Dynamics equations

2024-12-13

salmonMSE utilizes an age-structured model. The population is tracked by age and year but various dynamics correspond to the salmon life stages as described below.

Definitions

Definition of variable names and the corresponding slots in either the input (SOM) or output (SMSE) objects in salmonMSE.

Name	Definition	Type	Class	Slot
$\textrm{NOS}$	Natural origin spawners	Natural production	SMSE	NOS
$\textrm{HOS}$	Hatchery origin spawners	Hatchery	SMSE	HOS
$\textrm{HOS}_\textrm{eff}$	Effective number of HOS, spawning output discounted by $\gamma$	Hatchery	SMSE	HOSeff
$\textrm{Fry}^\textrm{NOS}$	Fry production by natural origin spawners, assumed to be equal to egg production	Natural production	SMSE	Fry_NOS
$\textrm{Fry}^\textrm{HOS}$	Fry production by hatchery origin spawners, assumed to be equal to egg production	Hatchery	SMSE	Fry_HOS
$\textrm{Smolt}^\textrm{NOS}$	Smolt production by natural origin spawners, density-dependent	Natural production	SMSE	Smolt_NOS
$\textrm{Smolt}^\textrm{HOS}$	Smolt production by hatchery origin spawners, density-dependent	Hatchery	SMSE	Smolt_HOS
$C_\textrm{smolt}$	Carrying capacity of smolts (Beverton-Holt stock-recruit parameter)	Natural production + Habitat	SOM	capacity_smolt
$S_\textrm{max}$	Spawning output that maximizes smolt production (Ricker stock-recruit parameter)	Natural production + Habitat	SOM	Smax
$\kappa$	Maximum recruitment production rate (stock-recruit parameter), units of recruit per spawner	Natural production + Habitat	SOM	kappa
$\phi$	Unfished egg production rate, units of egg per smolt	Natural production	SOM	phi
$r$	Maturity at age, i.e., recruitment rate	Natural production	SOM	p_mature
$u^\textrm{PT}$	Pre-terminal fishery harvest rate	Harvest	SOM	u_preterminal
$u^\textrm{T}$	Terminal fishery harvest rate	Harvest	SOM	u_terminal
$\delta$	Mortality from catch and release (proportion)	Harvest	SOM	release_mort
$v$	Relative vulnerability by age to the fishery	Harvest	SOM	vulPT, vulT
$\textrm{Fec}$	Fecundity of spawners (eggs per female)	Natural production	SOM	fec
$\textrm{Fec}^\textrm{brood}$	Fecundity of broodtake (eggs per female)	Natural production	SOM	fec_brood
$p^\textrm{female}$	Proportion of female spawners in broodtake and spawners	Natural production + Hatchery	SOM	p_female
$\textrm{SAR}$	Smolt-to-adult recruit survival	Natural production + Hatchery	-	-
$M$	Instantaneous natural mortality of juvenile, corresponding to either the freshwater or marine environment by age class	Natural production + Hatchhery	SOM	Mjuv_NOS, Mjuv_HOS
$s_\textrm{enroute}$	Survival of escapement to spawning grounds and hatchery	Natural production	SOM	s_enroute
$\textrm{NOB}$	Natural origin broodtake	Hatchery	SMSE	NOB
$\textrm{HOB}$	Hatchery origin broodtake	Hatchery	SMSE	HOB
$s_\textrm{yearling}$	Survival of hatchery eggs to yearling life stage	Hatchery	SOM	s_egg_smolt
$s_\textrm{subyearling}$	Survival of hatchery eggs to subyearling life stage	Hatchery	SOM	s_egg_subyearling
$p_\textrm{yearling}$	Proportion of hatchery releases as yearling (vs. subyearling)	Hatchery	Internal state variable	-
$s_\textrm{prespawn}$	Survival of adult broodtake in hatchery	Hatchery	SOM	s_prespawn
$n_\textrm{yearling}$	Target number of hatchery releases as yearlings	Hatchery	SOM	n_yearling
$n_\textrm{subyearling}$	Target number of hatchery releases as subyearlings	Hatchery	SOM	n_subyearling
$m$	Mark rate of hatchery fish	Hatchery	SOM	m
$p_\textrm{NOB}$	Proportion of the total broodtake of natural origin (vs. hatchery origin)	Hatchery	Internal state variable	-
$p_\textrm{HOB}$	Proportion of the total broodtake of hatchery origin (vs. natural origin)	Hatchery	Internal state variable	-
$p^\textrm{esc}_\textrm{max}$	Maximum proportion of total escapement (after en-route mortality) to use as broodtake	Hatchery	SOM	pmax_esc
$p^\textrm{NOB}_\textrm{target}$	Target proportion of the natural origin broodtake from the escapement (after en-route mortality), i.e., NOB/NOS ratio	Hatchery	SOM	ptarget_NOB
$p^\textrm{NOB}_\textrm{max}$	Maximum proportion of the natural origin broodtake from the escapement (after en-route mortality), i.e., NOB/NOS ratio	Hatchery	SOM	pmax_NOB
$\textrm{NOR}$	Natural origin return	Natural production	SMSE	Return_NOS
$\textrm{HOR}$	Hachery origin return	Hatchery	SMSE	Return_HOS
$p^\textrm{hatchery}$	Proportion of hatchery origin escapement to hatchery, available for broodtake	Hatchery	SOM	phatchery
$p^\textrm{HOS}_\textrm{removal}$	Proportion of hatchery origin escapement removed from spawning grounds, not available for broodtake	Hatchery	SOM	premove_HOS
$\gamma$	Reduced reproductive success of HOS (relative to NOS)	Hatchery	SOM	gamma
$\bar{z}$	Mean phenotypic value of cohort in natural and hatchery environments	Fitness	Internal state variable and SOM	zbar_start
$\theta$	Optimal phenotypic value for natural and hatchery environments	Fitness	SOM	theta
$\sigma^2$	Variance of phenotypic traits in population	Fitness	SOM	fitness_variance
$\Omega$	Selection strength: fitness variance relative to phenotypic variance	Fitness	SOM	selection_strength
$h^2$	Heritability of phenotypic traits	Fitness	SOM	heritability
$\bar{W}$	Population fitness in the natural and hatchery environments	Fitness	SMSE	fitness
$\ell_i$	Relative fitness loss at the i-th life stage (egg, fry, smolt)	Fitness	SOM	rel_loss
$\textrm{PNI}$	Proportionate natural influence	Fitness	SMSE	PNI
$m$	Mark rate of hatchery fish (affects fishery retention of hatchery fish relative to natural fish)	Harvest	SOM	m
$p_\textrm{NOS}$	Proportion of natural origin spawners (vs. effective HOS)	Population dynamics	Internal state variable	-
$p_\textrm{HOSeff}$	Proportion of effective hatchery origin spawners (vs. NOS)	Population dynamics	SMSE	NOS, HOS_effective
$p_\textrm{HOScensus}$	Proportion of hatchery origin spawners (vs. NOS)	Population dynamics	SMSE	NOS, HOS

Natural production

First, we consider natural production in the absence of fitness effects from the hatchery improvement.

Egg production

From the spawners (NOS and HOS) of age $a$ in year $y$, the corresponding egg production of the subsequent generation is calculated as:

$$\begin{align} \textrm{Egg}^\textrm{NOS}_y &= \sum_a\textrm{NOS}_{y,a} \times p^\textrm{female} \times \textrm{Fec}_a\\ \textrm{Egg}^\textrm{HOS}_y &= \sum_a\textrm{HOS}_{\textrm{eff}y,a} \times p^\textrm{female} \times \textrm{Fec}_a \end{align}$$

where $\textrm{HOS}_{\textrm{eff}} = \gamma \times \textrm{HOS}$ and the superscript denotes the parentage of the progeny.

Fry

Fry production is assumed to be equal to egg production, i.e., $\textrm{Fry}^\textrm{NOS}_{y+1} = \textrm{Egg}^\textrm{NOS}_y$ and $\textrm{Fry}^\textrm{HOS}_y = \textrm{Egg}^\textrm{HOS}_y$.

Smolts

Survival to the smolt stage is density-dependent. With the Beverton-Holt stock-recruit relationship, the age-1 smolt production is

$$\begin{align} \textrm{Smolt}^\textrm{NOS}_{y+1} &= \frac{\alpha \times \textrm{Fry}^\textrm{NOS}_{y+1}}{1 + \beta(\textrm{Fry}^\textrm{NOS}_{y+1} + \textrm{Fry}^\textrm{HOS}_{y+1} + n^\textrm{sub}_{y+1})}\\ \textrm{Smolt}^\textrm{HOS}_{y+1} &= \frac{\alpha \times \textrm{Fry}^\textrm{HOS}_{y+1}}{1 + \beta(\textrm{Fry}^\textrm{NOS}_{y+1} + \textrm{Fry}^\textrm{HOS}_{y+1} + n^\textrm{sub}_{y+1})} \end{align}$$

where $\alpha = \kappa/\phi$, $\beta = \alpha/{C_\textrm{smolt}}$, the unfished egg per smolt $\phi = \sum_a\left(\prod_{i=1}^{a-1}\exp(-M^\textrm{NOS}_i)(1-r_i)\right)\times r_a \times p^\textrm{female} \times \textrm{Fec}_a$, with $r_a$ as the maturity at age.

The density-independent component of the survival equation is controlled by $\alpha$ and the density-dependent component of survival is controlled by $\beta$ and scaled by the total number of fry, as well as subyearling hatchery releases (see Hatchery section), in the system.

If there is knife-edge maturity, i.e., all fish mature at the same age, then the equation simplifies to $\phi = \textrm{SAR} \times p^\textrm{female} \times \textrm{Fec}$, with $\textrm{SAR}$ as the marine survival (between 0-1).

With the Ricker stock-recruit relationship, smolt production is

$$\begin{align} \textrm{Smolt}^\textrm{NOS}_{y+1} &= \alpha \times \textrm{Fry}^\textrm{NOS}_{y+1}\times\exp(-\beta[\textrm{Fry}^\textrm{NOS}_{y+1} + \textrm{Fry}^\textrm{HOS}_{y+1} + n^\textrm{sub}_{y+1}])\\ \textrm{Smolt}^\textrm{HOS}_{y+1} &= \alpha \times \textrm{Fry}^\textrm{HOS}_{y+1}\times\exp(-\beta[\textrm{Fry}^\textrm{NOS}_{y+1} + \textrm{Fry}^\textrm{HOS}_{y+1} + n^\textrm{sub}_{y+1}]) \end{align}$$

where $\alpha = \kappa/\phi$ and $\beta = 1/{S_\textrm{max}}$.

Habitat improvement

Habitat improvement can improve either the productivity parameter, capacity parameter, or both.

Base terms $\alpha$ and $\beta$ are used in the historical period of the model, e.g., conditioning from data. In the projection, the stock recruit parameters $\alpha^\prime$ and $\beta^\prime$ are updated with the improvement parameters, $I_\kappa$ or $I_C$, as specified in the operating model.

The corresponding parameters in the projection is:

$$\begin{align} \alpha^\prime &= \dfrac{\kappa}{\phi}\times I_\kappa\\ \beta^\prime &= \begin{cases} \dfrac{\alpha^\prime}{C_\textrm{smolt} \times I_C} & \textrm{Beverton-Holt}\\ \dfrac{1}{S_\textrm{max} \times I_C} & \textrm{Ricker}\\ \end{cases} \end{align}$$

Comparison of the change in two density-dependent stock-recruit functions if the improvement parameter is applied to either the productivity (kappa) or capacity parameter. The base value of kappa is 3, Beverton-Holt capacity is 1000, and Ricker Smax is 500. The dotted line is the one-to-one unfished replacement line (corresponding to $1/\phi$).

Hatchery production

Hatchery production is controlled by several sets of variables specified by the analyst.

The first consideration is to specify the target number of annual releases of sub-yearlings $n^\textrm{subyearling}_\textrm{target}$ and yearlings $n^\textrm{yearling}_\textrm{target}$. Going backwards, the corresponding number of eggs needed to reach the target number depends on the egg survival to those life stages. Finally, the corresponding number of broodtake is dependent on the brood fecundity and hatchery survival of broodtake.

Additional considerations are the composition (natural vs. hatchery origin) of the broodtake. To minimize genetic drift of the population due to hatchery production, it is desirable to maintain a high proportion of natural origin broodtake. This is controlled by $p^\textrm{NOB}_\textrm{target}$, the desired proportion of natural broodtake relative to all broodtake, but can be exceeded if there is insufficient escapement of natural origin fish.

Second, it is also desirable to maintain high spawning of natural origin fish. This is controlled by $p^\textrm{NOB}_\textrm{max}$, the maximum allowable proportion of the natural origin escapement to be used as broodtake. This value is never exceeded.

The following equations then generate the annual broodtake and hatchery production from the state variables given these constraints.

Broodtake

The annual target egg production for the hatchery is calculated from the target releases as

$$\textrm{Egg}_\textrm{target,broodtake} = \dfrac{n^\textrm{yearling}_\textrm{target}}{s^\textrm{yearling}} + \dfrac{n^\textrm{subyearling}_\textrm{target}}{s^\textrm{subyearling}}$$

where $s$ is the corresponding survival term from the egg life stage.

The broodtake is back-calculated from the target egg production. The composition of natural and hatchery origin broodtake (NOB and HOB, respectively) is dependent on the mark rate $m$ and the target proportion of NOB $p^\textrm{NOB}_\textrm{target}$. When the mark rate is 1, then the realized pNOB should be equal to $p^\textrm{NOB}_\textrm{target}$ provided there is sufficient escapement. If the mark rate is less than one, then $p^\textrm{NOB}_\textrm{target}$ reflects the proportion of unmarked fish in the broodtake, some which are hatchery origin. Thus, the realized pNOB is less than $p^\textrm{NOB}_\textrm{target}$. If the mark rate is zero, then broodtake is non-selective with pNOB equal to the proportion of natural origin escapement.

From the escapement in year $y$, some proportion $p^\textrm{broodtake}$ is used as broodtake:

$$\begin{align} \textrm{NOB}_{y,a} &= p^\textrm{broodtake,unmarked}_y \times \textrm{NOR}^\textrm{escapement}_{y,a} \times s_\textrm{enroute} \times p^\textrm{esc}_\textrm{max}\\ \textrm{HOB}^\textrm{unmarked}_{y,a} &= p^\textrm{broodtake,unmarked}_y \times (1-m) \times p^\textrm{hatchery} \times \textrm{HOR}^\textrm{escapement}_{y,a} \times s_\textrm{enroute} \times p^\textrm{esc}_\textrm{max}\\ \textrm{HOB}^\textrm{marked}_{y,a} &= p^\textrm{broodtake,marked}_y \times m \times p^\textrm{hatchery} \times \textrm{HOR}^\textrm{escapement}_{y,a} \times s_\textrm{enroute} \times p^\textrm{esc}_\textrm{max} \end{align}$$

The availability of both natural and hatchery origin fish depends on the escapement reduced by en-route mortality and can be capped by some proportion denoted by the $p^\textrm{esc}_\textrm{max}$ parameter.

The proportion of the available hatchery fish for broodtake is also reduced by $p^\textrm{hatchery}$, which can include fish swimming back to the hatchery or removed from spawning grounds.

The realized hatchery egg production is

$$\begin{align} \textrm{Egg}_\textrm{y}^\textrm{NOB} &= \sum_a \textrm{NOB}_{y,a} \times s^\textrm{prespawn} \times p^\textrm{female} \times \textrm{Fec}^\textrm{brood}_a\\ \textrm{Egg}_\textrm{y}^\textrm{HOB} &= \sum_a (\textrm{HOB}^\textrm{marked}_{y,a} + \textrm{HOB}^\textrm{unmarked}_{y,a}) \times s^\textrm{prespawn} \times p^\textrm{female} \times \textrm{Fec}^\textrm{brood}_a \end{align}$$

where egg production is subject to a survival term $s^\textrm{prespawn}$ for the broodtake.

The proportion $p^\textrm{broodtake}_y$ is solved annually to satisfy the following conditions:

$\dfrac{\sum_a(\textrm{NOB}_{y,a} + \textrm{HOB}^\textrm{unmarked}_{y,a})}{\sum_a\textrm{NOB}_{y,a} + \sum_a\textrm{HOB}_{y,a}} = p^\textrm{NOB}_\textrm{target}$

$0 < p^\textrm{broodtake,marked}_y \le 1$

$0 < p^\textrm{broodtake,unmarked}_y \le p^\textrm{NOB}_\textrm{max}$

$\textrm{Egg}_\textrm{y}^\textrm{NOB} + \textrm{Egg}_\textrm{y}^\textrm{HOB} = \textrm{Egg}_\textrm{broodtake}$

The target ratio $p^\textrm{NOB}_\textrm{target}$ reflects the objective to maintain a high proportion of natural origin fish in the broodtake, where its implementation is dependent on the mark rate. The maximum removal rate of natural origin fish $p^\textrm{NOB}_\textrm{max}$ or escapement $p^\textrm{esc}_\textrm{max}$ ensures that there is high abundance of natural origin spawners.

The total egg production in a given year can fail to reach the target if there is insufficient unmarked escapement. In this case, the unmarked take is set to the maximum removal rate ($p^\textrm{broodtake,unmarked}_y = p^\textrm{NOB}_\textrm{max}$), and the remaining deficit in egg production is met using HOB.

Smolt releases

After the total hatchery egg production is calculated, the production of yearlings and subyearlings is calculated to ensure the annual ratio is equal to the target ratio. To do so, the parameter $p^\textrm{egg,yearling}_y$ is solved subject to the following conditions:

$\textrm{Egg}_\textrm{brood,y} = \textrm{Egg}_\textrm{y}^\textrm{NOB} + \textrm{Egg}_\textrm{y}^\textrm{HOB}$

$n^\textrm{yearling}_{y+1} = p^\textrm{egg,yearling}_y \times \textrm{Egg}_\textrm{brood,y} \times s^\textrm{yearling}$

$n^\textrm{subyearling}_{y+1} = (1 - p^\textrm{egg,yearling}_y) \times \textrm{Egg}_\textrm{brood,y} \times s^\textrm{subyearling}$

$\frac{n^\textrm{yearling}_y}{n^\textrm{subyearling}_y + n^\textrm{yearling}_y} = \frac{n^\textrm{yearling}_\textrm{target}}{n^\textrm{subyearling}_\textrm{target} + n^\textrm{yearling}_\textrm{target}}$

From the total broodtake, the smolt releases is calculated as

$$\textrm{Smolt}^\textrm{Rel}_{y+1} = n^\textrm{yearling}_{y+1} + \frac{\alpha \times n^\textrm{subyearling}_{y+1}}{1 + \beta(\textrm{Fry}^\textrm{NOS}_{y+1} + \textrm{Fry}^\textrm{HOS}_{y+1} + n^\textrm{subyearling}_{y+1})}$$

or

$$\textrm{Smolt}^\textrm{Rel}_{y+1} = n^\textrm{yearling}_{y+1} + \alpha \times n^\textrm{subyearling}_{y+1} \times \exp(-\beta(\textrm{Fry}^\textrm{NOS}_{y+1} + \textrm{Fry}^\textrm{HOS}_{y+1} + n^\textrm{subyearling}_{y+1}))$$

Subyearlings are subject to density-dependent survival in competition with natural production of fry.

Pre-terminal fishery

Let $N^\textrm{juv}_{y,a}$ be the juvenile abundance in the population and $N^\textrm{juv,NOS}_{y,a=1} = \textrm{Smolt}^\textrm{NOS}_y + \textrm{Smolt}^\textrm{HOS}_y$ and $N^\textrm{juv,HOS}_{y,a=1} = \textrm{Smolt}^\textrm{Rel}$. The superscript for the smolt variable corresponds to the parentage while the superscript for $N$ denotes the origin of the current cohort.

Harvest $u^\textrm{PT}$ in the pre-terminal ($\textrm{PT}$) fishery, assuming no mark-selective fishing, is modeled as a seasonal process. The kept catch $K$ is

$$\begin{align} K^\textrm{NOS,PT}_{y,a} &= \left(1 - \exp(-v^\textrm{PT}_a F^\textrm{PT}_y)\right)N^\textrm{juv,NOS}_{y,a}\\ K^\textrm{HOS,PT}_{y,a} &= \left(1 - \exp(-v^\textrm{PT}_a F^\textrm{PT}_y)\right)N^\textrm{juv,HOS}_{y,a}\\ \end{align}$$

The instantaneous fishing mortality is solved so that the ratio of the total catch and the total vulnerable abundance is equal to the specified harvest rate:

$$u^\textrm{PT} = \dfrac{\sum_a K^\textrm{NOS,PT}_{y,a} + \sum_a K^\textrm{HOS,PT}_{y,a}}{\sum_a v^\textrm{PT}_a (N^\textrm{juv,NOS}_{y,a} + N^\textrm{juv,HOS}_{y,a})}$$

Recruitment and maturity

The recruitment is calculated from the survival of juvenile fish after pre-terminal harvest and maturation:

$$\begin{align} \textrm{NOR}_{y,a} &= N^\textrm{juv,NOS}_{y,a}\exp(-v_aF^\textrm{PT}_y)r_{y,a}\\ \textrm{HOR}_{y,a} &= N^\textrm{juv,HOS}_{y,a}\exp(-v_aF^\textrm{PT}_y)r_{y,a} \end{align}$$

The juvenile abundance in the following year consists of fish that did not mature and subsequently survived natural mortality $M$:

$$\begin{align} N^\textrm{juv,NOS}_{y+1,a+1} &= N^\textrm{juv,NOS}_{y,a}\exp\left(-[v_aF^\textrm{PT}_y + M^\textrm{NOS}_{y,a}]\right)(1 - r_{y,a})\\ N^\textrm{juv,HOS}_{y+1,a+1} &= N^\textrm{juv,HOS}_{y,a}\exp\left(-[v_aF^\textrm{PT}_y + M^\textrm{HOS}_{y,a}]\right)(1 - r_{y,a}) \end{align}$$

Natural mortality is specified by age class. Accordingly, this mortality corresponds to either the freshwater or marine survival depending on age class.

Terminal fishery

Assuming no mark-selective fishing, the retained catch of the terminal ($\textrm{T}$) fishery is calculated from the harvest rate similarly as with the pre-terminal fishery:

$$\begin{align} K^\textrm{NOS,T}_{y,a} &= \left(1 - \exp(-v^\textrm{T}_a F^\textrm{T}_y)\right)\textrm{NOR}_{y,a}\\ K^\textrm{HOS,T}_{y,a} &= \left(1 - \exp(-v^\textrm{T}_a F^\textrm{T}_y)\right)\textrm{HOR}_{y,a}\\ \end{align}$$

subject to

$$u^\textrm{T} = \dfrac{\sum_a K^\textrm{NOS,T}_{y,a} + \sum_a K^\textrm{HOS,T}_{y,a}}{\sum_a v^\textrm{T}_a(\textrm{NOR}_{y,a} + \textrm{HOR}_{y,a})}$$

Escapement and spawners

The escapement consists of the survivors of the terminal fishery:

$$\begin{align} \textrm{NOR}^\textrm{escapement}_{y,a} &= \textrm{NOR}_{y,a}\exp(-v_aF^\textrm{T}_y)\\ \textrm{HOR}^\textrm{escapement}_{y,a} &= \textrm{HOR}_{y,a}\exp(-v_aF^\textrm{T}_y) \end{align}$$

The abundance of natural origin spawners consists of the escapement that survive migration to the spawning ground ($s_\textrm{enroute}$) and are not removed for brood:

$$\textrm{NOS}_{y,a} = \textrm{NOR}^\textrm{escapement}_{y,a} \times s_\textrm{enroute} - \textrm{NOB}_{y,a}$$

The hatchery origin spawners is the escapement that survive migration, do not return to the hatchery (either by swim-in facilities or in-river collection), and are not removed from the spawning ground (through proportion $p^\textrm{HOS}_\textrm{removal}$, these animals are not available for brood).

$$\textrm{HOS}_{y,a} = (1 - p^\textrm{hatchery}) (1 - p^\textrm{HOS}_\textrm{removal}) \textrm{HOR}^\textrm{escapement}_{y,a} \times s_\textrm{enroute}$$

Fitness effects on survival

Reproductive success of first generation hatchery fish has been observed to be lower than their natural counterparts, and is accounted for in the $\gamma$ parameter (see review in Withler et al. 2018).

Through genetic and epigenetic factors, survival of hatchery juveniles in the hatchery environment selects for fish with a phenotype best adapted for that environment, and likewise for juveniles spawned in the natural environment. Since these traits are heritable, the fitness of the natural population can shift away from the optimum for the natural environment towards that of the hatchery environment on an evolutionary time scale, i.e., over a number of generations, when hatchery fish are allowed to spawn.

As described in Ford 2002 and derived in Lande 1976, the fitness loss function $W$ for an individual with phenotypic trait value $z$ in a given environment is

$$W(z) = \exp\left(\dfrac{-(z-\theta)^2}{2\omega^2}\right)$$

where $\theta$ is the optimum for that environment and $\omega^2$ is the fitness variance.

If the phenotypic trait value $z$ in the population is a random normal variable with mean $\bar{z}$ and variance $\sigma^2$, then the mean fitness of the population in generation $g$ is $\bar{W}(z) = \int W(z) f(z) dz$, where $f(z)$ is the Gaussian probability density function. The solution is proportional to

$$\bar{W}(z) \propto \exp\left(\dfrac{-(\bar{z}-\theta)^2}{2(\omega+\sigma)^2}\right)$$

The mean phenotype $\bar{z}$ is calculated iteratively, where the change $\Delta\bar{z}$ from generation $g-1$ to $g$ is

$$\begin{align} \Delta\bar{z} &= \bar{z}_g - \bar{z}_{g-1} = (\bar{z}^\prime_{g-1} - \bar{z}_{g-1})h^2\\ \bar{z}_g &= \bar{z}_{g-1} + (\bar{z}^\prime_{g-1} - \bar{z}_{g-1})h^2\\ \end{align}$$

where $h^2$ is the heritability of $z$ and $\bar{z}^\prime_{g-1}$ is the trait value after applying the fitness function, defined as:

$$\begin{align} \bar{z}^\prime_{g-1} &= \dfrac{1}{\bar{W}_{g-1}}\int W_{g-1}(z)\times zf(z)dz\\ &= \dfrac{\bar{z}_{g-1}\omega^2 + \theta \sigma^2}{\omega^2 + \sigma^2} \end{align}$$

Let $\bar{z}^\prime_{g-1}(\theta)$ be a function that returns the mean trait value after selection in an environment with optimum value $\theta$. With a hatchery program, the mean trait value of the progeny in the natural environment is a weighted average of the mean trait value in natural and hatchery origin spawners, with selection in the natural environment, i.e., with optimum trait value $\theta^\textrm{natural}$:

$$\begin{align} \bar{z}^\textrm{natural}_g = & (1 - p^\textrm{HOSeff}_{g-1}) \times \left(\bar{z}^\textrm{natural}_{g-1} + [\bar{z}^{\prime\textrm{natural}}_{g-1}(\theta^\textrm{natural}) - \bar{z}^\textrm{natural}_{g-1}] h^2\right) +\\ & p^\textrm{HOSeff}_{g-1} \times \left(\bar{z}^\textrm{hatchery}_{g-1} + [\bar{z}^{\prime\textrm{hatchery}}_{g-1}(\theta^\textrm{natural}) - \bar{z}^\textrm{hatchery}_{g-1}] h^2\right) \end{align}$$

where $p^\textrm{HOSeff} = \gamma\times\textrm{HOS}/(\textrm{NOS} + \gamma\times\textrm{HOS})$.

Similarly, the mean trait value in the hatchery environment $\bar{z}^\textrm{hatchery}_g$ is a weighted average of the mean trait value of the natural and hatchery broodtake, with selection in the hatchery environment, i.e., with optimum trait value $\theta^\textrm{hatchery}$:

$$\begin{align} \bar{z}^\textrm{hatchery}_g = & p^\textrm{NOB}_{g-1} \times \left(\bar{z}^\textrm{natural}_{g-1} + [\bar{z}^{\prime\textrm{natural}}_{g-1}(\theta^\textrm{hatchery}) - \bar{z}^\textrm{natural}_{g-1}] h^2\right) +\\ & (1 - p^\textrm{NOB}_{g-1}) \times \left(\bar{z}^\textrm{hatchery}_{g-1} + [\bar{z}^{\prime\textrm{hatchery}}_{g-1}(\theta^\textrm{hatchery}) - \bar{z}^\textrm{hatchery}_{g-1}] h^2\right) \end{align}$$

where $p^\textrm{NOB} = \textrm{NOB}/(\textrm{NOB} + \textrm{HOB})$.

The fitness variance $\omega^2$ and phenotype variance $\sigma^2$ are assumed constant between the two environments. The fitness variance is parameterized relative to the variance of the phenotype, i.e., $\omega = \Omega \sigma$ where $\Omega$ is the “selection strength”.

The mean fitness of generation $g$ in the natural environment is then:

$$\bar{W}^\textrm{natural}_g = \exp\left(\dfrac{-(\bar{z}^\textrm{natural}_g-\theta^\textrm{natural})^2}{2(\omega+\sigma)^2}\right)$$

Mixed brood-year return

If a mixed-brood year return in year $y$ across several ages $a$ produces the smolt cohort in year $y+1$, then the mean trait value in the progeny is calculated from a weighted average by brood year and age class fecundity:

$$\begin{align} \bar{z}^\textrm{natural}_{y+1} = & \sum_a p^\textrm{NOS}_{y,a} \times \left(\bar{z}^\textrm{natural}_{y-a+1} + [\bar{z}^{\prime\textrm{natural}}_{y-a+1}(\theta^\textrm{natural}) - \bar{z}^\textrm{natural}_{y-a+1}] h^2\right) +\\ & \sum_a p^\textrm{HOSeff}_{y,a} \times \left(\bar{z}^\textrm{hatchery}_{y-a+1} + [\bar{z}^{\prime\textrm{hatchery}}_{y-a+1}(\theta^\textrm{natural}) - \bar{z}^\textrm{hatchery}_{y-a+1}] h^2\right) \end{align}$$

$$\begin{align} \bar{z}^\textrm{hatchery}_{y+1} = & \sum_a p^\textrm{NOB}_{y,a} \times \left(\bar{z}^\textrm{natural}_{y-a+1} + [\bar{z}^{\prime\textrm{natural}}_{y-a+1}(\theta^\textrm{hatchery}) - \bar{z}^\textrm{natural}_{y-a+1}] h^2\right) +\\ & \sum_a p^\textrm{HOB}_{y,a} \times \left(\bar{z}^\textrm{hatchery}_{y-a+1} + [\bar{z}^{\prime\textrm{hatchery}}_{y-a+1}(\theta^\textrm{hatchery}) - \bar{z}^\textrm{hatchery}_{y-a+1}] h^2\right) \end{align}$$

where

$p^\textrm{NOS}_{y,a} = \dfrac{\textrm{Fec}_a \times \textrm{NOS}_{y,a}}{\sum_a\textrm{Fec}_a(\textrm{NOS}_{y,a} + \gamma \times \textrm{HOS}_{y,a})}$

$p^\textrm{HOSeff}_{y,a} = \dfrac{\textrm{Fec}_a \times \gamma \times \textrm{HOS}_{y,a}}{\sum_a\textrm{Fec}_a(\textrm{NOS}_{y,a} + \gamma \times \textrm{HOS}_{y,a})}$

$p^\textrm{NOB}_{y,a} = \dfrac{\textrm{Fec}^\textrm{brood}_a \times \textrm{NOB}_{y,a}}{\sum_a\textrm{Fec}^\textrm{brood}_a(\textrm{NOB}_{y,a} + \textrm{HOB}_{y,a})}$

$p^\textrm{HOB}_{y,a} = \dfrac{\textrm{Fec}^\textrm{brood}_a \times \textrm{HOB}_{y,a}}{\sum_a\textrm{Fec}^\textrm{brood}_a(\textrm{NOB}_{y,a} + \textrm{HOB}_{y,a})}$

Effective proportions, i.e., weighting by age-class fecundity, accounts for older age classes that are more fecund and more likely to contribute to the production of next cohort.

Fitness loss

Fitness can reduce survival in the egg, fry, and immature life stages:

$$\begin{align} \textrm{Fry}^\textrm{NOS}_{y+1} &= \textrm{Egg}^\textrm{NOS}_y \times (W^\textrm{nat.}_{y+1})^{\ell_\textrm{egg}}\\ \textrm{Fry}^\textrm{HOS}_{y+1} &= \textrm{Egg}^\textrm{HOS}_y \times (W^\textrm{nat.}_{y+1})^{\ell_\textrm{egg}} \end{align}$$

$$\begin{align} \textrm{Smolt}^\textrm{NOS}_{y+1} &= \frac{\alpha^{\prime\prime} \times \textrm{Fry}^\textrm{NOS}_{y+1}}{1 + \beta^{\prime\prime} (\textrm{Fry}^\textrm{NOS}_{y+1} + \textrm{Fry}^\textrm{HOS}_{y+1})}\\ \textrm{Smolt}^\textrm{HOS}_{y+1} &= \frac{\alpha^{\prime\prime} \times \textrm{Fry}^\textrm{HOS}_{y+1}}{1 + \beta^{\prime\prime} (\textrm{Fry}^\textrm{NOS}_{y+1} + \textrm{Fry}^\textrm{HOS}_{y+1})} \end{align}$$

$$M^\textrm{NOS}_{y,a} = -\log(\exp(-M^\textrm{base,NOS}_{y,a}) \times (W^\textrm{nat.}_{y-a})^{\ell_\textrm{smolt}})$$

with $\ell_i$ is the proportion of the fitness loss apportioned among the three life stages, $\sum_i \ell_i = 1$, and density-dependent parameters $\alpha^{\prime\prime} = \frac{\kappa}{\phi} \times I_\kappa \times (W^\textrm{nat.}_{y+1})^{\ell_\textrm{fry}}$, $\beta^{\prime\prime} = \alpha^{\prime\prime}/[C_\textrm{smolt} \times I_C \times (W^\textrm{nat.}_{y+1})^{\ell_\textrm{fry}}]$.

In the marine environment, age-specific natural survival is reduced proportional to the fitness loss term and modeled as a cohort effect.

With the Ricker density-dependent survival,

$$\begin{align} \textrm{Smolt}^\textrm{NOS}_{y+1} &= \alpha^{\prime\prime} \times \textrm{Fry}^\textrm{NOS}_{y+1} \times \exp(-\beta^{\prime\prime} [\textrm{Fry}^\textrm{NOS}_{y+1} + \textrm{Fry}^\textrm{HOS}_{y+1}])\\ \textrm{Smolt}^\textrm{HOS}_{y+1} &= \alpha^{\prime\prime} \times \textrm{Fry}^\textrm{HOS}_{y+1} \times \exp(-\beta^{\prime\prime} [\textrm{Fry}^\textrm{NOS}_{y+1} + \textrm{Fry}^\textrm{HOS}_{y+1}]) \end{align}$$

with $\beta^{\prime\prime} = 1/[S_\textrm{max} \times I_C \times (W^\textrm{nat.}_{y+1})^{\ell_\textrm{fry}}]$.

PNI

PNI (proportionate natural influence) is an approximation of the rate of gene flow from the hatchery to the natural environment, calculated for the progeny in year $y+1$ from the parental composition of year $y$:

$$\textrm{PNI}_{y+1} = \dfrac{\sum_a p^{\textrm{NOB}}_{y,a}}{\sum_a p^{\textrm{NOB}}_{y,a} + \sum_a p^{\textrm{HOSeff}}_{y,a}}$$

Generally, a combination of minimizing hatchery releases, increasing natural broodtake, and reducing the number hatchery origin spawners maintains high PNI, i.e., low rate of gene flow from the hatchery to natural environment.

Wild salmon

With single brood-year returns, the proportion of wild salmon, natural origin spawners whose parents were also natural spawners, can be calculated as

$$p^\textrm{WILD}_g = (1 - p^\textrm{HOScensus}_g) \times \dfrac{(1 - p^\textrm{HOScensus}_{g-1})^2} {(1 - p^\textrm{HOScensus}_{g-1})^2 + 2 \gamma \times p^\textrm{HOScensus}_{g-1}(1 - p^\textrm{HOScensus}_{g-1}) + \gamma^2 (p^\textrm{HOScensus}_{g-1})^2}$$

where $p^\textrm{HOScensus} = \textrm{HOS}/(\textrm{HOS} + \textrm{NOS})$.

The first term is the proportion of natural spawners in the current generation $g$.

The ratio comprising the second term discounts the proportion of the current generation to include natural spawners whose parents were both natural spawners. Assuming non-assortative mating, the three terms in the denominator gives the composition of generation $g$ whose parents who are both natural origin, mixed origin (one parent in natural origin and the other is hatchery origin), and both hatchery origin.

To generalize for mixed-brood year return, we calculate the probability weighted across brood-years and age class fecundity:

$$p^\textrm{WILD}_y = \sum_a \dfrac{\textrm{NOS}_{y,a}}{\sum_{a'}(\textrm{NOS}_{y,a'} + \textrm{HOS}_{y,a'})} \times \dfrac{(\sum_{a'} p^\textrm{NOScensus}_{y-a,a'})^2} {(\sum_{a'} p^\textrm{NOScensus}_{y-a,a'})^2 + 2 \gamma \times (\sum_{a'}p^\textrm{NOScensus}_{y-a,a'})(\sum_{a'}p^\textrm{HOScensus}_{y-a,a'}) + \gamma^2 (\sum_{a'}p^\textrm{HOScensus}_{y-a,a'})^2}$$

where

$p^\textrm{NOScensus}_{y,a} = \dfrac{\textrm{Fec}_a \times \textrm{NOS}_{y,a}}{\sum_a{\textrm{Fec}_a (\textrm{NOS}_{y,a}} + \textrm{HOS}_{y,a})}$

$p^\textrm{HOScensus}_{y,a} = \dfrac{\textrm{Fec}_a \times \textrm{HOS}_{y,a}}{\sum_a{\textrm{Fec}_a (\textrm{NOS}_{y,a}} + \textrm{HOS}_{y,a})}$

The probability of finding a wild salmon in year $y$ is the sum of probabilities of finding a wild salmon over all ages. For each age $a$, the first ratio is the probability of finding a natural spawner in year $y$. The second ratio is the probability of mating success from two parental natural spawners in year $y-a$ using a Punnett square, assuming non-assortative mating across age and origin. The summation across dummy age variable $a'$ calculates the total proportion of spawners in a given year.

Effective proportions, i.e., weighting by age-class fecundity, in the parental composition accounts for older age classes that are more fecund and more likely to contribute to the production of offspring.

Mark-selective fishing

If the mark rate $m$ of hatchery fish is greater than zero, then mark-selective fishing is implemented for both the pre-terminal and terminal fisheries. The mark rate is a proxy for retention and the harvest rate $u^\textrm{harvest}$ corresponds to the ratio of the kept catch and abundance. The exploitation rate $u^\textrm{exploit}$ is calculated from kept catch and dead releases. Exploitation rates differ between hatchery and natural origin fish because there is no retention of the latter.

Let the instantaneous fishing mortality for kept catch and released catch be

$$\begin{align} F^\textrm{kept} &= mE\\ F^\textrm{rel.} &= (1 - m)\delta E \end{align}$$

where $\delta$ is the proportion of released fish that die.

$E$ is an index of fishing effort, also referred to as the encounter rate by the fishery, that links together $F^\textrm{kept}$ and $F^\textrm{rel.}$. Intuitively, fishing effort can increase in a mark-selective fishery compared to a non-selective fishery. For example, if the mark rate is 20 percent, then the fishing effort could be 500 percent higher than in a non-selective fishery in order to attain the kept quota or bag limit. Additional catch and release mortality then occurs for un-marked fish, according to $\delta$.

In the pre-terminal ($\textrm{PT}$) fishery, $E$ is solved to satisfy the following equation for hatchery fish:

$$u^\textrm{harvest,HOS,PT} = \dfrac{\sum_aK^\textrm{HOS,PT}_{y,a}}{\sum_a v^\textrm{PT}_a N^\textrm{juv,HOS}_{y,a}}$$

where the kept catch $K$ is

$K^\textrm{HOS,PT}_{y,a} = \dfrac{F^\textrm{kept,PT}_y}{F^\textrm{kept,PT}_y + F^\textrm{rel,PT}_y}\left(1 - \exp(-v^\textrm{PT}_a[F^\textrm{kept,PT} + F^\textrm{rel,PT}])\right)N^\textrm{juv,HOS}_{y,a}$.

The exploitation rate for natural origin fish is calculated from dead discards. The exploitation rate for hatchery origin fish is calculated from kept catch and dead discards:

$$\begin{align} u^\textrm{exploit,NOS,PT}_y &= \dfrac{\sum_a (1 - \exp(-v_a F^\textrm{rel.,PT}_y))N^\textrm{juv,NOS}_{y,a}}{\sum_a v^\textrm{PT}_a N^\textrm{juv,NOS}_{y,a}}\\ u^\textrm{exploit,HOS,PT}_y &= \dfrac{\sum_a (1 - \exp(-v_a[F^\textrm{kept,PT}_y + F^\textrm{rel.,PT}_y]))N^\textrm{juv,HOS}_{y,a}}{\sum_a v^\textrm{PT}_a N^\textrm{juv,HOS}_{y,a}} \end{align}$$

Similarly, in the terminal fishery, the fishing effort satisfies the equation

$$u^\textrm{harvest,HOS,T} = \dfrac{\sum_aK^\textrm{HOS,T}_{y,a}}{\sum_a v^\textrm{T}_a \textrm{HOR}_{y,a}}$$

with the corresponding exploitation rates:

$$\begin{align} u^\textrm{exploit,NOS,T}_y &= \dfrac{\sum_a (1 - \exp(-v_a F^\textrm{rel,T}_y))\textrm{NOR}_{y,a}}{\sum_a v^\textrm{T}_a \textrm{NOR}_{y,a}}\\ u^\textrm{exploit,HOS,T}_y &= \dfrac{\sum_a (1 - \exp(-v_a[F^\textrm{kept,T}_y + F^\textrm{rel.,T}_y]))\textrm{HOR}_{y,a}}{\sum_a v^\textrm{T}_a \textrm{HOR}_{y,a}} \end{align}$$