Fits the re-parameterized von Bertalanffy growth equation of Francis (1988) by using nonlinear least-squares

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`age` |
Vector of ages of individual fish. |

`L` |
Vector of lengths of individual fish. |

`agephi` |
Arbitrary reference age phi |

`agepsi` |
Arbitrary reference age psi. agepsi>agephi. |

`graph` |
Logical specifiying whether observed versus predicted, and residual plots should be drawn. Default=TRUE. |

`gestimate` |
Logical specifying whether automatic generation of starting
values of |

`Lphiparms` |
If gestimate=FALSE, starting value, lower limit and upper limit
of |

`Lchiparms` |
If gestimate=FALSE, starting value, lower limit and upper limit
of |

`Lpsiparms` |
If gestimate=FALSE, starting value, lower limit and upper limit
of |

`control` |
see |

Francis (1988) re-parameterized the von Bertalanffy growth equation for age-length
in order to make equivalent comparison of parameters to parameters of a common model
used to estimate growth from tagging data. Three parameters, *lphi*, *lchi*
and *lpsi*, are estimated. The re-parameterization also has better
statistical properties than the original equation.

The formulae to get the conventional von Bertalanffy parameters are:

Linf = lphi + (lpsi-lphi)/(1-r^2) where r = (lpsi-lchi)/(lchi-lphi)

K = -(2*log(r))/(agepsi-agephi)

t0 = agephi + (1/K)*log((Linf-lphi)/Linf)

If gestimate=TRUE, unconstrained nonlinear least-squares (function *nls*) is used
to fit the model. If gestimate=FALSE, constrained nonlinear least-squares
is used (algorithm "port" in *nls*).

*nls* object of model results. Use *summary* to extract results.

Gary A. Nelson, Massachusetts Division of Marine Fisheries gary.nelson@state.ma.us

Francis, R. I. C. C. 1988. Are growth parameters estimated from tagging and age-length data comparable? Can. J. Fish. Aquat. Sci. 45: 936-942.

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