Mass extinction in a dynamical system of evolution with variable dimension

Introducing the effect of extinction into the so-called replicator equation s in mathematical biology, we construct a general model where the diversity of species, i.e., the dimension of the equation, is a time-dependent varia ble. The system shows very different behavior from the original replicator equation, and leads to mass extinction when the system initially has high d iversity. The present theory can serve as a mathematical foundation for the paleontologic theory for mass extinction. This extinction dynamics is a pr ototype of dynamical systems where the variable dimension is inevitable. [S 1063-651X(99)06707-0].