000; Caswell, 2001; Dobson and Oli, 2001; Oli et al., 2001). When age-specific demographic data are available, the age-structured Leslie matrix model provides the best estimates of λ, and sensitivity or elasticity of λ to changes in age-specific demographic variables because it adequately incorporates age-specific demographic information. However, parameterization of the Leslie matrix model requires ≤2n − 1 age-specific parameters (where n = number of age classes). For example, one must estimate up to 10 age-specific fertility rates and 9 age-specific survival rates to fully parameterize an age-structured model for a population with 10 age classes. Collection of data for estimating age-specific demographic parameters is difficult, and such data are seldom available for rare or endangered species or for species with long lifespans, and ecologists frequently have to rely on incomplete demographic data for addressing basic and applied ecological problems. Theoretical as well as empirical studies have shown that age at first reproduction (the age at which reproduction begins) is an important life-history variable with substantial potential for influencing population dynamics and fitness (Cole, 1954; Lewontin, 1965; Stearns, 1992; Oli and Dobson, 1999; Oli et al., 2002). However, this variable does not appear as an explicit model parameter in the Leslie matrix model, and absolute or proportional sensitivity of λ to changes in age at first reproduction cannot be estimated using standard techniques. Although some age-specific information is lost, the post-breeding census partial life-cycle model of Oli
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and Zinner (2001a) has several desirable properties. First, variables such as α and ω explicitly appear as model parameters in the characteristic equation, and the sensitivity analyses as well as analyses of life-table response experiments (LTRE analyses; Caswell, 1989, 1996, 2001) involving these life-history variables can be directly estimated using standard analytical techniques (Levin et al., 1996; Heppell et al., 2000; Sæther and Bakke, 2000; Dobson and Oli, 2001; Oli et al., 2001). Second, the partial life-cycle model can be fully parameterized with five parameters (α, ω, Pj , Pa , and F) regardless of the number of age classes (whereas the Leslie matrix model requires ≤2n−1 parameters). Consequently, the partial life-cycle model allows demographic analyses based on incomplete demographic data, but it can also be parameterized with age-specific data when available. Third, because the number of parameters in the partial life-cycle model is fixed regardless of the number of age classes, this model allows direct comparison of the life-history or population dynamics among species with vastly different life histories or age structure (Silvertown et al., 1993; Heppell et al., 2000; Sæther and Bakke, 2000; Oli and Dobson, 2003). Finally, results presented in this paper suggest that the dynamical properties of the age-structured model are generally retained in the partial life-cycle model. When age-specific demographic data are available and when perturbation analyses involving ages at first and last reproduction are not desired, the age-structured model is the best option because it adequately considers age-specific demographic information. If perturbation analyses involving ages at first and last reproduction are desired, then the partial life-cycle model is preferable even when age-specific demographic data are available. If only partial demographic data are available, then the partial life-cycle model is the best option for demographic analyses. Finally, I note that timing of a census relative to the birth-pulse has important implications in the construction and analysis of matrix population models (Caswell, 2001; Oli and Zinner, 2001a,b). Consequently, estimation of parameters and choice of an appropriate model should be based on the timing of a census relative to the birth-pulse. If demographic data are collected immediately after the birth-pulse (post-breeding censuses), the post-breeding census partial life-cycle model of Oli and Zinner (2001a) should be used. If data are
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collected just before the birth-pulse (i.e. pre-breeding census), then the pre-breeding census partial life-cycle model of Oli and Zinner (2001b) is appropriate (see Slade et al., 1998 and Caswell, 2001 for similar models). Results presented here suggest that the partial life-cycle analysis will not generally compromise precision of the analyses nor conclusions of an investigation if parameters are estimated adequately and an appropriate model is used. Thus, the partial life-cycle model of Oli and Zinner (2001a) may be used with confidence for demographic analyses of populations with minimal demographic data.
Acknowledgements I thank Dr. Boris Schröder and an anonymous reviewer for insightful comments on the manuscript. This research was supported in part by a grant from the National Science Foundation (DEB-0224953) and the Florida Agricultural Experiment Station, and approved for publication as Journal Series No. R-09495.
Appendix A References for demographic data used in the analyses reported in this paper. Within each order, species are sorted in alphabetical order. When reproductive data were not reported in the original source, estimates of average fecundity compiled by Purvis and Harvey (1995) were used. Order Artiodactyla Aepyceros melampus: Spinage (1972); Alces alces: Boer (1988); Capreolus capreolus: Pielowski (1984); Cervus elaphus: Clutton-Brock et al. (1988), Lowe (1969); Connochaetes taurinus: Watson (1969), Attwell (1982); Damaliscus lunatus: Mertens (1985); Hemitragus jemlahicus: Caughley (1966, 1977); Kobus defassa ugandae: Spinage (1970); Kobus ellipsiprymnus: van Sickle et al. (1987); Kobus kob: Mertens (1985); Moschus berezovskii: Yang et al. (1990); Odocoileus hemionus: Medin and Anderson (1979); Odocoileus virginianus: Jensen (1995); Ovis aries: Clutton-Brock et al. (1997); Ovis canadensis: Woodgerd (1964); Ovis dalli: Hoefs and Bayer
(1983), Simmons et al. (1984); Phachochoerus aethiopicus: Rodgers (1984); Procapra gutturosa: Jiang et al. (1993); Pseudois nayaur: Wegge (1979); Rangifer t. tarandus: Messier et al. (1988); Rangifer tarandus: Leader-Williams (1988); Rupicapra rupicapra: Caughley (1970); Sus scrofa: Jezierski (1976), Ahmad et al. (1995); Syncerus caffer: Sinclair (1977); Tayassu pecari: Gottdenker and Bodmer (1998); Tayassu tajacu: Hellgren et al. (1995). Order Carnivora Acinonyx jubatus: Kelly et al. (1998); Ailuropoda melanoleuca: Wei et al. (1989); Alopex lagopus: Macpherson (1969); Canis lupus: Parker and Luttich (1986); Felis catus: Warner (1985); Helogale parvula: Waser et al. (1995); Lutra canadensis: Stephenson (1977); Lynx rufus: Crowe (1975), Rolley (1985); Martes pennanti: Kohn et al. (1993); Martes zibellina: Monakhov (1983); Meles meles: Rogers et al. (1997); Mephitis mephitis: Casey and Webster (1975); Mungos mungo: Waser et al. (1995); Mustela putorius: Weber (1989); Nyctereutes procyonoides: Helle and Kauhala (1993); Pathera leo: Packer et al. (1988, 1998); Urocyon cinereoargenteus: Michod and Anderson (1980), Fritzell et al. (1985); Ursus americanus: McLean and Pelton (1994, two populations), Yozdis and Kolenosky (1986); Ursus arctos: Knight and Eberhardt (1985); Vulpes vulpes: Nelson and Chapman (1982), Harris and Smith (1987, two populations). Order Chiroptera Carollia perspicillata: Flemming (1988); Eptesicus fuscus: Gaisler (1979); Myotis lucifugus: Humphrey and Cope (1976); Myotis myotis: Gaisler (1979); Pipistrellus pipistrellus: Thompson (1987); Pipistrellus subflavus: Davis (1966); Rhinolophus ferrumequinum: Corbet and Harris (1991). Order Insectivora Talpa europea: Lodal and Grue (1985). Order Lagomorpha Lepus europaeus: Kovacs (1983); Ochotona princeps: Smith (1974); Ochotona princeps: Millar and
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Zwickel (1972); Oryctolagus cuniculus: Cowan and Roman (1985); Sylvilagus floridanus: Lord (1961a,b). Order Perissodactyla Equus asinus: Perryman and Muchlinski (1987); Equus asinus: Choquenot (1991, two populations); Equus burchelli: Spinage (1972); Equus caballus: Garrott and Taylor (1990); Equus hemionus: Saltz and Rubenstein (1995). Order Pinnipedia Arctocephalus australis: Lima and Páez (1997); Callorhinus ursinus: Barlow and Boveng (1991); Callorhinus ursinus: Landers (1981); Eumetopias jubatus: Calkins and Pitcher (1982); Halichoerus grypus: Mansfield and Beck (1977); Leptonychotes weddelli: Croxall and Hiby (1983); Mirounga angustirostris: Le Boeuf and Reiter (1988); Mirounga angustirostris: Hindell (1991); Mirounga angustirostris: Bester and Wilkinson (1994); Phoca vitulina: Reijnders (1978). Order Primates Cebus olivaceus: Robinson and O’Brien (1991, two populations); Cercopithecus aethiops: Cheney et al. (1988); Macaca fascicularis: Crockett et al. (1996); Macaca fuscata: Richard (1985); Macaca mulatta: Richard (1985); Macaca mulatta: Jiang et al. (1989); Macaca sinica: Gage and Dyke (1988); Macaca thibetana: Li et al. (1995); Pan troglodytes: Courtenay and Santow (1989); Papio cynocephalus: Packer et al. (1998); Theropithecus gelada: Richard (1985); Theropithecus gelada: Dunbar (1980). Order Proboscida Loxodonta africana: Laws (1966, 1969). Order Rodentia Apodemus flavicollis: Bobek (1973); Castor canadensis: Payne (1984), Larson (1967); Clethrionomys glareolus: Bobek (1973); Cynomys gunnisoni: Cully (1997, two populations); Cynomys ludovicianus: Hoogland (1995); Dipodomys spectabilis: Waser and Jones (1991); Dipodomys stephensi: Price and Kelly
321
(1994); Histrix africaeaustralis: van Aarde (1987); Liomys adspersus: Flemming (1971); Liomys salvini: Flemming (1974); Marmota flaviventris: Schwartz et al. (1998); Microtus oeconomus: Johannensen and Andreassen (1998); Myocastor coypus: Willner et al. (1983); Oryzomys capito: Flemming (1971); Proechimys semispinosus: Flemming (1971); Sciurus carolinensis: Barkalow et al. (1970); Spermophilus armatus: Slade and Balph (1974, three populations); Spermophilus beldingi: Sherman and Morton (1984); Spermophilus columbianus: Zammuto (1987, six populations); Spermophilus d. alaschani: Chen (1991); Spermophilus dauricus: Luo and Fox (1990); Spermophilus lateralis: Bronson (1979, four populations); Spermophilus townsendii: Smith and Johnson (1985); Tachyoryctes splendens: Jarvis (1973); Tamias striatus: Tryon and Snyder (1973); Tamiasciurus hudsonicus: Kemp and Keith (1970); Zapus hodsonius: Hoyle and Boonstra (1986); Zapus princeps: Falk and Millar (1987). Order Sirenia Trichechus manatus: Eberhardt and O’Shea (1995, Blue Spring). References Akçakaya, H.R., Burgman, M.A., Ginzburg, L.R., 1997. Applied Population Ecology. Sinauer, Sunderland. Ahmad, E., Brooks, J.E., Hussain, I., Khan, M.H., 1995. Reproduction in Eurasian wild boar in central Punjab. Pakistan Acta Theriol. 40, 163–173. Attwell, C.A.M., 1982. Population ecology of the blue wildebeest Connochaetes taurinus taurinus in Zululand, South Africa. Afr. J. Ecol. 20, 147–168. Barkalow, F.S., Hamilton, R.B., Soots, R.F., 1970. The vital statistics of an unexploited gray squirrel population. J. Wildl. Manage. 34, 489–500. Barlow, J., Boveng, P., 1991. Modeling age-specific mortality for marine mammal populations. Marine Mammal Sci. 7, 50–65. Bester, M.N., Wilkinson, I.S., 1994. Population ecology of southern elephant seals at Marion Island. In: Le Boeuf, B.J., Laws, R.M. (Eds.), Elephant Seals: Population Ecology, Behaviour, and Physiology. University of California Press, Berkeley, pp. 85–97. Bobek, B., 1973. Net production of small rodents in a deciduous forest. Acta Theriol. 18, 403–434. Boer, A.H., 1988. Mortality rates of moose in New Brunswick: a life table analysis. J. Wildl. Manage. 52, 21–25. Bronson, M.T., 1979. Altitudinal variation in the life history of golden-mantled ground squirrel (Spermophilus lateralis). Ecology 60, 272–279.
322
M.K. Oli / Ecological Modelling 169 (2003) 313–325
Calkins, D.G., Pitcher, K.W., 1982. Population assessment, ecology, and trophic relationships of Steller sea lions in the Gulf of Alaska. Alaska Department of Fish and Game. Final Report RU 243. Alaska Department of Fish and Game, Anchorage. Casey, G.A., Webster, W.A., 1975. Age and sex determination of stripped skunks (Mephitis mephitis) from Ontario, Manitoba, and Quebec. Can. J. Zool. 53, 223–226. Caswell, H., 1989. The analysis of life table response experiments. I. Decomposition of treatment effects on population growth rate. Ecol. Model. 46, 221–237. Caswell, H., 1996. Analysis of life table response experiments. 2. Alternative parameterizations for size- and stage-structured models. Ecol. Model. 88, 73–82. Caswell, H., 2001. Matrix Population Models: Construction, Analysis, and Interpretation. Sinauer, Sunderland. Caughley, G., 1966. Mortality patterns in mammals. Ecology 47, 906–918. Caughley, G., 1970. Population statistics of Chamois. Mammalia 34, 194–199. Caughley, G., 1977. Analysis of Vertebrate Populations. Wiley, New York. Chen, J., 1991. Life table and reproduction of Alashan ground squirrel (Chinese). Acta Theriol. Sin. 11, 138–142. Cheney, D., Seyfarth, R.M., Andelman, S.J., Lee, P.C., 1988. Reproductive success in vervet monkeys. In: Clutton-Brock, T.H. (Ed.), Reproductive Success. Studies of Individual Variation in Contrasting Breeding Systems. University of Chicago Press, Chicago, pp. 384–402. Choquenot, D., 1991. Density-dependent growth, body condition, and demography in feral donkeys: testing the food hypothesis. Ecology 72, 805–813. Clutton-Brock, T.H., Albon, S.D., Guinness, F.E., 1988. Reproductive success in male and female red deer. In: Clutton-Brock, T.H. (Ed.), Reproductive Success. Studies of Individual Variation in Contrasting Breeding Systems. University of Chicago Press, Chicago, pp. 325–343. Clutton-Brock, T.H., Illius, A.W., Wilson, K., Grenfell, B.T., MacColl, A.D.C., Albon, S.D., 1997. Stability and instability in ungulate populations: an empirical analysis. Am. Nat. 149, 195–219. Cole, L., 1954. The population consequences of life-history phenomena. Q. Rev. Biol. 29, 103–137. Corbet, G.B., Harris, S., 1991. The Handbook of British Mammals. Blackwell, Oxford. Courtenay, J., Santow, G., 1989. Mortality of wild and captive chimpanzees. Folia Primat. 52, 167–177. Cowan, D.P., Roman, E.A., 1985. The construction of life-tables with special reference to the European wild rabbit (Oryctolagus cuniculus). J. Zool. (Lond.) 207, 607–609. Crockett, C.M., Kyes, R.C., Sajuthi, D., 1996. Modeling managed monkey populations: sustainable harvest of longtailed macaques on a natural habitat island. Am. J. Primat. 40, 343–360. Crowe, D.M., 1975. Aspects of ageing, growth, and reproduction in bobcats from Wyoming. J. Mammal. 56, 177–198. Croxall, J.P., Hiby, L., 1983. Fecundity, survival and site fidelity in Weddel seals, Leptonychotes weddelli. J. Appl. Ecol. 20, 19–32. Cully Jr., J.F., 1997. Growth and life history changes in Gunnison’s prairie dogs after a plague epizootic. J. Mammal. 78, 146–157.
Davis, W.H., 1966. Population dynamics of the bat Pipistrellus subflavus. J. Mammal. 47, 383–396. Dobson, F.S., Oli, M.K., 2001. The demographic basis of population regulation in Columbian ground squirrels. Am. Nat. 158, 236–247. Dunbar, R.I.M., 1980. Demographic and life history variables of a population of gelada baboons (Theropthecus gelada). J. Anim. Ecol. 49, 485–506. Eberhardt, L.L., O’Shea, T.J., 1995. Integration of Florida manatee life history data and population modeling. In: O’Shea, T.J., Ackerman, B.B., Percival, H.F. (Eds.), Population Biology of the Florida Manatee. US Department of the Interior, Washington, DC, pp. 269–279. Falk, J.W., Millar, J.S., 1987. Reproduction by female Zapus princeps in relation to age, size, and body fat. Can. J. Zool. 65, 568–571. Flemming, T.H., 1971. Population ecology of three species of Neotropical rodents. Miscellaneous publications of the museum of zoology. Univ. Michigan 143, 1–77. Flemming, T.H., 1974. The population ecology of two species of Costa Rican Heteromyid rodents. Ecology 55, 493–510. Flemming, T.H., 1988. The Short-Tailed Fruit Bat: A Study in Plant Animal Interactions. University of Chicago Press, Chicago. Fritzell, E.K., Hubert, G.F., Meyen, B.E., Sanderson, G.C., 1985. Age-specific reproduction in Illinois and Missouri raccoons. J. Wildl. Manage. 49, 901–905. Gage, T.B., Dyke, B., 1988. Model life tables for the larger Old World monkeys. Am. J. Primat. 16, 305–320. Gaisler, J., 1979. Ecology of bats. In: Stoddart, D.M. (Ed.), Ecology of Small Mammals. Chapman & Hall, London, pp. 281–342. Garrott, R.A., Taylor, L., 1990. Dynamics of feral horse population in Montana. J. Wildl. Manage. 54, 603–612. Gottdenker, N., Bodmer, R.E., 1998. Reproduction and productivity of white-lipped and collared peccaries in the Peruvian Amazon. J. Zool. (Lond.) 245, 423–430. Harris, S., Smith, G.C., 1987. Demography of two urban fox (Vulpes vulpes) populations. J. Appl. Ecol. 24, 75–86. Helle, E., Kauhala, K., 1993. Age structure, mortality, and sex ratio of the raccoon dog in Finland. J. Mammal. 74, 936–942. Hellgren, E.C., Synatzske, D.R., Oldenburg, P.W., Guthery, F.S., 1995. Demography of a collared peccary population in south Texas. J. Wildl. Manage. 59, 153–163. Heppell, S.S., Caswell, H., Crowder, L.B., 2000. Life histories and elasticity patterns: perturbation analysis for species with minimal demographic data. Ecology 81, 654–665. Hindell, M.A., 1991. Some life history parameters of a declining population of southern elephant seals, Mirounga leonina. J. Anim. Ecol. 60, 119–134. Hoefs, M., Bayer, M., 1983. Demographic characteristics of an unhunted Dall sheep (Ovis dalli dalli) population in southwest Yukon, Canada. Can. J. Zool. 61, 1346–1357. Hoogland, J., 1995. Black Tailed Prairie Dogs. University of Chicago Press, Chicago. Hoyle, J., Boonstra, R., 1986. Life history traits of the meadow jumping mouse, Zapus hudsonius, in Southern Ontario. Can. Field Nat. 100, 537–544. Humphrey, S.R., Cope, J.B., 1976. Population ecology of the little brown bat, Myotis lucifugus, in Indiana and North-central Kentucky. Am. Soc. Mammal. Special Publ. 4, 38–53.
M.K. Oli / Ecological Modelling 169 (2003) 313–325 Jarvis, J.U.M., 1973. The structure of a population of naked mole-rats, Tachyoryctes splendens (Rodentia: Rhyzomyidae). J. Zool. (Lond.) 171, 1–14. Jensen, A.L., 1995. Simple density dependent matrix model for population projection. Ecol. Model. 77, 43–48. Jezierski, W., 1976. Longevity and mortality rate in a population of wild boar. Acta Theriol. 22, 337–348. Jiang, H., Liu, Z., Yuan, X., Wang, H., 1989. Population structure of rhesus monkeys at Nanwan peninsula of Hainan Island (Chinese). Acta Theriol. Sin. 9, 254–261. Jiang, Z., Ma, Y., Gao, Z., 1993. A study on population structure and dynamic trend of Mongolian gazelle in China (Chinese). Acta Theriol. Sin. 13, 16–20. Johannensen, E.H.P., Andreaseen, H.P., 1998. Survival and reproduction of resident and immigrant female root voles (Microtus oceconomus). Can. J. Zool. 76, 763–766. Kelly, M.J., Laurenson, M.K., Fitzgibbon, C.D., Collins, D.A., Durant, S.M., Frame, W.F., Bertram, B.C.R., Caro, T.M., 1998. Demography of Serengeti cheetah (Acinonys jubatus) population: the first 25 years. J. Zool. (Lond.) 244, 473–488. Kemp, G.A., Keith, L.B., 1970. Dynamics and regulation of red squirrel (Tamiasciurus hudsonicus) populations. Ecology 51, 763–779. Knight, R.R., Eberhardt, L.L., 1985. Population dynamics of Yellowstone grizzly bears. Ecology 66, 323–334. Kohn, B.E., Payne, N.F., Ashbrenner, J.E., Creed, W.A., 1993. The fisher in Wisconsin. Wisconsin Dept. Nat. Resour. Tech. Bull. 183, 1–24. Kovacs, G., 1983. Survival patterns in adult European hares. Acta Zool. Fenn. 174, 69–70. Lande, R., 1988. Demographic models of the northern spotted owl (Strix occidentalis caurina). Oecologia 75, 601–607. Landers, R.H., 1981. A life table and biomass estimate for Alaskan fur seals. Fisheries Res. (Amsterdam) 1, 55–70. Larson, J.S., 1967. Age structure and sexual maturity within a western Maryland beaver (Castor canadensis) population. J. Mammal. 48, 408–413. Laws, R.M., 1966. Age criteria for the African elephant, Loxodanta a. africana. East Afr. Wildl. J. 4, 1–37. Laws, R.M., 1969. Aspects of reproduction in the African elephants, Loxodonta africana. J. Repr. Fert. (Suppl.) 6, 193– 218. Le Boeuf, B.J., Reiter, J., 1988. Life time reproductive success in northern elephant seals. In: Clutton-Brock, T.H. (Ed.), Reproductive Success. University of Chicago Press, Chicago, pp. 344–362. Leader-Williams, N., 1988. Reindeer on South Georgia. Cambridge University Press, New York. Leslie, P.H., 1945. On the use of matrices in certain population mathematics. Biometrica 33, 183–212. Leslie, P.H., 1948. Some further notes on the use of matrices in population mathematics. Biometrica 35, 213–245. Levin, L., Caswell, H., Bridges, T., DiBacco, C., Cabrera, D., Plaia, G., 1996. Demographic responses of estuarine polychaetes to pollutants: life table response experiments. Ecol. Appl. 6, 1295– 1313. Lewontin, R.C., 1965. Selection for colonizing ability. In: Baker, H.G., Stebins, G.L. (Eds.), The Genetics of Colonizing Species. Academic Press, New York, pp. 79–94.
323
Li, J., Wang, Q., Li, M., 1995. Studies on the population ecology of Tibetan monkeys (Macaca thibetana). III. Age structure and life table of Tibetan monkeys (Chinese). Acta Theriol. Sin. 15, 31–35. Lima, M., Páez, E., 1997. Demography and population dynamics of South American fur seal. J. Mammal. 78, 914–920. Lodal, J., Grue, H., 1985. Age determination and age distribution in populations of moles (Talpa europea) in Denmark. Acta Zool. Fenn. 173, 279–281. Lord, R.D.J., 1961a. Magnitudes of reproduction in cottontail rabbits. J. Wildl. Manage. 25, 28–33. Lord, R.D.J., 1961b. Mortality rates of cottontail rabbits. J. Wildl. Manage. 25, 33–40. Lowe, V.P.W., 1969. Population dynamics of the red deer (Cervus elaphus L.) on Rhum. J. Anim. Ecol. 38, 425–458. Luo, J., Fox, B.J., 1990. Life table comparison between two ground squirrels. J. Mammal. 71, 364–370. Macpherson, A.H., 1969. The dynamics of Canadian arctic fox populations. Can. Dept. Environ. Fisheries Marine Services Tech. Rep. 8, 1–48. Mansfield, A.W., Beck, B., 1977. The grey seal in eastern Canada. Can. Dept. Environ. Fisheries Marine Services Tech. Rep. 704, 1–81. McLean, P.K., Pelton, M.R., 1994. Estimates of population density and growth of black bears in the Smoky mountains. Int. Conf. Bear Res. Manage. 9, 254–261. Medin, D.E., Anderson, A.E., 1979. Modeling the dynamics of a Colorado mule deer population. Wildl. Monogr. 68, 1–77. Mertens, H., 1985. Structures de population et tables de survie de buffles, topis et cobs de buffon au parc national des Virunga. Zaire. Rev. Ecol. Terre Vie 40, 33–51. Messier, F., Huot, J., Le Henaff, D., Luttich, S., 1988. Demography of the George River Caribou herd: evidence of population regulation by forage exploitation and range expansion. Arctic 41, 279–287. Michod, R.E., Anderson, W.W., 1980. On calculating demographic parameters from age frequency data. Ecology 61, 265–269. Millar, J.S., Zwickel, F.C., 1972. Determination of age, age structure, and mortality of the pika, Ochotona princeps Richardson. Can. J. Zool. 50, 229–232. Monakhov, V.G., 1983. Age structure of the sable (Martes zibellina) populations (Russian). Z. Zhurnal 62, 1398–1406. Nelson, B.B., Chapman, J.A., 1982. Age determination and population characteristics of the wolf (Canis lupus labradorius Goldman) in northern Quebec and Labrador. Z. Säugetierkunde 47, 296–311. Oli, M.K., 2003. Partial life-cycle analysis: a simplified model for post-breeding census data. Ecol. Model., in press. Oli, M.K., Dobson, F.S., 1999. Population cycles in small mammals: the role of age at sexual maturity. Oikos 86, 557– 565. Oli, M.K., Dobson, F.S., 2001. Population cycles in small mammals: the ␣-hypothesis. J. Mammal. 82, 573–581. Oli, M.K., Dobson, F.S., 2003. The relative importance of life-history variables to population growth rate: Cole’s prediction revisited. Am. Nat. 161, 422–440. Oli, M.K., Slade, N.A., Dobson, F.S., 2001. Effect of density reduction on Uinta ground squirrels: an analysis of life table response experiments. Ecology 82, 1921–1929.
324
M.K. Oli / Ecological Modelling 169 (2003) 313–325
Oli, M.K., Hepp, G.R., Kennamer, R.A., 2002. Fitness consequences of delayed maturity in female wood ducks. Evol. Ecol. Res. 4, 563–576. Oli, M.K., Zinner, B., 2001a. Partial life cycle analysis: a model for birth-pulse populations. Ecology 82, 1180–1190. Oli, M.K., Zinner, B., 2001b. Partial life cycle analysis: a model for pre-breeding census data. Oikos 93, 376–387. Packer, C., Herbst, L., Pusey, A.E., Bygott, J.D., Cairns, S.J., Mulder, M.B., 1988. Lifetime reproductive success in lions. In: Clutton-Brock, T.H. (Ed.), Reproductive Success: Studies of Individual Variation in Contrasting Breeding Systems. University of Chicago Press, Chicago, pp. 363–383. Packer, C., Tatar, M., Collins, A., 1998. Reproductive cessation in female mammals. Nature 392, 807–811. Parker, G.R., Luttich, S., 1986. Characteristics of the wolf (Canis lupus labradorius) in northern Quebec and Labrador. Arctic 39, 145–149. Payne, N.F., 1984. Mortality rates of beaver in Newfoundland. J. Wildl. Manage. 48, 117–126. Perryman, P., Muchlinski, A., 1987. Population dynamics of feral burros at the Naval Weapons Center, China Lake, California. J. Mammal. 68, 435–438. Pielowski, Z., 1984. Some aspects of population structure and longevity of field roe deer. Acta Theriol. 29, 17–33. Price, M.V., Kelly, P.A., 1994. An age-structured demographic model for the endangered Stephen’s Kangaroo rat. Conserv. Biol. 8, 810–821. Purvis, A., Harvey, P.H., 1995. Mammal life history evolution: a comparative test of Charnov’s model. J. Zool. (Lond.) 237, 259–283. Reijnders, P.J.H., 1978. Recruitment in the harbour seal (Phoca vitulina) population in the Dutch Wadden Sea. Netherland J. Sea Res. 12, 164–179. Richard, A., 1985. Primates in Nature. W.H. Freeman, New York. Robinson, J.G., O’Brien, T.G., 1991. Adjustment in birth sex ratio in wedge-capped capuchin monkeys. Am. Nat. 138, 1173–1186. Rodgers, W.A., 1984. Warthog ecology in south east Tanzania. Mammalia 48, 327–350. Rogers, L.M., Cheeseman, C.L., Mallinson, P.J., Clifton-Hadley, R., 1997. The demography of a high density badger (Meles meles) population in the west of England. J. Zool. (Lond.) 242, 705–728. Rolley, R.E., 1985. Dynamics of a harvested bobcat population in Oklahoma. J. Wildl. Manage. 49, 283–292. Sæther, B.E., Bakke, Ø., 2000. Avian life history variation and contribution of demographic traits to the population growth rate. Ecology 81, 642–653. Saltz, D., Rubenstein, D.I., 1995. Population dynamics of a reintroduced asiatic wild ass (Equus hemionus) herd. Ecol. Appl. 5, 327–335. Schwartz, O.A., Armitage, K.B., Van Vuren, D., 1998. A 32-year demography of yellow-bellied marmots (Marmota flaviventris). J. Zool. (Lond.) 246, 337–346. Sherman, P.W., Morton, M.L., 1984. Demography of Belding’s ground squirrels. Ecology 65, 1617–1628. Silvertown, J., Franco, M., Pisanty, I., Mendonza, I., 1993. Comparative plant demography—relative importance of life-cycle components to finite rate of increase in woody and herbaceous perennials. J. Ecol. 81, 465–476.
Simmons, N.M., Bayer, M.B., Sinkey, L.O., 1984. Demography of Dall’s sheep in the Makenzie Mountains, Northwest Territories. J. Wildl. Manage. 48, 156–162. Sinclair, A.R.E., 1977. The African Buffalo. A Study in Resource Limitation of Populations. University of Chicago Press, Chicago. Slade, N.A., Balph, D.F., 1974. Population ecology of Uinta ground squirrels. Ecology 55, 989–1003. Slade, N.A., Gomulkiewicz, R., Alexander, H.M., 1998. Alternatives to Robinson and Redford’s method of assessing overharvest from incomplete demographic data. Conserv. Biol. 12, 1–8. Smith, A.T., 1974. The distribution and dispersal of pikas: consequences of insular population structure. Ecology 55, 1112– 1119. Smith, G.W., Johnson, D.R., 1985. Demography of a Townsend ground squirrel population in southwestern Idaho. Ecology 66, 171–178. Spinage, C.A., 1970. Population dynamics of the Uganda defassa waterbuck (Kobus defassa ugandae Neumann) in Queen Elizabeth Park. J. Anim. Ecol. 39, 51–78. Spinage, C.A., 1972. African ungulate life tables. Ecology 53, 645–652. Stearns, S.C., 1992. The Evolution of Life Histories. Oxford University Press, Oxford. Stephenson, A.B., 1977. Age determination and morphological variation of Ontario otters. Can. J. Zool. 55, 1577–1583. Thompson, M.J.A., 1987. Longevity and survival of female pipistrelle bats (Pipistrellus pipistrellus) on the Vale of York. Engl. J. Zool. (Lond.) 211, 209–214. Tryon, C.A., Snyder, D.P., 1973. Biology of the eastern chipmunk, Tamias striatus: life tables, age distributions, and trends in population numbers. J. Mammal. 54, 145–168. van Aarde, R.J., 1987. Demography of a Cape porcupine, Hystrix africaneaustralialis, population. J. Zool. (Lond.) 213, 205–212. van Sickle, J., Attwell, C.A.M., Craig, G.C., 1987. Estimating population growth rate from an age distribution of natural death. J. Wildl. Manage. 51, 941–948. Warner, R.E., 1985. Demography and movements of free ranging domestic cats in rural Illinois. J. Wildl. Manage. 49, 340–346. Waser, P.M., Jones, W.T., 1991. Survival and reproductive effort in banner-tailed kangaroo rats. Ecology 72, 771–777. Waser, P.M., Elliott, L.F., Creel, N.M., Creel, S.R., 1995. Habitat variation and mongoose demography. In: Sinclair, A.R.E., Arcese, P. (Eds.), Serengeti II. University of Chicago Press, Chicago, pp. 421–447. Watson, R.M., 1969. Reproduction in wildebeest, Connochaetes taurinus albojubatus Thomas, in the Serengeti region, and its significance to conservation. J. Reprod. Fert. Suppl. 6, 287–310. Weber, V.D., 1989. Zur Populationsbiologie schweizerischer Iltisse (Mustela putorius L.). Z. Jagdwissenschaften 35, 86–99. Wegge, P., 1979. Aspects of the population ecology of blue sheep in Nepal. J. Asian Ecol. 1, 10–20. Wei, F., Hu, J., Xu, G., Jiang, M., Deng, Q., Zhong, Z., 1989. A study on the life table of wild giant pandas (Chinese). Acta Theriol. Sin. 9, 81–86. Willner, G.R., Dixon, K.R., Chapman, J.A., 1983. Age determination and mortality of the nutria (Myocastor coypus) in Maryland. Z. Säugetierkunde 48, 19–34.
M.K. Oli / Ecological Modelling 169 (2003) 313–325 Woodgerd, W., 1964. Population dynamics of bighorn sheep on Wildhorse Island. J. Wildl. Manage. 28, 381–391. Yang, Q., Hu, J., Peng, J., 1990. A study on the population ecology of forest musk deer (Muschus berezovskii) at the north of Hengduan mountains (Chinese). Acta Theriol. Sin. 10, 255– 262.
325
Yozdis, P., Kolenosky, G.B., 1986. A population dynamics model of black bears in eastcentral Ontario. J. Wildl. Manage. 50, 602–612. Zammuto, R.M., 1987. Life histories of mammals: analyses among and within Spermophilus columbianus life tables. Ecology 68, 1351–1363.