**Vol. 4 (No. 3), pp. 95-104, 2011** |
doi:10.5047/absm.2011.00403.0095 |

## The Growth Rates of Population Projection Matrix Models in Random Environments

Tatsuro Akamine and the late Maki Suda

National Research Institute of Fisheries Science, Fisheries Research
Agency, Fukuura 2-12-4, Kanazawa, Yokohama 236-8648, Japan

e-mail: akabe@affrc.go.jp

(Received on June 1, 2011; Accepted on August 3, 2011; Online published on October 15, 2011)

**Abstract: **
Population projection matrix models in random environments are random walk models. The growth rate of the mean population size, which is equal to the maximum eigenvalue of the mean matrix, is better than the average of the intrinsic rates of natural increase calculated by computer simulations, because the population size is more important than the growth rate. The arithmetic mean of the maximum eigenvalues of matrices for all permutations converges to the maximum eigenvalue of the mean matrix. The periodicity of environments is more important than the correlation between environments. Simple matrices and three numerical models are used as examples.

**Keywords: **
eigenvalue, growth rate, Lefkovitch, Leslie, permutation, projection matrix