Leopard gecko eggs need warm temperatures and high humidity to hatch. Providing both of these by furnishing an incubation container and building an incubator are important.
Fill several plastic food storage containers halfway with vermiculite. A single female gecko can be served adequately with a single container, but having backups is always a good idea.
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Weigh the vermiculite with a food or postal scale and add an equal amount of nonchlorinated water, by weight, to the vermiculite by spraying it on top of the vermiculite with a spray bottle.
Choose a container to turn into an incubator based on the amount of eggs you will be caring for. The more eggs you have, the bigger container you will need, but bigger containers will cost more to heat. Small Styrofoam coolers can be used for small colonies, while large operations may require you to repurpose an old refrigerator or icebox.
Prepare a heat element, a heat rock, heating tape or other similar heat source for your incubator. The size and output will depend on the size of the container and will, in any case, require trial, error and calibration. In flat incubators like coolers, secure the heat source to the bottom of the container. Drill a hole or use an available drainage port to feed the power wire out of the incubator.
Place a sheet of egg crate light diffuser over the heating element. Cut cardboard tubes to support the egg crate and keep it from directly touching the heat source.
Remove interior squares of the egg crate with pliers to create a spot for a small electric fan to sit. Make sure the fan is close to the heat source and orient it so it sucks air from the heat source and dispenses it upward and outwards into the incubator. Feed the power cable from the fan out through the same hole as before.
Place an electronic thermometer into the incubator. Set it so that it remembers maximum and minimum temperatures.
Plug in your heat source and fan, close your incubator and let the entire system come to temperature. Check the thermometer often to see what temperatures it has been reading. Adjust the heating elements until the incubator can consistently read 90 degrees or above. The temperature controller can turn off the heat to make the temperature drop, but it can't turn up the heat any more than the "on" setting for the device you're using. Move the thermometer probe around to make sure that the entire incubator is reaching the same temperature and adjust the fan if it isn't.
Plug a nonsurge power strip into the wall and plug an incubation temperature controller into the strip. Feed the controller's temperature probe into the incubator and place it at a point where your eggs will be incubating, on top of the egg crate. Unplug your heat element from the wall and plug it into the appropriate port on the temperature controller.
Follow the manufacturer's directions for calibrating your temperature controller. Set the controller to 82 degrees Fahrenheit to produce mostly females. Choose 90 degrees for mostly males. A temperature of 86 to 87 will produce a mix of males and females. Set temperature alarms, if available, to 74 degrees on the low side and 95 degrees on the high side. Temperatures outside of this range can be lethal for eggs and many temperature controllers have audible alarms that can alert you to a critical failure in heating.
Make a mark on top of the eggs, when your female lays them, with a permanent marker. Rotating eggs can kill them, so making this mark will help you keep the eggs at the correct orientation while moving them.
Dig small indentations in the vermiculite for each egg. Place the eggs into these indentations gently and shore them up so they don't move around.
Seal the lids to these containers and place them into the incubator. Open the containers weekly to check on the eggs and ventilate the containers. Reapply water, as necessary, to keep the vermiculite moist. If any eggs show signs of mold, decay or discoloration, remove them and place them into separate containers -- they may still be viable but shouldn't be given the chance to contaminate other eggs.
Wait patiently. Leopard gecko eggs may hatch anywhere from 30 to 105 days after being laid.
If you have a reptile room or similar place where the temperature is consistently in the mentioned range, an incubator is unnecessary. Simply place the food storage container with the eggs on a shelf where it can't be disturbed and keep a thermometer handy to record the temperature and make sure it stays in the appropriate range.