Analysis of an age structured model for tick populations subject to seasonal effects

We investigate an age-structured hyperbolic equation model by allowing the birth and death functions to be density dependent and periodic in time with the consideration of seasonal effects. By studying the integral form solution of this general hyperbolic equation obtained through the method of integration along characteristics, we give a detailed proof of the uniqueness and existence of the solution in light of the contraction mapping theorem. With additional biologically natural assumptions, using the tick population growth as a motivating example, we derive an age-structured model with time-dependent periodic maturation delays, which is quite different from the existing population models with time-independent maturation delays. For this periodic differential system with seasonal delays, the basic reproduction number R0 is defined as the spectral radius of the next generation operator. Then, we show the tick population tends to die out when R0<1 while remains persistent if R0>1. When there is no intra-specific competition among immature individuals due to the sufficient availability of immature tick hosts, the global stability of the positive periodic state for the whole model system of four delay differential equations can be obtained with the observation that a scalar subsystem for the adult stage size can be decoupled. The challenge for the proof of such a global stability result can be overcome by introducing a new phase space, based on which, a periodic solution semiflow can be defined which is eventually strongly monotone and strictly subhomogeneous.