Contributed Paper
Allee Effect and the Uncertainty of Population Recovery ANNA KUPARINEN,∗ † DAVID M. KEITH,‡ AND JEFFREY A. HUTCHINGS‡§ ∗
Department of Environmental Sciences, P.O. Box 65, 00014 University of Helsinki, Finland, email [email protected] †Ecological Genetics Research Unit, Department of Biosciences, P.O. Box 65, 00014 University of Helsinki, Finland ‡Department of Biology, Dalhousie University, 1355 Oxford Street, P.O. Box 15000, Halifax, NS B3H4R2, Canada §Centre for Ecological and Evolutionary Synthesis, Department of Biosciences, University of Oslo, NO-0316, Oslo, Norway
Abstract: Recovery of depleted populations is fundamentally important for conservation biology and sustainable resource harvesting. At low abundance, population growth rate, a primary determinant of population recovery, is generally assumed to be relatively fast because competition is low (i.e., negative density dependence). But population growth can be limited in small populations by an Allee effect. This is particularly relevant for collapsed populations or species that have not recovered despite large reductions in, or elimination of, threats. We investigated how an Allee effect can influence the dynamics of recovery. We used Atlantic cod (Gadus morhua) as the study organism and an empirically quantified Allee effect for the species to parameterize our simulations. We simulated recovery through an individual-based mechanistic simulation model and then compared recovery among scenarios incorporating an Allee effect, negative density dependence, and an intermediate scenario. Although an Allee effect significantly slowed recovery, such that population increase could be negligible even after 100 years or more, it also made the time required for biomass rebuilding much less predictable. Our finding that an Allee effect greatly increased the uncertainty in recovery time frames provides an empirically based explanation for why the removal of threat does not always result in the recovery of depleted populations or species. Keywords: Allee effect, Atlantic cod, depensation, overfishing, population recovery, sustainable harvesting El Efecto Allee y la Incertidumbre de la Recuperaci´ on de Poblaciones
Resumen:
La recuperaci´ on de las poblaciones disminuidas es fundamentalmente importante para la biolog´ıa de la conservaci´ on y la cosecha sustentable de recursos. Con poca abundancia, generalmente se asume que la tasa de crecimiento poblacional, un determinante primario de la recuperaci´ on de poblaciones, es relativamente r´ apida porque la competencia es baja (p. ej.: dependencia de densidad negativa) pero el crecimiento poblacional puede estar limitado por el efecto Allee en poblaciones peque˜ nas. Esto es particularmente relevante para las poblaciones o especies colapsadas que no se han recuperado a pesar de grandes reducciones en, o la eliminaci´ on de, las amenazas. Investigamos c´ omo el efecto Allee puede influenciar las din´ amicas de recuperaci´ on. Usamos al bacalao del Atl´ antico (Gadus morhua) como organismo de estudio y un efecto Allee para las especies cuantificado emp´ıricamente para hacer par´ ametros en nuestras simulaciones. Simulamos la recuperaci´ on por medio de un modelo de simulaci´ on mecanicista basado en un individuo y despu´es comparamos la recuperaci´ on entre escenarios incorporando un efecto Allee, dependencia de densidad negativa y un escenario intermedio. Aunque el efecto Allee hizo significativamente m´ as lenta la recuperaci´ on, tal incremento en la poblaci´ on puede ser despreciable incluso despu´es de 100 a˜ nos o m´ as; tambi´en hizo que el tiempo requerido para la reconstrucci´ on de la biomasa fuera mucho menos predecible. Nuestro descubrimiento de que el efecto Allee incrementa mucho la incertidumbre en los marcos de tiempo de la recuperaci´ on proporciona una explicaci´ on con base emp´ırica de por qu´e la remoci´ on de la amenaza no siempre resulta en la recuperaci´ on de poblaciones o especies empobrecidas.
Palabras Clave: Bacalao del Atl´antico, cosecha sustentable, depensaci´on, efecto Allee, recuperaci´on de poblaciones, sobrepesca
Paper submitted June 12, 2013; revised manuscript accepted September 5, 2013.
790 Conservation Biology, Volume 28, No. 3, 790–798 C 2014 Society for Conservation Biology DOI: 10.1111/cobi.12216
Kuparinen et al.
791
Introduction The ability of a depleted population to recover to a higher abundance is fundamentally important to wildlife conservation because of the inextricable link that exists between recovery ability and a well-established (negative) correlate of extinction risk—per capita population growth rate. All else being equal, a depleted but fastgrowing population experiences a lower risk of extinction than a similarly depleted slow-growing population (Dulvy et al. 2004; Mace et al. 2008). The basic theory of density-dependent population dynamics suggests that population growth is inversely related to density, such that small populations grow faster, because of reduced intraspecific competition, than large populations (Gotelli 2008). However, this might not always be the case (Bertram 1994). At very low abundance, individual survival and (or) reproductive success can be reduced, jointly leading to reduced per capita population growth—a phenomenon called the demographic Allee effect (as opposed to component Allee effects which take place at the level of individual fitness components) (Stephens et al. 1999; Gascoigne & Lipcius 2004). The theory underlying Allee effects is well established and empirical evidence has been detected in various systems (Kramer et al. 2009) (but see Gregory et al. 2010). Generally, it is well understood that the demographic Allee effect impedes the growth of small natural populations, but affected populations are also vulnerable to even small increases in mortality (Courchamp et al. 1999) and to demographic or environmental stochasticity (Lande 1998; Dennis 2002), which can easily drive populations to extinction. Despite this knowledge, the consequences of Allee effects for practical conservation decisions, recovery plans, and harvesting strategies have rarely been implemented (Stephens & Sutherland 1999). A primary reason for this is that it can be difficult to document an Allee effect in populations that have declined or for which there are few data. Compounding this empirical challenge is the fact that the rapidly required decision making associated with conservation action does not allow for long-term data collection (Gilroy et al. 2012). In the context of sustainable harvesting, the future production of harvestable biomass by depleted populations depends on recovery potential (Rosenberg et al. 1993; Worm et al. 2009). In a fisheries context, harvest strategies and management rely heavily on the traditional assumption that fish population productivity is inversely related to population density (i.e., compensatory dynamics; Hilborn & Walters 1992). Consequently, harvest-induced depletions are not usually considered unduly problematic given the expectation that such populations will grow rapidly when fishing is relaxed (Hilborn & Walters 1992). This contention has been fueled by a general lack of empirical evidence of an Allee effect in fish populations that
have declined (e.g., Myers et al. 1995; Liermann & Hilborn 1997; Gascoigne & Lipcius 2004), many fish populations that have declined substantially show little evidence of recovery despite large reductions in fishing pressure (Pauly et al. 1998; Hutchings et al. 2010), thereby challenging the traditional view about the recovery ability of depleted populations. This underscores the need to both understand and properly account for the processes that can negatively influence the recovery of fish populations (Denney et al. 2002; Hutchings & Reynolds 2004). Specifically, based on basic population dynamics theory, a wellmotivated question to ask is whether the slow recovery of many exploited fish populations might be attributable to a demographic Allee effect (also called depensation) (Frank & Brickman 2000). The reductions of Northwest Atlantic cod (Gadus morhua) in the early 1990s are among the most dramatic fish population depletions ever documented (Hutchings & Myers 1994). Despite significant reductions in fishing pressure, most populations have exhibited few signs of recovery (Hutchings & Rangeley 2011). In some areas, such as the southern Gulf of St. Lawrence, increased natural mortality may be preventing cod populations from growing (Swain 2011). A recently published metaanalysis of 207 exploited marine fish populations (104 species) detected signs of a demographic Allee effect in Atlantic cod: estimates of per capita offspring production declined when population size was at the lowest levels of abundance, a pattern opposite that predicted by traditionally assumed compensatory population dynamics (Keith & Hutchings 2012). This finding lends empirical support to suggestions (Shelton & Healey 1999; Hutchings & Rangeley 2011) that an Allee effect plays a considerably greater role in the dynamics of depleted cod populations, and on their ability to recover following collapse, than has been previously considered (Myers et al. 1995). Apart from its relevance to marine conservation and fisheries, Atlantic cod is an excellent study system for investigating the influence of Allee effects on population recovery. First, for this species, a demographic Allee effect has been estimated from extensive analyses of several populations, thus providing an empirically defensible basis for population recovery projections. Second, among vertebrates, Atlantic cod has experienced exceptionally large and deep population depletions, such that the Allee effect is likely to play a nontrivial role in the dynamics of local populations. We used the empirically quantified estimate of the demographic Allee effect in Atlantic cod reported by Keith and Hutchings (2012) and investigated how it affected the recovery of depleted cod populations. To this end, we incorporated the Allee effect into an individual-based simulation model of cod population dynamics. This model has been applied previously to study Conservation Biology Volume 28, No. 3, 2014
792
of adaptation of cod life histories to different mortality regimes (Kuparinen et al. 2012) and the role of fisheriesinduced evolution on population recovery (Kuparinen & Hutchings 2012), and it has been proven that this model can reliably mimic ecological and evolutionary dynamics in cod. We compared the dynamics of population recovery resulting from overfishing (i.e., biomass rebuilding) under the scenarios of Allee effect, traditionally assumed compensatory dynamics, and a scenario intermediate to these extremes.
Methods Model Description Cod population dynamics were simulated using an individual-based mechanistic model that characterizes individual life histories by incorporating quantitative genetics, ecological processes, and empirical data on cod growth and fecundity. Although the model has been fully described elsewhere (Kuparinen et al. 2012; Kuparinen & Hutchings 2012), we provide a description of its main features. In the model, individual fish in a population are tracked at annual time steps. At each time step, the processes of mortality, growth, and reproduction are simulated on an individual basis; thus, demographic stochasticity in each of these processes is accounted for. At its core, the model is founded on the von Bertalanffy (VB) growth curve, L(t) = L∞ − (L∞ − L0 )e−kt , and the well-known correlations (i.e., life-history invariants) that exist between VB curve parameters k (metric of intrinsic growth rate) and L∞ (maximum or asymptotic length) and between L∞ and length at maturity (Charnov 1993). The growth trajectories are assumed heritable so that at birth each individual is assigned its own k and L∞ parameters, and during its lifetime an individual then grows along its VB growth trajectory and matures when it has reached a body length of 66% of its L∞ . The L0 (average length at time step 0) was set to 4.0 cm for each individual. Given that quantitative traits are generally coded on the basis of a large number of loci, each assumed to have small additive effects (Roff 2002), L∞ was determined with 10 diploid loci each having 2 alleles. The values of the alleles were coded with 0 and 1, and the impacts of the loci were assumed equal and additive. Inheritance of the alleles was assumed to follow classical Mendelian inheritance such that at each locus an offspring received one randomly sampled allele from its mother and one from its father. The sum of allelic values (ranging from 0 to 20), coupled with a normally distributed random number to allow some phenotypic variation around the genotype, was then linearly translated into values of L∞ . The amount of phenotypic variation was calibrated to yield realistic heritability of approximately 0.2–0.3 for the phenotypic traits (Mousseau & Roff 1987). The range in Conservation Biology Volume 28, No. 3, 2014
Allee Effect and Population Recovery
L∞ was set to 30–130 cm, based on empirical data on 258 cod growth trajectories (Kuparinen et al. 2012). The association between L∞ and k was based on the same data, through a regression fit of log(k) = −0.609 − 0.013 × L∞ (with residual SE 0.305). At each time step, the survival of each individual was simulated by drawing random numbers from a binomial distribution to determine whether an individual died or not. Juvenile survival up to age 3 was 1.13 × 10−6 , and after that annual mortality was set to 0.12. Mature individuals experienced survival cost of reproduction that was set to 0.1. These values were chosen to provide best match between empirical data and simulated cod life histories (Kuparinen et al. 2012; Kuparinen & Hutchings 2012). Those individuals that did not die grew along their individual growth trajectory (defined by an individual’s VB parameters). Density dependence of growth was implemented such that in a sparse population the annual progress along the individual growth curve was greater than that of a dense population. In practice, this was implemented such that the annual time available for growth ranged from 0 to 1. If the population was far from its carrying capacity, the time spent on growth within 1 year was very close to 1. However, if the population was very close to or above its carrying capacity, we assumed that the time available for growth was reduced according to a logistic equation, e15 – 17.6 × c (1 + e15 – 17.6 × c )−1 , where c is the ratio of the population’s biomass and its carrying capacity. Those individuals that reached 66% of their L∞ were considered mature. Mating occurred by drawing randomly, for each mature female, a mate from among the mature males. The number of juveniles produced by each mature female depended positively on female body size and further on population density (see below). The relationship between female body size and fecundity was derived from Hutchings (2005), such that the number of eggs produced was predicted to be eggs = (0.48 × [(female weight + 0.37)/1.45] + 0.12) × 106 . The length– weight relationship was estimated based on the same data as VB parameters and set to be weight = 3.52 × 10−6 × length3.19 (Kuparinen et al. 2012). Inheritance of genetic traits was modeled as described above, and the sex of each juvenile was drawn randomly from a Bernoulli trial with the probability of 0.5. Maximum lifetime of individuals was set to 25 years. Additional details of the model and its parameterization are in Kuparinen et al. (2012) and Kuparinen and Hutchings (2012). Our primary objective was to investigate the role of an Allee effect on population recovery in isolation from possible evolutionary processes induced by fishing. It is important that these influences on recovery be distinguished, given that fisheries-induced evolution almost certainly alters the equilibrium biomass of a harvested population (Enberg et al. 2009; Kuparinen & Hutchings 2012), thereby affecting the population’s dynamics and
Kuparinen et al.
Implementation of the Allee Effect The Allee effect scenario we applied was based empirically on recruit-per-spawner ratios estimated for cod across discrete bins of population abundance (i.e., estimated for 0–10%, 10–20%, . . . of maximum observed spawning stock biomass [SSBmax ]; Keith & Hutchings 2012). The estimates were averaged over 19 cod populations (data available from http://ramlegacy. marinebiodiversity.ca/srdb/updated-srdb) and thus provide a generic relationship between the population abundance and reproductive success for the species. These estimates suggest that if the spawning stock (population) biomass (SSB) is greater than 10% of its maximum (SSBmax ), then the reproductive success (recruits-perspawner) correlates negatively with population abundance but the pattern reverses at low abundance. Reproductive success at an SSB lower than 10% of SSBmax is reduced compared with that at 10–20% of SSBmax . The Allee effect scenario was contrasted with a scenario of compensation (reproductive success consistently increases as abundance declines) estimated using a Ricker recruit-per-spawner model fit to cod population data restricted to abundances above 40% of SSBmax (i.e., encompassing a range of population sizes for which population growth usually shows strong negative density dependence; Keith & Hutchings 2012). Compensation at low abundances (
Allee Effect and the Uncertainty of Population Recovery ANNA KUPARINEN,∗ † DAVID M. KEITH,‡ AND JEFFREY A. HUTCHINGS‡§ ∗
Department of Environmental Sciences, P.O. Box 65, 00014 University of Helsinki, Finland, email [email protected] †Ecological Genetics Research Unit, Department of Biosciences, P.O. Box 65, 00014 University of Helsinki, Finland ‡Department of Biology, Dalhousie University, 1355 Oxford Street, P.O. Box 15000, Halifax, NS B3H4R2, Canada §Centre for Ecological and Evolutionary Synthesis, Department of Biosciences, University of Oslo, NO-0316, Oslo, Norway
Abstract: Recovery of depleted populations is fundamentally important for conservation biology and sustainable resource harvesting. At low abundance, population growth rate, a primary determinant of population recovery, is generally assumed to be relatively fast because competition is low (i.e., negative density dependence). But population growth can be limited in small populations by an Allee effect. This is particularly relevant for collapsed populations or species that have not recovered despite large reductions in, or elimination of, threats. We investigated how an Allee effect can influence the dynamics of recovery. We used Atlantic cod (Gadus morhua) as the study organism and an empirically quantified Allee effect for the species to parameterize our simulations. We simulated recovery through an individual-based mechanistic simulation model and then compared recovery among scenarios incorporating an Allee effect, negative density dependence, and an intermediate scenario. Although an Allee effect significantly slowed recovery, such that population increase could be negligible even after 100 years or more, it also made the time required for biomass rebuilding much less predictable. Our finding that an Allee effect greatly increased the uncertainty in recovery time frames provides an empirically based explanation for why the removal of threat does not always result in the recovery of depleted populations or species. Keywords: Allee effect, Atlantic cod, depensation, overfishing, population recovery, sustainable harvesting El Efecto Allee y la Incertidumbre de la Recuperaci´ on de Poblaciones
Resumen:
La recuperaci´ on de las poblaciones disminuidas es fundamentalmente importante para la biolog´ıa de la conservaci´ on y la cosecha sustentable de recursos. Con poca abundancia, generalmente se asume que la tasa de crecimiento poblacional, un determinante primario de la recuperaci´ on de poblaciones, es relativamente r´ apida porque la competencia es baja (p. ej.: dependencia de densidad negativa) pero el crecimiento poblacional puede estar limitado por el efecto Allee en poblaciones peque˜ nas. Esto es particularmente relevante para las poblaciones o especies colapsadas que no se han recuperado a pesar de grandes reducciones en, o la eliminaci´ on de, las amenazas. Investigamos c´ omo el efecto Allee puede influenciar las din´ amicas de recuperaci´ on. Usamos al bacalao del Atl´ antico (Gadus morhua) como organismo de estudio y un efecto Allee para las especies cuantificado emp´ıricamente para hacer par´ ametros en nuestras simulaciones. Simulamos la recuperaci´ on por medio de un modelo de simulaci´ on mecanicista basado en un individuo y despu´es comparamos la recuperaci´ on entre escenarios incorporando un efecto Allee, dependencia de densidad negativa y un escenario intermedio. Aunque el efecto Allee hizo significativamente m´ as lenta la recuperaci´ on, tal incremento en la poblaci´ on puede ser despreciable incluso despu´es de 100 a˜ nos o m´ as; tambi´en hizo que el tiempo requerido para la reconstrucci´ on de la biomasa fuera mucho menos predecible. Nuestro descubrimiento de que el efecto Allee incrementa mucho la incertidumbre en los marcos de tiempo de la recuperaci´ on proporciona una explicaci´ on con base emp´ırica de por qu´e la remoci´ on de la amenaza no siempre resulta en la recuperaci´ on de poblaciones o especies empobrecidas.
Palabras Clave: Bacalao del Atl´antico, cosecha sustentable, depensaci´on, efecto Allee, recuperaci´on de poblaciones, sobrepesca
Paper submitted June 12, 2013; revised manuscript accepted September 5, 2013.
790 Conservation Biology, Volume 28, No. 3, 790–798 C 2014 Society for Conservation Biology DOI: 10.1111/cobi.12216
Kuparinen et al.
791
Introduction The ability of a depleted population to recover to a higher abundance is fundamentally important to wildlife conservation because of the inextricable link that exists between recovery ability and a well-established (negative) correlate of extinction risk—per capita population growth rate. All else being equal, a depleted but fastgrowing population experiences a lower risk of extinction than a similarly depleted slow-growing population (Dulvy et al. 2004; Mace et al. 2008). The basic theory of density-dependent population dynamics suggests that population growth is inversely related to density, such that small populations grow faster, because of reduced intraspecific competition, than large populations (Gotelli 2008). However, this might not always be the case (Bertram 1994). At very low abundance, individual survival and (or) reproductive success can be reduced, jointly leading to reduced per capita population growth—a phenomenon called the demographic Allee effect (as opposed to component Allee effects which take place at the level of individual fitness components) (Stephens et al. 1999; Gascoigne & Lipcius 2004). The theory underlying Allee effects is well established and empirical evidence has been detected in various systems (Kramer et al. 2009) (but see Gregory et al. 2010). Generally, it is well understood that the demographic Allee effect impedes the growth of small natural populations, but affected populations are also vulnerable to even small increases in mortality (Courchamp et al. 1999) and to demographic or environmental stochasticity (Lande 1998; Dennis 2002), which can easily drive populations to extinction. Despite this knowledge, the consequences of Allee effects for practical conservation decisions, recovery plans, and harvesting strategies have rarely been implemented (Stephens & Sutherland 1999). A primary reason for this is that it can be difficult to document an Allee effect in populations that have declined or for which there are few data. Compounding this empirical challenge is the fact that the rapidly required decision making associated with conservation action does not allow for long-term data collection (Gilroy et al. 2012). In the context of sustainable harvesting, the future production of harvestable biomass by depleted populations depends on recovery potential (Rosenberg et al. 1993; Worm et al. 2009). In a fisheries context, harvest strategies and management rely heavily on the traditional assumption that fish population productivity is inversely related to population density (i.e., compensatory dynamics; Hilborn & Walters 1992). Consequently, harvest-induced depletions are not usually considered unduly problematic given the expectation that such populations will grow rapidly when fishing is relaxed (Hilborn & Walters 1992). This contention has been fueled by a general lack of empirical evidence of an Allee effect in fish populations that
have declined (e.g., Myers et al. 1995; Liermann & Hilborn 1997; Gascoigne & Lipcius 2004), many fish populations that have declined substantially show little evidence of recovery despite large reductions in fishing pressure (Pauly et al. 1998; Hutchings et al. 2010), thereby challenging the traditional view about the recovery ability of depleted populations. This underscores the need to both understand and properly account for the processes that can negatively influence the recovery of fish populations (Denney et al. 2002; Hutchings & Reynolds 2004). Specifically, based on basic population dynamics theory, a wellmotivated question to ask is whether the slow recovery of many exploited fish populations might be attributable to a demographic Allee effect (also called depensation) (Frank & Brickman 2000). The reductions of Northwest Atlantic cod (Gadus morhua) in the early 1990s are among the most dramatic fish population depletions ever documented (Hutchings & Myers 1994). Despite significant reductions in fishing pressure, most populations have exhibited few signs of recovery (Hutchings & Rangeley 2011). In some areas, such as the southern Gulf of St. Lawrence, increased natural mortality may be preventing cod populations from growing (Swain 2011). A recently published metaanalysis of 207 exploited marine fish populations (104 species) detected signs of a demographic Allee effect in Atlantic cod: estimates of per capita offspring production declined when population size was at the lowest levels of abundance, a pattern opposite that predicted by traditionally assumed compensatory population dynamics (Keith & Hutchings 2012). This finding lends empirical support to suggestions (Shelton & Healey 1999; Hutchings & Rangeley 2011) that an Allee effect plays a considerably greater role in the dynamics of depleted cod populations, and on their ability to recover following collapse, than has been previously considered (Myers et al. 1995). Apart from its relevance to marine conservation and fisheries, Atlantic cod is an excellent study system for investigating the influence of Allee effects on population recovery. First, for this species, a demographic Allee effect has been estimated from extensive analyses of several populations, thus providing an empirically defensible basis for population recovery projections. Second, among vertebrates, Atlantic cod has experienced exceptionally large and deep population depletions, such that the Allee effect is likely to play a nontrivial role in the dynamics of local populations. We used the empirically quantified estimate of the demographic Allee effect in Atlantic cod reported by Keith and Hutchings (2012) and investigated how it affected the recovery of depleted cod populations. To this end, we incorporated the Allee effect into an individual-based simulation model of cod population dynamics. This model has been applied previously to study Conservation Biology Volume 28, No. 3, 2014
792
of adaptation of cod life histories to different mortality regimes (Kuparinen et al. 2012) and the role of fisheriesinduced evolution on population recovery (Kuparinen & Hutchings 2012), and it has been proven that this model can reliably mimic ecological and evolutionary dynamics in cod. We compared the dynamics of population recovery resulting from overfishing (i.e., biomass rebuilding) under the scenarios of Allee effect, traditionally assumed compensatory dynamics, and a scenario intermediate to these extremes.
Methods Model Description Cod population dynamics were simulated using an individual-based mechanistic model that characterizes individual life histories by incorporating quantitative genetics, ecological processes, and empirical data on cod growth and fecundity. Although the model has been fully described elsewhere (Kuparinen et al. 2012; Kuparinen & Hutchings 2012), we provide a description of its main features. In the model, individual fish in a population are tracked at annual time steps. At each time step, the processes of mortality, growth, and reproduction are simulated on an individual basis; thus, demographic stochasticity in each of these processes is accounted for. At its core, the model is founded on the von Bertalanffy (VB) growth curve, L(t) = L∞ − (L∞ − L0 )e−kt , and the well-known correlations (i.e., life-history invariants) that exist between VB curve parameters k (metric of intrinsic growth rate) and L∞ (maximum or asymptotic length) and between L∞ and length at maturity (Charnov 1993). The growth trajectories are assumed heritable so that at birth each individual is assigned its own k and L∞ parameters, and during its lifetime an individual then grows along its VB growth trajectory and matures when it has reached a body length of 66% of its L∞ . The L0 (average length at time step 0) was set to 4.0 cm for each individual. Given that quantitative traits are generally coded on the basis of a large number of loci, each assumed to have small additive effects (Roff 2002), L∞ was determined with 10 diploid loci each having 2 alleles. The values of the alleles were coded with 0 and 1, and the impacts of the loci were assumed equal and additive. Inheritance of the alleles was assumed to follow classical Mendelian inheritance such that at each locus an offspring received one randomly sampled allele from its mother and one from its father. The sum of allelic values (ranging from 0 to 20), coupled with a normally distributed random number to allow some phenotypic variation around the genotype, was then linearly translated into values of L∞ . The amount of phenotypic variation was calibrated to yield realistic heritability of approximately 0.2–0.3 for the phenotypic traits (Mousseau & Roff 1987). The range in Conservation Biology Volume 28, No. 3, 2014
Allee Effect and Population Recovery
L∞ was set to 30–130 cm, based on empirical data on 258 cod growth trajectories (Kuparinen et al. 2012). The association between L∞ and k was based on the same data, through a regression fit of log(k) = −0.609 − 0.013 × L∞ (with residual SE 0.305). At each time step, the survival of each individual was simulated by drawing random numbers from a binomial distribution to determine whether an individual died or not. Juvenile survival up to age 3 was 1.13 × 10−6 , and after that annual mortality was set to 0.12. Mature individuals experienced survival cost of reproduction that was set to 0.1. These values were chosen to provide best match between empirical data and simulated cod life histories (Kuparinen et al. 2012; Kuparinen & Hutchings 2012). Those individuals that did not die grew along their individual growth trajectory (defined by an individual’s VB parameters). Density dependence of growth was implemented such that in a sparse population the annual progress along the individual growth curve was greater than that of a dense population. In practice, this was implemented such that the annual time available for growth ranged from 0 to 1. If the population was far from its carrying capacity, the time spent on growth within 1 year was very close to 1. However, if the population was very close to or above its carrying capacity, we assumed that the time available for growth was reduced according to a logistic equation, e15 – 17.6 × c (1 + e15 – 17.6 × c )−1 , where c is the ratio of the population’s biomass and its carrying capacity. Those individuals that reached 66% of their L∞ were considered mature. Mating occurred by drawing randomly, for each mature female, a mate from among the mature males. The number of juveniles produced by each mature female depended positively on female body size and further on population density (see below). The relationship between female body size and fecundity was derived from Hutchings (2005), such that the number of eggs produced was predicted to be eggs = (0.48 × [(female weight + 0.37)/1.45] + 0.12) × 106 . The length– weight relationship was estimated based on the same data as VB parameters and set to be weight = 3.52 × 10−6 × length3.19 (Kuparinen et al. 2012). Inheritance of genetic traits was modeled as described above, and the sex of each juvenile was drawn randomly from a Bernoulli trial with the probability of 0.5. Maximum lifetime of individuals was set to 25 years. Additional details of the model and its parameterization are in Kuparinen et al. (2012) and Kuparinen and Hutchings (2012). Our primary objective was to investigate the role of an Allee effect on population recovery in isolation from possible evolutionary processes induced by fishing. It is important that these influences on recovery be distinguished, given that fisheries-induced evolution almost certainly alters the equilibrium biomass of a harvested population (Enberg et al. 2009; Kuparinen & Hutchings 2012), thereby affecting the population’s dynamics and
Kuparinen et al.
Implementation of the Allee Effect The Allee effect scenario we applied was based empirically on recruit-per-spawner ratios estimated for cod across discrete bins of population abundance (i.e., estimated for 0–10%, 10–20%, . . . of maximum observed spawning stock biomass [SSBmax ]; Keith & Hutchings 2012). The estimates were averaged over 19 cod populations (data available from http://ramlegacy. marinebiodiversity.ca/srdb/updated-srdb) and thus provide a generic relationship between the population abundance and reproductive success for the species. These estimates suggest that if the spawning stock (population) biomass (SSB) is greater than 10% of its maximum (SSBmax ), then the reproductive success (recruits-perspawner) correlates negatively with population abundance but the pattern reverses at low abundance. Reproductive success at an SSB lower than 10% of SSBmax is reduced compared with that at 10–20% of SSBmax . The Allee effect scenario was contrasted with a scenario of compensation (reproductive success consistently increases as abundance declines) estimated using a Ricker recruit-per-spawner model fit to cod population data restricted to abundances above 40% of SSBmax (i.e., encompassing a range of population sizes for which population growth usually shows strong negative density dependence; Keith & Hutchings 2012). Compensation at low abundances (
