MODELS FOR VARIABLE RECRUITMENT (continued)

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Andra Nichols
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1 ODL FOR VARIABL RCRUITNT (ontinue) The other moel ommonly use to relate reruitment strength with the size of the parental spawning population is a moel evelope by Beverton an Holt (957, etion 6), whih is on the Reommene Reaing list. The Beverton & Holt pawner-reruit oel We will use the same notation as before. k is the total number of eggs, whih then beome larvae, then juveniles, an then aults. are the number of spawning aults, the parents of. is the average number of eggs lai per spawning ault. is the total number of eggs lai. k The Beverton an Holt stok-reruit moel omes from the solution to the following simple ifferential equation. t where is the ensity-epenent rate of natural mortality. is the ensity-inepenent rate of natural mortality. Here the ensity epenent term is proportional to the ohort abunane, not to the parental abunane (as in the Riker moel). This form of mortality oul arise from proesses suh as ompetition for foo or other sare resoures. This ifferential equation is muh more ompliate to solve than the one that le to the Riker moel. t ( ) As before, we start by separating the variables. ( ) t But, now what o we o? We an solve this equation by separating the left han sie, whih is the prout of two frations, into an equivalent sum of two frations. This is sometimes esribe as the metho of partial frations. The following example illustrates the general metho. ( X a) ( X b) A X a B X b We an integrate the sum on the right. The integral of a sum is the sun of the integrals. We nee to fin values for A an B that satisfy the above equation. To o so, multiply together the two terms on the right han sie an then einate the enominator. ( X a) ( X b) A( X b) B( X a) ( X a) ( X b) A ( X b) B( X a) A X B X Ab The left han sie of this equation ontains no terms in X, whih implies that B a FW43/53 Copyright 2008 by Davi B. ampson Reruitment2 - Page 82

2 0 A X B X an A b > B A an A b Aa A( a b) Ba > A a b an B b a Given values for a, b, an, we an use these equations to solve for A an B. For our problem we want to expan the fration ( ) > ( 0) We have a -/, b 0, an /. If we substitute these into the equations for A an B, we get A an B > ( ) o, we an write the ifferential equation as t Fator out / an multiply the first term by /. t Integrate. t X / ( ab X ) (/b) ln( ab X ) ln ln ln t C t C ln(x) - ln(y) ln( X / Y ) FW43/53 Copyright 2008 by Davi B. ampson Reruitment2 - Page 83

3 exp( t) exp( C) The general solution. For initial onitions, speify that at t0. exp( 0) exp( C) exp( C) exp( t) The partiular solution. This is the solution to the ifferential equation, but we want it in the form f(t). To put it in this form we nee to solve for. ( ) 0 exp( t) To simplify the notation let Ι an substitute into the equation to get Ιexp( t) Ιexp( t) Ιexp( t) Ιexp( t) Collet terms with on the left. Ιexp( t) Ιexp( t) Ι Ι exp( t) exp( t) Ι exp t Put in the full expression for Ι. ultiply the numerator an enominator by exp( t) 0 an ivie the numerator an enominator by. exp t exp( t) exp( t) exp( t) ( exp( t) ) FW43/53 Copyright 2008 by Davi B. ampson Reruitment2 - Page 84

4 Now set t t, the age at spawning, an assume that all spawners ie after spawning. The "eggs" that survive to beome spawners are the reruits. ( t ) R exp( t ) ( exp ( t ) ) Finally, set k... k R Divie numerator an exp( t ) ( exp ( t enominator by k. ) ) k R k exp t exp t ( )... an let k exp ( t ) ( ) exp t We en up with the Beverton an Holt stok-reruit moel R. The quantity / represents the total number of eggs that woul survive if the only soure of mortality was. The ratio / is the fration surviving the ensity inepenent mortality. kexp t Numerial xample of Beverton an Holt's R oel i R i Reruits pawners FW43/53 Copyright 2008 by Davi B. ampson Reruitment2 - Page 85

5 If R, then the spawning population will just replae itself with new reruits an the population will be at equilibrium from one generation to the next. Algebraially this means that R > > Parameter is imensionless an has units [/fish]. olving for we get the population size at equilibrium eq. For our example we have eq What is the behavior of R() when is large? R We nee to use L'Hôpital's rule to evaluate this inefinite form. R ( ) The graph of R() has a horizontal asymptote at R/. For our numerial example The slope of R() is R ( ) 2 Y U V - Y U V -2 ( -V ) U V - R ( ) 2 ( ) 2 ( ) 2 etting to zero gives the slope of R() at the origin, ( 0) 2. This is the ratio of reruits to spawners when there is no ensity epenene. Also, notie that the slope of R() is never equal to zero, exept in the it as approahes infinity. This means that the graph of R() has no istint point at whih R() is either a maximum or a minimum. xplore the effets of parameters an on eq, on the slope of R() at the origin, an on the horizontal asymptote using the xel emonstration. FW43/53 Copyright 2008 by Davi B. ampson Reruitment2 - Page 86

6 A athematial Digression The Beverton an Holt stok-reruit moel has a form that is very similar to the equation for a ath proess with hanling, whih we enountere previously. Both equations are retangular hyperbolas. Both are of the form X Y onstant If we start with the graph of Y X -, whih has the X axis as a horizontal asymptote an the Y axis as a vertial asymptote, we an obtain the graph orresponing to the Beverton an Holt moel (or the ath proess moel) by shifting the loation of the origin. When measure on the original axes (X,Y), the point (Xa,Y-b) orrespons to the origin of the axes (X',Y'). Retangular Hyperbola, Y -/X. 0 a 2 3 X a Y b X Y X' 0 Y' 0 a 0 X' b 0 Y' X X' a Y Y' b 2 3 b Also, the point (Xa,Y-b) must satisfy the relationship Y -/X. b > ab a The relationship X Y - orrespons to the new relationship ( X' a) ( Y' b) ab whih has a vertial asymptote at X' -a, an a horizontal asymptote at Y' b. In terms of the Beverton an Holt stok-reruit moel we have a The vertial asymptote an b The horizontal asymptote > R whih is equivalent to R R 2 > 2 R FW43/53 Copyright 2008 by Davi B. ampson Reruitment2 - Page 87

7 > R Dynami Behavior of the Beverton an Holt R oel What happens to the number of spawners from one generation to the next? The reruits proue by one generation of spawners beomes the next generation of spawners. R t t t t tart with 0 parents. 0 They proue R 0 reruits, equivalent to parents. The parents proue R reruits, equivalent to 2 parents, an so on. Reruits Unlike the Riker moel, for the Beverton an Holt stok-reruit moel the equilibrium point eq is always stable. pawners xplore the effets of hanging 0 an an on the system's ynami behavior using the xel emonstration.. FW43/53 Copyright 2008 by Davi B. ampson Reruitment2 - Page 88

Some facts you should know that would be convenient when evaluating a limit:

Some facts you should know that would be convenient when evaluating a limit: Some fats you should know that would be onvenient when evaluating a it: When evaluating a it of fration of two funtions, f(x) x a g(x) If f and g are both ontinuous inside an open interval that ontains