Population Momentum Across Vertebrate Life Histories

ecological modelling 197 (2006) 418–430

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Population momentum across vertebrate life histories

David N. Koons a,∗, James B. Grand b, Jennifer M. Arnold a,1 a Alabama Cooperative Fish and Wildlife Research Unit, School of Forestry and Wildlife Sciences, Auburn University, AL 36849, United States b USGS, Alabama Cooperative Fish and Wildlife Research Unit, School of Forestry and Wildlife Sciences, Auburn University, AL 36849, United States article info abstract

Article history: Population abundance is critically important in conservation, management, and demo- Received 14 November 2005 graphic theory. Thus, to better understand how perturbations to the life history affect Received in revised form 22 long-term population size, we examined population momentum for four vertebrate classes February 2006 with different life history strategies. In a series of demographic experiments we show that Accepted 28 March 2006 population momentum generally has a larger effect on long-term population size for organ- Published on line 3 May 2006 isms with long generation times than for organisms with short generation times. However, patterns between population momentum and generation time varied across taxonomic Keywords: groups and according to the life history parameter that was changed. Our ﬁndings indi- Age structure cate that momentum may be an especially important aspect of population dynamics for Iteroparity long-lived vertebrates, and deserves greater attention in life history studies. Further, we Life history discuss the importance of population momentum in natural resource management, pest Population momentum control, and conservation arenas. Semelparity © 2006 Elsevier B.V. All rights reserved.

1. Introduction al., 1992, 1994; United Nations, 2003); however, the relevant tools are rarely used in population ecology (Koons et al., 2006), Population size and growth rate are central aspects of biol- and theory describing the behavior of population momentum ogy that are commonly estimated with tools that assume across species is lacking. stability of population structure (i.e., age, stage, or size struc- Population momentum occurs in structured populations ture) through time (e.g., see papers within Heppell et al., because there is often a time-lag between a change in the vital 2000; Sibly et al., 2002). Ecologists, however, realize that this rate (i.e., a life history parameter such as survival, fertility, etc.) assumption may rarely be met in nature (Bierzychudek, 1999; and the actual observed impact on the population. For exam- Fox and Gurevitch, 2000; Clutton-Brock and Coulson, 2002; ple, imagine a population with high per capita fertility rates, Nichols and Hines, 2002; Hastings, 2004). An unstable pop- where offspring have a high likelihood of surviving to matu- ulation structure can have a strong inertial effect on future rity. If fertility suddenly dropped to the stationary level (i.e., population size, which is known as population momentum 1 = 1, the level of lifetime individual replacement), a popu- (Keyﬁtz, 1971). In human demography, studies of population lation would keep growing because overabundance of young momentum have inﬂuenced international policy (e.g., Bos et individuals would ensure high net fertility rates long after

∗ Corresponding author at: Max Planck Institute, Konrad-Zuse Str. 1, D-18057 Rostock, Germany. Tel.: +49 381 2081 226; fax: +49 381 2081 526. E-mail addresses: [email protected] (D.N. Koons), [email protected] (J.B. Grand), [email protected] (J.M. Arnold). 1 Current address: USGS Patuxent Wildlife Research Center, 12100 Beech Forest Road, Laurel, MD 20708, United States. Tel.: +1 301 497 5720; fax: +1 301 497 5545. 0304-3800/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2006.03.034 ecological modelling 197 (2006) 418–430 419

the transition to stationary per capita fertility (sensu Keyfitz, Table 1). Species within each vertebrate class were selected 1971). In structured populations, changes in any vital rate such that population momentum could be examined across will alter population structure, causing transient dynamics life histories exhibiting a large range in age at maturity (˛), a (i.e., unstable short-term dynamics), and ultimately, popula- close surrogate to generation time (Cole, 1954; Lewontin, 1965). tion momentum (Caswell, 2001). Activities that directly alter Because first-year and sub-adult survival are poorly studied population structure, like commercial fishing (Hall, 1999), will for many vertebrate organisms, we had to impose assump- do the same. Understanding population momentum and its tions about these vital rates to generate a complete life cycle effect on population dynamics is important to increase our for selected species (see Table 1). understanding of life history evolution and the efficacy of We parameterized the life history variables for each species conservation, natural resource management, and pest control into a partial life-cycle projection matrix using a 1-yr time practices. step (shown below, A is an iteroparous and B a semelparous Recent studies across vertebrates have found that asymp- life-cycle matrix), which can accommodate reproductive delay totic population dynamics (e.g., sensitivities and elasticities) (Oli and Zinner, 2001), but do not capture the potential impor- are directly related to values of individual vital rates (Heppell, tance of seasonal variation in reproductive effort. We assumed 1998; Heppell et al., 2000b; Sæther and Bakke, 2000), or ratios birth-pulse reproduction and used a pre-birth census, where of vital rates (Oli and Dobson, 2003). To truly understand pop- fertility (F) equalled the product of fecundity (m, the number of ulation dynamics, however, it is necessary to look at the entire female offspring produced per female each year) and age-class life history, not just selected vital rates. Generation time, mea- 1 survival (F = m × P1)(Caswell, 2001), Psa equalled sub-adult sured as the mean age of offspring production, is a function survival, and Pa equalled adult survival: of all demographic vital rates. The inverse of generation time ⎡ ⎤ ⎡ ⎤ ··· F ··· F measures population turnover, and is expected to provide a 0 0 0 0 ⎢ P ··· ⎥ ⎢ P ··· ⎥ reliable measure of a given life history along the “slow-fast” ⎢ sa 00⎥ ⎢ sa 00⎥ A = , B = . ⎣ .. ⎦ ⎣ .. ⎦ continuum (Gaillard et al., 2005) where slow species are typi- 0 . 00 0 . 00 fied by late maturation, high annual survival, and low repro- 0 ··· Psa Pa 0 ··· Psa 0 ductive rates, whereas fast species have low annual survival, early maturation, and high reproductive rates (sensu Gaillard Partial life-cycle matrices are amenable to comparative life et al., 1989; Charnov, 1993). Not surprisingly, the rate at which history studies, and have been shown to accurately portray population structure approaches the stable state (e.g., stable the dynamics of many age-structured vertebrate populations age distribution) is related to generation time (Tuljapurkar, (Heppell et al., 2000b; Sæther and Bakke, 2000; Oli and Dobson, 1986). In addition, Koons et al. (2005) have recently shown that 2003; Oli, 2003). transient dynamics of vertebrates with long generation times To develop our demographic experiments, we created are more reactive to changes in population structure than growing (1 = 1.1) populations by numerically changing fertil- they are for vertebrates with short generation times. Thus, ity, which often causes fast population growth when at high population momentum should be related to generation time levels in nature. Because quickly declining populations are as well; however, it is possible for large transient increases often caused by suppressed survival probabilities (Ricklefs, and decreases in population size to effectively cancel each 1990), we created declining (1 = 0.9) populations by numeri- other out over time and have little effect on long-term pop- cally changing adult survival for iteroparous species, and sub- ulation size. Interspecific studies of population momentum adult survival for semelparous species. For simplicity, we lim- have not been conducted to elucidate these possibilities. Here, ited our study to these special cases of growing and declining we use extensions of Keyfitz’s formula (1971) and simulation populations, but acknowledge that fertility or survival could to calculate and examine population momentum across both cause populations to change in either direction. semelparous and iteroparous vertebrate life history strategies. We predict that the magnitude of population momentum will 2.2. Demographic experiments increase with generation time, and discuss some of the under- lying demographic mechanisms that may be responsible for Here, we consider scenarios where populations are growing variation in population momentum across life histories. so rapidly that they could cause environmental damage and where populations are declining at a rate that could yield extinction (i.e., the growing and declining populations defined 2. Methods above). Our objective was to examine population size following changes in vital rates designed to halt the growth or decline. 2.1. Life history data Although vital rates are expected to vary over time in most environments, we purposefully ignored such stochastic varia- To reveal how life history and population momentum are tion to isolate and study the effects of historical age structure related, we explored life histories where reproduction can on long-term population size following isolated changes in occur several times within a lifetime (i.e., iteroparous), and a vital rate (i.e., population momentum). Furthermore, large, where reproduction occurs once and is followed by death (i.e., temporally isolated changes in environmental conditions do semelparous). We attained life history data from the peer- occur in nature and are of interest in ecology and evolution reviewed literature for 10 iteroparous species within each of (e.g., hurricanes, large floods, petroleum catastrophes, etc.). 4 vertebrate classes: Aves, Mammalia, Reptilia, Osteichthyes; In the demographic experiments, we first started with a and for 5 semelparous Osteichthyes species (all pacific salmon; growing population (1 = 1.1) in its stable population structure 420 ecological modelling 197 (2006) 418–430 Source Heppell et al. (2000b) Heppell et al. (2000b) Heppell et al. (2000b) Heppell et al. (2000b) Heppell et al. (2000b) Heppell et al. (2000b) Heppell et al. (2000b) Heppell et al. (2000b) Heppell et al. (2000b) Heppell et al. (2000b) Sæther and Bakke (2000) Shine and Charnov (1992) Shine and Charnov (1992) Shine and Charnov (1992) Shine and Charnov (1992) ) of the vertebrate 1 Heppell (1998) Heppell (1998) Heppell (1998) Heppell (1998) Heppell (1998) and and and and and Source and and and and and 1 Source Crawford (1980) and Weimerskirch et al. (1997) ), and generation time ( Dunham and Miles (1985) Dunham and Miles (1985) Dunham and Miles (1985) Dunham and Miles (1985) Heppell (1998) Wilbur and Morin (1994) Wilbur and Morin (1994) Wilbur and Morin (1994) Wilbur and Morin (1994) Wilbur and Morin (1994) , a Sæther and Bakke (2000) P a P and Burton (1959) , 1 Weimerskirch (1992) Cooch et al. (2001) Kiel (1955) Sæther and Bakke (2000) Russell (1999) Sæther and Bakke (2000) Russell (1999) Sæther and Bakke (2000) Sæther and Bakke (2000) Sæther and Bakke (2000) sa P a P 0.700.50 5.33 3.00 1 ), adult survival probability ( sa P sa P a a a P ˛ 0.950.94 31.00 0.940.80 23.67 0.84 20.38 0.71 8.00 8.25 4.45 ˛ 3 2 1.51 – – 0.20 0.24 1.42 1.32 sa P a a a a a a b b b b F 0.95 1 – 0.20 1.25 1.09 2 0.44 0.52 3.08 0.23 2 0.96 0.79 5.76 0.18 3 0.75 0.90 12.00 0.15 4 0.90 0.91 14.11 0.24 5 0.87 0.86 11.14 0.09 8 0.98 0.95 27.00 0.09 10 0.95 0.94 25.67 0.08 14 0.92 0.95 33.00 0.07 14 0.98 0.94 29.67 F ˛ 1.00 1.33 0.76 0.61 0.130.25 7 4 0.76 0.67 0.81 0.88 10.38 11.06 0.40 8 0.77 0.96 32.00 1.88 14 0.81 0.94 29.95 1.04 19 0.75 0.97 47.41 51.64 22 0.72 0.81 40.75 ), sub-adult survival probability ( ˛ F 0.320.68 2 1 – 0.40 1.67 0.210.05 100.440.08 6 0.32 0.86 4 3 0.88 2 0.93 1.5 0.83 23.71 – 0.83 0.43 6.92 2.06 0.16 8 0.10 12 Thamnophis sirtalis Lacerta vivipara Takydromus takydromoides Agkistrodon contortrix Trachemys scripta Kinosternum subrubrum Chrysemys picta Gopherus agassizi Chelydra sperpentina Caretta caretta Lepus americanus Myocastor coypus Rangifer tarandus Calorhinus ursinus Trichechus manatus Ursus americanus Gorilla gorilla Hippopotamus amphibius Pan troglodytes Loxodonta africana ), modal age at maturity ( F Falco naumanni Parus caeruleus Diomedea exulans Aptenodytes patagonica Stercorarius parasiticus Strix occidentalis Chen caerulescens Fulica Americana Fulmarus glacialis Phoebetria fusca fertility rate ( Snowshoe Hare Coypu Reindeer Northern Fur Seal Manatee Common Gartersnake Viviparous Lizard Black Bear Lesser Kestrel Blue Tit Wandering Albatross King Penguin Parasitic Jaeger Northern Spotted Owl Lesser Snow Goose American Coot Northern Fulmar Gorilla Japanese Grass Lizard Hippopotamus species used in our demographic experiments (see Section 2 for calculation of fertility and generation time) Chimpanzee Broad-Banded Copperhead Slider Turtle Common Mud Turtle Painted Turtle Desert Tortoise Snapping Turtle Common nameSooty Albatross Species, Aves Common nameAfrican Elephant Species, Mammalia Common nameLoggerhead Sea Turtle Species, Reptilia Table1–The ecological modelling 197 (2006) 418–430 421 ). Quinn (2005) Quinn (2005) Quinn (2005) Quinn (2005) and and and and = 1. Estimated values 1 Source Koons, 2005 ( Heppell et al. (1999) King and McFarlane (2003) King and McFarlane (2003) King and McFarlane (2003) King and McFarlane (2003) King and McFarlane (2003) Frisk et al. (2001) King and McFarlane (2003) King and McFarlane (2003) Source b and and and and and and and and and and a Frisk et al. (2001) and al if sub-adult survival estimates were not Groot and Margolis (1991) Groot and Margolis (1991) Groot and Margolis (1991) Groot and Margolis (1991) Wilson (2003) Gunderson (1997) Gunderson (1997) Roff (1984) Gunderson (1997) Gunderson (1997) Gunderson (1997) Gunderson (1997) Gunderson (1997) Cohen et al. (1983) Gunderson (1997) ory (for original source of data please see the cited 1 1 a P a P 0.910.960.820.83 33.62 0.95 44.50 0.87 19.52 0.82 16.78 0.69 30.50 0.60 16.65 0.37 10.52 7.23 5.50 3.58 sa P sa a a a a a a a a a a P ˛ ). Simulation has shown that our results are robust to assumptions ˛ 6 5 4 3 23 20 15 12 11 10 Stearns, 1992 F b b b b b b b b b b 22.67 2 0.12 – 2.00 25.90 3 0.32 – 3.00 85.37 4 0.31 – 4.00 38.37 5 0.60 – 5.00 15.84 5 0.48 – 5.00 F 4.67 0.08 2.98 0.08 0.46 0.49 1.36 3.81 1.40 0.62 , if estimates of first-year survival were not available for noted species, we numerically calculated the value that gives (Iteroparous) (Semelparous) Osteichthyes Osteichthyes Heppell et al. (2000b) adult survival, which agrees with life history theory ( and Ammodytes hexapterus Sebastes aleutianus Hippoglossoides platessoides Sebastes alutus Sebastes flavidus Melanogrammus aegelfinus Gadus macrocephalus Morone saxatilis Hippoglossus stenolepis Squalas acanthias Oncorhynchus gorbuscha Oncorhynchus kisutch Oncorhynchus keta Oncorhynchus nerka Oncorhynchus tshawytscha ) Heppell (1998) Continued of first-year survival were available for the noted species. Because first-year and sub-adult survival are poorly studied for manyAccording vertebrate to organisms, we assumed sub-adult survival to be equal to adult surviv Sand Lance Rougheye Rockfish Long Rough Dab Pacific Ocean Perch Yellowtail Rockfish Haddock Pacific Cod Striped Bass Pink Salmon Pacific Halibut Coho Salmon Chum Salmon Sockeye Salmon Common nameDogfish Species, Common nameChinook Salmon Species, references). Here, life history parameters are rounded to the second decimal. Table 1 ( In most cases, life history parameters were attained directly,a or calculated from data provided within interspecific studies ofb vertebrate life hist 422 ecological modelling 197 (2006) 418–430

as the initial condition and then instantaneously decremented point. However, semelparous organisms have positive fertility fertility by the necessary amount to attain stationary growth in only one age class; therefore, projection matrices for semel-

(1 = 1) (i.e., an instantaneous transition; sensu Keyﬁtz, 1971). parous life histories are imprimitive and have co-dominant A gradual transition in fertility to the replacement level can eigenvalues. Thus, unless the population structure begins in substantially increase population momentum (Schoen and the stable state, the asymptotic population size and (st)age Kim, 1998; Li and Tuljapurkar, 1999). Yet, unlike in human distribution of semelparous organisms are cyclic with a period populations where it can take decades for fertility rates to equal to the number of eigenvalues that share the largest mag- approach the replacement level (Li and Tuljapurkar, 1999), nitude (d)(Caswell, 2001). Still, a running average of the (st)age wild vertebrate populations can experience relatively rapid distribution over d converges to w1 and grows at the rate 1 changes in environmental conditions and vital rates, either (Cull and Vogt, 1973). For semelparous life histories, we always naturally (Young, 1993) or via anthropogenic factors (e.g., Piatt measured population momentum with the limit of these run- and Lensink, 1989). Furthermore, management and conserva- ning average values. tion objectives are often centered on near-term outcomes (Fox and Gurevitch, 2000). Accordingly, we conducted a second set 2.2.2. Measuring the impacts of gradual change in a vital of experiments where we decremented fertility in a monotonic rate on population momentum fashion over 5 years by the amounts necessary to eventually To compute population momentum for gradual changes in attain stationary growth (i.e., we reduced fertility linearly from vital rates, we calculated the numerator of Eq. (1) with a the initial level to the ﬁnal level over 5 years). Markov chain of gradually changing vital rates (in the follow- In the second group of experiments, we started with a ing equations, A can also be replaced with B): declining population (1 = 0.9) in its stable population structure as the initial condition and then instantaneously augmented nt = At−1At−2···A1A0n0. (3) adult survival (sub-adult survival for semelparous species) by the amount necessary to attain stationary growth. We then Thus, a numerical calculation of population momentum for conducted a similar experiment, but augmented adult survival gradual change in any vital rate to the stationary level can be (sub-adult survival for semelparous species) in a monotonic written as fashion over 5 years by the amounts necessary to eventually attain stationary growth. || ··· || M = At−1At−2 A1A0n0 1 grad lim (4) t→∞ ||n0||1 2.2.1. Calculation of population momentum

Population momentum (M) is always calculated according to where At represents the time-speciﬁc projection matrix of Keyﬁtz (1971): vital rates (e.g., Schoen and Kim, 1998; Li and Tuljapurkar, 1999, 2000; Goldstein, 2002). All measures of population momentum ||nt|| M = lim 1 (1) are centered on 1. Values of momentum above 1 indicate that t→∞ ||n || 0 1 the population will grow to a larger ultimate size following || || = |n | change in a vital rate to the stationary level, and values below where n 1 i i is the 1-norm, which gives total popula- 1 indicate that the population will decline to a smaller ulti- tion size. This is simply the ratio of the ultimate population mate size. size following a transition to the stationary level, to the size When a vital rate gradually changes to the stationary immediately before the transition. For discrete-time models, level, two factors cause population momentum. First, gradu- Eq. (1) can be approximated with ally changing vital rates will contribute to population growth

(new) (initial) (new) or decline regardless of population structure. Second, the eT(v × w )w M = 1 1 1 inst (2) actual population structure that acts on each new set of T × (initial) e w1 vital rates is not stable, which produces inertia in population size (Tuljapurkar and Lee, 1997; Schoen and Jonsson, 2003). when changes in vital rates are instantaneous, denoted by the This measurement of net population growth following gradual subscriptinst on M (Caswell, 2001:104). Here, e is a vector of change in a vital rate is not directly comparable to that follow- ones, w1 is the dominant right eigenvector of the projection ing an instantaneous change, in which only the second factor matrix (A or B) that describes the stable population structure causes momentum (Keyﬁtz, 1971). To explicitly understand || || = (scaled such that w1 2 1), v1is the dominant left eigenvec- the two components contributing to population momentum tor of the projection matrix that describes reproductive value, following gradual change in a vital rate to the stationary level, ‘initial’ refers to the projection matrix for the initial condi- we projected a hypothetical stable population (SP) for the set tions (growing or declining population), and ‘new’ refers to the of changing vital rates using another Markov chain: projection matrix following change in fertility or survival probability. The left and right eigenvectors of a projection matrix SP =||(At−1ut−1||At−2ut−2||1· · ·||A1u1||1||A0u0|| 1)||1 (5) satisfy vi, wi=1 and vi, wj=0 for i=j where is the scalar product. where ut = w ,t/||w ,t|| is the stable (st)age distribution at Because iteroparous organisms can reproduce in succes- 1 1 1 time t. Eq. (5) can be written as sive time steps, projection matrices for iteroparous life histories are primitive, meaning that there exists one dominant = ··· eigenvalue and the asymptotic dynamics approach a stable SP 1,t−1 1,t−2 1,1 1,0 (6) ecological modelling 197 (2006) 418–430 423

where 1,t is the dominant real eigenvalue of At. Placed on tion (1 = 1.1) to actually decrease in abundance before reach- the same number scale as population momentum (see above), ing the stationary level (Minst <1; Fig. 1, solid circles; see SP simply describes the ultimate population size that would Table 1 for generation times of speciﬁc species). Further- be expected if population structure were ignored or some- more, population momentum declined linearly with gener- how remained stable throughout time. Although the latter will ation time (SIC ≥ 3.04 units better than null model) among 2 2 rarely be true, one can compare Mgrad to SP in order to better the iteroparous Aves (R = 0.59; Fig. 1a), Mammalia (R = 0.41; understand how historical population structure contributes Fig. 1b), and Reptilia (R2 = 0.98; Fig. 1c) species and for the 2 to Mgrad.IfMgrad = SP, then population structure has no effect semelparous paciﬁc salmon (R = 0.81; Fig. 1e). Among the on the ultimate population size. However, if Mgrad < SP, this iteroparous Osteichthyes, population momentum decreased implies that population structure limits the population from until a generation time of approximately 30 years but then becoming as large as it could, given the vital rates. Conversely, levelled out (quadratic model 3.09 SIC units better than null 2 if Mgrad > SP, this implies that population structure causes a model; R = 0.54; Fig. 1d). Thus, the magnitude of population larger ultimate population size than would be expected from momentum |M − 1| increased with generation time, lending the vital rates alone. Demographic experiments and calcula- support to our hypothesis (Fig. 1, solid circles). tions were conducted in MATLAB (The MathWorks, 2005). Population momentum following gradual changes in fertility (i.e., 5-year transitions) declined with generation time 2.3. Statistical analysis as well. However, population momentum following gradual change was generally >1, indicating that momentum usually To test our prediction that the magnitude of population caused net population growth (Fig. 1, open circles). Yet, in all momentum would increase with generation time, we used cases the actual population structure that experienced the 5- Proc REG (SAS Institute, Inc. 2000) to examine null, linear, year transition in fertility restricted populations from growing and quadratic models of the relationship between popula- as large as they could have given the surplus levels of fertility tion momentum and generation time. Generation time (1) (Mgrad < SP), especially amongst vertebrates with long genera- was measured from the stationary populations as the mean tion times (Fig. 1, comparison of open circles to plus signs). In age at which members of a cohort (of newborns) produce off- fact, the effect of population structure was so strong in Rep- spring (Coale, 1972; Caswell, 2001). Although consideration of tilia and Osteichthyes species with long generation times that body size and phylogenetic relatedness would be of interest it actually caused a net population decline (Mgrad <1; Fig. 1c in an evolutionary analysis, statistically controlling for these and d). variables in the models would only obscure the current demo- After instantaneously augmenting adult survival probabil- graphic relationship between population momentum and life ity (sub-adult survival for semelparous species) in the declin- history (Price, 1997; Kelly and Price, 2004), which is the focus ing population experiments (1 = 0.9), the magnitude of popu- here. lation momentum was small, causing slight decline in abun- We used Schwarz’s Information Criterion (SIC; Schwarz, dance before ultimately reaching stationary growth for most

1978) to evaluate the amount of support in our data for each vertebrates that we studied (Minst <1;Fig. 2, solid circles). How- model in our candidate list (see above) because it has been ever, population structure of the declining loggerhead sea tur- shown to more accurately estimate the order of the underly- tle (Caretta caretta), snapping turtle (Chelydra sperpentina), dog- ing model than Akaike’s Information Criterion (Akaike, 1973; fish (Squalas acanthias), long rough dab (Hippoglossoides plates- Stone, 1979; Hooten, 1995). We considered the best approxi- soides), and pacific halibut (Hippoglossus stenolepis) populations mating model to be that with the lowest SIC value. If the differ- actually reversed the direction of population growth before ence (SIC) between information criteria values of competing eventually reaching the stationary level (Minst >1; Fig. 2c–d, models is ≤2, then the models are statistically indistinguish- solid circles; see Table 1 for generation times of specific able (Sakamoto et al., 1986). Thus, models with generation species). Population momentum declined linearly with gener- time as a predictor variable had to beat the null model by >2 ation time (SIC ≥ 2.70 units better than null model) among SIC units to provide support for a relationship between popu- the iteroparous Aves species (R2 = 0.41; Fig. 2a), increased lation momentum and generation time. slightly across semelparous pacific salmon species (R2 = 0.58; Fig. 2e), but was not related to generation time in the other vertebrate classes (no model > 2 SIC units better than null 3. Results model). Population momentum following gradual changes in sur- Our results allow for interspecific comparisons of popula- vival displayed similar patterns as that following instanta- tion momentum for iteroparous and semelparous vertebrates. neous change. However, with the exception of loggerhead For initially increasing and decreasing populations, popula- sea turtle, gradual changes in survival resulted in substan- tion momentum following instantaneous (Minst) and grad- tial declines in abundance before reaching the stationary level ual (Mgrad) change in a vital rate exhibited similar patterns (Mgrad 1; Fig. 2, open circles). Nevertheless, most of this across generation time within taxa. However, the relationship momentum was caused by the deficient survival probabilities between population momentum and generation time differed rather than population structure (Mgrad similar to SP). Thus, somewhat among the four vertebrate classes that we exam- population structure generally had a smaller effect on popu- ined. lation momentum following both instantaneous and gradual Following instantaneous decrements of fertility, popula- changes in survival probabilities when compared to changes tion momentum caused the historically increasing popula- in fertility. 424 ecological modelling 197 (2006) 418–430

Fig. 1 – Plots of population momentum for the growing population experiments (1 = 1.1) where fertility was decremented instantaneously (solid circles and solid lines) or gradually (open circles and dashed lines) to the stationary level across life history generation time of the iteroparous (a) Aves, (b) Mammalia, (c) Reptilia, and (d) Osteichthyes vertebrate classes as well as the semelparous (e) paciﬁc salmon (Class Osteichthyes). The light dotted line is a reference line for population momentum = 1. For comparison with population momentum following a gradual transition in fertility, the + symbols represent the hypothetical stable populations (SP) for each life history, which simply describe the ultimate population size that would be expected if population structure were ignored or somehow remained stable throughout the gradual transition. See Table 1 for generation times of speciﬁc species.

Because the stable population structure created by the ulation vector of the growing populations relative to that for increasing population’s vital rates (1 = 1.1) is not immediately the stationary populations is <1). The decrease in fertility also stable to the experimentally introduced stationary level of shifted some of the reproductive value away from adults and fertility (1 = 1), it takes some time to reach the new stable pop- toward younger age classes, especially amongst life histories ulation structure. For example, the stable population structure with long generation times (see Fig. 3, open circles; the ‘ratio of of growing populations had a surplus of young immature indi- adult reproductive value’ for the increasing populations rela- viduals, and a deﬁcit of mature adults relative to the stable tive to that in the stationary populations is >1, indicating that population structure associated with stationary levels of fer- adult reproductive value was higher, and immature reproduc- tility (see Fig. 3, solid circles; ‘ratio of adults’ in the stable pop- tive value lower, before the transition to stationary levels of ecological modelling 197 (2006) 418–430 425

Fig. 2 – Plots of population momentum for the declining population experiments (1 = 0.9) where adult survival (sub-adult survival for pacific salmon) was augmented instantaneously (solid circles and solid lines) or gradually (open circles and dashed lines) to the stationary level across life history generation time of the iteroparous (a) Aves, (b) Mammalia, (c) Reptilia, and (d) Osteichthyes vertebrate classes as well as the semelparous (e) pacific salmon (Class Osteichthyes). Lack of a prediction line indicates lack of evidence for a relationship between population momentum and generation time based on SIC model values (see Section 2). The light dotted line is a reference line for population momentum = 1. For comparison with population momentum following a gradual transition in survival, the + symbols represent the hypothetical stable populations (SP) for each life history, which simply describe the ultimate population size that would be expected if population structure were ignored or somehow remained stable throughout the gradual transition. See Table 1 for generation times of specific species.

fertility). The combined differences in population structure neous change in fertility (Minst < 1), limited population growth and reproductive value before and after the experiment led during gradual changes (Mgrad < SP), and caused the strong to transient population dynamics with greater net mortality relationships between population momentum and life history and lesser net reproduction than would have occurred under generation time (Fig. 1). asymptotic conditions for each life history. This caused the Although less pronounced than in the aforementioned reversal in direction of population growth following instanta- experiment, population structures of declining populations 426 ecological modelling 197 (2006) 418–430

Fig. 3 – The ratio between number of adults in the stable population structure of initially increasing populations relative to number of adults in the eventually stable populations with stationary levels of fertility (solid circles; values < 1 indicate a deficit of mature adults, and implicitly, a surplus of immature individuals in the former relative to the latter, while the converse would be true for values > 1), and the ratio between adult reproductive value in increasing populations relative to that when fertility is at the stationary level (open circles; values > 1 indicate that when the initially increasing population undergoes a transition to stationary levels of fertility, the adult reproductive value decreases, and implicitly, net reproductive value of immature individuals increases; the converse is true for values < 1) across life history generation time of the iteroparous (a) Aves, (b) Mammalia, (c) Reptilia, and (d) Osteichthyes vertebrate classes as well as the semelparous (e) pacific salmon (Class Osteichthyes). (See Table 1 for generation times of specific species.)

(1 = 0.9) also had a deﬁcit of mature adults relative to the tal changes in survival produced smaller and more variable stable population structure associated with stationary (1 =1) differences between historical (declining population) and ulti- survival probabilities (Fig. 4, solid circles). However, unlike mate (stationary population) stable population structure and results following experimental change in fertility, the increase reproductive value across the life histories when compared to in survival shifted reproductive value towards adults (Fig. 4, experimental changes in fertility (Figs. 3 and 4), which is why open circles; see paciﬁc salmon for exception), which usually population momentum was not related to generation time in resulted in population momentum in the same direction as the declining population experiments for several vertebrate historical population growth (Fig. 2). Furthermore, experimen- classes. ecological modelling 197 (2006) 418–430 427

Fig. 4 – The ratio between number of adults in the stable population structure of declining populations relative to number of adults in the eventually stable populations with stationary-growth levels of adult survival (sub-adult survival for pacific salmon) (solid circles; values < 1 indicate a deficit of mature adults, and implicitly, a surplus of immature individuals in the former relative to the latter, while the converse would be true for values > 1), and the ratio between adult reproductive value in declining populations relative to that when survival probabilities are at the stationary level (open circles; values < 1 indicate that when the initially decreasing population undergoes a transition to stationary levels of survival, the adult reproductive value increases, and implicitly, net reproductive value of immature individuals decreases; the converse is true for values > 1) across life history generation time of the iteroparous (a) Aves, (b) Mammalia, (c) Reptilia, and (d) Osteichthyes vertebrate classes as well as the semelparous (e) pacific salmon (Class Osteichthyes). (See Table 1 for generation times of specific species.)

speciﬁc vital rate that was changed, the magnitude of that change, and overall life history strategy. We used life histo- 4. Discussion ries representative of a wide variety of vertebrate species to examine this latter ﬁnding in greater depth, and discovered Our understanding of population momentum is limited interesting relationships between life history and population because it has not been examined for most animals. Recently, momentum. Koons et al. (2006) examined three animal life histories and Li and Tuljapurkar (1999) discovered that population found that population momentum varied according to the momentum increased exponentially with the time over which 428 ecological modelling 197 (2006) 418–430

a gradual transition in fertility to the stationary level occurred, lation size (Fig. 1, Minst) or limited populations from becoming and was generally much larger than that following instan- as large as they could have (Fig. 1, comparison of Mgrad to SP). taneous change. Yet, momentum following gradual change Experimental augmentations of survival probability did not in a vital rate is caused by historical population structure as affect reproductive value and asymptotic population structure well as gradually changing, non-stationary vital rates, whereas as much as reductions in fertility did. Furthermore, change in momentum following instantaneous change in a vital rate is survival had an opposite effect on the reproductive value of only caused by historical population structure (Bongaarts and adults when compared to reductions in fertility (Figs. 3 and 4, Bulatao, 1999; Schoen and Jonsson, 2003). By comparing pop- open circles), which seemed to diminish the effects of histor- ulation momentum following gradual changes in a vital rate ical population structure and transient dynamics on popula-

(Mgrad) to the size of a hypothetical stable population (SP), we tion momentum for most vertebrates that we examined (Minst found that population structure actually had the same effect similar to 1, Mgrad similar to SP; Fig. 2). Still, a few Reptilia and on momentum following both gradual (Mgrad) and instanta- Osteichthyes species with long generation times did not fit this neous (Minst) change in a vital rate. Hence, population momen- generalization, indicating an imperfect relationship between tum following gradual change in a vital rate displayed the generation time and population momentum. same pattern across life history generation time as popula- Furthermore, patterns between population momentum tion momentum following instantaneous change. The differ- and generation time differed amongst the vertebrate classes ences between Mgrad and Minst were primarily caused by the because stable population structures and the allocation of set of non-stationary vital rates that populations experienced reproductive value across age classes are inherently different during gradual, monotonic change to the stationary level. It among the vertebrate classes. For example, the stable popula- remains to be seen whether this finding holds for transitions tion structure of fish species is heavily skewed towards young taking a longer time to complete and in stochastic environ- and the distribution of reproductive value is heavily skewed ments. toward adults. This may be why the relationship between pop- Notwithstanding, population momentum following ulation momentum and generation time was more complex changes in fertility always declined with generation time, across iteroparous Osteichthyes species than in the other ver- and following instantaneous changes, caused historically tebrate classes where these distributions are less skewed. Yet, increasing populations to decrease in size. As a result of to better understand the demographic connection between instantaneous changes in fertility, the magnitude of momen- population momentum and population structure, reproduc- tum |M − 1| increased with generation time, which lends tive value, and life history parameters, a method for measur- support to our original hypothesis. This relationship was ing the sensitivity of population momentum to equal unit or most pronounced in the Reptilia and Osteichthyes classes proportional changes in life history parameters will be needed. (Fig. 1c and d, solid circles). However, because the effect These tools are currently being developed. of population structure was to reduce population size, the Population size is important in ecology, conservation, magnitude of momentum decreased with generation time pest control, and harvest management. Thus, ecologists and across Aves, Mammalia, and pacific salmon species following resource managers should consider population momentum gradual changes to the stationary level. In fact, the effect in population projections that are used to make management of population structure was so strong in Reptilia and Oste- decisions or when quantifying the ecological causes of histori- ichthyes species with long generation times that it actually cal population dynamics (Caswell, 2001). Our results indicated caused a reduction in abundance, despite the surplus levels that population momentum will generally have the strongest of fertility during the gradual change (Mgrad <1;Fig. 1c and d). effect on population size of vertebrates with long generation Demographically, these patterns can be explained by popu- times, especially amongst Reptilia and Osteichthyes species, lation structure and reproductive value, which directly affect the latter of which are often harvested for commercial or sport population momentum (e.g., see Eq. (2)). Because the func- purposes. Recently, C. Hauser, E.G. Cooch, and J.-D. Lebreton tional contributions of fertility and survival to 1 (measured found that population momentum could limit the ability of with elasticities) vary with life history (Heppell et al., 2000b; managers to regulate populations with harvest techniques Sæther and Bakke, 2000; Oli and Dobson, 2003), larger change (unpublished data). Thus, we highly recommend implemen- in a vital rate was generally required to achieve the stationary tation of population momentum into harvest-management level for species with long generation times than for species models to better assess the effects of harvest on population with short generation times. Larger changes in a vital rate dynamics. can cause larger changes in stable population structure and The direction of population momentum (>1 or <1) will cause larger shifts in reproductive value (e.g., Figs. 3 and 4). be critically important to consider in pest control and con- As a result, a historical population structure that acts on the servation. Human demographers have always found popu- newly changed vital rates will effectively have either a large lation momentum to follow the direction of historical pop- surplus or deficit of breeding adults. This produces transient ulation growth (e.g., Fischer and Heilig, 1997). Yet, popula- population dynamics with greater or lesser net reproduction, tion momentum for a stage-structured population of Calathea recruitment to adulthood, or both, than would occur under ovandensis was in the opposite direction of historical growth asymptotic conditions. Following our experimental reductions (Caswell, 2001:106), and Koons et al. (2006) found similar in fertility, adult reproductive value was changed in such a results for pseudo age-structured animal populations. In way that the population structure of the increasing popula- Physics, momentum of an object that changes direction is tions (1 = 1.1) was not favorable to reproduction and produced known as an ‘impulse’ (Buckwalter and Riban, 1987). Similar to transient dynamics that led to a reduction in long-term popu- the aforementioned studies, impulses occurred in our study, ecological modelling 197 (2006) 418–430 429

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