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.
Right, but what most people mean by stable population is a population that doesn’t fluctuate outside of a range. A population that grows and shrinks but has some kind of negative feedback that causes it to oscillate around a value, is stable enough, even if the local behavior of the curve is exponential.
That sort of sinusoidally oscillating population is certainly possible, but be careful what you wish for. The negative feedback, as far as population is concerned, is "death". Before the exponentially-growing post-Industrial population, we'd have periodic wars/famines/plagues that result from the population of an area outgrowing its local carrying capacity. That's not generally considered a good thing.
Death is not the only thing to inevitably reduce population - there's also taxes. As in, give subsidies to childbearers in eras where the population is too low, and take them away when the population gets too high.
The problem with taxes & subsidies is that their discriminatory power is not as great as policy makers would like to believe. They tend to either not work at all, or they work too well and overshoot the target. (This is the issue with many other economic interventions as well.)
Economists like to pretend that everything is a continuous function of price, but this very rarely reflects real markets. Most utility functions are step functions; there is some point at which changing your behavior becomes economical, and above it everybody switches, while below it nobody switches. People can either afford to raise a child in the lifestyle they expect, or they can't - and many of the sacrifices in lifestyle don't move the needle in fundamental living costs. For having a child, that point appears to be in the hundreds of thousands of dollars. If you make childcare and college free and give 2 years of maternity leave and ensure that you can buy a 3BR+ house on an average income, then sure middle-class people will start having kids again. All the middle class people will start having kids again, because they all face the same economic constraints. And then you'll have fertility > 2 again, and the population will start exponentially growing.
You could have a large per-child subsidy and cap at 2, but this also is a lot harder to generate the desired effect than expected. Accidental pregnancies are a thing. Twins are a thing. Infertility is a thing. Careers are a thing. People who want to remain childfree regardless of cost are a thing. If you cap at 2, you'll get a fertility rate under 2 because of all these other considerations.
And this is what's playing out empirically across many countries right now. A lot of European countries have generous subsidies for childbearing; they still find that they can't raise the fertility rate above 1.5 or so. China lifted its one-child only policy, moving it to 2 children in 2016 and 3 children in 2021, and finds that fertility still collapsed (even after instituting subsidies) and they can't convince people to have babies. There are significant non-monetary costs to parenthood; it's a whole other lifestyle, and the money involved needs to be pretty large to alter that.
Luckily there are many fertility rates that are indistinguishable from "stable population" over a span of a couple generations, say 100 years or so. That means there is plenty of time for policy shifts to incentivize or disincentivize having children.
At current fertility rates, Europe's population is going to halve in 3 generations, and who remains is going to be quite old on average. I don't know your age, but if you're young enough this will happen within your lifetime. It's not a distant and abstract problem.
I am not accounting for immigration of course, but from the wider point of view, immigration is musical chairs. Sri Lanka is well below replacement and still falling fast. Who is going to immigrate there? Ditto for most other countries.
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.