Abstract
 Top of page
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Acknowledgements
 Literature cited
 Appendices
Sizeselective mortality caused by fishing can impose strong selection on harvested fish populations, causing evolution in important lifehistory traits. Understanding and predicting harvestinduced evolutionary change can help maintain sustainable fisheries. We investigate the evolutionary sustainability of alternative management regimes for lacustrine brook charr (Salvelinus fontinalis) fisheries in southern Canada and aim to optimize these regimes with respect to the competing objectives of maximizing mean annual yield and minimizing evolutionary change in maturation schedules. Using a stochastic simulation model of brook charr populations consuming a dynamic resource, we investigate how harvesting affects brook charr maturation schedules. We show that when approximately 5% to 15% of the brook charr biomass is harvested, yields are high, and harvestinduced evolutionary changes remain small. Intensive harvesting (at approximately >15% of brook charr biomass) results in high average yields and little evolutionary change only when harvesting is restricted to brook charr larger than the size at 50% maturation probability at the age of 2 years. Otherwise, intensive harvesting lowers average yield and causes evolutionary change in the maturation schedule of brook charr. Our results indicate that intermediate harvesting efforts offer an acceptable compromise between avoiding harvestinduced evolutionary change and securing high average yields.
Introduction
 Top of page
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Acknowledgements
 Literature cited
 Appendices
Lacustrine populations of the brook charr, Salvelinus fontinalis Mitchill, from the Canadian Shield represent a promising model system for studying the effects of selective harvesting on the evolution of lifehistory traits in fish populations. Previous modeling work on this species has shown that fishing can cause evolution in the migratory behavior in exploited populations (Thériault et al. 2008). Moreover, in a study comparing 17 populations, Magnan et al. (2005) found that charr from fished lakes mature significantly earlier than those from unfished lakes. As brook charr exhibit a short life cycle in these lakes, with lifespans of 3–7 years, even relatively recent harvesting could impose strong selective pressures on reproductive traits such as the size at maturation, leading to significant selection responses within a few generations. Genetic change caused by harvesting may thus explain the smaller sizes at maturation observed in fished lakes. However, previous studies on unfished brook charr populations found associations between early maturation and rapid growth rates (e.g., Hutchings 1993). It is also known that intraspecific competition from adults can depress juvenile growth rates, and therefore possibly delay maturation in fish populations (e.g., van Kooten et al. 2007). Hence, growthmediated maturation plasticity might suffice to explain observed differences in maturation schedules between harvested and unharvested lakes.
To assess how fishing potentially impacts brook charr life history at the genetic level, we follow the example of earlier research (e.g., Olsen et al. 2004 and Dunlop et al. 2007) by examining how fishing may cause evolution in the probabilistic maturation reaction norm (PMRN) for age and size at maturation. A PMRN for age and size at maturation describes the probability that an immature individual undergoes maturation within a given time interval, depending on its age and size (e.g., Heino et al. 2002).
Indeed, genetic change in the maturation reaction norm caused by fishing will often be more difficult to reverse by the simple cessation of fishing than growthmediated phenotypic plasticity in maturation (e.g., Law and Grey 1989 and Dunlop et al. 2009b). While fishery managers routinely seek to implement sustainable management regimes (e.g., Getz and Haight 1989; Fenichel et al. 2008), most such attempts currently do not explicitly address fisheriesinduced evolution (but see Law and Grey 1989 and Heino 1998).
Here we use the ecogenetic modeling approach (Dunlop et al. 2007 and Dunlop et al. 2009b) to describe the effects of harvesting on the genetic determinants of the PMRN in a harvested population of brook charr. Ecogenetic models seek to integrate key ecological processes, such as resource consumption and somatic growth, with an explicit treatment of changes in the distribution of genotypic traits.
Here we examine how such models could help develop advantageous management regimes for the brook charr fishery in Southern Quebec. We incorporate sizespecific fishing mortality explicitly, and compare harvest regimes according to their mean annual yield, as well as to the amount of evolutionary change that they cause. Within this comparative framework, we seek to address how fisheries managers can regulate fishing effort to reduce future evolutionary change in fish populations, while also maintaining acceptably high annual yields. Although our model is tailored to brook charr in Southern Quebec, we believe that our approach and predictions will also be applicable to other fisheries.
Methods
 Top of page
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Acknowledgements
 Literature cited
 Appendices
Somatic mass is divided into irreversible mass and reversible mass. An individual's irreversible, or structural, mass X is determined by components such as organ and skeletal tissue that cannot be starved away (de Roos and Persson 2001). In contrast, an individual's reversible mass is determined by energy reserves such as fat, other lipids, and gonadal tissue in mature individuals, that can be marshaled for basic metabolic functions during starvation and hence can be starved away. We partition reversible mass further into the mass of storage tissue, such as fat and other lipids, Y, and that of gonadal tissue, G (e.g., Broekhuizen et al. 1994; Persson et al. 1998; de Roos and Persson 2001). Thus,
 (1)
In males, gonadal mass G is interpreted as the amount of reversible mass expended on reproduction – for example, through the loss of somatic mass incurred by fighting other males over access to a redd, or spawning nest.
The model's dynamics are iterated on an annual time step. In each time step, the model cycles through all individuals to determine their fates, with processes occurring in the following sequence:
 1
Reproduction
 2
Natural mortality, resource consumption, and growth
 3
When harvesting occurs, fishing mortality
Model parameters were determined through analyzing data previously investigated in Magnan et al. (2005), calibrating the model to reproduce the observed data (Appendices A and B), and incorporating values available from the literature (Table 1).
Table 1. Demographic parameters and their numerical values in the model. Parameter  Description  Unit  Value  Equation  Source 


p  Length–mass coefficient  mm g^{−u}  77  –  Based on P. Magnan and M. Plante, unpublished data, and R. Curry et al. (1993) 
u  Length–mass exponent  –  0.30  –  Based on P. Magnan and M. Plante, unpublished data, and R. Curry et al. (1993) 
d  PMRN width  mm  110.3  (2)  Based on P. Magnan and M. Plante, unpublished data 
ν_{0}  Initial mean value of PMRN intercept  mm  200  (3)  Based on P. Magnan and M. Plante, unpublished data 
σ_{0}  Initial mean value of PMRN slope  mm year^{−1}  7.68  (3)  Based on P. Magnan and M. Plante, unpublished data 
W_{0}  Average mass of an egg  g  0.01  (4)  Based on P. Magnan and M. Plante, unpublished data R. Curry et al. (1993) 
υ_{f}  Nonlinearity of female reproductive success  –  2  (5)  Model assumption 
υ_{m}  Nonlinearity of male reproductive success  –  2  (6)  Model assumption 
κ_{1}  Beverton–Holt constant  –  0.0076  (7)  Chosen to be consistent with M. Power and G. Power (1995) 
κ_{2}  Beverton–Holt constant  –  1 117 391  (7)  Chosen to be consistent with M. Power and G. Power (1995) 
β_{s}  Strength of conditionspecific effects on mortality  –  4  (8)  N. Broekhuizen et al. (1994) 
β_{1}  Strength of sizeunspecific effects on mortality  –  −1.90  (8)  Based on P. Magnan and M. Plante, unpublished data 
β_{2}  Strength of sizespecific effects on mortality  cm^{−1}  0.011  (8)  Based on P. Magnan and M. Plante, unpublished data 
α  Sizespecific consumption coefficient  g^{1−γ} day^{−1}  0.07  ((9)  Based on P. Magnan and M. Plante, unpublished data 
γ  Sizespecific consumption exponent  –  0.68  (9)  Based on P. Magnan and M. Plante, unpublished data 
ϑ_{1}  Maximum consumption rate  liter^{−1} day^{−1}  164  (10)  M. L. Koski and B. M. Johnson (2002) 
ϑ_{2}  Halfsaturation resource density in consumption rate  liter^{−1}  42.2  (10)  M. L. Koski and B. M. Johnson (2002) 
L  Volume of shared habitat  liter  1.8 × 10^{7}  (10)  Chosen to be consistent with P. Magnan et al. (2005) 
r  Intrinsic growth rate of resource  day^{−1}  0.1  (11)  D. Claessen et al. (2000) and consistent with F. Marchand et al. (2002) 
K  Carrying capacity of unharvested resource  –  1.8 × 10^{9}  (11)  Chosen to be consistent with F. Marchand et al. (2002)and P. Magnan et al. (2005) and D. Claessen et al. (2000) 
λ  Product of zooplankton mass (50 μg and a dimensionless trophic conversion factor (0.61)  g  0.00003  (11)  D. Claessen et al. (2000) and A. E. Gamble et al. (2006) 
δ  Metabolic rate  day^{−1}  0.005  (13)  Based on P. Magnan and M. Plante, unpublished data 
T  Length of the year  day  365  (16)  
σ  Standard deviation of the stochastic component of the growth equation  g^{1−γ}  0.700  (14)  Based on P. Magnan and M. Plante, unpublished data 
ρ_{J}  Juvenile maximum condition  –  1.6  –  G. Broekhuizen et al. (1994) 
ε  Fraction of energy intake allocated to reversible mass used for reproduction  –  0.32  (17)  Chosen to be consistent with P. Magnan and M. Plante, unpublished data 
Demography
Reproduction
The probability that an immature individual matures during a year is determined by its PMRN in conjunction with its age A and total body length l. An individual's body length l(X) = pX^{u} is allometrically determined by its irreversible mass X. Estimation of the parameters p and u is described in Appendix A.1. An individual's PMRN describes the length and agespecific probabilities of maturation between one season and the next. We assume that these PMRNs have logistic shape (as illustrated in Fig. 1A–C) and that their widths are constant across ages (as illustrated in Fig. 1D),
 (2)
where d is the PMRN width, measuring the difference between lengths leading to 25% and 75% maturation probability. We assume that the length l_{p50,A} at which the maturation probability equals 50% at age A can be described by the linear function (Fig. 1D)
 (3)
with a PMRN intercept ν and a PMRN slope σ that are specific to each individual (Dunlop et al. 2007; Dunlop et al. 2009b).
Brook charr breed once per year, and the model assumes random mating between males and females, conditioned on gonadal mass. The number F of fertilized eggs in the population is proportional to the total gonadal mass of the female population,
 (4)
where n_{f,t} is the total number of mature females in year t, G_{j} is the gonadal mass of female j, and W_{0} is the average mass of an egg (Appendix A.1). For each egg, the mother and father are drawn at random from mature males and females in the population. The probability that the egg comes from mature female i is a nonlinear function of the female's relative gonadal mass G_{i} in the population,
 (5)
The nonlinearity in reproductive value induced by υ_{f} > 1 reflects the positive correlation between gonadal mass and body size, as well as the superior ability of larger females to ensure offspring survivorship by, for example, identifying superior redd sites or remaining on spawning grounds longer to find suitable mates (e.g., Blanchfield and Ridgway 1997). Moreover, larger females also produce larger eggs, which in turn improve offspring survival (Mann and Mills 1985; Sehgal and Toor 1991; Maruyama et al. 2003).
Similarly, the probability that mature male i fertilizes a given egg is a function of its structural and gonadal mass, X_{i} + G_{i}, relative to the mass of other mature males in the population. To reflect the common observation that body length is positively correlated with reproductive value in males (e.g., Power 1980), a male's structural mass is incorporated into calculating its probability of fertilization,
 (6)
where n_{m,t} is the total number of mature males and υ_{m} determines how strongly a male's reproductive value increases with its relative irreversible mass.
Natural mortality
In any given year, the number Φ of newborns that survive to the growing season is related to the population's total egg production (Power and Power 1995). Here we model the recruitment Φ according to a BevertonHolt recruitment function (e.g., Beverton and Holt 1957),
 (7)
After the surplus newborns have died, individual survivorship from one year to the next is assumed to be related to total body mass W and condition Y/W at the beginning of the year,
 (8)
This functional form describes an exponential increase in survival with improved condition, and a sigmoidal increase in survival with larger body mass, and thus captures the typeIII survival relationships consistently found for salmonids (e.g., Power 1980). The estimation of parameters β_{1} and β_{2} is described in Appendix A.1, while the value of β_{s} was taken from the literature (Table 1).
Resource consumption
To describe resource competition among brook charr, we focused on zooplankton, one of the brook charr's major prey in some situations, such as when competitors (e.g., the white sucker Catostomus commersoni) are present, or when benthic organisms are rare (e.g., Magnan 1988; Tremblay and Magnan 1991, and Magnan et al. 2005). We also focused on zooplankton because they were the prey item for which sufficient data were available to parameterize our model.
The resourceconsumption rate E_{g,i,τ} of individual i at time τ during a year is a function of its body weight W and of the resource (zooplankton) number R,
 (9)
where h(R) describes the proportion by which an individual's resource consumption is diminished when the resource density R falls short of the maximum daily consumption rate, based on a typeII functional response (e.g., Koski and Johnson 2002),
 (10)
where ϑ_{1} is the maximum number of zooplankton that can be eaten in a day, ϑ_{2} is the half saturation constant, and L is the volume of the habitat shared by zooplankton and fish.
We assume that the resource changes according to semichemostat dynamics (e.g., Claessen et al. 2000) and predation by charr,
 (11)
where λ is the product of the average weight of an individual zooplankton (e.g., Gamble et al. 2006) and a conversion factor that takes into account assimilation efficiency (e.g., Claessen et al. 2000), E_{g,i,0} is the consumption rate of zooplankton by individual i at the beginning of the year, and n_{t} is the population size of brook charr at the beginning of year t. We assume that the resource's population dynamics are much faster than the charr's population dynamics and that the impact of the charr population on the resource is roughly constant throughout the year. Thus, in each year t the resource density quickly attains an equilibrium,
 (12)
Somatic growth
A brook charr with mass W grows according to the instantaneous growth equation (e.g., West et al. 2001)
 (13)
where δ describes the brook charr's metabolic rate. Thus, we assume that when the mass derived from consumption exceeds metabolic costs, the surplus mass is invested in somatic growth. Since R is assumed to be constant throughout the year (once it has equilibrated to R_{t}) and since γ < 1, the growth increment ΔW(R_{t}) from year t to year t + 1 can be obtained as
 (14)
where ζ is a normally distributed random variable with mean 0 and standard deviation σ, and
 (15)
and
 (16)
(Hiyama and Kitahara 1993), where T is the length of the year. Individual brook charr are assumed to be born with the maximum ratio ρ_{J} between reversible mass and irreversible mass. As long as resources are not limiting, juvenile brook charr will maintain this ratio of reversible to irreversible mass. The fraction F_{A}(X_{t},Y_{t}) of ΔW(R_{t}) that is allocated to irreversible mass is an empirically derived function of irreversible and reversible mass at the beginning of the growing season (Appendix C).
In mature individuals, a fixed fraction ε of (1−F_{A} (X_{t},Y_{t}))ΔW(R_{t}) is set aside for reproduction. In females, this takes the form of allocation to gonadal mass G, while in males this includes, for example, the loss in mass associated with searching and securing redds. For simplicity, we assume that gonadal mass in juveniles is negligible. Thus, at the end of the growing season, reversible, irreversible, and gonadal mass are given by
 (17)
where I_{f} equals 0 if the individual is immature, and equals 1 if the individual is mature, and [x]_{+} = max(x,0). The growth equations above are based on the assumption that all gonadal weight from the previous year is spent on reproduction. The estimation of parameters of the growth model is described in Appendix A.1. A Mann–Whitney test indicated a good fit between the modelpredicted size distribution and the observed sizefrequency data (Appendix A.2), suggesting that the model's structural assumptions and parameters are appropriate for describing the modeled populations of brook charr.
Genetics
We adopt the ecogenetic modeling approach, and hence its reliance on the principles of quantitative genetics, to characterize the evolution of the PMRN as a result of harvesting in our simulated brook charr populations (Dunlop et al. 2007 and Dunlop et al. 2009b). We model both males and females, with sex being determined randomly assuming an even sex ratio at birth and in the initial population.
Trait inheritance
An individual's expressed size at maturation is a function of genetic and environmental effects. Each individual charr i possesses two quantitative traits that determine its PMRN, the PMRN slope σ_{i} and the PMRN intercept ν_{i}. These traits undergo diploid inheritance and expression through a sequence of two steps.
First, each parent produces a haploid gamete that is envisaged to contain, at random, half the parental alleles. As these alleles are not explicitly modeled in quantitative genetics, deviations in gametic genetic values from parental genetic values as a result of the combined effects of recombination, segregation, and mutation are described by random deviates. The genetic values of a gamete produced by parent i thus equal σ_{i} + ϱ and ν_{i} + ϱ, where ϱ is randomly drawn from the normal distribution N(0,(zσ_{i})^{2}) with probability μ and equals σ_{i} with probability 1 − μ, where μ describes the probability that the gamete's genetic value is recognizably different from the parent's genetic value; ϱ is drawn analogously. Although z, the coefficient of variation of the gametic genetic value from the parental genetic value is assumed to be constant throughout time, the magnitude of the recombinationsegregationmutation effect varies as the parental genetic values evolve.
Second, the offspring's genetic values are obtained as the midparental values resulting from the union of two gametes, given by the arithmetic means of the gamete's genetic values. When these midparental values are phenotypically expressed, environmental variance is added in the form of a normally distributed random deviate with a mean of 0 and a standard deviation equal to a fraction k of the midparental value. The coefficient k of environmental variation was chosen to yield heritabilities of 0.15, consistent with common observations for numerous lifehistory traits (e.g., Mousseau and Roff 1987; Roff 1997). Values for z, μ, and k are given in Table 2.
Table 2. Genetic parameters and their numerical values in the model. Parameter  Description  Value  Source 


z  Coefficient of recombinationsegregationmutation variation in PMRN traits  0.05  Approximate midpoint of the range in Claessen and Dieckmann (2002) for an analogous parameter for an asexually reproducing population 
μ  Probability that a gametic genetic value is recognizably different from its parental genetic value  0.01  Based on the range in Bürger (2000) for a similar parameter 
��  Coefficient of initial genetic variation in PMRN traits  0.2  Doubled from values in Houle 1992 for lifehistory traits 
k  Coefficient of environmental variation in PMRN traits  0.05  Chosen to result in a narrowsense heritability for the PMRN traits of approximately 15%, which is commonly observed for lifehistory traits (e.g., Roff 1997) 
Initial genetic structure
We chose the initial populationlevel means σ_{0} and ν_{0} (Table 1) of the two genetic PMRN traits in accordance with the average PMRN of the modeled brook charr populations in unharvested lakes, using the estimation method described by Barot et al. (2004) and matching the estimation results to a linear PMRN with constant width (Appendix B). Figure 1 shows the PMRN thus obtained. The effect of age on maturation probability is relatively small compared to the effect of body length, indicating that the latter is the dominant indicator of maturation probability for brook charr from unharvested lakes.
To describe genetic variation in the PMRN traits,which allows selection to occur with and without harvesting, we initialized populations with combinations of the two genetic traits with variances (��σ)^{2} and (��ν)^{2} around the population's initial mean PMRN slope and intercept, σ_{0} and ν_{0}, respectively. The standard deviations of these two normal random distributions were set to a fifth of the respective mean genetic trait values (�� = 0.2; Table 2) to ensure that the initial genetic coefficients of variation (e.g., Houle 1992 and Bürger 2000) were of an order of magnitude that is comparable to values reported for lifehistory traits in Houle (1992). These genetic coefficients of variation determine a population's ability to respond to selection. The values given in Houle (1992) were doubled to reduce the effect of the initial distribution of breeding values on subsequent evolutionary trajectories. Moreover, by increasing the genetic coefficient of variation, we also sought to ensure that by the time fishing began, there was still sufficient genetic variation in the PMRN traits on which selection could act. Most importantly, when fishing began, the genetic coefficient of variation in the model (mean coefficient of variation of 14% with a standard deviation of 2% for both PMRN traits) approached values reported in Houle (1992) for other lifehistory traits.
Genetic assumptions
The model choices described above are consistent with empirical work on the genetics of lifehistory traits in other taxa. For instance, lifehistory traits are believed to generally have low additive genetic variances (e.g., Mousseau and Roff 1987; Price and Schluter 1991, and Bürger 2000). Thus, we assume the proportion of genetic variance in a particular trait's genetic value attributable to the effects of recombination and segregation to be small. Moreover, existing research suggests that most mutations have negligible effects (e.g., Lynch and Walsh 1998), indicating that the contribution of mutation to genetic variation is restricted. Most mutational variance appears to be attributable to a few mutations of large effect in, for example, Drosophila (e.g., Bürger 2000 and references therein). Given this limited potential for genetic differences between parental genetic values and gametic genetic values for lifehistory traits to arise through recombination, mutation, and segregation, we introduced the parameter μ to control for how frequently one can expect the combined effect of these processes to produce gametic genetic values of a trait that are recognizably distinct from the corresponding parental genetic value.
Our model also assumes no interaction effects and free recombination between the loci controlling the PMRN intercept and slope, even though some degree of pleiotropy or linkage may well exist between these quantitative traits. In addition, genetic correlations and genetic or developmental constraints could limit the degree of evolutionary change in the modeled brook charr populations. However, in the absence of studies revealing the degrees of genetic interactions and genetic correlations between these two traits, we feel that the case of no interactions and free recombination provides a basis on which to develop further work examining the effects of relaxing these simplifying assumptions by investigating more complicated genetic settings. Indeed, our model, phrased in terms of individual quantitative traits, is formally equivalent to a singlelocus, infiniteallele model for each trait. On that basis, one could readily extend our analyses to include complications such as pleiotropy and linkage between the PMRN traits.
For simplicity, we also assumed that lifehistory traits other than the two PMRN traits – such as offspring size, investment in gonads, and rates of somatic growth – are not subject to significant evolutionary change over the modeled time frame. This could apply because of low selection pressures or low genetic variation, with the former assumption being supported by results in Dunlop et al. (2009b), Dunlop et al. (2009a)), and Enberg et al. (2009).
Harvesting
The goal of this study is to identify harvest regimes that best mitigate harvestinduced evolutionary change. We compare a range of potential harvest regimes, and characterize each by three parameters, a, M_{s}, and b_{H}. The first two parameters describe the sizeselectivity of harvesting. Specifically, the selectivity to which a fish of mass W is subjected is assumed to be given by , where I_{x}(a,b) describes the cumulative distribution function in x of a beta distribution with shape parameters (a, b) (Fig. 2) and M_{s} describes the length, as a fraction of the maximum observed length for brook charr (700 mm), at which selectivity equals 50%. For comparison, the length at which maturation probability equals 50% for 2yearolds is approximately 30% of the maximum length in the population. The parameter a controls the steepness of the selectivity curve around M_{s} (Fig. 2), and thus, in effect, how sizeselective the harvest regime is. For example, if the 50th percentile of the sizeselectivity function is interpreted as the minimum catch size, a can be interpreted as the stringency with which the minimum catch size is enforced. The third parameter, b_{H}, governs the density dependence of harvesting. In particular, the total allowable catch in year t is assumed to be given by
 (18)
(e.g., Hilborn and Walters 1992), where
 (19)
is the total harvestable biomass in year t, with the sum extending over all n_{t} brook charr in year t and a_{H} is the total allowable catch for H_{t} = 0. The small nonzero constant a_{H} describes rare poaching and bycatch mortality (e.g., Hilborn and Walters 1992). We assumed that the fraction of biomass harvested when b_{H} = 0 was minimal, and therefore a_{H} was set equal to 50 g, the approximate mass of a single charr at full condition with a maturation probability of 50%. For virtually all harvest regimes considered here, this meant that the densitydependent component of harvesting, b_{H}, approximately equaled the harvest probability, i.e., the fraction of the total harvestable biomass H_{t} designated as total allowable catch ��_{t} in year t. The parameters used to characterize the harvest regimes are summarized in Table 3.
Table 3. Harvesting parameters. Parameter  Description  Value or range  Unit  Comments on range 

M_{s}  Length at which selectivity equals 50%, as a fraction of the maximum observed length  0.17–0.43  –  Chosen to cover approximately the sizes at 25% through 75% maturation probability across the age ranges examined (e.g., Fig. 1D) 
a  Steepness of the sizespecific selectivity curve around M_{s}  0.5–25  –  Chosen to cover a range of shapes of the selectivity curve (e.g., Fig. 2) 
b_{H}  Density–dependent component of harvesting  0.01–0.2  –  Chosen to cover a range of harvest regimes not resulting in extinctions 
a_{H}  Total allowable catch for H_{t}=0  50  g  Chosen to minimize the fraction of biomass harvested when b_{H}=0 
ϕ  Relative importance managers attach to avoiding harvestinduced evolutionary change  0.1–2  –  Chosen to cover a range of distinct levels importance attached to avoiding harvestinduced evolutionary change 
The fishing season is assumed to run concurrently with the growing season, but the probability that an individual charr is harvested depends on its size at the beginning of the fishing season. Thus, the annual probability μ_{H} with which a brook charr at weight W_{i} is harvested is given by
 (20)
We highlight that this probabilistic treatment of harvesting, when applied to a finite brook charr population, implies sampling variation in annual yield ��_{t}.
Evaluation of harvest regimes
Choosing suitable time horizons
Each model run was initiated with 1500 individuals. To evaluate the effect and desirability of different harvest regimes, the model described above was run for 100 years without harvesting (e.g., Tenhumberg et al. 2004) to allow the population to reach a demographic steady state and allow the buildup of an endogenous genetic structure. In the 101st year of model runs, harvesting began, and the brook charr population became subject to the harvesting mortality μ_{H}. The model runs proceeded for another 50 years, or until the brook charr population went extinct. Thus, we investigated a management time frame of 50 years during which the same harvest regime was consistently applied. We focused on combinations (a,M_{s},b_{H}) of parameters of the harvest regime that did not lead to deterministic extinction of the brook charr population during the first 50 years of harvesting. Focusing on a 50year time horizon for harvesting is desirable for providing insights to decision makers: the model is parameterized to reflect the current life history of the brook charr populations and aims at providing decision makers with relevant information about the effects of different harvest regimes within a time frame that can be deemed relevant for current management.
Although the current genetic distributions of the PMRN coefficients in brook charr populations are unknown, using the distribution of genotypes after running the model for 100 years can be justified on a number of grounds. First, the coefficients of genetic variation for the PMRN coefficients resembled values of the genetic coefficients of variation observed for lifehistory traits in other taxa (e.g., Houle 1992 and Houle 1996). Second, the distribution of genotypes after 100 years was unimodal, approximately symmetric, and qualitatively similar to the distribution of observed lifehistory traits in other taxa, as well as to the normal distribution of traits assumed in many other studies (e.g., Houle 1996; Lynch and Walsh 1998; Bürger 2000). Finally, the mean trait values after the model was run without harvesting for 100 years, or even for 15 000 years, were not significantly different from values measured in the field. Thus, we feel the distribution of genotypes after 100 years provided a reasonable approximation of natural genetic variation in the brook charr populations.
Indeed, an earlier study that also examined the impact of different management regimes on harvestinduced evolution found that, for an explicit multilocus model, simulating 100 years without harvesting allowed the distribution of allele frequencies to stabilize (Tenhumberg et al. 2004). Because our model does not involve complications that result from explicit multilocus genetics (such as linkage disequilibrium between loci resulting from the model's initialization), we decided that running the model for 100 years prior to harvesting was sufficient to lessen the effects of transient population dynamics and of the initial genetic distribution of PMRN traits.
To assess the plausibility of the model's evolutionary equilibrium, we also ran 100 replicates of the model without harvesting for 15 000 years, i.e., for the approximate number of years since the brook charr colonized the study area (e.g., Angers and Bernatchez 1998). The average values of the predicted PMRN traits after 15 000 years without harvesting (283 mm and 7.15 mm year^{−1} for the PMRN intercept and slope, respectively) were well within two standard deviations of the PMRN coefficients estimated for unharvested populations in the field. We therefore conclude that our model evolved to reasonable values for unharvested brook charr.
Because it could possibly take longer than 15 000 years for the PMRN traits to reach evolutionary equilibrium, we sought to assess how quickly evolutionary change was occurring in the PMRN traits in the absence of harvesting. For the 100 replicates of the model that were run without harvesting for 15 000 years, we fit an exponential decay model (e.g., Ritz and Streibig 2005) to the difference between the PMRN traits in a given year and the PMRN traits after 15 000 years. The resultant fits suggest a deterministic rate of relative evolutionary change, in the absence of harvesting, of 0.08% per year for the PMRN intercept and of 0.03% per year for the PMRN slope.
Assessing harvestinduced evolution under residual trends and genetic drift
Residual deterministic trends and stochastic genetic drift affecting the evolving traits under natural selection cause uncertainty in assessing the evolutionary effects of harvesting. In particular, genetic drift could predominate when intensive harvesting strongly reduces population abundance. Even in large marine stocks, the effective population size of exploited fish stocks can be several orders of magnitude smaller than their census population size (e.g., Hauser et al. 2002). The potential for genetic drift resulting from harvesting is exacerbated in freshwater stocks, where census population sizes are considerably smaller than in marine stocks even in the absence of harvesting. For instance, Fraser et al. (2004) used microsatellites to estimate the number of breeders in seven populations of brook charr, and found that results ranged between 57 and 200 individuals in six of the seven populations. Because the brook charr populations considered in our study inhabit relatively small lakes, with an average surface area of approximately 13 ha, these stocks have relatively low population sizes compared to many marine stocks. As harvesting can further diminish these population sizes, it is desirable to adopt a probabilistic framework for comparing evolutionary trends in harvested and unharvested populations. Indeed, quantifying the probability that a particular harvest regime causes evolutionary change addresses this inherent uncertainty. Comparing the magnitude of evolutionary changes under a particular harvest regime to the average magnitude of changes expected in the absence of harvesting provides one approach to achieving this.
To quantify how much evolutionary change we expect in the absence of harvesting, we ran 2500 replicates of our model for the full 150 years without harvesting. For each replicate model run without harvesting, we calculated the differences in the population's mean values of the PMRN traits between the end (year 150) and the beginning (year 0) of the simulation. We thus determined the empirical distribution of the amount of evolutionary change that would occur in the PMRN traits over 150 years in the absence of harvesting (Fig. 3).
For a particular harvest regime ℋ, we evaluated in each model run the differences in the population's means of the evolving traits between the simulation's end (σ_{ℋ,150} and ν_{ℋ,150} for the PMRN intercept and slope, respectively) and its beginning (σ_{ℋ,0} and ν_{ℋ,0} for the PMRN intercept and slope, respectively). We then calculated their twosided empirical Pvalues, that is, the probability that the magnitude of evolutionary change in the absence of harvesting would be at least as large as the predicted magnitude of evolutionary change resulting from a particular harvest regime. These Pvalues were calculated by comparing the changes in the population means of the PMRN traits for a particular harvest regime with the distribution of such changes without harvesting. The measured amount of evolutionary change a harvest regime caused in a given model run was characterized by the smaller of the two empirical Pvalues thus obtained for changes in the two PMRN traits. The goal here was not to categorically accept or reject the null hypothesis of no evolutionary change. Rather, we sought to use the smaller of the two empirical Pvalues to quantify the probability of a given harvest regime causing evolutionary change as extreme as or more extreme than would be expected in the absence of harvesting. Thus, if σ_{��,t} and ν_{��,t} describe the mean PMRN traits in populations without harvesting, then the probability ℰ that a given harvest regime causes evolutionary change as extreme or more extreme than would be expected in the absence of harvesting was quantified as
 (21)
Based on this construction, 1 − ℰ can be interpreted as the confidence with which an observer can conclude that harvestinduced evolution is occurring. The probabilities in the expression above were obtained by integrating the relative areas within the corresponding tails of the frequency distributions in Fig. 3. We used a large sample of 2500 model runs to adequately characterize the distribution of genotypes in the absence of harvesting. However, for each harvest regime, we were not ultimately interested in the distribution of genetic values, but rather in their means. Trial and error indicated that about 15 replicate model runs were sufficient to estimate these means consistently. For each harvest regime, we also evaluated the mean annual yield during the years in which harvesting took place. We ran 15 replicates of each harvest regime, and characterized the evolutionary change caused by a harvest regime as the mean of ℰ across the 15 replicates. Similarly, the yield of a given harvest regime was estimated by taking the average of the mean annual yield across the 15 replicates. In this manner, our assessment of the evolutionary effect of a harvest regime accounts not only for residual evolutionary trends potentially occurring in the absence of harvesting, but also for uncertainty in predicted evolutionary outcomes in the absence and presence of harvesting.
Evaluating the desirability of harvest regimes
We expect fisheries managers and other stakeholders to vary in their concern for the risk of fisheriesinduced evolution in a particular fishery. We therefore introduce a parameter ϕ to describe the relative importance that the avoidance of harvestinduced evolution has to management decision making. Low values of ϕ describe a situation in which managers are comfortable implementing a harvest regime that risks significant evolutionary change, whereas high values of ϕ magnify the relative importance of evolutionary change to management decision making, and thus describe situations in which managers or other stakeholders are averse to inducing evolutionary change. On this basis, we characterize the desirability of a particular harvest regime by the product of mean annual yield with ℰ^{ϕ}. Consequently, regimes that cause a lot of evolutionary change (corresponding to consistently low ℰ) have their mean annual yields penalized considerably relative to regimes that cause little evolutionary change. Our goal is to identify those regimes that maintain a high yield while simultaneously limiting the amount of harvestinduced evolutionary change.
Discussion
 Top of page
 Abstract
 Introduction
 Methods
 Results
 Discussion
 Acknowledgements
 Literature cited
 Appendices
The magnitudes of harvestinduced evolutionary changes in probabilistic maturation reaction norms (PMRNs) found in our study were broadly consistent with qualitative predictions made in the existing literature on harvestinduced maturation evolution. In particular, as in Ernande et al. (2004); Dunlop et al. (2007), and Dunlop et al. (2009b), we found that harvesting resulted in evolutionary shifts of PMRN traits that caused earlier maturation at smaller size. Our results suggest that intense harvesting of brook charr extending below the size at 50% maturation of 2yearold fish readily causes evolutionary changes in the PMRN. As we systematically averaged results across replicate model runs and extensively sampled the range of potential harvest regimes, we believe that these results are robust. In particular, were random genetic drift, rather than directional selection, a major driver of the modelpredicted evolutionary changes, differences between harvested and unharvested populations would not exhibit directional trends. The directional evolutionary trends we found in PMRN traits under a range of intense harvest regimes and across many replicate model runs therefore suggest that genetic drift is not a primary cause of the described harvestinduced evolution of PMRN traits.
Our results suggest that to reliably avoid harvestinduced maturation evolution in lacustrine brook charr, harvest regimes should be designed to limit the harvest to less than approximately 15% of the population's biomass, or restrict harvesting to individuals larger than 50% of the observed maximum length. Moreover, intensive harvesting extending to smaller brook charr proved undesirable from the perspectives of minimizing harvestinduced evolution and maximizing annual yield. One feature of our results that underscores the tradeoff inherent in the choice of harvest regimes is that, regardless of the harvest's sizeselectivity, less intense harvesting caused minimal evolutionary change (Fig. 4), but also resulted in low yields (Fig. 6). The tradeoff between annual yield and evolutionary change is aggravated when managers (or other stakeholders) are very concerned about harvestinduced evolution (i.e., when ϕ, measuring the relative importance of avoiding harvestinduced evolution to management decision making, is greater than 1; Fig. 9C,D). Conversely, the tradeoff is relaxed when managers are relatively unconcerned about harvestinduced evolution (ϕ < 1, Fig. 9A,B). In the latter case, risking harvestinduced evolution by intensively harvesting smaller individuals becomes an acceptable outcome.
Our results also underscore that from the perspective of maximizing mean annual yield, intensively harvesting immature brook charr is, in any event, undesirable. Low yields need not be a consequence of the evolutionary effects of harvesting, as models that do not include evolution also show that increased juvenile mortality can result in much reduced population biomass (e.g., Chesson 1998) and, hence, in reduced yields. Our model accounts for the length–dependence of maturation probabilities, so that, as in reality, harvesting ever smaller individuals implies that more juveniles are harvested. Even in the absence of any evolutionary effects of harvesting, yields can be lowered through recruitment overfishing, i.e., through the demographic consequences of increased juvenile mortality. Indeed, it is widely understood that additional mortality imposed on small, immature fish can have strongly negative effects on the sustainability of fisheries, and therefore ultimately also on yields (e.g., Beverton and Holt 1957; Getz and Haight 1989; Beck et al. 2001; Roberts et al. 2005). Hence, there are good reasons, apart from the risk of harvestinduced evolution, to eschew harvest regimes that intensely exploit small, immature fish. It is therefore important that our approach to evaluating the desirability of alternative harvest regimes accounted for the demographic as well as the evolutionary effects of harvesting.
We chose to commence harvesting after running our model for 100 years without harvesting, and we initialized the model with the PMRN traits observed in the field. As, in the model as well as in nature, residual evolutionary transients may exist as a result of natural directional selection pressures that have not yet had enough time to run their course, and as resultant trends in genetic traits may thus be superimposed on the response of populations to harvesting, we devised a general probabilistic approach to quantifying the likelihood of harvestinduced evolutionary change. We achieved this by comparing the magnitudes of evolutionary changes in unharvested populations with those in harvested populations across multiple model runs, so as to estimate the probabilities, separately for each evolving trait, that the former exceed the latter. We then took the smallest of these probabilities to quantify the confidence (Equation 21) with which an observer may conclude that the evolutionary changes encountered during the harvest period are merely the consequence of genetic drift and/or residual evolutionary transients. In this manner, our probabilistic approach can also deal with environmental trends, such as with those implied by climate change, and their evolutionary consequences. We therefore expect this approach to be applicable under a broad range of management scenarios.
While we have assumed throughout our analysis that managers will seek to avoid harvestinduced maturation evolution, some fisheries managers may regard such evolution as desirable under certain circumstances. For example, harvest regimes could potentially select for increased growth rates (e.g., Conover and Munch 2002), or fisheries may target mature individuals to select for later maturation at larger size (e.g., Heino 1998). In some cases, such change may be considered beneficial if it enhances a stock's resilience to harvesting in the short term (e.g., Enberg et al. 2009). We expect our approach to be applicable to situations in which harvestinduced evolution is valued differently. Technically, this may be achieved as easily as by replacing ℰ with 1 − ℰ in the definition of a harvesting regime's desirability.
The model used in this study did not consider gene flow from unharvested populations, which can counteract local adaptation in harvested lakes towards smaller maturation size (e.g., Kawecki and Ebert 2004). Gene flow between oligotrophic lakes in a boreal forest is probably quite limited on the short temporal scale considered here. Nevertheless, depending on the strength of selection, even extremely limited amounts of gene flow can homogenize populations (e.g., Hartl and Clark 1989). Transplanting brook charr from unharvested to harvested lakes could therefore be a viable option for mitigating harvestinduced evolution.
Some other simplifying assumptions could also affect our estimation of the shortterm effects of harvesting regimes. For instance, if there were strong interactions between the evolutionary effects of harvesting and residual evolutionary transients, our comparative approach could misestimate the former. For example, if strong stabilizing selection operates on the PMRN traits independently of harvesting, and if this stabilizing selection has not yet run its course when harvesting commences, the effects of even mild changes in the harvesting regime on the PMRN traits after 50 years could be amplified by the residual stabilizing selection. By initializing the model with trait values observed in the field and by commencing harvesting after 100 years of natural selection, we might therefore be underestimating the predicted evolutionary effects of milder harvesting regimes. Furthermore, the relatively high genetic coefficient of variation used to initialize the model elevates the propensity of the PMRN traits to respond to harvestinduced selection. If that coefficient were lower, the evolutionary response to harvesting over the time frame considered here would not be as pronounced (e.g., Dunlop et al. 2007).
In spite of some simplifying assumptions adopted in this study, we expect that the framework developed here can help guide the design of harvest regimes in brook charr. The comparative evaluation of different harvest regimes is mandated by a precautionary approach to fisheries management (e.g., Fenichel et al. 2008). The defining feature of the precautionary approach is the development of riskaverse objectives (Food and Agricultural Organization of the United Nations 1995; Peterman 2004). Moreover, harvest regimes may need to be designed to comply with legal mandates or ethical imperatives to conserve standing genetic variation (e.g., Humphries et al. 1995 and Balmford et al. 2005). Additionally, fisheriesinduced evolution could impede recovery efforts for a fishery or have cascading ecosystem effects (e.g., Enberg et al. 2009). Hence, when harvestinduced evolution poses unacceptable risks to yields, fish stocks, or ecosystems, or when there are pressing legal or ethical guidelines to minimize fisheriesinduced evolution, stakeholders should carefully assess ‘worstcase’ scenarios that are particularly likely to induce evolutionary change. Indeed, a sound, transparent, and wellcommunicated understanding of such scenarios can contribute to avoiding them.
We expect the results of our modeling work to be generalizable to formulating management strategies for other harvested fish species with similar life histories. By integrating sizespecific lifehistory processes with elements of harvestinduced lifehistory evolution and management strategy evaluation, we expect our approach and results to shed new light on the causes and consequences of harvestinduced evolution in fish and thereby aid the development of sustainable harvest regimes.