
Mathematical Biosciences and Engineering, 2012, 9(2): 313346. doi: 10.3934/mbe.2012.9.313.
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The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments
1. Graduate School of Mathematical Sciences, University of Tokyo, 381 Komaba Meguroku, Tokyo 1538914
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Since the mid1990s, 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.
Keywords: Malthusian parameter; weak ergodicity; periodic environments; Basic reproduction number; uniform primitivity; exponential solutions.
Citation: Hisashi Inaba. The Malthusian parameter and $R_0$ for heterogeneous populations in periodic environments. Mathematical Biosciences and Engineering, 2012, 9(2): 313346. doi: 10.3934/mbe.2012.9.313
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