Linking Growth, Survival, and Heterogeneity through Vitality
Abstract:
We model the cross‐stage effect of juvenile growth on future cohort survival with vitality, a single stochastic measure of an organism’s survival capacity that results in death when it reaches 0. In this construct, the distribution of vitality at the end of a growth treatment stage, which is a measure of survival capacity heterogeneity, determines a cohort’s susceptibility to starvation in a subsequent challenge stage. The model predicts that the treatment‐stage duration and mass gain determine the mean and variance of the initial vitality distribution of the challenge stage, which in turn determine the effect of a challenge‐stage stressor on survival. Studies linking the effect of juvenile growth on time to starvation for chinook salmon and yellow perch are compared to model predictions. The feasibility of predicting survival and heterogeneity in overwintering fish populations from first‐year growth is considered. Some limitations and potential extensions of the model to other scenarios are discussed.
Submitted September 26, 2006; Accepted September 19, 2007; Electronically published December 7, 2007
Keywords: vitality, heterogeneity, growth, survival, chinook salmon, yellow perch.
The pattern of growth in juvenile stages affects the short‐term and long‐term survival potential of organisms across a range of taxa (Arendt 1997). The linkage is complex, with specific patterns of growth having both beneficial and detrimental consequences. For example, rapid growth allows organisms to escape from the vulnerable early stages and builds reserves that protect against overwinter starvation (Arendt 1997; Sogard 1997). However, rapid growth requires increased foraging, which increases vulnerability to predators (Mangel and Stamps 2001) and the rate of metabolic damage (Finkel and Holbrook 2000). In addition, animals on fast growth trajectories may be more susceptible to starvation (Blanckenhorn 2000). Fast growth, and catch‐up growth in particular, appears to reduce life span in organisms, whereas caloric restriction increases life span. The evolutionary connections between growth and life span are also complex. Life span tends to increase with body size across animal species and decrease with body size within species (Metcalfe and Monaghan 2003).
Modeling these complex interactions is difficult. A recent model (Mangel and Bonsall 2004) linked the rate of growth to the rate of oxidative damage in cells. The cumulative damage was then linked to the rate of mortality. The framework produced a variety of survival trajectories and was used to explore the possible value of compensatory, or catch‐up, growth in periods of resource deprivation (Mangel and Munch 2005). One important conclusion of the study was that growth‐associated damage was crucial to producing compensatory growth. Yearsley et al. (2005) developed a similar approach, in which somatic damage from resource acquisition was linked to the mortality rate. They conclude that at early ages, resource acquisition is a primary path for regulating life‐history costs. These studies and others (e.g., Munch and Mangel 2006) express a link between growth over one period and survival in another. However, they do not address how the amount of damage from growth differs among members of a cohort. This is an equally important issue because variability in survival ability is an important factor in the dynamics of mortality (Vaupel et al. 1979; Benton et al. 2006) and in life‐history patterns of mortality and fertility (Vaupel et al. 1998). Including variability can be difficult and can result in a large number of parameters; for example, the Mangel and Munch (2005) model used 11 dimensions to model the average response to growth compensation. One way to consider variability without parameter explosion is to express the stochastic loss of a single process, “vitality,” that leads to senescence and death (Anderson 1992; Weitz and Fraser 2001). However, such models have generally considered processes occurring within one life stage, such as survival over the adult stage (Aalen and Gjessing 2001) or within dose response experiments (Anderson 2000; Hamel 2001; Springman et al. 2005).
In this article, our goal is to link the cross‐life‐stage effects of growth to heterogeneity in survival by extending the vitality model into a cross‐stage framework. We develop approximations describing the effect of growth on the loss of vitality over a treatment stage and then express the effects of decreased vitality on survival in a subsequent challenge stage. We evaluate the model with two studies of the effects of growth on time to starvation in fish and consider the application of the model to studying the effect of first‐year growth on overwinter survival.
Model
Vitality Distribution and Survival Equation
Representing aging and death as the result of small, random accumulations of physiological damage was proposed a half‐century ago (Sacher 1956; Strehler and Mildvan 1960), and the idea has been explored intermittently in demography (Vaupel et al. 1979; Piantanelli 1986; Gavrilov and Gavrilova 2003; Steinsaltz and Evans 2007). Recent studies have identified possible causes of this damage, including accumulation of faulty cells due to the mistranscription of messenger RNA (Wiegel et al. 1973), free radicals that produce oxidative damage (Beckman and Ames 1998; Ashok and Ali 1999), minute impairments to the immune and neuronal endocrine systems (Yin and Chen 2005), shortening of telomeres important for chromosome replication and protection (Passos and von Zglinicki 2005), and activation of a gene controlling proliferation of stem cells involved with tissue repair and regeneration (Janzen et al. 2006; Krishnamurthy et al. 2006). In addition, several studies connected growth to increased oxidative damage and reduced life span (Olson and Shine 2002; Ruel and Whitham 2002; Munch and Conover 2003).
Mathematically, incremental accumulations of damage resulting in a threshold process, death, have been represented as a Wiener‐process model by a number of authors (Anderson 1992; Weitz and Fraser 2001; Aalen and Gjessing 2001; Steinsaltz and Evans 2004, 2007). Using the notation of Anderson (2000), we consider the accumulation of damage in terms of the loss of survival capacity, designated “vitality.” Each individual begins with an initial vitality, 
, and dies when its vitality reaches 0 (fig. 1A). The random trajectory of vitality, 
, between 
and 0 is described by the Wiener process, 
where ρ is the mean loss rate of vitality, σ is the magnitude of the variability in the rate, and εt is a white‐noise process characterizing the variability. The initial vitality density is described by a Dirac delta function, that is, a unit impulse at 
with infinite height, zero width, and unit area. This function gives all individuals in a cohort the same initial vitality. Death is represented by an absorbing boundary at 
, which ensures that individuals reaching the zero boundary are removed from the cohort. With these conditions, the vitality density (pv) at time t is (Chhikara and Folks 1989) 
Figure 1: A, Individual vitality trajectories (eq. [1]) and survival (eq. [3]). B, From equation (2), vitality density evolves from a Dirac distribution at day 0 into a Gaussian distribution over days 2–6 and into a gammalike distribution by day 9. With time, the area under the curve diminishes by loss of vitality into the zero boundary.
The vitality density evolves as a traveling, widening, unimodal distribution typical of an advection‐diffusion process. It begins as a Dirac delta function spike and spreads into a Gaussian‐like shape and then into a quasi‐stable gammalike distribution that is absorbed into the zero‐vitality boundary (Aalen and Gjessing 2001; fig. 1B). The area under the curve is the fraction of the cohort remaining, that is, not having died from loss of vitality. Integrating equation (2) over the range (0, ∞) gives the vitality‐based survival (fig. 1A), 
where Φ is a cumulative normal distribution, 
(units of t−1) is the normalized vitality drift rate, and 
(units of t−1/2) is the normalized vitality spread rate. Derivation of equation (3) is detailed in appendix A.
Equation (3) is a model of mortality resulting from age‐accumulating factors. However, mortality can also occur from accidental causes independent of the animal condition or age. We represent this mortality by a Poisson process, 
in which the rate coefficient, k (units of t−1), is the same in young and old individuals and is equivalent to the age‐independent coefficient in the Gompertz‐Makeham survival model. See appendix A for description of the Gompertz‐Makeham model and a comparison to the vitality model.
When the age‐dependent (eq. [3]) and age‐independent (eq. [4]) survival functions are combined, the fraction of a cohort surviving to time t becomes 
where r, s, and k are essential model parameters of interest and the cohort is assumed to have a homogeneous initial vitality, 
.
The coefficients r, s, and k, standard error estimates, and a goodness of fit can be estimated with a maximum likelihood estimator developed by Salinger et al. (2003) that fits interval‐censored mortality data, in which mortalities are counted at the end of each time period rather than continuously. Goodness‐of‐fit tests are based on an analogue of Pearson’s C test, and the estimates’ standard errors are calculated from the inverse of the Hessian of the negative log likelihood evaluated at the parameter estimates. The algorithm, in S‐PLUS or R, is available at http://www.cbr.washington.edu/vitality/.
To describe the cross‐stage effects of growth on survival, we designate two adjacent stages: a treatment stage and a challenge stage. We first characterize the effect of growth on the evolution of vitality over the treatment stage, which is by definition the initial vitality for the challenge stage. Specifically, starting with the vitality density (eq. [2]), we develop equations in which the treatment‐stage growth rate and duration define the mean and variance at the end of the treatment stage. Second, the challenge‐stage survival curve is determined with equation (5), where r, s, and k are corrected for the mean and variance in the initial challenge‐stage vitality distribution. In essence, tracking the evolution of vitality across treatment and challenge stages links the effects of growth or environmental factors in one stage to survival in another. We test the model with studies on the effects of fish growth on a subsequent starvation challenge. In principle, the framework is equally suited to characterize the effects of spring/summer growth size on overwinter survival of animals in their natural habitats.
Relating Treatment‐Stage Growth to Challenge‐Stage Initial Vitality
To relate treatment‐stage growth to the challenge‐stage initial vitality, we require an equation relating how growth affects the evolution of vitality over the treatment stage. Assuming that growth, or reduced growth, acts as a stressor, we evaluated articles in which the vitality model was applied to 16 dose response experiments covering the effects of temperature, feeding rates, bacteria, and toxicants on survival in insects and fish (Anderson 2000; Hamel 2001; Springman et al. 2005). In all studies, the vitality parameters could be related to stressors with linear functions. However, linear forms are mathematically unsuitable for our model, so we assume that treatment‐stage vitality parameters are related to growth rate, x, as 
where 
, 
, a, and b characterize the immediate effects of growth on vitality and the subscript asterisk designates a treatment‐stage parameter. Besides being more tractable, the exponential forms fit the data as well as, or better than, the linear forms used in the cited articles. Statistical fits with equation (6) gave 
or greater. Also, note that data fitting the linear form 
also fit 
.
We characterize the final treatment‐stage vitality distribution in terms of its mean and variance, designated 
and 
, respectively. Assuming that a cohort’s initial treatment‐stage vitality distribution is homogeneous and then using equation (2), we calculate the mean vitality at the stage end, T, which is also the initial challenge‐stage vitality, as 
By using equation (6) in equation (7), we express the challenge‐stage 
in terms of treatment stage of x, T, and coefficients 
, 
, a, and b. However, the equation is unwieldy, and so we derived an approximation for the condition in which 
is Gaussian‐like, that is, when treatment‐stage mortality is less than 0.2. The approximation was derived using 
, 
, a, and b values representative of the parameter ranges in the 16 dose response studies with different exposure times, T, across a range of doses, x. The challenge‐stage mean initial vitality is 
, where 
is a constant that produces the best approximation to equation (7), T is the treatment‐stage duration, 
is the treatment‐stage initial vitality, and α and β characterize the evolution of vitality over the stage. Derivation of equation (7) and an illustration of the approximation are described in appendix A.
We approximate the final treatment‐stage vitality variance, 
, as the sum of a treatment‐stage initial variance, 
, and a variance that increases with time, 
, so 
. Because we restrict the analysis to Gaussian‐like vitality distributions, absorption into the 
boundary is not significant, and the evolving variance increases linearly with time as 
, where 
is the vitality spread rate over the treatment stage. If the effect of growth on the spread parameter is defined using equation (6), then 
, and the variance becomes 
. Next, expressing the exponential as the first term of its series gives 
, and when the initial and evolving variances are combined, the treatment‐stage variance as a function of time and growth rate is 
.
Taking x to be the average growth rate over T and assuming exponential growth, so mass increases as 
, where M0 is the initial treatment‐stage mass, then 
, and the initial challenge‐stage vitality mean and variance are approximated as 
respectively, where 
. Thus, equation (8) links treatment‐stage growth characterized by the initial (M0) and final (M) treatment‐stage masses and stage duration (T) to the mean and variance of vitality at the beginning of the challenge stage.
Effects of 
and 
on Challenge‐Stage Mortality
To link treatment‐stage growth to challenge‐stage mortality, we need to characterize the effect of 
on the normalized challenge‐stage vitality parameters. Recall that equation (5) assumes that the challenge‐stage initial vitality has a Dirac delta function distribution; that is, 
. However, treatment‐stage growth makes 
, so the challenge‐stage vitality parameters estimated with the Salinger et al. (2003) algorithm are biased, and we must characterize how fitted challenge‐stage r and s change with 
. Our approach has three steps. First, we simulate challenge‐stage vitality trajectories with different initial distributions and drift and spread rates. Second, we fit the Salinger et al. (2003) algorithm to the resulting challenge‐stage survival curves to obtain fitted vitality parameters. Third, we characterize the relationships between the fitted and actual challenge‐stage parameters in terms of the mean and variance of the initial distributions.
For each of 3,100 (ρ, σ) pairs, we simulated vitality trajectories with a numerical equivalent of equation (1): 
, where W is a normally distributed random number with zero mean and unit variance and i is the individual’s age in simulation time increments. For each parameter pair, we simulated 1,000 trajectories for 51 initial‐vitality Gaussian distributions with mean values of 1 and standard deviations covering the range over which the vitality distribution is Gaussian‐like (Aalen and Gjessing 2001). Time to mortality occurred either by loss of vitality, that is, when 
, or by accidental mortality, represented by a Poisson process with rate k. Parameter ranges were 
and 
and with k set at 0.005, 0.01, 0.015, or 0.02.
The simulations produced 632,400 individual survival curves across variations of ρ, σ, k, and 
with 
. For each simulated survival curve, we obtained fitted parameters 
, 
, and 
. Across the parameter combinations, the regression 
had relatively constant coefficients: the (1/3, 1/2, 2/3) quantiles of the distribution of coefficients were (0.999, 1.001, 1.003) for a0 and (−0.686, −0.622, −0.562) for a1, with R2 quantiles of (0.79, 0.85, 0.89). Over the range in which the initial distribution is Gaussian, that is, 
, the equation gives 
, so we assume that initial challenge‐stage variance does not affect the challenge‐stage fitted drift rate. The effective challenge‐stage normalized drift rate is then 
where ρ is the actual drift rate for the challenge stage.
Over the parameter combinations, the relationship of the spread rate ratio to variance was expressed as 
, where the (1/3, 1/2, 2/3) quantiles of the regression coefficient distributions were (0.118, 0.127, 0.136) for b0 and (6.69, 6.88, 7.06) for b1, with R2 quantiles of (0.86, 0.87, 0.89). We set 
, so the dimensional form of the challenge‐stage spread rate was 
. Over the range of initial variances, this results in 
, and so the effective normalized challenge‐stage spread rate can be approximated as 
where σ is the actual vitality spread rate in the challenge stage.
When equations (8), (9), and (10) are combined, the relationships of treatment‐stage growth to fitted challenge‐stage drift (
) and spread (
) rates are 
where 
, 
, 
, 
, and 
with units 
(
), 
(
), and T (t). Note that the biases in equations (9) and (10) are absorbed into the regression coefficients r0 and s0 and thus do not materially affect the relationship between the treatment and challenge stages.
The simulations also revealed that 
was independent of 
when 
. Therefore, if the real value of k were insignificant, within the assumptions of the model, it would be feasible to describe challenge‐stage survival in terms of treatment‐stage growth. However, equation (5) simply partitions the mortality between vitality‐dependent and vitality‐independent factors. Conditions that increase the loss of vitality could also increase the age‐independent rate of mortality. We explored several possible relationships between r and k, and for our data, a power function suffices to characterize a possible coupling, 
where ak and bk are constants derived from a log‐log regression.
To review, equations (11) and (12) are approximations, derived from stochastic properties of vitality, that link treatment‐stage growth to challenge‐stage vitality parameters, which in turn, through equation (5), link treatment‐stage growth to challenge‐stage survival. Mathematically, the link is developed in terms of the mean and variance of a cohort’s vitality distribution from the beginning of the treatment stage (subscript 0) to the beginning of the challenge stage (subscript asterisk) and through the challenge stage (no subscript). Diagrammatically, this can be represented as 
. Growth rate, x, and stage duration, T, characterize the evolution of vitality over the treatment stage, and the actual challenge‐stage drift (ρ) and spread (σ) rates and stage duration (t) characterize the evolution of vitality over the challenge stage. Note also that when exponential growth is assumed, x is expressed in terms of the initial and final treatment‐stage masses (M0, M) and stage duration (T).
Fitting the Model to Growth‐Starvation Studies
To evaluate how well the model predicts cross‐stage effects of growth on mortality, we require experimental data relating growth over a treatment stage to mortality in a subsequent challenge stage. We tested the model’s predictions with three steps. First, the vitality parameters 
, 
, and 
were estimated by fitting equation (5) to challenge‐stage survival curves using the algorithm of Salinger et al. (2003). To evaluate the relationships between growth and challenge‐stage parameters, we regressed the fitted vitality coefficients against 
and T with equations (11) and (12), and in regression forms, 
where the coefficients are 
, 
, 
, 
, 
, 
, 
, and 
. With T fixed, the regressions are 
where the coefficients are 
, 
, 
, and 
.
Equation (13) and (14) fits were judged with P and r2. We next predicted 
, 
, and 
, using the regression coefficients and treatment‐stage growth measures. Finally, we used the predicted parameters in equation (5) to determine survival. Predictions on groups used to derive the regression coefficients represent pseudovalidations, and predictions on groups not used in the calibrations represent true validations. Model parameters, variables, and regression coefficient definitions and units are summarized in table 1.
Results
Chinook Salmon
Cobleigh (2003) determined the effect of growth on survival during a subsequent temperature/starvation challenge. For the growth treatment, approximately 1,000 juvenile chinook salmon (Oncorhynchus tshawytscha), reared in the University of Washington hatchery from brood year 2001, were fed 3% of their weight daily for one month under ambient light, resulting in an average mass of 3.6 g (
). The fish were then divided into 18 10‐L tanks with flow‐through fresh water and maintained on one of three feeding treatments. Treatment 1 fish were fed daily for 4 weeks, treatment 2 fish were fed daily for 2 weeks and starved for 2 weeks, and treatment 3 were starved for 4 weeks. Weights at the end of the treatments differed by 3 g (table 2). No fish died over the treatment period. After the treatment, water temperature was increased from 11.3° to 20.1°C over an 8‐h period. As a result of the higher temperature, oxygen supersaturation occurred in some tanks (mean 
), producing significant gas‐bubble‐disease mortality in two out of six tanks for treatment 1, five out of six tanks for treatment 2 and three out of six tanks for treatment 3. Supersaturation was removed (
), and subsequently, mortality rates declined in all tanks. To lessen the effect of supersaturation, we combined the six replicates in each treatment and defined the start of the challenge period as 4 days after the supersaturation event. By this time, the mortality rate had dropped, suggesting that the effects of supersaturation had dissipated. We defined the 30‐day growth period and the 10 days of supersaturation and recovery as the total treatment, so 
days.
Challenge‐stage vitality parameters were estimated by fitting equation (5) to the challenge‐stage survival data (table 2). The model fitted the treatment 3 survival curve at the 
level, but treatments 1 and 2 fits were rejected at 
(fig. 2, dashed lines). We suggest this difference was the result of greater supersaturation in treatments 1 and 2 than in treatment 3. We then regressed the fitted challenge‐stage vitality parameters from table 2 against fish mass (eq. [14]): 
and 
fits to mass were significant at the 
level, while 
was not significant (
; table 3). Survivals generated with vitality parameters derived from equation (14) are depicted by solid lines in figure 2. The difference between the two estimates of survival is wholly the result of the poor fit of 
in equation (14).
Figure 2: Vitality model fit to juvenile chinook salmon survival data under a temperature/starvation challenge with growth treatments 1, 2, and 3 (table 2). Points represent observed survival; lines are generated with equation (5) using parameters in table 2 (dashed lines) or parameters estimated from treatment mass (eq. [14]) using coefficients in table 3 (solid lines).
Yellow Perch
Letcher et al. (1996) determined the effect of 18 feeding schedules on time to starvation in larvae and juvenile yellow perch (Perca flavescens). Newly hatched perch larvae were held in a tank maintained in a 12L:12D photoperiod cycle and fed brine shrimp, dry commercial feed, and a beef liver mixture. On days 
, 21, and 35, corresponding to fork lengths of 10‐, 15‐, and 20‐mm fish, six groups of fish (∼75 fish in each 10‐mm group and ∼50 in each 15‐ and 20‐mm group) were transferred to flow‐through, temperature‐controlled (
) aquaria for 0–4 starvation days followed by 0–8 additional feeding days before a starvation challenge beginning on day tc. Growth duration was 
. Initial and final treatment‐stage masses were measured on day 13 and tc, respectively (table 4).
The fish that were transferred to aquaria at 10 and 15 mm and fed continuously grew exponentially (
, 
, and 
), and results for them lay on a straight line on a log mass–versus–growth duration plot (fig. 3). Groups of 10‐ and 15‐mm fish starved between 1 and 4 days lost weight and were below the line, as were all the 20‐mm fish (fig. 3). The authors suggested that the 20‐mm fish's growth anomalies resulted from inadequate amounts of food for their larger size.
Figure 3: Log mass (M; μg) of yellow perch over time. Line depicts growth of continuously fed fish. Filled circles are for 10‐ and 15‐mm fish used in calibration of equation (5). Open circles are for 20‐mm fish, and numbers indicate feeding treatments in table 4.
We estimated the challenge‐stage vitality parameters 
, 
, and 
and standard errors (table 4) by fitting equation (5) to starvation‐stage survival normalized to 
at day tc. Model fits for each group are illustrated by dashed lines in figures B1–B3. In general, the models fitted the data well: 12 were significant at 
, and two were not, while in four, the sample size was insufficient for the χ2 test.
We evaluated the relationship between treatment‐stage growth and fitted challenge‐stage (i.e., starvation challenge) vitality parameters, using growth and survival data from the 10‐ and 15‐mm groups in table 4. The fitted vitality parameters were then regressed against the treatment‐stage variables (initial and final masses and stage duration) to obtain equation (13) coefficients (table 5). For the 10‐ and 15‐mm groups, the 
regression to treatment‐stage mass and duration was highly significant, 
−07. The 
regression was less significant, 
, and parameter ds was not significant, 
. However, cr and cs, which by definition are equivalent, were within one standard deviation. Dropping 
from the regression improved the fit, 
, and the Akaike Information Criterion (AIC) score decreased from 2.33 to 0.79, while as, bs, and cs did not change significantly. However, with this three‐parameter regression, the cs and cr were no longer within a standard deviation (table 5). Thus, the three‐ and four‐parameter forms of equation (13) give equivalent growth‐survival relationships, and while the four‐parameter version is more complete in terms of the theory, the three‐parameter regression is more parsimonious. In addition, we repeated 
and 
regressions with values of n between 0 and 1, and, based on AIC scores, 
provided the best fit. This corresponds to 
, which is the exponent in equation (A24) that gives the best approximation of average vitality (app. A). The regression for 
also was significant (
).
For 10‐ and 15‐mm fish, the challenge‐stage vitality parameters obtained by fitting equation (5) to survival data and predicted from equation (13) using treatment‐stage mass and duration, in general, had one‐to‐one relationships (fig. 4). In addition, survival curves generated with the predicted parameters were close to the observed challenge‐stage survival and the fitted survivals (figs. B1, B2). We refer to the regressions in figure 4 as a pseudovalidation because both predicted and fitted values were derived from the 10‐ and 15‐mm fish. We validated the model by comparing 20‐mm fish's vitality parameters predicted from equation (13) with treatment‐stage growth (table 4) and coefficients derived from the 10‐ and 15‐mm fish (table 5) with parameters derived by fitting equation (5) to survival curves. The prediction errors were larger for this group (fig. 4), but in general, challenge‐stage survivals predicted from treatment‐stage growth fitted the observed survivals reasonably well (fig. B3).
Figure 4: Yellow perch vitality parameters predicted from treatment‐stage growth (eq. [13]) versus parameters fitted to challenge‐stage survival data (eq. [5]). Filled circles represent predicted values for 10‐ and 15‐mm fish, which were used in calibration. Open circles represent predicted values for the 20‐mm fish. Numbers correspond to table 4 feeding treatments of 20‐mm fish.
The effect of fish age and growth on treatment‐stage vitality variance is depicted in figure 5. Variance, scaled to the initial treatment‐stage vitality, 
, was estimated from equation (8), where the scaled effects of age, 
, and growth, 
, were obtained from table 4, with 
. Variance increases linearly with fish age at the start of the challenge stage (fig. 5A, line), while for a given age treatment, growth may increase or decrease variance, depending on the sign of ds. Because ds has high uncertainty (table 4), the age‐independent contribution of growth is unresolved for yellow perch. In figure 5A, points depicting the upper and lower estimates of variance, with 
, fall above (
) and below (
) the linear effect of age. Figure 5B illustrates the initial challenge‐stage variance as a function of challenge‐stage initial weight. For fish growing exponentially, the variance tends to an asymptote with weight, while for fish growing suboptimally, variance is larger than in exponentially growing fish of the same weight. It is noteworthy that when 
, the initial vitality distribution is no longer Gaussian and a fundamental model assumption is violated. We suggest that the difficulty in predicting 20‐mm yellow perch vitality parameters (treatments 13–18 in fig. 5B) was a result of the larger initial challenge‐stage variance for this group.
Figure 5: Growth‐stage variance 
versus yellow perch age (A) and weight (B) at the beginning of starvation challenge. The line in A depicts variance associated with fish age, characterized by bs in equation (13). Data points represent upper and lower contributions of growth to variance derived with 
(eq. [13]): treatments with optimal growth (filled circles), that is, on the exponential‐growth line (fig. 3), and suboptimal growth (open circles). Numbers in B indicate treatments with suboptimal growth.
Letcher et al. (1996) concluded that among yellow perch fed for the same number of days, the times to 50% mortality (TM50%) were not significantly different. However, we found that both feeding treatment and fish size affected TM50% (table 4). From the mathematical properties of vitality, 
should have a linear relationship with TM50%, and this was indeed found. For challenge‐stage 
obtained by fitting 10‐ and 15‐mm fish survival curves, 
, with 
and 
−013. For challenge‐stage 
predicted from treatment‐stage growth with equation (13) using all 18 treatments, TM50% =
, with 
and 
across (fig. 6). Therefore, the average time to starvation of the fish in a challenge stage depends on both the duration of the treatment stage and the amount of growth over the stage.
Figure 6: Observed days to 50% mortality (TM50%) versus the reciprocal of the vitality drift rate (
) derived by fitting equation (5) to 10‐ and 15‐mm yellow perch survival data (open circles) and with fish growth from table 4 in equation (13) (filled circles). The line is a linear regression of the survival data points.
Discussion
Cohort mortality has been described in terms of the stochastic decline of organism vitality to an absorbing boundary representing death (Anderson 1992, 2000; Aalen and Gjessing 2001; Weitz and Fraser 2001; Steinsaltz and Evans 2004, 2007), and the effects of natural and xenobiotic stressors on mortality have been modeled through their effects on vitality (Anderson 2000; Hamel 2001; Springman et al. 2005). In addition, a Poisson process has also been included to account for vitality‐dependent mortality (Anderson 2000). Here we extend the theory to characterize the effect of growth in one stage on mortality in the next stage: growth, by affecting vitality in the treatment stage, determines the initial vitality in a following challenge stage and therefore the amount of vitality remaining to withstand a stressor.
We apply the model to experiments that showed that starvation time in yellow perch and chinook salmon depended on the growth rate and duration before a starvation challenge and conclude that the cross‐stage linkage results from the effect of treatment‐stage growth on the initial vitality of the starvation challenge. The experiments also revealed that suboptimal growth increased starvation time variability: for example, exponentially growing yellow perch died within a 3‐day interval when starved (figs. B1, B2), while fish growing suboptimally died over a 10‐day interval (fig. B3). We explain these interactions in terms of competing effects of treatment‐stage duration and mass gain on the mean and variance of the initial challenge‐stage vitality. For exponentially growing fish, the positive effect of mass gain on average vitality dominates the negative effect, in which stage duration increases vitality variance. Mass gain decreases 
and correspondingly increases the time to starvation in the challenge stage (fig. 6). For fish not growing exponentially, the initial challenge‐stage average vitality is lower and the variance is higher than under optimal growth (fig. 5). As a result, the effective challenge‐stage spread rate (
) increases, and variability in time to starvation increases.
A relationship between growth and vitality could reflect genetic differences in the ability of individuals to withstand starvation, factors involving competition in foraging, or both. Differences in growth capacity are common and important. For example, in salmonids, individual differences in growth efficiency affect habitat selection: metabolically efficient individuals tend to occupy low‐energy environments, while inefficient individuals tend to occupy high‐energy environments (Morinville and Rasmussen 2002, 2006). Also, the year of migration of juvenile salmon (Salmo salar L.) appears to be related to the rate of protein turnover (Morgan et al. 2000). Thus, when these salmon are placed together in an aquarium, we expect efficient individuals to grow at a faster rate, which would further increase their competitive advantage over inefficient individuals; that is, growth would increase heterogeneity. This property emerges from equation (8). The effect of mass gain on vitality variance, 
, is characterized by 
, and so if 
, then faster growth increases cohort heterogeneity, as measured by the initial challenge‐stage variance. For chinook salmon, 
(table 3), and for yellow perch, 
(table 4). Therefore, the model suggests that faster growth increases heterogeneity, although the proximate mechanisms are unclear.
The model provides a quasi‐mechanistic explanation for how growth affects survival capacity heterogeneity but does not consider details of growth trajectories or the effects of compensatory growth explored by Mangel and Munch (2005). In addition, it does not explain why lower growth correlates with increased life span in many species (Metcalfe and Monaghan 2003). To capture these complexities, additional terms are required in the vitality drift and spread rates (eq. [6]). However, because at this level of the model the equations are essentially empirical, further extensions would have to be justified by experiment.
One possible application of the model would be to characterize the effect of first‐year growth on overwinter survival, an important process in fish recruitment dynamics (Healy 1982; Arendt 1997; Chambers and Trippel 1997; Sogard 1997; Matthews et al. 2001; Sutton and Ney 2001; Beamish et al. 2004; Pangle et al. 2004; Moss et al. 2005). From hatch date and summer growth, 
and T can be estimated and used in equation (13) with suitable coefficients to estimate the overwinter vitality drift and spread rates. Equation (5) then predicts the overwinter survival curve of the cohort. The applicability of such a prediction depends on the degree to which the model coefficients are transferable across environments and taxa. The growth‐stage coefficients α, β, η, and 
should have physiological bases and therefore may be similar across taxa and independent of the environment. Tables 3 and 5 provide some support for this conjecture. The effect of mass on vitality loss (β) is 0.25 for chinook salmon (order Salmoniformes) and 0.29 for yellow perch (order Perciformes). In both species, the effect of treatment‐stage mass gain on challenge‐stage initial vitality is more important than the effect of treatment‐stage duration on initial challenge‐stage variance. Additional support for generic coefficients comes from the observation that the percentage of mass lost during starvation was similar across 11 species (
; Letcher et al. 1996). Thus, for a first approximation, we might apply table 4 values for br, bs, cr, cs, and ds. Alternatively, the coefficients could be estimated in treatment‐challenge studies on the species of interest. The coefficients ar and as reflect the effect of overwinter conditions on vitality‐dependent mortality. These might be estimated by experiments characterizing the effect of temperature on starvation time (e.g., McCollum et al. 2003; Braaten and Guy 2004). The vitality‐independent terms (ak, bk) might also be generic and applicable across years. In any case, we surmise that vitality‐dependent processes should dominate overwinter mortality.
While the model provides a framework in which to quantify the effects of growth on the survival capacity of fish, in a more general sense, it can be used to characterize how cohort heterogeneity and survival capacity evolve over time or in different environments. The approach is straightforward. Performing challenge tests on a single cohort at different ages or on multiple cohorts of the same age from different environments yields respective drift and spread rates. The vitality mean and variance from test i relative to j are then defined as 
and 
, and so the relative change in cohort heterogeneity across time or the relative heterogeneity in cohorts across environments can be inferred. In essence, we suggest that challenge‐survival curves contain valuable information for exploring a variety of ecological issues, such as the effect of sublethal contaminants on population survival, the effect of resource variability on heterogeneity, or effects of a stressor on the rate of natural selection. However, in its current form, the model assumes a Gaussian distribution of vitality over the treatment stage. This presents problems if significant vitality‐dependent mortality occurs over the treatment. For example, in the 20‐mm yellow perch, the variance was near 0.4 (fig. 5) and therefore not Gaussian, which, we believe, contributes to the difficulty in fitting this group. As a note, a form of the model with a non‐Gaussian initial distribution is being developed.
In closing, we note that many researchers have considered heterogeneity an important issue in ecology, demography, and evolution (Vaupel et al. 1998; Zens and Peart 2003; Benton et al. 2006; Saccheri and Hanski 2006), and studying its dynamics and consequences has proceeded with some difficulty through two approaches (DeAngelis and Mooij 2005). At one end, heterogeneity is represented in differential‐equation population models by including additional dimensions, such as size and age. At the other end, it is explicitly represented by modeling individual organisms, that is, individual‐based models (IBMs). Both approaches have strengths and weaknesses. Differential‐equation models have fewer parameters and so are more amenable to comparison to data, but their realism is limited. The IBMs readily incorporate biological mechanisms but at the expense of additional dimensions, which makes fitting them to data difficult. Vitality lies in the borderland between IBMs and population‐based models. The arguable premise, that heterogeneity, characterized by vitality, can be treated as a dynamic process, was proposed a half‐century ago (Sacher 1956). However, studies of its mathematical properties are recent (Anderson 2000; Aalen and Gjessing 2001; Steinsaltz and Evans 2004), and experimental tests of the theory are few (Anderson 2000; Hamel 2001; Springman et al. 2005), so the approach has received little attention (Mangel 2006). Nonetheless, we believe that vitality theory, combined with treatment‐challenge experiments, offers a new and potentially powerful framework in which to explore ecological and evolutionary processes.
Acknowledgments
This work was supported by Bonneville Power Administration and the Northwest Fisheries Science Center of the National Oceanographic and Atmospheric Agency. The authors gratefully acknowledge the assistance of N. Beer, J. Emlen, E. Gurarie, J. Murphy, J. Nestler, K. Rose, A. Steele‐Feldman, J. Tran, and R. Zabel.
Appendix A Mathematical Derivations
Derivation of Vitality
The vitality model derivation below follows from Anderson (2000). Cohort survival over time, l, can be expressed as the product of the probability of survival according to the organism’s vitality, lv, and the probability of survival in avoiding random mortality events, la, so that 
The random evolution of vitality, 
, between 
and 0 is described by the Wiener process, 
where ρ is the mean loss rate of vitality, σ is the magnitude of the variability in the rate, and εt is a white‐noise process characterizing the variability. Since vitality varies randomly over time, an individual’s vitality can be described only through the conditional probability density of 
at time t. This depends on the initial vitality, 
, at time 0: 
The conditional probability distribution defined by equation (A3) is expressed by a Fokker‐Planck equation describing the change in probability of 
according to the rate of growth of its mean and variance (for reference, see Gardiner 1985). With ρ and σ representing the average vitality rate processes over a stage, they are constant, and the Fokker‐Planck equation for the rate of change of the conditional probability density is 
Mortality from vitality‐related causes is defined as entering a zero‐vitality state, and mathematically it is expressed by the absorbing boundary condition, 
Vitality has no upper limit and is expressed 
With these conditions, the vitality density at time t is (Chhikara and Folks 1989) 
where 
is the initial vitality, ρ is the deterministic rate of change of vitality or drift rate, and σ is the intensity of the stochastic rate of change of vitality, or spread rate. The probability density function can be interpreted as the probability distribution of vitality in a cohort at age t. The probability that an organism has not reached 0 at t is determined by integrating equation (A7) over the possible range of vitality (0, ∞): 
Noting the identity 
where μ and σ2 are mean and variance of the distribution, then vitality‐based survival is 
where Φ is a cumulative normal distribution, 
(units of t−1) is the normalized vitality drift rate, and 
(units of t−1/2) is the normalized vitality spread rate.
Organisms that do not die of vitality‐related causes die of accidental causes, which are independent of their history. Assuming that accidental deaths are randomly distributed, the rate of accidental mortality can be defined by a Poisson process, and the probability that an organism is alive, excluding the vitality process, is the probability of observing zero mortality events to age t and can be expressed 
where k is the accidental‐mortality rate coefficient, with a dimension of t−1.
When the age‐dependent (eq. [A10]) and age‐independent (eq. [A11]) processes into equation (A1) are combined, the fraction of a cohort surviving to time t becomes 
where r, s, and k are essential model parameters of interest and the population is assumed to have homogeneous initial vitality, 
.
Comparison of Vitality and Gompertz‐Makeham Survival Models
The first parametric model of mortality was proposed by Gompertz (1825) and was subsequently modified by Makeham (1867). Variants of the Gompertz‐Makeham (GM) model are widely used in demography, and they generally fit survival curves for humans between the ages of 15 and 85 (Gavrilov and Gavrilova 1991; Toupance et al. 1998). In the model, an aging, or senescent, component of mortality is described with two parameters: a sets the overall level of adult mortality, and b determines how mortality accelerates with age. A third term, the Makeham component, k, expresses the age‐independent mortality rate. The force of mortality becomes 
and survival is 
In general, the model is not as flexible as the vitality model in fitting survival data, as is demonstrated in figure A1, which compares the fit of GM and vitality models in four yellow perch feeding treatments. The GM coefficients were estimated by fitting the survival data to equation (A14) using the nonlinear least squares fitting routine in S‐PLUS (“nls”). The model does not fit the 10‐ and 15‐mm groups easily, and the fits to the 20‐mm group are problematic. In particular, to represent age‐independent mortality, we require 
; however, with a GM model, the fit often produces 
, as is seen in treatments 13 and 17 (fig. A1).
Figure A1: Comparison of the vitality (dashed lines) and Gompertz‐Makeham (GM; solid lines) model fits to yellow perch feeding treatments (circles). The GM coefficients for treatments are given in each panel, and the corresponding vitality coefficients are given in table 5.
Derivation of Average Vitality
To derive the average normalized vitality for which 
, first express equation (A7) as 
Hereafter, 
refers to 
, and the average vitality 
is 
The denominator of equation (A16) is given by equation (A10). Solving for the numerator of equation (A16) involves a number of steps. First, it is divided into parts: 
For the first component of equation (A17), if 
, then 
, so 
and 
. Substituting x for 
gives 
Aside 1 of equation (A18). Given that 
, then 
Aside 2 of equation (A18). From the previous substitution of x for 
, 
Then switch the bounds and sign and negate the bounds, so the negative signs are canceled, giving 
Using identity (A9), it follows that 
Combining the two asides into the first component of the numerator yields 
For the second component of equation (A17), substitute 
, so that 
, and it follows that 
and 
. Then, substitute x for 
, and 
Aside 1 of equation (20). Given that 
, then 
Aside 2 of equation (A20). From the previous substitution of x for 
, 
Switch and negate the bounds, which results in no sign change; then, with identity (A9), 
Combining asides 1 and 2 into equation (A20) yields 
Now, combining equations (A19) and (A21) and simplifying the numerator yields 
When equations (A7) and (A22) are combined, the average vitality at t is 
The average vitality evolves over dose and time, beginning with an initial value of 1 at 
, and declines to 0 as time and dose increase. In our model, the vitality coefficients in the treatment stage are related to treatment‐stage dose by equation (6). The resulting shape of average vitality with time and dose is illustrated in figure A2. This form is characteristic when r and s have exponential relationships with dose.
Figure A2: Evolution of average vitality, according to equation (A23), for r and s fitted to juvenile pink salmon exposure to sulfite waste liquor (Anderson 2000).
Average Vitality Approximation
Because equation (A23) is unwieldy, we derived an approximation that expresses average vitality as a function of dose and time. The form is 
where T is the treatment duration and x is the dose or growth rate, which is a property that affects the evolution of vitality over the treatment stage. For the sulfite waste liquor example depicted in figure A2, equation (A24) suitably fits the analytical solution given by equation (A23). The coefficient m affects the curvature, and for our purpose 
results in a more linear fit for 
. Ultimately, the value of m should be selected that, according to the model, fits the growth data. Using other r and s values in equation (6) results in average vitality surfaces similar to that in figure A3. Across a range of r and s, equation (A24) is a suitable approximation for average vitality.
Figure A3: Comparison of average vitality from equation (A23) and estimated average vitality from equation (A24) for r and s fitted to juvenile pink salmon exposure to sulfite waste liquor (Anderson 2000). Line indicates a one‐to‐one relationship.
Appendix B Fit of Model to Yellow Perch Survival Data
Figure B1: Yellow perch survival curves of 10‐mm fish. Circles represent data from Letcher et al. (1996). Dashed lines show fits using the survival model (eq. [5]). Solid lines show fits using vitality coefficients predicted from growth (eq. [14]) with coefficients estimated using 10‐ and 15‐mm groups.
Figure B2: Yellow perch survival curves of 15‐mm fish. Circles represent data from Letcher et al. (1996). Dashed lines show fits using the survival model (eq. [5]). Solid lines show fits using vitality coefficients predicted from growth (eq. [14]) with coefficients estimated using 10‐ and 15‐mm groups.
Figure B3: Yellow perch survival curves of 20‐mm fish. Circles represent data from Letcher et al. (1996). Dashed lines show fits using the survival model (eq. [5]). Solid lines show fits using vitality coefficients predicted from growth (eq. [14]) with coefficients estimated using 10‐ and 15‐mm groups.
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* Corresponding author; e‐mail: jjand@u.washington.edu.
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† Present address: Department of Biology, University of Rochester, River Campus Box 270211, Rochester, New York 14627‐0211; e‐mail: molly.gildea@rochester.edu.
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‡ E‐mail: dwhittlesey@gmail.com.
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§ E‐mail: ltkitty@u.washington.edu