Answer: There are two main types of grass grown here in the low desert: a warm-season grass, which is Bermuda; it is sometimes overseeded for the cool-season with ryegrass. There is no grass species that can survive both temperature extremes here. However, many people install the Bermuda grass for the warm months and then let it go dormant in the winter. It's usually brown from about Thanksgiving until temps. start warming up in February. It is not essential to overseed your Bermuda with ryegrass. Many people choose to take a break from lawn chores, saving time and water, and let the Bermuda go dormant. If dormant, you need to water no more than once per month, and fertilizing and mowing is unnecessary until about April. If you do choose to overseed, it takes place from about mid-October to mid-November so that seed can germinate before soil gets too cold.
Without more detail, I can't say why your front and back yard grow at different rates. There are no grasses that perform well in the shade in the low desert. Is your front yard shady, or the grass beneath tree canopies? If so, it might be better to remove the grass and do something else, as it will never thrive. You may need to aerate and/or dethatch the Bermuda next summer to improve its vigor.
If you choose to overseed, here's the proper procedure. It is taken from the chapter Growing a Healthy Lawn written by Sharon Dewey, a local lawn expert, in the book Desert Landscaping for Beginners by Arizona Master Gardener Press, 0-9651987-3-1.
In September, when temperatures start to drop below 100, gradually mow Bermuda progressively lower to less than one-half inch to prepare for overseeding. Don't fertilize Bermuda 4 to 6 weeks before overseeding. You don't want it to be too vigorous to compete with the rye. 10 to 14 days before overseeding, cut the watering frequency in half. Stop mowing completely 1 week in advance.
The day of overseeding, mow the grass three times. The first cut should be at the regular height, the next cut at the next lower height on your mower. Collect the clippings. If using a rotary mower, decrease the height and mow again. (You don't want to cut exceedingly low just one time because it damages the grass.) If using a reel mower, you can seed in betwen the second and third mowings, leaving the uncollected clippings of the third cutting on the lawn as a mulch.
Spread the ryegrass seed, raking to ensure good contact with the soil. Cover the seed with no more than one-quarter inch of organic matter such as composted steer manure or forest mulch, or the grass clippings as stated above.
You can use annual or perennial ryegrass. Both will need to be sown following years. Don't mix the two. Annual needs more maintenance than perennial, costs less, but more seed is required for the same coverage. Perennial is darker green and has a finer blade. For each 1000 square feet of lawn, use 10 to 20 pounds of annual or 10 to 15 pounds of perennial seed. Water two times daily, for 5 to 10 minutes each (about 9 am and 2 pm) to maintain moist soil until seeds germinate and seedlings establish. When seedlings reach 1 inch, reduce watering to 1 time daily. When grass is 2 inches high, mow to a height of 1.5. After the second mowing, reduce watering to an as-needed basis, depending on weather and rain. This may be every 3 to 7 days or every 5 to 10 days. Water should soak to a depth of 4 to 6 inches, which is how deep ryegrass roots reach.
I recommend you read the chapter mentioned above. It is the most complete and detailed information on the topic, also describing the different grass choices, how to install them and how to water, fertilize, mow, dethatch and aerify. The book is available at the Phoenix libraries, bookstores or through the Maricopa County Cooperative Extension. Good luck with your new home's landscaping!
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