Bulletin of the Japan Society for Industrial and Applied Mathematics

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In this survey, we introduce the relatively new mathematical idea "permanence" which implies robust coexistence for all elements (that is, the species or cells et al.) of the biological systems. During last two decades, analytical method to prove permanence as well as the idea itself has been rapidly developed. Also the study on permanence proves its importance both for mathematicians and biologists to cooperate together. That is, it has proposed many interesting mathematical problems and stimulated to acquire new mathematical ideas. Simultaneously, it gave the deep understanding of the biological world. First, we sketch some history on permanence and give the similar framework of the models for epidemics, HIV, autoimmune. Next, we survey the results on permanence for ODE models with or without time delays. We argue also average extinction time for some stochastic process. We gave many references including some Japanese books for the readers who are interested in mathematics on permanence.

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Impact craters observed on the planetary surfaces have been formed by celestial body impacts. Such impacts can also disrupt celestial bodies if their energy is sufficiently large. In this short paper, mathematical and physical models of celestial body impacts are reviewed, focusing on the impact cratering process which is commonly divided into three stages: the contact and compression stage, the excavation stage, and the modification stage. The contact and compression stage determines the initial thermodynamic state immediately after impact. During the excavation stage, transient cavity formation and target material ejection are caused by excavation flow. The transient cavity is modified by gravity in the modification stage, and then a final crater is formed. Shock wave and gravity play an important role during these stages because celestial body impacts are characterized by their high speed(>〜km/s) and large scale. Scaling laws derived from dimensional analysis are also described.

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It was shown by Ruell and Takens that fluid motion can be chaotic with three distinct frequencies being aquired due to instabilities of flow. Thus, the instability of a steady flow is the first step to chaos and hence to turbulence. After overviewing the scenario of transitions of flow to turbulence briefly, we show four numerical methods to analyze the linear stability of steady flows: numerical simulation of a set of linear equations for a disturbance, successive over relaxation method, Arnoldi method, and shift and inverse power method. We discuss the merit of each numerical method together with some issues arising in numerical stability analyses of flow past a circular cylinder, taking it as an example.

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