Unfortunately your newly seeded lawn and pets are not a match made in heaven. Do you have cats or dogs? They are usually the worst offenders when it comes to damaging lawns, whether newly seeded or not.
It is for this reason, that I prefer to use turf when making a lawn, although it still requires care while it is becoming established, it does have a better chance of survival. Another point, are these your pets or the neighbors? If your neighbor has cats, it will be very difficult to stop them damaging a new lawn as they love to dig in earth.
You could cat-proof your garden of course, it does mean going to some trouble to do so, but it is possible. Complaining to the neighbor will not help much either, unless he is prepared to do the cat-proofing, he won't be able to stop them coming into your garden. If you have dogs they may be less of a problem, as you can exercise them off site, you could also try puting strings attached to pegs across the area, that might even help keep the cats off.
When your lawn is established, it will be a good idea to keep any female dog you have from passing water on the grass, there is nothing worse than this for killing the grass, it is worse even than male dogs. None of this is very encouraging, I'm afraid, but unless you can keep the pets away until the grass is growing strongly, you will not have much of a lawn.
If you are talking about your pet dog then give them a little Apple Cider Vingar in their drinking water. Aside from it having many health benefits it regulates their ph levels and their urine won't yellow your new lawn. Most dogs don't care for the taste of the apple cider vinegar right away.
You begin with just a drop or 2 in their water. Eventually add a little more as they get accustomed to the taste. You only need about a spoonful or two of the vinegar per gallon of water.
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