The Malthusian parameter and R0 for heterogeneous populations in periodic environments.

Since the classical stable population theory in demography by Sharpe and Lotka, the sign relation sign(λ0)=sign(R0-1) between the basic reproduction number R0 and the Malthusian parameter (the intrinsic rate of natural increase) λ0 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 R0 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 R0 so that it can be applied to population dynamics in periodic environments. In particular, the definition of R0 in a periodic environment by Bacaer and Guernaoui (J. Math. Biol. 53, 2006) is most important, because their definition of R0 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 R0 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 R0 and the Malthusian parameter λ0 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 R0 given by the spectral radius of the next generation operator by Bacaer and Guernaoui's definition.