Talla de Madurez
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Talla de Madurez
Benjamin Suarez
1) Determinar talla media de primera madurez (\( L_{50} \)) mediante el calculo de la ojiva de madurez sexual para Jurel en distintas zonas aplicando verosimilitud.
Para calcular la proporcion estimada de ejemplares maduros se utilizo la siguiente formula:
\[ P_{L} = \frac{1}{(1 + exp^{-(B_{0} + B_{1} \ast L_{t})})} \]
La funcion de verosimilitud se define como:
\[ LogLike = M \ast log(P_{L}) + (I \ast log(1 - P_{L})) \]
donde \( M \) es la cantidad de ejemplares maduros, \( I \) la cantidad de ejemplares inmaduros y \( P_{L} \) la proporcion estimada de ejemplares maduros.
2) Determinar por que la talla media de primera madurez es igual al cuociente negativo de los parametros de la ojiva de madurez.
\[ L_{50} = \frac{-B_{0}}{B_{1}} \]
Esta ecuacion resulta del despeje realizado en la ecuacion mediante la que se obtiene la proporcion estimada de ejemplares maduros. Se ha establecido que la talla media de primera madurez sexual corresponde al momento en que el 50% de los ejemplares muestreados se encuentran maduros (Cerna & Oyarzun, 1998; Leal & Oyarzun 2003; Leal et al. 2011), por lo que el despeje resulta de la siguiente manera:
\[ P_{L} = \frac{1}{(1 + exp^{-(B_{0} + B_{1} \ast L_{t})})} \]
\[ 0.5 = \frac{1}{(1 + exp^{-(B_{0} + B_{1} \ast L_{t})})} \]
\[ 0.5 \ast (1 + exp^{-(B_{0} + B_{1} \ast L_{t})}) = 1 \]
\[ 0.5 + 0.5 \ast exp^{-(B_{0} + B_{1} \ast L_{t})} = 1 \]
\[ 0.5 \ast exp^{-(B_{0} + B_{1} \ast L_{t})} = 1 - 0.5 \]
\[ 0.5 \ast exp^{-(B_{0} + B_{1} \ast L_{t})} = 0.5 \]
\[ exp^{-(B_{0} + B_{1} \ast L_{t})} = \frac{0.5}{0.5} \]
\[ exp^{-(B_{0} + B_{1} \ast L_{t})} = 1 \]
\[ - B_{0} + B_{1} \ast L_{t} = Ln(1) \]
\[ - B_{0} + B_{1} \ast L_{t} = 0 \]
\[ B_{1} \ast L_{t} = B_{0} \]
\[ L_{t} = \frac{B_{0}}{B_{1}} \]
A continuacion se procede a desarrollar el calculo de la \( L_{50} \):
Incidencia de hembras de jurel maduras e inmaduras microscopicamente por estrato de talla, colectadas en la zona norte y sur de Chile durante 1999.
Lt <- c(20:35) frec <- c(2, 2, 52, 102, 151, 216, 250, 208, 83, 24, 32, 21, 17, 9, 7, 3) m <- c(0, 0, 17, 39, 70, 144, 207, 185, 80, 23, 32, 21, 17, 9, 7, 3) I <- rep(NA, 16) mad.obs <- rep(NA, 16) for (i in 1:16) { I[i] <- frec[i] - m[i] mad.obs[i] <- m[i]/frec[i] } data.frame(Lt, frec, m, I, mad.obs)
## Lt frec m I mad.obs ## 1 20 2 0 2 0.0000 ## 2 21 2 0 2 0.0000 ## 3 22 52 17 35 0.3269 ## 4 23 102 39 63 0.3824 ## 5 24 151 70 81 0.4636 ## 6 25 216 144 72 0.6667 ## 7 26 250 207 43 0.8280 ## 8 27 208 185 23 0.8894 ## 9 28 83 80 3 0.9639 ## 10 29 24 23 1 0.9583 ## 11 30 32 32 0 1.0000 ## 12 31 21 21 0 1.0000 ## 13 32 17 17 0 1.0000 ## 14 33 9 9 0 1.0000 ## 15 34 7 7 0 1.0000 ## 16 35 3 3 0 1.0000
plot(Lt, mad.obs, xlim = c(10, 35), ylim = c(0, 1), xlab = "Longitud total (cm)", ylab = "Proporcion Madura") # Modelo no lineal con verosimilitud negloglike <- function(parms) { a <- parms[1] b <- parms[2] pl <- 1/(1 + exp(-(a + b * Lt))) -sum(m * log(pl) + (I * log(1 - pl))) } m1 <- optim(par = c(22, -2), fn = negloglike, method = "BFGS", hessian = TRUE) LM1 <- -m1$par[1]/m1$par[2] lt <- seq(10, 35, 0.1) mad.est <- 1/(1 + exp(-(m1$par[1] + m1$par[2] * lt))) lines(lt, mad.est) text(16, 0.8, expression(P(L) == frac(1, 1 + exp(-(a + b * L))))) text(16, 0.6, expression(LM1 == 23.86))
Incidencia de hembras de jurel maduras e inmaduras microscopicamente por estrato de talla, colestadas en la zona norte durante 1999.
Lt <- c(20:35) frec <- c(2, 2, 46, 42, 51, 78, 115, 103, 60, 13, 16, 17, 11, 7, 5, 3) m <- c(0, 0, 17, 25, 40, 65, 111, 103, 60, 13, 16, 17, 11, 7, 5, 3) I <- rep(NA, 16) mad.obs <- rep(NA, 16) for (i in 1:16) { I[i] <- frec[i] - m[i] mad.obs[i] <- m[i]/frec[i] } data.frame(Lt, frec, m, I, mad.obs)
## Lt frec m I mad.obs ## 1 20 2 0 2 0.0000 ## 2 21 2 0 2 0.0000 ## 3 22 46 17 29 0.3696 ## 4 23 42 25 17 0.5952 ## 5 24 51 40 11 0.7843 ## 6 25 78 65 13 0.8333 ## 7 26 115 111 4 0.9652 ## 8 27 103 103 0 1.0000 ## 9 28 60 60 0 1.0000 ## 10 29 13 13 0 1.0000 ## 11 30 16 16 0 1.0000 ## 12 31 17 17 0 1.0000 ## 13 32 11 11 0 1.0000 ## 14 33 7 7 0 1.0000 ## 15 34 5 5 0 1.0000 ## 16 35 3 3 0 1.0000
plot(Lt, mad.obs, xlim = c(10, 35), ylim = c(0, 1), xlab = "Longitud total (cm)", ylab = "Proporcion Madura") # Modelo no lineal con verosimilitud negloglike <- function(parms) { a <- parms[1] b <- parms[2] pl <- 1/(1 + exp(-(a + b * Lt))) -sum(m * log(pl) + (I * log(1 - pl))) } m2 <- optim(par = c(22, -2), fn = negloglike, method = "BFGS", hessian = TRUE) LM2 <- -m2$par[1]/m2$par[2] lt <- seq(10, 35, 0.1) mad.est <- 1/(1 + exp(-(m2$par[1] + m2$par[2] * lt))) lines(lt, mad.est) text(16, 0.8, expression(P(L) == frac(1, 1 + exp(-(a + b * L))))) text(16, 0.6, expression(LM2 == 22.71))
Incidencia de hembras de jurel maduras e inmaduras microscopicamente por estrato de talla, colestadas en la zona sur durante 1999.
Lt <- c(22:34) frec <- c(6, 60, 100, 138, 135, 105, 23, 11, 16, 4, 6, 2, 2) m <- c(0, 14, 30, 79, 96, 82, 20, 10, 16, 4, 6, 2, 2) I <- rep(NA, 13) mad.obs <- rep(NA, 13) for (i in 1:13) { I[i] <- frec[i] - m[i] mad.obs[i] <- m[i]/frec[i] } data.frame(Lt, frec, m, I, mad.obs)
## Lt frec m I mad.obs ## 1 22 6 0 6 0.0000 ## 2 23 60 14 46 0.2333 ## 3 24 100 30 70 0.3000 ## 4 25 138 79 59 0.5725 ## 5 26 135 96 39 0.7111 ## 6 27 105 82 23 0.7810 ## 7 28 23 20 3 0.8696 ## 8 29 11 10 1 0.9091 ## 9 30 16 16 0 1.0000 ## 10 31 4 4 0 1.0000 ## 11 32 6 6 0 1.0000 ## 12 33 2 2 0 1.0000 ## 13 34 2 2 0 1.0000
plot(Lt, mad.obs, xlim = c(10, 35), ylim = c(0, 1), xlab = "Longitud total (cm)", ylab = "Proporcion Madura") # Modelo no lineal con verosimilitud negloglike <- function(parms) { a <- parms[1] b <- parms[2] pl <- 1/(1 + exp(-(a + b * Lt))) -sum(m * log(pl) + (I * log(1 - pl))) } m3 <- optim(par = c(22, -2), fn = negloglike, method = "BFGS", hessian = TRUE) LM3 <- -m3$par[1]/m3$par[2] lt <- seq(10, 35, 0.1) mad.est <- 1/(1 + exp(-(m3$par[1] + m3$par[2] * lt))) lines(lt, mad.est) text(16, 0.8, expression(P(L) == frac(1, 1 + exp(-(a + b * L))))) text(16, 0.6, expression(LM3 == 24.86))
Zona <- c("Ambas", "Norte", "Sur") B0 <- c(m1$par[1], m2$par[1], m3$par[1]) B1 <- c(m1$par[2], m2$par[2], m3$par[2]) LMad <- c(LM1, LM2, LM3) data.frame(Zona, B0, B1, LMad)
## Zona B0 B1 LMad ## 1 Ambas -16.20 0.6789 23.86 ## 2 Norte -21.87 0.9632 22.71 ## 3 Sur -17.34 0.6974 24.86
Referencias:
Cerna J. & C. Oyarzun. 1998. Talla de primera madurez sexual y fecundidad parcial de la merluza común (Merluccius gayi, Guichenot 1848) del área de la pesquería industrial de la zona de Talcahuano, Chile. Investigaciones Marinas, Valparaíso 26: 31-40.
Leal E. & C. Oyarzun. 2003. Talla de madurez y época de desove de la reineta (Brama australis Valenciennes, 1836) en la costa central de Chile. Investigaciones Marinas, Valparaíso 31(2): 17-24.
Leal E., Canales M., Aranis A. & M. Gonzales. 2011. Actividad reproductiva y longitud de madurez de sardina austral Sprattus fuegensis en el mar interior de Chiloé, Chile. Revista de Biología Marina y Oceanografía 46(1): 43-51.













