I believe your assessment of the situation is fairly accurate where the breeding of animals is concerned. Typically animals breed according to the carry capacity of their environment. An abundance of resources means more births, whereas a lack of resources means fewer births. To have more offspring than the environment will support means death through starvation and/or disease. What we've done to the poor in this country is give them a false ceiling through welfare. Actually, to take it a step further, we encourage the poor to have more children because it means an increase in welfare subsidies. Of course our situation is much more complex than that of the animal kingdom because so many things not controlled by nature, such as immigration and welfare, have been introduced into the scheme. One thing remains constant though, overpopulation degrades the quality of life for all .... be it wildlife or human.
With all this said, I'm still of the firm belief that immigration is our #1 problem, not births. We can't do anything about the births until we get a handle (reduction) on immigration. That includes all immigration, legal, illegal, and refugees.