This preview shows page 1. Sign up to view the full content.
Unformatted text preview: RECRUITMENT MODELS
A. Recruits  for our purposes “Recruitment" is not reproduction (as eggs or larvae), but
those entering the population as usable stock
a) KNIFE  EDGED recruitment  all fish of an age enter at the same time of year
b) By PLATOONS  the vulnerability of an age class increases gradually over 2 or more
years, but those that enter come in at 1 time of year
c) CONTINUOUS  vulnerability increases gradually over 2 or more years as the fish grows.
 in applying the models below on an annual basis then we describe situation a above (Knife
 edged recruitment) best, the others successively less well DESCRIBING THE STOCK  RECRUIT RELATIONSHIP
RICKER DESCRIBES DESIRABLE PROPERTIES of a STOCK RECRUIT CURVE
1) No intercept
2) Should not cross X axis at higher population levels
3) Recruitment rates ( R ) should decrease continuously over P similar to
P
decreasing as N increases in the Schaeffer model
4) at some part of the range we should observe R P
(when in equivalent units) otherwise, the stock cannot replenish itself ˜N
N EXST025  Biological Population Statistics Page 2 DERIVATION OF RICKERS MODEL  a scenario (later generalized)
a) Suppose that earliest recruitment is INDEPENDENT of the density of R (i.e. no
density dependence) so it depends only on P (environmental influence is a source
of random variation)
then
R>" = "! P>" = (b  d) P>"
where b and d are constants similar to the previously defined
˜ N> = rN> = (b  d) N> Rt1 Pt1
we are not interested in time for recruitment and will not integrate (we are interested
in only one season at a time)
b) now suppose that later some fraction of recruit mortality IS density dependent
i.e. suppose that the final number of recruits (at time t) decreases proportionally to
the number of recruits at time (t 1). This could probably be a decay type curve. 1
Rt
Rt1
0 Rt1 = Pt1 NOTE: that the X axis is not “time", it is a value of stock EXST025  Biological Population Statistics Page 3 the model is;
R>
R>" = e# R>" or R> = R>" e# R>" This would require two measures of recruits (or life cycle stages), 1 early and 1 late
(if taken literally, or the two types of mortality may be working simultaneously)
R> is the recruitment we are actually interested in,
R>" is actually a number of prerecruits.
Let's suppose that for some fish
a) egg production at sea is a strict linear function of parental stock
Eggs = bP b = “births" we could even include some density independent deaths, since this could also be
expressed as a function of parental stock, which might get us up to a later life stage
with only density dependent mortality.
b) most density independent mortality occurs early
1) either larvae don't survive due to environmental conditions
2) or don't reach nursery grounds
3) predation is density independent because most predator stocks have not yet
had time to respond so, to this point deaths are a constant (i.e. random
variable, independent of R) Rt1 Pt1
Then, the stock reaching nursery ground is essentially a constant function of parental stock
let this be (arbitrarily) “R>" ", where R>" = (b  d) P = "! P
since at this point there is no density dependence, any term above is a constant multiple of
the others EXST025  Biological Population Statistics now suppose that the factors affecting a prolonged stay in the nursery areas are principally
density dependent
1) predators stocks respond
2) relatively restricted area results in increased parasitism
3) competition for food and space
which can be described by
R> = R>" e# R>" Rt Rt1
then since R>" = "! P>" substitute this into the model above, and
R> = "! P>" e#"! P>"
let the constants #"! = "" , and P represent Parental stock at some previous time
(eg. P>" ), so we simplify
R> = "! P> e"" P>
in fact, the subscript on P is likely to be the same as on R, as a statistical model indicating
which P matches with which R. However, we understand Biologically that this
must be some previous time.
we have a recruitment curve which contains both density independent ("! ) and
density dependent ("" ) parameters
End of Scenario: In fact, the process of density dependent and density independent
mortalities take place simultaneously. From the final model it is apparent that
several lags are not necessary, all resulting recruitment (R> ) can be described on
the basis of the Parental stock (P) at some previous time. Page 4 EXST025  Biological Population Statistics Page 5 THE RICKER MODEL is not linear Rt Pt
 it can be fitted as either a non  linear equation or as a linearized model (which have
different variance assumptions)
best linearized version is by Rounsefell (1958) see Ricker
R> = "! Pe"" P %>
1) log(R> ) = log("! ) + log(P)  "" P + log(%> )
log(R> )  log(P) = log("! )  "" P + log(%> ) log R> = log("! )  "" P + log(%> )
P where R> is measured at time t and P measured at time t1 (possibly the preceding
season, depending on the species)
or take
R> = "! Pe"" P %>
and divide through by P
R>
P and then take logs = "! e"" P %> log R> = log("! )  "" P + log(%> )
P EXST025  Biological Population Statistics Page 6 FAMILY OF RICKER CURVES Rt Pt
a “diagonal" line is generally enter to indicate what recruitment is necessary for
exact replacement of the parental stock
this will be “diagonal" such that R> = P if we are talking about reproducing recruits,
Effect of varying "! and "" (see Pitcher and Hart) 0 constant
1 varying 0 varying
1 constant EXST025  Biological Population Statistics Page 7 For the model
R> = "! Pe"" P %>
The maximum number of recruitment occurs when (from 1st derivative)
Pmax = 1
"" and
Rmax œ "!
"" e œ 0.3679"!
"" This calculation can be made to get the Parental stock for Maximum recruitment, and the
maximum recruitment regardless of the units of Parental stock and recruits. (see
Ricker)
However, If R and P are in the same units, then we can calculate a point where the
parental stock precisely replace themselves.
P = R = Pr
then substitute Pr into the original model (for both R and P) and solve for
Pr = ln("! )
"" and note that
"! = e"" Pr
"" = Pr ln("! )
and define a new constant
" = "" Pr EXST025  Biological Population Statistics Page 8 If we take the original Ricker curve, and have R = P, and consider the relationship for "!
above, after completing the replacement of terms the result can be rewritten as
Pr R> = e"" Pr Pe"" P Pr
Pr R> = Pe"" Pr "" P Pr P R> = Pe"" Pr "" Pr Pr
R> = Pe"" Pr
R> = Pe" 1 P Pr P
1 P r Replacement adjusted form of the Ricker curve.
which occurs when the parental stock and recruits are expressed in the same units (eg.
Eggs, Recruits, etc.).
This is a model in which Pr is a parameter (useful if Pr ) can be estimated, and whose shape is
determined by a single parameter " . Replacement adjusted form EXST025  Biological Population Statistics Page 9 MAXIMUM SURPLUS occurs at a lower level than maximum Recruitment,
just as the MEY is less than the MSY Rt Pt
The Maximum surplus yield is another estimate of MSY, since if we could manage a
stock to maximize recruitment, we would also maximize stock levels and yield.
Ricker has a Appendix III which gives numerous estimates and characteristics for the two
models above. The MSY for the 2 models occurs when the Recruitment line is
parallel to the replacement line. The equations are;
For the general recruitment model
R> = "! Pe"" P %>
MSY occurs when the slope equals 1, or
(1 "" P)"! e"" P = 1 And for the Replacement model
R> = Pe" P
1 P r MSY occurs when the slope equals 1, or
P
(1 " Pr )e" P
1 P r = 1 EXST025  Biological Population Statistics Page 10 RECALL THE DESIRABLE PROPERTIES OF A RECRUITMENT CURVE GIVEN BY
RICKER
1) No intercept
2) Does not cross X axis at higher population levels
3) Recruitment rates ( R ) should decrease continuously over P similar to
P
decreasing as N increases in the Schaeffer model ˜N
N 4) at some part of the range we should observe R P
(when in equivalent units)
otherwise, the stock cannot replenish itself
œ note that his curve fulfills all of the criteria NOTES ON RECRUITMENT MODEL
1) Assumption for regression apply as usual
2) ERROR IN ORIGINAL MODEL IS ASSUMED MULTIPLICATIVE if additive,
then it is Non  linear
R> = "! Pe"" P %>
RICKER STATES that the log version has the “advantage of stabilizing" the variance, I
assume by this that he expects nonhomogeneous variance
NON LINEAR MODELS IN SAS  Derivative free version (now available) get prior
estimates from the linearized version
PROC NLIN;
PARAMETERS B0 = 1.0
B1 = 0.1;
MODEL R = B0 * P * EXP (B1 * P);
Note: iterative solution
see handout  derivative free method did not give same results as with derivatives for
this data
it will also be less efficient EXST025  Biological Population Statistics Page 11 ANOTHER RECRUITMENT CURVE  HYPERBOLIC
HYPERBOLA  general
b" = Xi Yi
which describes a RECTANGULAR HYPERBOLA Y X
asymptotic on both X and Y
fit as
1
Yi = b" Xi inverse relationship
or as
Yi = 1
bw X i
" EXST025  Biological Population Statistics OTHER HYPERBOLAS
1
Yi = b! + b" Xi asymptote on the intercept, and on the Y axis b goes up b goes down
all above have additive error assumed, etc.
Hyperbolic curve with an intercept
1
Yi = "! + "" Xc this has to be fitted as a nonlinear curve, or an estimate of “c" is required for linear fitting
(subtract from X) Page 12 EXST025  Biological Population Statistics Page 13 BEVERTON  HOLT RECRUITMENT CURVE
R œ 1
"! ""
P œ P
"! P "" note the problem if the first equation calculating when P = 0
not the same intercept ("! ) and slope ("" ) as for Rickers' curve
Shape of the Curve Asymptote 1
b0 2 transformations to linearize
a) 1
R 1
= "! "" P invert both sides getting asymptote = 1
"! note that in this form, P cannot equal 0 (undefined)
b) P
R = "" "! P Paulick (1973)
note interchange of coefficients EXST025  Biological Population Statistics Page 14 ASSUMPTIONS
Note that version (a) and (b) have incompatible in terms of error
if 1
R but, if 1
œ "! "" P %
1
R 1
œ "! "" P then
%
P P
R œ "" "! P %P then P
R œ "" "! P % we saw this with other model presented as a correction for non homogeneity when
the model was additive
Other assumptions as usual for linear regression
See example for fit
Biological assumptions 1) Obviously, the index of parental stock must be producing the recruits, so the assumption
of a closed stock is indicated.
2) Also, unless the production of all parental units is equal (ie. all ages produce the same
number of eggs) an assumption of constant age structure may be needed (or an
age adjustment). EXST025  Biological Population Statistics Page 15 Effect of varying "! and "" (see Pitcher and Hart) 0 constant
1 varying 0 varying
1 constant EXST025  Biological Population Statistics Page 16 As with the Ricker curve, there are various values of interest which can be derived from the
Beverton  Holt recruitment equation (see Ricker Appendix III).
a) Basic Equation
R œ P
"! P "" b) The maximum recruitment rate is
Rmax œ 1
"! which occurs as P p ∞
c) First derivative of the basic equation
R œ ""
"! P "" # which gives the MSY when solved for 1
d) Replacement Adjusted form of the Beverton  Holt Recruitment Curve
R œ P
1 " 1 P Pr where " is a new, single constant Replacement adjusted form
see Ricker (Appendix III) for other values of interest EXST025  Biological Population Statistics Page 17 Which of the two Recruitment models is Best? (see Ricker and Pitcher & Hart)
The Ricker curve is best when
a) there is cannibalism by the adults
b) increased prerecruit density results in slower growth and a longer vulnerability
c) high initial densities result in increased predation (population response by the
predator)
d) scramble competition exists  high level of competition exists between the recruits
results in diminished condition and increased mortality. The whole population is
affected.
This type of situation results in a HIGH domed Ricker curve.
e) contest competition (for riffle areas or safe refuges) will result in a low domed
Ricker curve.
This type of competition results in increased mortality for the loosers of contest competition,
but the winners are relatively unaffected.
The Beverton Holt recruitment curve is better when
a) Extreme situations of contest competition exist, such that the number of survivors
is determined (and asymptotic on) the number of save refuges.
Note that the LOW domed Ricker curve approaches the shape of the hyperbolic, asymptotic
curve.
b) when a ceiling is placed on recruitment level by available food or habitat
c) predator attack rates are continuously adjusted to changes in the prey abundance
eg. A good Bluegill recruitment year will result in a good Bass recruitment year, and the
bass feed continuously on Bluegills, resulting in a Ricker curve(M> responding to
R! ). However, if the Bass switched the preferred prey species to Crappie or
Pumkinseed as Bluegill levels fell below a certain level, an asymptotic curve would
result (M> responding to R>" ). EXST025  Biological Population Statistics Page 18 NOTES on Recruitment Curves.
Getting Parental Stock and Recruits in the same units is not as easy as it sounds.
Very often the abundances are measured only by indices, and rarely does one year
of Parental stock produce one year of Recruits (as in shrimp).
Linearization of the Ricker curve (using logs) results in geometric means, while
direct fitting with NLIN results in arithmetic means. This will cause differences in
addition to the nature of the error term differing. Geometric means are smaller
than arithmetic means.
Recall also the asymmetric confidence intervals resulting from the detransformation of
geometric means.
Linearization of the Beverton Holt model with inverses results in harmonic means
which are smaller than BOTH geometric and arithmetic means.
Several versions should be tried for each model (especially the Beverton Holt) as
unlikely results are possible
eg. Not expected Expected possible
I tried several cases on two data sets from Ricker, and got unacceptable results for both on
all linearized models. EXST025  Biological Population Statistics Page 19 The Beverton Holt model shares many of the desirable properties described by Ricker for a
Recruitment Curve.
1) No intercept  BUT REMEMBER THAT FOR ONE FORM OF THE
EQUATION, P = 0 DOES NOT EXIST.
2) Does not cross X axis at higher population levels
3) Recruitment rates ( R ) should decrease continuously over P similar to
P
decreasing as N increases in the Schaeffer model ˜N
N 4) at some part of the range we should observe R P
(when in equivalent units) Beverton Holt model in SAS
Set up any inverses needed in the DATA step.
For the NLIN fit, if spurious results occur (parameters going negative should not be
negative, BOUNDS can be placed to get the best fit with the restriction that the
parameter be positive.
eg.
PROC NLIN;
PARAMETERS B0 = 0.16 B1 = 0.06;
BOUNDS B0 0 B1 0;
MODEL R = P / (B1 * P B0); EXST025  Biological Population Statistics Page 20 Other hyperbolic models
One model we have seen previously can also produce a hyperbolic fit.
Recall
Yi = "! Xi""
Used for the Length  Weight relationship, and various applications of meristic relationships 30 b1=negative 25 b1=0 20
15 0<b1<1 10 b1=1 5 b1>1 0
0 5 10 This model can have some unexpected behaviors.
Know what to look for. 15 20 25 30 ...
View
Full
Document
This note was uploaded on 12/29/2011 for the course EXST 7025 taught by Professor Geaghan,j during the Spring '08 term at LSU.
 Spring '08
 Geaghan,J

Click to edit the document details