ABSTRACT
This supplementary material is an appendix to Chapters 7 and 12 that can help to understand useful Ricker stock-recruitment concepts.
The Ricker stock-recruitment relationships (see Fig. 7.3 in Chapter 7) stems from the hypothesis that the per capita mortality rate (due to predation, disease, cannibalism, etc.) of larvae during their growing phase (0 < t < T ) is linearly dependent on the population size of spawners (St) with slope and intercept function of the fluctuating environment ([244]):
Nt
dNt dt
= k1(t)− k2(t)× St
Hypothesizing:
1. no mortality of adults during the larval growing phase,i.e.,St = S0 = S,
2. initial condition that the eggs are proportional to the number of spawners N0 ∝ S,
3. terminal condition yielding the recruitment R = Nt,
the previous equation can be solved as
NT = N0 × exp {∫ T
(k1,t − k2,t × S) dt }
= N0 × exp (∫ T
k1,tdt− S × ∫ T
k1,tdt
)
leading to the classical Ricker form already given by Eq. (7.4)
R = αS × exp(−β × S) (C.1)
Therefore an elementary model with nonoverlapping generations can be made of a simple deterministic Ricker stock-recruitment cycle with recruits R, surviving as adults with a natural mortality pi and becoming spawners S after catch C. This is depicted by the following equations:{
Rt = αSt−1 × exp(−βSt−1) St = piRt − Ct
Rearranging the terms it comes:
Rt+1 = α× ( Rt − Ct
pi
) × exp
( log(pi)− βpi
( Rt − Ct
pi
)) Thus
βpi
log(α) + log(pi) Rt+1 =
βpi
log(α) + log(pi) × ( Rt − Ct
pi
) × α× exp
( log(pi)×
( 1− βpi
log(α) + log(pi)
( Rt − Ct
pi
))) and rescaling variables x = R × βpilog(α)+log(pi) , r = α × exp(log pi) and u = C× βlog(α)+log(pi) for the ease of notations, one highlights the special role played by the intrinsic log-survival rate r:
xt+1 = (xt − ut)× e−r×{1−(xt−ut)} (C.2) Note that this simplified deterministic salmon cycle with nonover-
lapping generations is a particular member of the general class of deterministic models in discrete time ([24]):
xt+1 = F (xt − ut) (C.3)
with F (z) = z × e−r×(1−z).
