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The optimal subsampling is an statistical methodology for generalized linear models (GLMs) to make inference quickly about parameter estimation in massive data regression. Existing literature only considers bounded covariates. In this paper, the asymptotic normality of the subsampling M-estimator based on the Fisher information matrix is obtained. Then, we study the asymptotic properties of subsampling estimators of unbounded GLMs with nonnatural links, including conditional asymptotic properties and unconditional asymptotic properties.

In this paper, a stage-structured jellyfish model with two time delays is formulated and analyzed, the first delay represents the time from the asexually reproduced young polyp to the mature polyp and the second denotes the time from the developed polyp to ephyra (incipient medusa). Global dynamics of the model are obtained via monotone dynamical theory: the jellyfish populations go extinct and the trivial equilibrium is globally asymptotically stable if the survival rate of polyp during cloning and the survival rate of the incipient medusa during strobilation are less than their death rates. And if the survival rate of polyp during cloning and the survival rate of the incipient medusa during strobilation are larger than their death rates, a unique positive equilibrium is globally asymptotically stable. Moreover, it is proved that the only stage of polyps will continue without growing into medusa and the boundary equilibrium is globally asymptotically stable if the survival rate of polyp is larger than its death rate during cloning and if there is no survival of the incipient medusa. Numerical simulations are performed to verify our analytical results and to explore the dynamics with/without delays.