All natural stocks incorporate a truncated Ricker Spawner/Recruit Relationship (SRR) relating spawners to adult equivalent recruitment. The general form of the Ricker SRR is:

[4.7]

where

• Spawners = number of adult spawners on the spawning grounds
• Recruits = number of adults recruiting to the fishery
• = productivity parameter
• = capacity parameter.

The general form of the Ricker SRR is illustrated in Fig. 4.4. The slope of the curve at the origin is and is the spawning level at the point where the SRR intersects the exact replacement line (in most cases, this is equivalent to the equilibrium condition in the absence of harvesting).

Fig. 4.4 Typical Ricker spawner-recruit relationship. is the spawning level that produces maximum sustainable harvest (i.e., maximum difference between recruitment and exact replacement line), produces the maximum number of recruits, and is the equilibrium spawning level in the absence of harvest.

For each stock, the input data files provide , but not . Instead the model inputs the estimated optimum number of spawners (), as determined from historical data and field observations. The Ricker parameter is then computed from and using the approximation given by Hilborn (1985):

[4.8]

Maximum production () is given by

[4.9]

Note that Recruits in eq (4.7) includes ages two through five. For modeling purposes, it is necessary to simulate the production of AgeOneFish, not the mature fish recruiting to the fishery. For chinook salmon stocks, several year classes may contribute to the spawning stock. In the equilibrium condition with no fishing, the age distribution is stationary and there is a constant linear relationship between adult recruitment and AgeOneFish (Fig. 4.5).

Fig. 4.5 Illustration of abundance and recruitment when there is no fishing mortality and the age distribution is stationary.

For ages three, four and five, the abundance at the start of each year and the number of mature fish returning to spawn are given by

N(s,a) = N(s,a-1)SurvRt(s,a-1)(1-MatRt(s,a-1)) [4.10]

Recruits(s,a) = N(s,a)SurvRt(s,a)(1-MatRt(s,a)) [4.11]

The model relates spawners to AgeOneFish by computing a constant scaling factor called RecAtAge1. This value is computed by setting N(s,1) equal to one and recursively computing N(s,a) using eq (4.10) and summing Recruits(s,a) for ages two through five:

[4.12]

When maturation rates are permitted to vary by year, new RectAtAge1 parameters are computed each year. The number of AgeOneFish is computed by:

[4.13]

For natural stocks without supplementation, the Ricker SRR is truncated at either or (Fig. 4.6).

Fig. 4.6 Truncated Ricker curves used for natural production with no enhancement.

Supplementation

The model allows for enhancement of natural stocks (also called supplementation) in which a portion of the natural spawners are removed for hatchery production. The number of spawners removed may not exceed a maximum allowable percentage of the adult spawners.

MaxBrood(s) = EnhProp(s)·Spawners(s) [4.14]

where

• MaxBrood(s) = maximum number of spawners that can be removed for supplementation for stock s.
• EnhProp(s) = maximum enhancement proportion for stock s.

When density independence is assumed, natural and hatchery production are computed independently and added together. The naturally produced portion of AgeOneFish is computed from the remaining natural spawners (i.e., Spawners(s) - EnhSpawn(s)) using the appropriate truncated Ricker curve (Fig. 4.6). Hatchery produced portion of AgeOneFish is computed as follows:

[4.15]

When density dependence is assumed, AgeOneFish is computed using a truncated Ricker curve (Fig. 4.6), but the "effective" size of the spawning stock is increased to reflect the fact that eggs from some of the spawners are reared in a hatchery. The enhancement efficiency of the hatchery is given by

[4.16]

In general, HatchProd(s) is greater than (s) so EnhEff(s) is usually greater than one. The effective number of spawners is given by

[4.17]

Fig. 4.7 Production functions for natural stocks with enhancement. Ricker curves truncated at are shown. Other stocks may have the Ricker curve truncated at .