🙂
The patrol involves Torah-level prohibitions. The goal is not only to protect the animals, but to prevent the extinction of wildlife from certain species, for example rhinos and elephants that are hunted for their tusks and horns. If the patrol does not operate on the Sabbath, the poachers will come in droves on the Sabbath when there is no supervision, and certain species will become extinct.

What I was thinking is that just as it is permitted to desecrate the Sabbath in order to defend oneself against a thief on the Sabbath, so too the public should be permitted to desecrate the Sabbath in order to defend itself against someone who threatens its cultural or environmental assets, such as biodiversity.

Is the result of the extinction of such species something practical? If so, one still has to discuss whether the impact of that extinction is significant for our lives. If both of those conditions are met, and if the only way to prevent it is through guarding that involves desecrating the Sabbath (and not, for example, by strictly regulating trade in tusks or other wildlife products and severely punishing anyone involved in it), then maybe you are right. But I am very doubtful. This is a very indirect and very non-necessary connection, and it is hard to permit desecration of the Sabbath for it.

You could discuss this from the standpoint of the animals' monetary value. Suppose a person keeps expensive exotic animals in his backyard for the enjoyment of their beauty. How is that different from the public wanting to preserve its wildlife in order to enjoy its beauty? Poachers who hunt them are literally stealing a public resource, like natural gas and oil reserves in the Mediterranean. Why shouldn't the public be able to protect such assets just as a private individual can protect his property on the Sabbath?

I think that this analogy is formal but not substantive. This is not theft of property, because the public does not possess the animals and does not use them; it only wants them to exist. Therefore they are ownerless, not public property. That does not mean a regulation forbidding hunting has no validity, like any law, but it is not a prohibition by an owner against use of his property.

It is proper to preserve the forests on the Sabbath
"Sabbath" is an acronym: On the Sabbath, desist in the forests