java, no need for insults. I've never claimed to be a math genius. I've not had differential equations, but I have taken population dynamics, a course taught through the Wildlife and Fisheries Sciences department at A&M and required of fisheries management majors. A good example of a J growth curve is the rabbit. The natality rate of the rabbit is such that its mortality can't keep up, either through predation or disease. The populations grows and grows to the point that it is well above carrying capacity, then it crashes through starvation or disease or both. This occurs in regular cycles with rabbits and the growth curve resembles a "J", thus the name. Since Human species has no natural predator, what is to keep our growth rate as an S or sigmoid rate? We are more like rabbits than deer in this regard.
My opinion remains that unrestricted human population growth is the root of most environmental problems. The more humans there are on the planet, the more they consume, and not just food, but energy resources, and the more impact they have on their environment. Limit human populations to reasonable numbers and the earth can handle the stress. It don't take a rocket scientist to figure this stuff out. It's really pretty simple. What is so onerous about limiting human natality rates? I could be done, it should be done, but no one has the will to do it. It's a political thing, I reckon. Do you want to be a rabbit in an overpopulated environment, searching for food, finding none, starving to death or dying of disease in a weakened state? There are only so much humans that can fit on this planet and if global warming is true, the carrying capacity could suddenly plummet.
None of that has a lot to do with the Whooping Crane, however, as it was man's fashion tastes that hurt their numbers a century ago. Were it not for lady's hats, they'd probably be a game species with all the benefits of game regulation today, just like their cousins the Sandhills.