**Vol. 2 (No. 3), pp. 1-45, 2009** |
doi:10.5047/absm.2009.00203.0001 |

## Non-linear and Graphical Methods for Fish Stock Analysis with Statistical Modeling

Tatsuro Akamine

National Research Institute of Fisheries Science, Fisheries Research Agency,

Fukuura 2-12-4, Yokohama 236-8648, Japan

(Received on March 4, 2008; Revised on March 13, 2009; Accepted on April 28, 2009; Published online on July 28, 2009)

**Abstract: **
Useful methods for growth curve fitting, body-size composition analysis, and estimation of population size in fish stocks are presented. These methods are statistically based on the maximum likelihood method and the likelihood ratio test. Mathematical explanation of the standard Richards growth formula with seasonal change, the generalized reproduction model, and the Awaya method for estimating implicit function models are given. Mathematical proofs of the iteration method, called the Hasselblad method, or the EM algorithm for estimating the mixture of normal distributions, and the Marquardt method for general optimization are shown. For population size estimation, the Petersen method for mark–recapture experiments, the quadrat method, and the DeLury removal method are discussed. These are based on the binomial distribution and the classical Bayesian statistical methods which are also discussed. Mathematical proofs of the sum formulae of the binomial and hyper-geometric distributions are given. The virtual population analysis using mortality rates, the Leslie matrix model, and the linear programming for discrete fishing models are also explained. All the methods stated here can be easily carried out using spread-sheet software.

**Keywords: **
body-size composition, growth curve, population size, reproduction, survival