Earlier quoted context omitted.
Due to the dynamics of demography, a population is either exponentially growing or exponentially shrinking. A stable population is an unrealistic goal.
You’ll have to provide a lot more context for this to be believable. Maybe it is an unrealistic goal for animals who aren’t interested in having a stable global population.
dPop / dt = fertility rate * pop (birth rate) - deaths
Note the "current population" factor in that. As long as the number of births depends on the current population (i.e. anyone has a chance to give birth), this equation will result in an exponential curve. If the fertility rate is greater than replacement rate (that "deaths" term that I've handwaved away), the population will exponentially grow. If it's less than the replacement rate, the population will exponentially decay. But it is necessarily exponential, because it's a constant rate multiplied by the current population, and if you remember your calculus, an exponential is defined by a constant growth rate.
There's a whole subfield of population dynamics, and I remember lots and lots of predator/prey models, resource constraint models, etc. from my applied calculus class. Some of those do have non-exponential growth curves, eg. a basic 2-species predator/prey model gives rise to sinusoidal population curves as the increase in prey leads to an increase in predators, which results in more of the prey being killed and eaten, which results in a decrease of food, which results in the predators dying off. But what they all have in common is death rates that are proportional to some function of the current population. In other words, you have to kill off proportionally more of the current population based on how many people they are, something which would be ethically unacceptable to most humans. If you just have natural deaths (i.e. a death rate proportional to Pop(t - life expectancy)), you always get either exponential growth or exponential decay, because that's the way the math works.
I suppose there was one other population curve that gives a sigmoid function (i.e. a logistic curve that asymptotically approaches a limit but never reaches it). This is what you get when there is a natural resource limit to giving birth, i.e. when instead of births being fertility rate * total population, it's fertility rate * (exponentially decreasing fraction of the population). Many people find this scenario ethically challenging as well: in plain English, it means that only the elite can afford to have kids, or alternatively - cap the number of houses so that only the top X families can have a house and raise a child, and then create social stigma around having kids when you cannot afford those increasingly scarce markers of stability. It is disturbingly close to reality, though.