In this paper, we perform a numerical study on the eigenvalues of the operator of a semi-discretized age-structured predator-prey model. The age-structuration is based on an agile modeling framework for population dynamics. We also prove a theoretical result on the distribution of the eigenvalues of a semi-discretized McKendrick equation. We show that, under regularity assumptions, and after a rescaling step, the eigenvalues converge to the unit circle as the number of age groups goes to infinity. A sufficient condition to get this convergence consists in having piecewise-constant birth and death rates; by using random values of birth and death rates, we also show numerically that having not enough regular values invalidates the result. We complete this study with properties on the original McKendrick equation, such as the compactness of its operator, the countability of its spectrum, and the fact that the eigenvectors form a complete basis of the function space.