The Growth Rates of Population Projection Matrix Models in Random Environments

N t N t t + ( )= ( ) ( ) ( ) where N is the population size, λ is the growth rate and t is discrete time. Thus, we obtain Suppose the value of λ(i) is α or β at random. The arithmetic mean of these is The t-th power is The right-hand side has 2 t permutations of α and β and the left-hand side is an arithmetic mean of all permutations on the right-hand side. This is an exponential model and δ is the mean growth rate. When α = 2, β = 0.5, we obtain δ = 1.25 > 1. Thus, the mean population size will increase. The Growth Rates of Population Projection Matrix Models in Random Environments 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