Recruitment in terms of survival
For some species it might be more intuitive to consider the stock recruitment relationship
in terms of survival rather than recruitment (although the same functions can be used to
represent pregnancy rates and other related processes).
R=SE
(eq. 1)
Where S is survival and E is the number of eggs.
The commonly used Ricker and Beverton-Holt stock-recruitment models can be put in
terms of survival
Beverton-Holt
S

1  D
(eq. 2)
Ricker model
S   exp  D
(eq. 3)
Where  is the maximum survival,  is a model parameter, D is the density measure
that controls density dependence and could be equal to E.
Both of these models show concave decreasing survival functions that describe survival
decreasing fastest at low stock size. For many species, particularly low fecund species, it
might be more reasonable to expect survival to decrease faster when the population
approaches carrying capacity.
The logistic recruitment model assumes a linear decline in survival with density.
S    D
(eq. 4)
Which can be reformulated in terms of the carrying capacity (see Punt XXXX and Taylor
200X)

D

S   1 
D
0 

(e.q. 5)
The move flexible three parameter stock-recruitment models may be more appropriate to
model a wider range of survival functions including both convex and concave functions
of density. The Deriso-Schnute is a general three parameter model that has special cases
that include the Beverton-Holt (   1 ), Ricker (   0 ), recruitment proportional to
spawners (    ) and logistic (   1 ). The model is convex decreasing when   1 and
 0.
S   1  D 
1/ 
(e.q. 6)
An alternative to the Deriso-Schnute model is the three parameter stock-recruitment
curve that has been applied to marine mammals (e.g. Pun XXXX; Breen et al. XXXX)
and is based on the generalized logistic or Pella-Tomlinson model. The survival function
for this stock-recruitment relationship is
  D 
S   1   
  





(eq. 7)
  0 and   0
And allows the survival to be either a convex (   1 ) or concave ( 0    1 ) decreasing
(   0 ) function of density.
To make the generalized logistic stock-recruitment model more consistent with
contemporary uses of stock-recruitment models in fisheries stock assessment. The
survival, S, is modeled using the generalized logistic equation formulated in terms of the
maximum survival rate , Smax, when the abundance measure, D, that represents the
density dependent effect approaches zero, and the survival rate that occurs when the
population is in a virgin state, S0. (note E and D may be the same or different and Smax
may be greater than 1 if E is only a relative index of “egg” production).
 Dz
S  1  z
 D0

S max  S 0   S 0

(eq. 8)
Where S0 can be calculated as a function of the virgin recruitment, R0, and virgin “egg”
production, E0.
S0 
R0
E0
(eq. 9)
Finally, the shape parameter of the generalized logistic can be reparameterized in terms
of survival, Sx, at a given depletion level x, where x =D/D0.

S  S0 
ln 1  x

S max  S 0 
z 
ln x
(eq. 10)
It also may be convenient to define Sx as a fraction of Smax, Sx = pSmax, so that p is the
model parameter to fix or estimate. For example, S50 = 0.8Smax and the parameters of the
model to estimate are Smax and R0. The parameter p or Sx could also be estimated. The
quantities D0 and E0 are calculated from R0 using the natural mortality at age and other
biological quantities.
Marine mammal applications parameterize the generalized logistic model in terms of the
maximum sustainable yield level (MSYL) and the maximum sustainable yield rate
(MSYR) (Punt XXXX).
Taylor (200X) suggests redefining Smax = Sfrac(1-S0) + S0 to avoid keeping Smax
higher than S0, as Sfrac is naturally defined between 0 and 1 (if E is eggs not some
relative term) and it may also reduce confounding between S0 and Smax.
 Dz 
S  1  z S frac 1  S 0   S 0
 D0 
(eq. 11)
Ian Taylor also suggested reparameterization of Sx in a way that would deal directly with
the constraint S0 < Sx < Smax, such as,
S x  S 0  pS max  S 0 
(eq. 12)
In terms of the reparameterization of Smax suggested above, this would be
S x  S 0  p S frac 1  S 0 
(e.q. 13)
Appendix 1: Reparameterization in terms of steepness.
The shape parameter (from Eq. 10) could also be parameterized in terms of steepness, h,
the recruitment as a percentage of R0 achieved at 20% of E0. This is commonly used to
reparameterize the Beverton-Holt stock-recruitment model, but is not as useful for stockrecruitment models that are dome-shaped. If E ≠ D then this is somewhat problematic
since E20 ≠ 0.2E when D/D0 = 0.2 and the value of E20 will depend on how D20 was
achieved.
 hR0

 S0 

E

ln 1  20
 S max  S 0 



z 
ln 0.2
(e.q. A1.1)
Appendix 2: Other interesting quantities.
The depletion level where maximum recruitment occurs (from Eq. 10) can be found and
could be used to fix the shape parameter. However, this calculation requires that E = D.

DR max 
S max

 
D0
 S max  S 0  z  1 
1/ z
(e.q. A2.1