Homing endonuclease genes are ''selfish'' mobile genetic elements whose endonuclease promotes the spread of its own gene by creating a break at a specific target site and using the host machinery to repair the break by copying and inserting the gene at this site. Horizontal transfer across the boundary of a species or population within which mating takes place has been thought to be necessary for their evolutionary persistence. This is based on the assumption that they will become fixed in a host population, where opportunities of homing will disappear, and become susceptible to degeneration. To test this hypothesis, we modeled behavior of a homing endonuclease gene that moves during meiosis through double-strand break repair. We mathematically explored conditions for persistence of the homing endonuclease gene and elucidated their parameter dependence as phase diagrams. We found that, if the cost of the pseudogene is lower than that of the homing endonuclease gene, the 2 forms can persist in a population through autonomous periodic oscillation. If the cost of the pseudogene is higher, 2 types of dynamics appear that enable evolutionary persistence: bistability dependent on initial frequency or fixation irrespective of initial frequency. The prediction of long persistence in the absence of horizontal transfer was confirmed by stochastic simulations in finite populations. The average time to extinction of the endonuclease gene was found to be thousands of meiotic generations or more based on realistic parameter values. These results provide a solid theoretical basis for an understanding of these and other extremely selfish elements.H oming endonucleases are a large class of proteins that are common in simple eukaryotes and in prokaryotes (1). Their coding regions (called genes here for convenience), are inserted into a specific DNA sequence and spliced out of either the transcribed mRNA as an intron, or out of the translated protein as an intein. Homing endonuclease genes act as ''selfish'' mobile genetic elements, because their gene product introduces a break at an ''empty'' target site and forces the host machinery to repair the break by copying the endonuclease gene. These genes are not essential to their hosts. Currently, much is known about the molecular structure and action of homing endonucleases and their genes, but little information is available on their population biology (2).From a population genetics standpoint, a homing endonuclease allele could easily become fixed in a host population and susceptible to degeneration because homing opportunities disappear. Therefore, the long-term survival of a homing endonuclease allele at a specific site is proposed to depend on horizontal transfer across the boundary of a species or population, within which mating takes place, to another species or population that do not yet contain the homing endonuclease allele at that particular site (3, 4). Specifically, they have been hypothesized to persist over a long evolutionary period only if they can transmit to other ...