06 Recruitment (Parts 1 & 2)

06 Recruitment - RECRUITMENT MODELS A Recruits for our purposes “Recruitment" is not reproduction(as eggs or larvae but those entering the

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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> Rt-1 Pt-1 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 Rt-1 0 Rt-1 = Pt-1 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 pre-recruits. 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) Rt-1 Pt-1 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 Rt-1 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 t-1 (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 non-homogeneous 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 = "! + "" X-c 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 pre-recruit 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.

Ask a homework question - tutors are online