Initiating a chain reaction in nuclear reactors?

In a reactor that has been in operation, there is already spent fuel in the core that has been irradiated and will produce neutrons. The normal plan is to replace one third of a core with new fuel, leaving 1/3 that has been in for 2 refuelings and 1/3 that has been in for one refueling.

But in an initial core with all new fuel, you have to add a source like

Californium that spontaneously fissions and produces neutrons to get the chain reaction going.

I believe the fissionable materials (U235 and Plutonium239) are naturally unstable and fission naturally. When you get enough the the material close enough together these neutrons impact other atoms to create more neutrons in a chain reaction that the energy release increases exponentially. For weapons (from Wikipedia), Neutron trigger or initiator: One of the key elements in the proper operation of a nuclear weapon is initiation of the fission chain reaction at the proper time. To obtain a significant nuclear yield, sufficient neutrons must be present within the supercritical core at just the right time. If the chain reaction starts too soon, the result will be only a 'fizzle yield', well below the design specification; if it occurs too late, there may be no yield whatsoever. Several ways to produce neutrons at the appropriate moment have been developed. Early neutron triggers consisted of a highly radioactive isotope of polonium (Po-210), which is a strong alpha emitter, combined with beryllium, which will absorb alphas and emit neutrons. This isotope of polonium has a half life of 138 days. Therefore, a neutron initiator using this material needs to have the polonium replaced frequently. The polonium is produced in a nuclear reactor. To supply the initiation pulse of neutrons at the right time, the polonium and the beryllium need to be kept apart until the appropriate moment and then thoroughly and rapidly mixed by the implosion of the weapon. This method of neutron initiation is sufficient for weapons utilizing the slower gun combination method, but the timing is not precise enough for an implosion-weapon design.