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A hyperparasitic system with prolonged diapause for host is investigated. It is assumed that host prolonged diapause occur at larval stage, and parasitoid attack is limited to egg stage before the initiation of host diapause. Such behavior has been reported for many ichneumons. Hyperparasite only attacks the parasitoids that parasitize the hosts. Hyperparasitic system is often used in biological control. The existence and stability of nonnegative fixed points are explored. Numerical simulations are carried out to explore the global dynamics of the system, which demonstrate appropriate prolonged diapause rate and appropriate intrinsic growth rate can stabilize the system. The reasons are explained according to the ecological perspective. Furthermore, many other complexities which include quasi-periodicity, period-doubling bifurcations leading to chaos, chaotic attractor, intermittent and supertransients are observed.
In this paper, we define β-Hausdorff operator on the unit polydisk
and study the boundedness of the operator on Lipschitz space. Firstly, we
translate the problem of coefficient into integral of weighted composition operator, then give the sufficient conditions of boundedness, and also obtain
an upper bound for the operator norm on Lipschitz space.