Our model is multi-generational and tracks the simulated population at annual time steps. Although the model describes individual-level fish dynamics, the model actually tracks ‘super-individuals’ with each super-individual potentially representing a multitude of individual fish (e.g., Scheffer et al. 1995). This approach is useful as it does not necessitate the elimination of simulated fish whenever there is a mortality event (mortality simply causes a reduction in the number of individuals represented by a super-individual), thereby allowing for efficient simulation of large populations. In addition to the number of individual fish (NI) represented by a super-individual, other individual-level (i.e., I-state) variables include: sex (binary), age (years), maturation status (binary), total length (mm), somatic weight (g), gonadal weight (g) and heritable growth and maturation parameters. Each year individual fish grow, potentially mature, and experience some mortality. Between annual time steps (y and y + 1), mature individuals reproduce. The resulting new individuals (which are characterized by heritable growth and maturation parameters obtained from reproductively successful adults) enter the population as age 1 fish at the beginning of annual time step, y + 2. Our inheritance design for growth and maturation parameters is similar to previous applications (e.g., Strand et al. 2002) and based on Mendelian genetics. For each maturation and growth parameter, a new individual receives one random heritable value from both its mother and father. Parameter expression is based on co-dominance (i.e., each parameter is expressed as the mean of paired values). Our inheritance design facilitates modeling genetics without having to assume that traits are normally distributed. Also, while we did not define heritability of traits in the model, maturation and growth traits are subject to both genetic- and density-dependent effects, thereby heritability <1. Moreover, while some studies suggest that growth and maturation traits may co-evolve (e.g., Hard et al. 2008), empirical measures of genetic structure or correlation for these traits are lacking. Thus, we assumed no dependence among growth and maturation parameters or between parameters of each trait. Consequently, expression of evolving traits in our model might be more flexible than a model that imposed covariance among genetic parameters. We specified a maximum age of 50 years, but none of our results presented here depended on the maximum age.
The growth model tracks two tissue types (somatic and gonadal) of equal energy density, i.e., growth of gonads occurs at a direct cost to growth of soma, and is a biphasic growth model (Quince et al. 2008a,b) building on the von Bertalanffy growth model. Spawning occurs at the end of each year, and consequently at the beginning of a year, individuals’ gonads weigh 0 g. Individual growth rates are dependent on population abundance, stochastic processes, and heritable growth parameters.
Each individual is characterized by 12 pairs of heritable growth values and means of each pair determine an individual’s 12 growth parameters (X1–X12; six parameters relate to growth of males, X1–X6, and females, X7–X12, respectively). These growth parameters can take values between 0.1 and 1, and are initially drawn from a uniform distribution between 0 and 1. The lower bound, 0.1, is necessary to ensure positive growth (i.e., K > 0; see below). Growth parameters, in turn, determine an individual’s initial somatic growth rate (K) and maximum length (Lmax). We assume a negative relationship between K and Lmax (e.g., Beauchamp et al. 2004) and these variables can take a defined range of values, i.e., for an individual male:
That is, increases in X1 and X2 (e.g., induced by selection) would correspond to a relatively large K and small Lmax for an individual male. We chose coefficients in eqns (1) and (2) such that K and Lmax would take values within the observed range for walleye and lake whitefish (e.g., Beauchamp et al. 2004).
During each annual time step, an individual’s potential somatic and gonadal growth is a function of its current length and genetically determined growth parameters (K and Lmax). However, an individual’s realized growth can be affected by stochastic and density-dependent processes which can cause realized growth to either exceed or fall below potential growth. Further, whereas mature or maturing individuals can allocate energy to both somatic and gonadal tissue, immature individuals can only allocate energy to somatic tissue. To facilitate these different growth possibilities, during each time step we calculate an individual’s potential growth and then use this as a basis for calculating realized growth of both somatic and gonadal tissue.
We assume a base allometric relationship between fish length (L, mm) and somatic weight (WS, g):
Further, we assume the expected gonadal weight (WG, g) to be a proportion (P) of expected somatic weight:
with different parameter values for males and females. We select parameters for eqn (4) (Table 1) to allow for greater gonad weight for females as compared to males and increased gonadal investment with length, i.e., patterns observed for a plethora of fish species (e.g., walleye; Moles et al. 2008).
Table 1. Parameters and coefficients used for baseline and initiation simulations.
|Parameters or coefficients ||Mean value (range)|
|MRNs (mm)||400 (150–750)|
|K (year−1)||0.3 (0.02–0.65)*|
|Lmax (mm)||622 (400–800)*|
|a (g mm−b)||0.00003|
|c, d, and f (mm−1) for males||0.03, 4.0, 0.008|
|c, d, and f (mm−1) for females||0.15, 20.0, 0.01|
|G (g year−1.5)||0.0005|
|β||5.5 × 10−12|
|εR||1.0 (0, 4.05)†|
|C (deterministic; g)||2.0 × 1011|
|C (stochastic; g)||1.5 × 1011–2.5 × 1011 (5 × 1010, 3.5 × 1011)|
We assume that total (somatic and gonadal) annual potential growth by an individual is a function of individual length, and that maturation status dictates the proportion of growth allocated toward somatic and gonadal tissue. Each year, we calculate a mature individual’s total growth potential (GPT, g) as the sum of potential somatic growth (GPS, g) and gonadal growth (GPG, g). A mature individual’s length is expected to increase (ΔL) as a function of its length at the beginning of a year (Lt):
Thus, the potential somatic growth for a mature individual is,
and its potential gonadal growth is,
Total realized growth (GRT, g) is then calculated by multiplying an individual’s total growth potential (GPT) by a term (ADJ) which encapsulates density dependent and stochastic processes:
where C is mean population carrying capacity (2 × 1011 g), εC is a uniform random number between 0.5 and 1.5, Wi and Ni are the somatic weight and number of individuals represented by super-individual i, respectively, and ψ is a fluctuating variable (a sine function varied from −5 × 1010 to +5 × 1010 g) representing inter-annual growth variation in 20-year cycles. With variation in ADJ, individual growth may be suppressed (ADJ < 1) or compensated (ADJ > 1). To bound potential growth rates, ADJ can take values between 0.25 and 2.5.
Total realized growth GRT is partitioned between somatic and gonadal growth. Allocation to somatic and gonadal growth is determined by (i) current length and weight of fish and (ii) total realized growth. If GRT is sufficient, growth of both soma and gonads occurs, the allometric relationship is maintained (eqn 3) and there is full allocation to gonadal growth (eqn 4). Otherwise, allocation to soma and/or gonads is reduced.
We compare the sum (WS+G, g) of an individual’s expected somatic weight (WS, i.e., given its length at the beginning of the year, eqn 3) and gonadal weight (WG, g, i.e., given its length at the beginning of the year, eqn 4) versus its new total weight (WT, g), i.e., the sum of its weight at the beginning of the year and GRT. If growth is sufficient, i.e., WT ≥ WS+G, then an individual would grow in somatic weight [WS(t+1), g], gonadal weight before spawning [WG(′), g], and length [L(t+1)]:
If WT does not exceed WS+G, then somatic weight and gonadal weight must equal some fraction of WS and WG, respectively (see eqns 12–18). We assume that a fish would not reduce its length when receiving insufficient energy, i.e., if WT < WS+G, then
If energy intake is very low (WT ≤ WS − WG), no allocation to gonads takes place:
If energy intake is somewhat greater, i.e., if WS > WT > WS − WG, then gonads grow preferentially,
Finally, if energy intake is moderate, i.e., if WS+G > WT ≥ WS, then gonadal growth is prioritized followed by somatic growth:
If the population contains at least one mature male and female fish, then reproduction occurs between annual time steps (y and y + 1), and the resulting new individuals enter the population as age 1 fish at the beginning of annual time step, y + 2. The number of new individuals (R) entering the population is a function of the population’s viable egg production (SEP; i.e., the sum of all female super-individuals’ egg production; EPi) and follows a Ricker-type stock recruitment curve:
where εR is a log-normally distributed random variable with a mean of 1.0. We select Ricker parameters resulting in moderately over-compensatory recruitment across the population-level egg production observed in simulations (Table 1). Each year, R is divided amongst 1000 new super-individuals. If R ≤ 103, then each new super-individual represents one fish. Similarly, if R ≥ 109, then each new super-individual represents 106 fish. The parents for these new super-individuals are drawn from pools of mature female and male fish as a function of individuals’ egg and milt production. The probability that an individual mature female will be the mother for a given new super-individual is simply the proportion of the population’s total egg production contributed by the individual (i.e., EPi/SEP), and the probability that an individual mature male will be the father for a given new super-individual is simply the proportion of the population’s total milt production contributed by the individual.
A female super-individual’s egg production (EPi) is calculated as a function of its WG(′),i, NIi, proportion of viable eggs (Vi), and mean egg weight (Ei; g),
A male super-individual’s milt production (MPi) is calculated in a similar manner, but we assume that for males Vi = 1.0 (i.e., sperm viability is not a function of age), and thus,
New super-individuals enter the model as immature, either male or female (0.5 probability) age 1 fish each representing R/1000 individuals. Initial gonadal weight is by definition 0 g (i.e., immature fish), and initial length and somatic weight are determined based on an individual’s inherited growth parameters.
To elucidate model behavior, we apply the model to simulate a series of scenarios. These baseline simulations can be grouped into four general categories: (i) maturation and growth parameters are fixed, (ii) only maturation parameters are heritable, (iii) only growth parameters are heritable, and (iv) both growth and maturation parameters are heritable. The initial population consists of 5000 super-individuals, with sex and age assigned randomly. The number of individuals initially represented by each super-individual is determined as a function of age and annual mortality rate. We specified distributions for heritable maturation and growth parameters to ensure that initial populations had suitable genetic variation. For simulations with heritable maturation parameters, initial parameter values for each super-individual are drawn from a normal distribution with a mean of 400 mm and a standard deviation of 300 mm (bounded by 150 and 750 mm). Similarly, for simulations with heritable growth parameters, initial parameter values for each super-individual are drawn from a uniform distribution with bounds of 0.1 and 1 (Table 1).
For baseline simulations with fixed maturation or growth parameters, we set M1–M18 to 400 mm and X1–X12 to 0.5, respectively. We also explore the effects of the magnitude of fixed maturation parameters on inheritance of growth, and vice versa, by running simulations with M1–M18 set at 300 or 500 mm and X1–X12 set at 0.3 or 0.7. Similarly, to explore the effects of mortality rate on inheritance patterns, we ran simulations with Z = 0.3, 0.5, or 0.7 year−1. We ran all simulations under three different scenarios: (i) to begin with, we ran all simulations assuming that recruitment is deterministic (εR = 1; recruitment is solely a function of spawner stock size) and carrying capacity is fixed (εC = 1 and ψ = 0); (ii) then, to explore the effects of stochasticity, we repeated simulations assuming that recruitment and carrying capacity vary stochastically from year to year (εR and εC vary stochastically and ψ fluctuates at 20-year cycles); and (iii) finally, to explore the effects of density-dependent individual growth on inheritance patterns we repeat all simulations with density-dependent growth effects removed from the model (i.e., ADJ = 1; a super-individual’s growth is solely a function of its growth parameters). Each simulation tracks the population for 200 years, and we extract population-level information at 20-year intervals. For each scenario, we ran 10 replicate simulations.
We extended model analyses to consider the joint ecological and evolutionary effects of fishing. In so doing, we assumed that growth and recruitment are density dependent and that individuals’ growth and maturation parameters are simultaneously heritable.
Prior to simulating fishing-induced effects, we conducted deterministic initiation simulations to evaluate how heritable parameters would evolve in our simulated environment in the absence of fishing. Initial maturation and growth parameters for these simulations are presented in Table 1. The population initially consisted of 5000 super-individuals. Initiation simulations were run for 1000 years with 10 replicates, and the mean values of evolved growth and maturation parameters were extracted at 20-year intervals. We found that 1000 years was a sufficient number of time steps for the population to evolve fairly stable heritable growth and maturation. Final growth and maturation mean values from the 10 baseline simulation replicates were then used as initial mean trait values for fishing simulations.
In fishing simulations, total instantaneous annual mortality rate (Z) is the sum of natural (M) and fishing (F) mortality rates. We assumed a uniform M = 0.2 year−1 for all individuals to emphasize responses to variable F. Preliminary simulations suggested that comparisons of population responses between size-limits or across different levels of F were not sensitive to level of M or differential M with age (also see Dunlop et al. 2007).
We considered two types of size-selective mortality: (i) targeting ‘small’ (more precisely, intermediate size) fish, we allowed fish <250 mm or >650 mm to escape fully from harvest, fish between 250 and 400 mm to recruit fully to fishing gear, and fish between 400 and 650 mm to recruit partially (i.e., F decreased linearly with length within this size range) and (ii) targeting large fish, we let fish <250 mm escape harvest, fish between 250 and 400 mm partially recruit (i.e., F increased linearly with length within this size range), and fish ≥400 mm to recruit fully to fishing gear. Again, our preliminary analyses showed that simulation results were not sensitive to the exact shapes of size-dependent mortality functions (but were sensitive to the differences in length ranges harvested).
We first evaluated effects of fishing under deterministic conditions, applying two size-limits (targeting large and small fish, respectively), each for 200 years at F = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 year−1, with 10 replicates (with the same number of initial super-individuals). For all simulations, we kept track of population mean values of sex- and age-specific maturation parameters, sex-specific K and Lmax, female length-at-age and maturation schedules, recruitment, and population abundance at 20-year intervals. Finally, to consider the effects of a more variable environment, we repeated initiation (prefishing) and fishing simulations under stochastic carrying capacity and recruitment, and we compared population responses to fishing under deterministic versus stochastic conditions.