© 2002 by CRC Press LLC CHAPTER 5 Population Models — Life History Steve Carroll Life-history models track characteristics of organisms as a function of age or stage. Stages can be defined by size, morphological state, or any classifying variable deemed to be demographically important. The most common characteristics that are tracked in such models are survival rates and fecundities, collectively referred to as vital rates. The typical endpoints for life-history models are: • Population abundance • Abundances of individual age or stage classes • Population growth rate or related parameters (e.g., sensitivity, elasticity) Life-history models are important for several reasons. In many populations, survival probabil- ities and fecundities vary among age or stage classes, and toxic chemicals may affect the various classes differently. The predictions made on the basis of life-history models generally differ from those made on the basis of simple population models in which individuals are assumed to be identical. Life-history models allow the exploration of age-specific or stage-specific management options. For the purposes of this review, matrix models were classified into four categories, with the last two representing software implementations of the first two (Table 5.1). These include: • Deterministic age- or stage-based matrix models (Caswell 2001) • Stochastic age- or stage-based matrix models (Caswell 2001) • RAMAS ® * Age, Stage, Metapop, or Ecotoxicology (Ferson 1993; Spencer and Ferson 1997b,c; Applied Biomathematics 2000) • ULM (unified life model) (Legendre and Clobert 1995) For an in-depth treatment of matrix population models, see Caswell (2001). Density dependence, or the lack thereof, is not explicit in this classification scheme. Thus, within a category, both density- dependent and density-independent models are included. DETERMINISTIC MATRIX MODELS (AGE OR STAGE BASED) These models assume that survival rate and fecundity are functions of the age class or stage to which an organism belongs. Age- and stage-structured models have been important tools in natural * RAMAS is a registered trademark of Applied Biomathematics. 1574CH05.fm Page 55 Tuesday, November 26, 2002 4:55 PM © 2002 by CRC Press LLC Table 5.1 Internet Web Site Resources for Life-History Models Model Name Description Reference Internet Web Site Deterministic age- or stage-based matrix models Life-history matrix models with fixed fecundity and survivorship parameters Caswell (2001) http://www.ramas.com/ramas1.htm; http://www.ento.vt.edu/~sharov/PopEcol/lec7/ leslie.html; http://www.ets.uidaho.edu/wlf448/Leslie1.htm Stochastic age- or stage-based matrix models Life-history matrix models incorporating random variability in fecundity and survivorship parameters Caswell (2001) http://www.ramas.com/ramas1.htm RAMAS Age, Stage, Metapop, or Ecotoxicology Software for life-history modeling of age- or stage-structured populations Ferson (1993); Spencer and Ferson (1997b,c); Applied Biomathematics (2000) http://www.ramas.com/ ULM (unified life model) Software for life-history modeling of age- or stage-structured populations Legendre and Clobert (1995) http://www.snv.jussieu.fr/Bio/ulm/ulm.html http://eco.wiz.uni-kassel. de/model_db/mdb/ulm.html 1574CH05.fm Page 56 Tuesday, November 26, 2002 4:55 PM © 2002 by CRC Press LLC resource management, especially for fish, since the 1950s (Barnthouse 1998). The simplest life- history model of this type is the Leslie population projection matrix (Leslie 1945): N(t) = LN(t – 1) where N(t) and N(t − 1) are vectors of the numbers of organisms in each age class (N 0 , . . . N k ). L = where s k = age-specific survivorship probability f k = average fecundity at age k Life stages, when not defined by age, are generally defined by size (height, weight, etc.) or morphological state. Individuals within a given age class or stage are assumed to have identical survival probabilities and fecundities. It is assumed that the vital rates do not vary as a result of random environmental fluctuations or demographic stochasticity. Depending on whether or not density dependence is included, the vital rates may vary (deterministically) as a function of population size. The vital rates, which comprise the population matrix, can be obtained either from laboratory experiments or from field observations. Realism — HIGH — These models reflect that vital rates generally change with age. In many cases, they incorporate all available demographic data because measures of variability in vital rates are often unavailable. Within this category of models, age- or stage-structured models with density dependence are more realistic than those without it. Relevance — HIGH — Potential endpoints include expected population size (age specific), lambda (the asymptotic population growth rate), or lambda-based measures such as sensitivity and elasticity. All of these endpoints are relevant to ecotoxicological assessments. Survivorship and fecundity parameters can be adjusted to reflect the effects of toxic chemicals observed in laboratory tests (e.g., Munns et al. 1997). Flexibility — HIGH — All parameters are species-specific. The number of age classes, as well as the vital rates, can vary among populations. Therefore, different population structures and life histories can be modeled. Treatment of Uncertainty — LOW — Neither environmental nor demographic stochasticity is incor- porated in the basic approach. However, generalizations are being developed to deal with environ- mental stochasticity (e.g., see below Stochastic Matrix Models and Caswell 2001). Degree of Development and Consistency — HIGH — Several software programs implement this model type, and documentation and technical support are available. Understanding the workings of the model is relatively easy. Ease of Estimating Parameters — MEDIUM — Parameters can in principle be obtained relatively easily, either from laboratory experiments or from field observations. Alternatively, fecundity and survivorship values may be approximated from standard survivorship and birth rate curves in the literature. However, particularly when laboratory experiments are not appropriate or feasible, assess - ing the magnitude of chemical effects on each vital rate may be very difficult. Regulatory Acceptance — HIGH — The model is being used by several regulatory agencies. Credibility — HIGH — The model type is well known within academia. Many applications of the model exist. Resource Efficiency — HIGH — Application of the model requires no programming because software is available. In many cases, available data are sufficient. However, when available data are not sufficient, considerable effort may be needed to obtain new data. 1574CH05.fm Page 57 Tuesday, November 26, 2002 4:55 PM sf sf sf s f 0 s0 0 0 0 0 0s 0 00 00s 00 000 s 0 01 02 23 k–1k 1 2 k L L L L L © 2002 by CRC Press LLC STOCHASTIC MATRIX MODELS (AGE OR STAGE BASED) Like the deterministic age- or stage-based matrix models, these models assume that survival rate and fecundity are functions of the age class or stage in which an organism resides. In contrast with their deterministic counterparts, the stochastic models incorporate environmental or demographic stochasticity or both in the estimates of survival probabilities and fecundities (Caswell 2001). Depending on whether density dependence is included, the vital rates may vary as a function of population size. The vital rates, which comprise the population matrix, can be obtained either from laboratory experiments or field observations. Realism — HIGH — In addition to reflecting that vital rates generally change with age (or stage), these models recognize that the rates may also vary as a result of random environmental fluctuations and demographic stochasticity. They generally incorporate all available demographic data. Within this category of models, age- or stage-structured models with density dependence are more realistic than those without it. Relevance — HIGH — Possible endpoints include expected population size (age- or stage-specific), risk of decline, risk of extinction, and expected crossing time (the time at which the population is expected to go either above or below a given size). These measures are all relevant to ecotoxicological assessments. As the deterministic version of this model, the survivorship and fecundity values can be adjusted to reflect the effects of toxic chemicals observed in laboratory tests. Flexibility — HIGH — All parameters are species-specific. The number of age classes or stages, the vital rates, and the variation in the vital rates can vary among populations. Therefore, different population structures and life histories can be modeled. Treatment of Uncertainty — HIGH — Both environmental and demographic stochasticity can be incorporated. Degree of Development and Consistency — HIGH — Several software programs implement this model type, and documentation and technical support are available. It is relatively easy to understand the workings of the model. Ease of Estimating Parameters — MEDIUM — Required parameters include average vital rates and their standard deviations. Furthermore, chemical impact on these parameters must be considered (this can be handled, for example, by estimating two matrices — one with toxins and one without toxins). Obtaining estimates of all such parameters can often be quite difficult. However, model parameters are intuitive and can be interpreted biologically. Regulatory Acceptance — HIGH — The model is used by several regulatory agencies. Credibility — HIGH — The model is well known within academia. There are many applications of the model. Resource Efficiency — MEDIUM — Application of the model requires no programming, as software is available. In some cases, data must be collected, but in many cases, available data are sufficient. RAMAS AGE, STAGE, METAPOP, OR ECOTOXICOLOGY These four computer programs (Spencer and Ferson 1997b; Ferson 1990; Ferson and Akçakaya 1988; Akçakaya 1998b) are collapsed into one category because they share a common source and they receive the same ratings. Each program applies matrix models for age- or stage-structured populations to estimate population-level parameters such as growth rate and extinction risk. In RAMAS Ecotoxicology, a model of population dynamics and toxicant kinetics can be constructed and linked to bioassay data. Using this program, a modeler can import data from standard laboratory bioassays, incorporate these data into the parameters of a population model, and perform a risk assessment by analyzing population-level differences between potentially affected and control samples. The user specifies the control survivorship and fecundity for each age class or stage. Density dependence in the form of ceiling, logistic, Ricker, or Beverton–Holt functions can be added for specific age classes or stages. A user simulates toxic effects by selecting 1574CH05.fm Page 58 Tuesday, November 26, 2002 4:55 PM © 2002 by CRC Press LLC and parameterizing dose–response models (Weibull, probit, or logit) linked to survivorship and fecundity values. Parameters can be specified as scalars, intervals, or distributions to account for environmental variability and uncertainty. Monte Carlo simulations are then used to predict future population trajectories and calculate the risk of adverse events such as extinctions or algal blooms. The software checks the validity of the data input and model structure specified by the user. Realism — HIGH — RAMAS Age models age-structured populations, and RAMAS Stage models stage-structured populations. RAMAS Metapop can model either age-structured or stage-structured populations. RAMAS Ecotoxicology can also model either age-structured or stage-structured pop - ulations and can also explicitly model the effects of toxic chemicals. All four programs can include density dependence as well as environmental and demographic stochasticity. Relevance — HIGH — Possible endpoints include expected population size (stage-specific), lambda, lambda-based measures such as sensitivity and elasticity, risk of decline, risk of extinction, and expected crossing time (the time at which the population is expected to exceed or to decrease to less than a given size). These measures are all potentially useful in ecotoxicological assessments. Effects of toxic chemicals are not explicitly modeled but can be incorporated by adjusting survivor - ship and fecundity values to reflect toxic effects (e.g., Munns et al. 1997). Flexibility — HIGH — All parameters are species-specific. The number of age classes or stages, the vital rates, and the variation in the vital rates can vary among populations. Therefore, different population structures and life histories can be modeled. In RAMAS Ecotoxicology, different toxic chemical dynamics and dose–response functions can be modeled. Treatment of Uncertainty — HIGH — Both environmental and demographic stochasticity can be incorporated. Degree of Development and Consistency — HIGH — These models are easy to use and easy to apply to different populations. Each program has a detailed user’s manual explaining the scientific basis of the model as well as the capabilities of the program. The programs include several internal checks for consistency. Ease of Estimating Parameters — MEDIUM — Required parameters include vital rates at the very least. In some cases, average vital rates and their standard deviations need to be estimated. Further - more, the chemical impact on these parameters may need to be considered (e.g., by choosing and parameterizing a dose–response function or by estimating two matrices — one with toxins and one without toxins). Obtaining estimates of all such parameters can often be quite difficult. Thus, ease of parameter estimation varies among cases. However, model parameters are intuitive and can be interpreted biologically. Regulatory Acceptance — HIGH — These models are being used by several regulatory agencies. Credibility — HIGH — These models are widely used within academia. Many applications of the models exist. Resource Efficiency — HIGH — No programming is necessary to use these programs. In some cases, data must be collected, but in many cases, available data are sufficient. UNIFIED LIFE MODEL (ULM) ULM (Legendre and Clobert 1995) is a computer program that implements deterministic or stochastic matrix models for analysis of population dynamics. ULM is somewhat similar to RAMAS, except for its interface and varied capabilities. ULM accommodates a wide range of populations with variable life-history characteristics. ULM can model any species life history as a matrix model with or without density dependence, environmental stochasticity, demographic stochasticity (as branching processes), inter- or intra-specific competition, parasitism, and metap - opulations. Results are expressed as population trajectories, distributions, growth rate, population stage- or age-structure, generation times, sensitivities to changes in parameters, probability of extinction, and extinction times. The stochastic models within ULM are implemented as Monte Carlo simulations. 1574CH05.fm Page 59 Tuesday, November 26, 2002 4:55 PM © 2002 by CRC Press LLC Realism — HIGH — ULM models age-structured and stage-structured populations. The number of age classes or stages is variable. Environmental stochasticity is included. Relevance — HIGH — Possible endpoints include expected population size (age- or stage-specific), lambda, lambda-based measures such as sensitivity and elasticity, and risk of extinction. These measures are all used in ecotoxicological assessments. Flexibility — HIGH — All parameters are species-specific. The number of age classes or stages, the vital rates, and the variation in the vital rates can vary among populations. Therefore, different population structures and life histories can be modeled. Treatment of Uncertainty — HIGH — Environmental stochasticity can be incorporated. Degree of Development and Consistency — MEDIUM — Some programming is required to use the model. This programming must use a language defined by the authors of the program. Legendre and Clobert (1995) explain the use of the model and provide examples. Ease of Estimating Parameters — HIGH — Parameters can be obtained relatively easily, either from laboratory experiments or field observations. Regulatory Acceptance — LOW — No information on the model’s regulatory acceptance is available. The model is not likely to be used by a regulatory agency at present because programming is required to use the model. Credibility — MEDIUM — Several (approximately 20) publications apply the model to different cases. Resource Efficiency — MEDIUM — Some programming is necessary to use this program. In some cases, data must be collected, but in many cases available data are sufficient. DISCUSSION AND RECOMMENDATIONS Life-history models (Table 5.2) are well developed and are already being used for ecotoxicological assessment (Levin et al. 1996; Munns et al. 1997; Crutchfield and Ferson 2000) (Table 5.3; also see Barnthouse [1993] for a review of other applications). These models have already been gener - alized to be spatially explicit within a metapopulation framework (RAMAS Metapop) and to explicitly include the dynamics and effects of toxic chemicals (RAMAS Ecotoxicology). Deter - ministic and stochastic matrix models and RAMAS Age, Stage, Metapop, or Ecotoxicology are therefore recommended for further evaluation and use in chemical risk assessment. Suggested future developments include development of software that includes both spatially explicit effects and the dynamics/effects of toxic chemicals. Software should be developed for calculating risk-based (probabilistic) sensitivities with respect to vital rates to answer questions such as How does the probability of decline change as a result of changes in the vital rates? This method would be an adaptation of the results of Uryasev (1995). This development would allow the conversion of sensitivity-based methods, such as decomposition of the change in lambda, into the language of risk. Such sensitivity-based applications are used to identify age-specific or stage-specific manage - ment strategies. The new methods will generalize widely used methods of lambda decomposition (Caswell 2001). 1574CH05.fm Page 60 Tuesday, November 26, 2002 4:55 PM © 2002 by CRC Press LLC Table 5.2 Evaluation of Population Models — Life History Models Evaluation Criteria Model Reference Realism Relevance Flexibility Treatment of Uncertainty Degree of Development Ease of Estimating Parameters Regulatory Acceptance Credibility Resource Efficiency Deterministic age- or stage-based matrix Caswell (2001) ◆◆◆ ◆◆◆ ◆◆◆ ◆ ◆◆◆ ◆◆ ◆◆◆ ◆◆◆ ◆◆◆ Stochastic age- or stage-based matrix Caswell (2001) ◆◆◆ ◆◆◆ ◆◆◆ ◆◆◆ ◆◆◆ ◆◆ ◆◆◆ ◆◆◆ ◆◆ RAMAS Age, Stage, Metapop, or Ecotoxicology Ferson (1993); Spencer and Ferson (1997c); Applied Biomathematics (2000) ◆◆◆ ◆◆◆ ◆◆◆ ◆◆◆ ◆◆◆ ◆◆ ◆◆◆ ◆◆◆ ◆◆◆ Unified life model (ULM) Legendre and Clobert (1995) ◆◆◆ ◆◆◆ ◆◆◆ ◆◆◆ ◆◆ ◆◆◆ ◆ ◆◆ ◆◆ Note: ◆◆◆ - high ◆◆ - medium ◆ - low 1574CH05.fm Page 61 Tuesday, November 26, 2002 4:55 PM © 2002 by CRC Press LLC Table 5.3 Applications of Life-History Models Model Species Location/Population Reference Deterministic age-based matrix Burrowing mayflies, Hexagenia spp. (H. limbata and H. rigida) Western Lake Erie Madenjian et al. (1998) Subtidal snail (Umbonium costatum) Hakodate Bay, northern Japan Noda and Nakao (1996) Mysid (Americamysis bahia) Laboratory Kuhn et al. (2000) Pea aphid (Acyrthosiphon pisum) Laboratory Walthall and Stark (1997) Northern sea lions (Eumetopias jubatus) Marmot Island, Alaska York (1994) Deterministic age-based matrix and deterministic stage-based matrix Polychaetes (Capitella sp. I and Streblospio benedicti) Estuaries and littoral wetlands throughout much of the United States Levin et al. (1996) Deterministic stage-based matrix Pea aphid (Aeyrthosiphon pisum) Laboratory Stark and Wennergren (1995) Soil mite (Platynothrus peltifer ), isopod (Porcellio scaber), and nematode (Plectus acuminatus) Laboratory Kammenga et al. (2001) Bluegill sunfish (Lepomis macrochirus) Generic lake in central Florida Bartell et al. (2000) Brook trout (Salvelinus fontinalis) Southern Appalachian mountain streams Marschall and Crowder (1996) Estuarine fish (Fundulus heteroclitus) New Bedford Harbor, Massachusetts Munns et al. (1997) Loggerhead sea turtles (Caretta caretta) Trawl fisheries of the southeastern United States Crowder et al. (1994) Yellow mud turtles (Kinosternon flavescens) and Kemp's ridley sea turtles (Lepidochelys kempi) Texa s Heppell et al. (1996) Killer whales (Orcinus orca) Pacific Northwest Brault and Caswell (1993) Savannah grass (Andropogon brevifolius Schwarz) Venezuelan savannas Canales et al. (1994) Deterministic stage-based matrix and stochastic stage-based matrix Snail kite (Rostrhamus sociabilis) Everglades Beissinger (1995) House sparrow (Passer domesticus) India, Pakistan Slade (1994) Steller sea lion (Eumetopias jubatus) Northeast Pacific Pascual and Adkinson (1994) American ginseng (Panax quinquefolium) and wild leek (Allium tricoccum) Canada Nantel et al. (1996) Mountain golden heather (Hudsonia montana) North Carolina Gross et al. (1998) Stochastic age-based matrix Brook trout (Salvelinus fontinalis) Montmorency County, Michigan McFadden et al. (1967) Striped bass (Morone saxatilis) Potomac River Cohen et al. (1983) 1574CH05.fm Page 62 Tuesday, November 26, 2002 4:55 PM © 2002 by CRC Press LLC Table 5.3 (cont.) Hawaiian stilt (Himantopus mexicanus knudseni) Hawaii Reed et al. (1998) Mediterranean monk seal (Monachus monachus) Atlantic Ocean (North Africa) and eastern Mediterranean Durant and Harwood (1992) Roan antelope (Hippotragus equinus) Parc National de l'Akagera in Rwanda Beudels et al. (1992) Asiatic wild ass (Equus hemionus) Negev Desert of southern Israel Solbreck (1991); Saltz and Rubenstein (1995) African elephant (Loxodonta africana) Tsavo National Park, Kenya Armbruster and Lande (1993) Stochastic stage-based matrix Algae (Ascophyllum nodosum) Swedish coast Aberg (1992) Marine bivalve (Yoldia notabilis) Otsuchi Bay, northeastern Japan Nakaoka (1997) Sea whip coral (Leptogorgia virgulata) Northeastern Gulf of Mexico Gotelli (1991) Grey seal (Halichoerus grypus) and ringed seal (Phoca hispida) Baltic Sea Kokko et al. (1997) Desert tortoise (Gopherus agassiz) Western Mojave Desert Doak et al. (1994) Red deer (Cervus elaphus) Rum, Western Isles, Scotland Benton et al. (1995) Redwood (Sequoia sempervirens) California, Oregon Namkoong and Roberds (1974) RAMAS Age Bluegill (Lepomis machrochirus) Hyco Reservoir, North Carolina Crutchfield and Ferson (2000) Striped bass (Morone saxatilis) Santee-Cooper system, South Carolina Bulak et al. (1995) Cod Atlantic Ginzburg et al. (1990) Marbled murrelet (Brachyramphus marmoratus) Oregon Oregon Department of Fish and Wildlife (1995) RAMAS Ecotoxicology Brook trout (Salvelinus fontinalis) Montmorency County, Michigan Spencer and Ferson (1997b) Lesser kestrel (Falco naumanni) Southern Spain Spencer and Ferson (1997b) Moose (Alces alces gigas) Northeastern Alberta Spencer and Ferson (1997b) RAMAS Stage Giant kelp (Macrocystis pyrifera) Southern California coastal waters Burgman and Gerard (1989) Bluegill sunfish (Lepomis macrochirus) North Carolina Crutchfield and Ferson (2000) Threadfin shad (Dorosoma petenense) South Carolina Barwick et al. (1994) Red-cockaded woodpecker (Picoides borealis) Georgia Piedmont Maguire et al. (1995) Bradshaw’s lomatinum (Lomatium bradshawii) Western Oregon Kaye et al. (1994) Sentry milk-vetch (Astragalus cremnophylax) Grand Canyon National Park, Arizona Maschinsky et al. (1997) ULM — stochastic stage-based matrix Grey partridge (Perdix perdix) Northwest and southeast France Bro et al. (2000) 1574CH05.fm Page 63 Tuesday, November 26, 2002 4:55 PM © 2002 by CRC Press LLC Table 5.3 (cont.) Model Species Location/Population Reference ULM — deterministic stage-based matrix Yellow-legged gull (Larus cachinnans) Medes Islands, northwestern Mediterranean Bosch et al. (2000) ULM — stochastic stage-based matrix Passerine songbirds User defined Legendre (1999) ULM — deterministic stage-based matrix and stochastic stage-based matrix Snake (Vipera ursinii ursinii) and raptor (Gyps fulvus fulvus) User defined Ferriere et al. (1996) 1574CH05.fm Page 64 Tuesday, November 26, 2002 4:55 PM . (19 95) ◆◆◆ ◆◆◆ ◆◆◆ ◆◆◆ ◆◆ ◆◆◆ ◆ ◆◆ ◆◆ Note: ◆◆◆ - high ◆◆ - medium ◆ - low 157 4CH 05. fm Page 61 Tuesday, November 26, 2002 4 :55 PM © 2002 by CRC Press LLC Table 5. 3 Applications of Life-History. life-history modeling of age- or stage-structured populations Legendre and Clobert (19 95) http://www.snv.jussieu.fr/Bio/ulm/ulm.html http://eco.wiz.uni-kassel. de/model_db/mdb/ulm.html 157 4CH 05. fm. it. Relevance — HIGH — Possible endpoints include expected population size (age- or stage-specific), risk of decline, risk of extinction, and expected crossing time (the time at which the population