The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments

Since the classical stable population theory in demography by Sharpe and Lotka, the sign relation ${\rm sign}(\lambda_0)={\rm sign}(R_0-1)$ between the basic reproduction number $R_0$ and the Malthusian parameter (the intrinsic rate of natural increase) $\lambda_0$ has played a central role in population theory and its applications, because it connects individual's average reproductivity described by life cycle parameters to growth character of the whole population. Since $R_0$ is originally defined for linear population evolution process in a constant environment, it is an important extension if we could formulate the same kind of threshold principle for population growth in time-heterogeneous environments.
    Since the mid-1990s, several authors proposed some ideas to extend the definition of $R_0$ so that it can be applied to population dynamics in periodic environments. In particular, the definition of $R_0$ in a periodic environment by Bacaër and Guernaoui (J. Math. Biol. 53, 2006) is most important, because their definition of $R_0$ in a periodic environment can be interpreted as the asymptotic per generation growth rate, so from the generational point of view, it can be seen as a direct extension of the most successful definition of $R_0$ in a constant environment by Diekmann, Heesterbeek and Metz ( J. Math. Biol. 28, 1990).
    In this paper, we propose a new approach to establish the sign relation between $R_0$ and the Malthusian parameter $\lambda_0$ for linear structured population dynamics in a periodic environment. Our arguments depend on the uniform primitivity of positive evolutionary system, which leads the weak ergodicity and the existence of exponential solution in periodic environments. For typical finite and infinite dimensional linear population models, we prove that a positive exponential solution exists and the sign relation holds between the Malthusian parameter, which is defined as the exponent of the exponential solution, and $R_0$ given by the spectral radius of the next generation operator by Bacaër and Guernaoui's definition.