Population dynamics of earthquakes and mathematical modeling

We present a mathematical model that describes temporal variations of earthquakes. This model is represented as $$dn(t)/dt = n(t)\left[ {\alpha - \beta n(t) - \int_{ - \infty }^t {n(s)h(t - s)ds} } \right].$$ Heren(t) shows the numberof earthquakes per unit time in a certain region. α and β are constants. The functionh(t) denotes the hysteresis effect of the earthquake occurrences and can take the following forms depending on the physical conditions of the crusts; (A)h(t)=0: the equation represents a logistic type increase or decrease and approaches a stationary state asymptotically. This describes aftershock series of large earthquakes and earthquake swarms of large scale such as the Wakayama and Matsushiro swarms in Japan; (B)h(t)=constant (β=0): frequencyn(t) increases initially and then decreases gradually and shows some kind of volcanic swarms; (C)h(t) = κ · {exp(−γ1 t) − exp(γ2 t)}, (γ2 > γ1): this denotes time delay effects and the model shows periodic patterns of bursts or “rhythms” of earthquakes, which are observed in earthquake swarms. When external effects are taken into consideration, the model is further generalized and can describe various seismic patterns. These effects represent various influences of the circumstances like the earth tide and fluctuations of plate motions, etc. Whenh(t) takes type (A) and the external effect is random, the equation displays repetitive random patterns with bursts. Particularly interesting cases may be those whenh(t) is type (C) and the external force is periodic like the earth tide. Various nonperiodic as well as periodic patterns of earthquakes appear. These are the phenomena of “chaos” and “entrainment”, etc. and can be commonly observed. Varieties of actual earthquake patterns seem to be, at least partly, explained by the nonlinear coupling between the tidal forces and autonomous rhythms of earthquakes.