Models for Population Dynamics

Numerical simulation almost always runs on continuous models — descriptions built from real-valued quantities that vary smoothly rather than jumping between discrete states. The reason is practical: once a quantity is real-valued and smooth, the whole apparatus of calculus becomes available, and calculus is what lets us write down how fast something is changing and then solve for its behavior over time.

Continuous models split into two classes, separated by a single question — does space matter?

• Space is ignored. The model tracks only how quantities evolve over time. This produces an ordinary differential equation (ODE): every derivative is taken with respect to the one independent variable, time.
• Space is resolved. The model tracks how quantities vary from place to place, not just over time. This produces a partial differential equation (PDE), with derivatives along each spatial direction entering alongside the time derivative.

An epidemic makes the split concrete. If all we care about is how many people are infected as the weeks pass, a headcount-over-time model is enough — and that is an ODE. But if the outbreak’s behavior depends on where it is unfolding — a dense city versus quiet countryside, one region locked down while its neighbor is not — then geography has to enter the model, and we need a PDE. COVID-19 was studied both ways, for exactly this reason.

Population dynamics — the study of how populations grow or decay over time — is the standard example of the ODE side, and the subject of this page. It has a long tradition in both biology and mathematics. A population is, strictly speaking, a count of discrete things: bacteria in a culture, animals in a habitat, people in a country, all of which come in whole numbers. The model nonetheless treats that count as a continuous, real-valued quantity — and that approximation is what buys us calculus. We will see shortly that an honest discrete model and the continuous one agree closely enough that the convenience costs almost nothing.

Two regimes are studied, in order of difficulty. The first is a population in isolation: a single species with no external influences, no interference, no competition imposed from outside — just the population and its own internal growth and decay. The second is coexistence: several species sharing an environment and influencing one another, whether peacefully (mutual benefit) or with hostility (one species preying on another, or two competing for the same resources). This page builds the single-species case first; multiple interacting species come afterward.

The hostile case has, in a sense, already appeared. The arms race of two superpowers introduced earlier is mathematically a two-”species” model — military budgets standing in for headcounts, escalating and restraining one another through the same kind of coupled ODEs — and it returns later on this page as a worked example.

The starting point for everything that follows is the oldest and simplest one-species model, introduced by Thomas Malthus in 1798.

The Malthus model

Malthus’s model is deliberately minimal. It follows a single species $P$ and tracks just one quantity: the number of individuals $p(t)$ alive at time $t$. It is worth being clear about what $p(t)$ is and isn’t. It is the size of the population at the instant $t$, the running headcount — not the number of newcomers, and not a rate of change. Everything the model predicts is the behavior of this one number as time runs.

The model is built from three rates.

A concrete reading: if each rabbit produces on average $\gamma = 0.1$ offspring per day, while rabbits are lost at $\delta = 0.02$ per rabbit per day, the population grows at $\lambda = 0.08$ per rabbit per day. The crucial word is per rabbit — each rate measures one individual’s worth of reproduction or loss. To get the behavior of the whole population, we still have to multiply by how many individuals are actually there. Malthus’s one modeling assumption is that all three rates are constant — they never drift, no matter how large the population grows or how much time passes.

So picture standing at time $t$ with $p(t)$ individuals, and ask what the population will be once a small time step $\Delta t$ has elapsed. The new headcount is the old one plus whatever was gained in between:

Read the change term $\lambda \cdot p(t) \cdot \Delta t$ as a product of three plain quantities: $\lambda$ is the net change per individual per unit time, $p(t)$ is how many individuals are contributing, and $\Delta t$ is how long they contribute for. Rate times headcount times elapsed time gives a number of individuals. This is what “growth is proportional to population size and time” means — double the population and the change doubles; double the interval and it doubles again. And note the change term is a count of new individuals, separate from $p(t)$, which is the running total they get added to.

This difference equation only tells us the population at instants spaced a step $\Delta t$ apart. To get a model that holds at every instant, we shrink that step toward zero, and the relation collapses into a differential equation.

The differential equation $\dot{p} = \lambda p$, which says a quantity’s rate of change is proportional to the quantity itself, is one of the most familiar in all of mathematics. Its solution is known outright, with no derivation work needed, which is why the exponential above can simply be written down.

The sign of the growth rate decides everything:

• $\lambda > 0$: births outpace deaths, and $p(t) = p_0\, e^{\lambda t}$ grows exponentially — not merely upward, but with an ever-steepening climb, doubling again and again over equal stretches of time.
• $\lambda < 0$: deaths win, and the population decays exponentially, sinking toward zero.
• $\lambda = 0$: births and deaths exactly cancel, and the population sits forever at its starting value $p_0$.

The figure shows all three at once, every curve started from the same population $p_0$. With $\lambda > 0$ the curve soars, steepening as it climbs; with $\lambda < 0$ it sinks toward zero; with $\lambda = 0$ it holds flat. Nothing but the sign of $\lambda$ separates a population that explodes from one that vanishes.

This is the model’s defining feature and equally its weakness: the behavior is purely exponential, and exponential change is violent — extremely fast growth or decay with nothing to temper it. Two limitations stand out. On the growth side, $p_0\, e^{\lambda t}$ rises without any ceiling, yet no real population grows forever: food, space, and other resources are finite, and sooner or later they bite. The refinements later on this page exist precisely to put a brake on this runaway growth.

On the decay side the flaw is subtler. An exponential with $\lambda < 0$ keeps shrinking toward zero without ever reaching it — the model would send a population of five rabbits through $0.3$ of a rabbit, then $0.01$, and onward forever. A real population is a whole-number count: five rabbits, then four, then — possibly — none at all. Treating the headcount as a continuous, real-valued quantity is exactly what made calculus available, but the price is that the model cannot represent the integer steps a population truly takes, nor outright extinction. Populations are discrete; the model is continuous, and this gap is the cost of that convenience.

A discrete model?

The derivation behind the Malthus model was, start to finish, a continuous one: we wrote down a difference equation, shrank the time step to zero, and handed the problem to calculus. That move sits a little uneasily next to something we already conceded: a population is a count of whole individuals, not a quantity that varies smoothly. So it is fair to push back. If the thing being modeled is inherently discrete, can we build a model that stays discrete throughout, never taking a limit at all? And if we can, does it disagree with the continuous one?

It turns out we can, and the discrete model rests on a single assumption — the very one Malthus made, only rephrased so that no rate of change appears in it. A constant growth rate and a fixed doubling time are two ways of saying the same thing: if a population grows by the same proportion in every equal stretch of time, then the time it needs to double is always the same.

So take “the population doubles every $\tau$ years” as the founding rule and follow it forward in steps of $\tau$. Starting from $p_0$ individuals at time $t = 0$:

• after $\tau$ years the population has doubled once, reaching $2\,p_0$;
• after $2\tau$ years it has doubled twice, reaching $4\,p_0$;
• after $k\tau$ years it has doubled $k$ times, reaching $2^k\, p_0$.

Writing $p(k\tau)$ for the population after $k$ doublings, the whole model is one line:

There is no differential equation here, no limit, no calculus — only repeated doubling, indexed by the whole number $k$. To set it next to Malthus’s $p(t) = p_0\, e^{\lambda t}$, all that is missing is to write this same rule with a base of $e$ rather than a base of $2$, which is a matter of algebra and not of modeling:

Now set the two models side by side. The continuous Malthus model gives $p(t) = p_0\, e^{\lambda t}$; the discrete model gives $p(k\tau) = p_0\, e^{\lambda\,(k\tau)}$. These are the same function of time. The discrete model has simply restricted where that function is allowed to be read.

That restriction is the one real difference between them, and it is worth stating plainly. The discrete model is defined only on a grid of times, $t = 0,\ \tau,\ 2\tau,\ 3\tau,\ \dots$, spaced one doubling time apart. With a doubling time of $\tau = 17$ years, for instance, it speaks about the population in year $0$, year $17$, year $34$, year $51$, and so on, and has nothing at all to say about year $25$, which is not a grid point. The continuous model, handed the same $\lambda$, returns a value at every real time $t$, the instants between the grid points included.

The two are joined in the natural way. Stop treating $k\tau$ as a whole number of completed doublings and instead let it range over every non-negative real value $t$, and $p_0\, e^{\lambda\,(k\tau)}$ becomes $p_0\, e^{\lambda t}$, the Malthus solution exactly. Filling in the instants between the grid points is the whole of what separates the discrete model from the continuous one.

This is a reassuring outcome. Two genuinely different starting points — one continuous and built on calculus, the other discrete and built on plain counting — converge on the very same formula. The continuous model is not a distortion accepted for the convenience of calculus; it is the natural interpolation of the discrete picture, the smooth curve threaded through the grid points that the discrete model pins down. That agreement is what lets us lean on continuous models for the rest of this page with a clear conscience.

An aside: interest rates

The exponential $p_0\, e^{\lambda t}$ that the Malthus model produced is worth pausing on, because it is not really a fact about populations at all. It is a fact about constant proportional growth: any quantity that grows by a fixed fraction of itself in each equal stretch of time obeys an exponential of exactly this shape. A population is one instance. Money in an interest-bearing bank account is another, and it turns out to be governed by the very same equation. The short detour through it is worth taking, because it makes a point that runs through all of modeling: a model is a mathematical structure, and one and the same structure can describe situations that have nothing physical in common.

Start with a principal $K$, an initial sum of money placed in an account, and suppose the bank offers a nominal interest rate of $R$ percent per year. It is tidier to carry the rate as a plain fraction than as a percentage, so write

turning a quoted rate of $R = 5\%$ into $r = 0.05$.

If the bank pays interest just once a year, each year multiplies the balance by $1 + r$. One year on, the account holds $K(1 + r)$; the year after, $K(1 + r)^2$; and after $n$ years,

Often, though, interest is added more than once a year — this is compound interest, the arrangement in which interest already credited goes on to earn interest of its own. Compounding monthly slices the annual rate into twelve equal parts of $r/12$ and applies one each month, which over $n$ years is $12n$ separate steps:

Compounding daily slices the year into $365$ steps of $r/365$ apiece, for $365n$ steps across $n$ years:

The pattern is plain: compounding more often means more steps, each one smaller. Carry that to its limit. Let $N$ be the number of compounding steps per year and let $N \to \infty$, so that interest is credited neither monthly nor daily but continuously, in infinitely many infinitely small installments. The balance after $n$ years is then a limit:

This limiting arrangement is called continuous compounding.

Now set this beside Malthus. Continuous compounding grows a principal to $K\, e^{rn}$ after $n$ years; the Malthus model grows a population to $p_0\, e^{\lambda t}$ after time $t$. These are the same formula. The principal $K$ stands exactly where the starting population $p_0$ stood, the interest rate $r$ where the growth rate $\lambda$ stood, and the elapsed years $n$ where the time $t$ stood. A continuously compounded bank account is, in every mathematical respect, a Malthusian population: money “reproduces” at a constant rate per unit held, just as Malthus’s individuals do, and it grows just as relentlessly, with no ceiling anywhere to slow it down. That is a comfortable arrangement for whoever is collecting the interest, and the banks have long known it.

Because the two models are identical, every consequence of one is a consequence of the other, the doubling time included. How long must a sum sit before it is worth twice the original $K$? Under annual compounding the balance has doubled once $K(1 + r)^\tau = 2K$; dividing by $K$ and taking logarithms,

Under continuous compounding the doubling condition is $K\, e^{r\tau} = 2K$ instead, and the same two moves give

This last one is precisely the population relation $\lambda \tau = \ln 2$ from before, now with $r$ playing the part of $\lambda$.

The two formulas look different, but for the modest rates a bank actually pays they very nearly agree, and both fold into a rule simple enough to do in your head. For small $r$ the logarithm $\ln(1 + r)$ sits very close to $r$ itself: at $r = 0.05$ it equals $0.0488$, barely different, so the annual-compounding $\tau$ all but lands on the continuous one. And $\ln 2 = 0.693\ldots$ is near enough to $0.7$ that

the final step using $r = R/100$. This is the rule of 70: to get the doubling time in years, divide $70$ by the interest rate written as a percentage. A balance at $R = 7\%$ doubles in about a decade; at $R = 2\%$ it takes about $35$ years. And since the population model is the same model, the shortcut carries over to it unchanged — a species growing at $2\%$ a year doubles roughly every $35$ years.

Is exponential growth realistic?

The Malthus model commits a population to pure exponential growth, climbing without limit for as long as time runs. Before refining it, the blunt question is worth asking head-on: does any real population behave like that? The honest answer is for a while, yes — but not forever.

Over a bounded stretch of time, real data can follow an exponential almost exactly. The standard illustration is world population between 1700 and 1961: across those two and a half centuries it tracked an exponential closely, at an annual growth rate of about $0.02$ (around $2\%$), which by the rule of 70 means the population doubled roughly every $35$ years, or $34.67$ to be precise. That clean fit is partly a fluke of the window. The period itself was anything but uniform — wars and the 1918 influenza pandemic held growth back while the industrial revolution pushed it forward — and the curve looks smoothly exponential only because those opposing forces happened to roughly cancel.

Stretch the window, though, and the exponential always fails. A steady $2\%$ rate would have doubled world population yet again by around 1996; it did not. No population grows without bound, and the reason is the one thing the Malthus model leaves out entirely: resources are finite. Food, water, space, air — none of them is unlimited, and as a population swells, its members compete ever harder for a supply that does not grow with them. That competition acts as a brake, dragging the growth rate down. So in general exponential growth is not realistic, and the next refinement is built precisely to give a population a ceiling it approaches rather than a climb that never ends.

A first refinement: Verhulstian saturation

The Malthus model came with a verdict attached: a real population tracks its exponential for a while, then finite resources intervene and growth has to slow. A model meant to be realistic has to build that slowdown in. The first refinement that does so is due to Pierre-François Verhulst and his contemporaries, working in the 19th century, and its organizing idea is saturation — growth that eases off as a population fills its environment, levelling out at a finite saturation limit rather than climbing without end.

The Malthus model cannot produce that behavior, and the reason traces to a single assumption: its growth rate is constant, pinned at one value whether the population is sparse or desperately overcrowded. That is the assumption Verhulst overturns.

In the Malthus model the birth rate $\gamma$ and the death rate $\delta$ were fixed numbers. Verhulst lets each one respond to the population itself: the more crowded the habitat, the harder reproduction becomes and the more dangerous life gets. The simplest way to build that in is to let each rate fall or rise along a straight line as the population changes.

Each rate is a constant baseline plus a term strictly proportional to $p(t)$, with no curvature and no higher powers. What those proportional terms do depends entirely on their sign, and the two signs are deliberately opposite:

• the birth rate carries a minus sign, $\gamma_0 - \gamma_1\, p(t)$, so a larger population lowers it — a crowded habitat reproduces less per unit of time;
• the death rate carries a plus sign, $\delta_0 + \delta_1\, p(t)$, so a larger population raises it — competition for finite food, water, and space makes a crowded habitat deadlier.

Both effects pull in the same direction: they brake growth. As $p(t)$ climbs, $\gamma(t)$ slides down while $\delta(t)$ rises, and the gap between them — the surplus of births over deaths that drives the population upward — keeps narrowing.

Those sign conditions are not arbitrary. That $\gamma_1$ and $\delta_1$ are positive is exactly what makes the feedback a brake and not an accelerator: a growing population then loses births and gains deaths, rather than the reverse. And $\gamma_0 > \delta_0$ makes the baseline birth rate beat the baseline death rate, so a small, uncrowded population still grows — without that, the population could only ever decline, leaving no saturation to speak of, just a slow slide to extinction.

With the rates set up this way, the population can no longer run away. As it grows the braking strengthens, with $\gamma(t)$ falling and $\delta(t)$ rising, until the two rates meet. At that population the births exactly offset the deaths, the net growth drops to zero, and the population stops changing. That balance point is a finite, positive number, and it is the saturation limit the whole refinement was built to produce — it has a name.

So the refinement amounts to a single substitution — the constant rates of Malthus become linear ones — and everything else follows from it. What that substitution does to the population over time is not visible yet: the linear rates still have to be folded into a differential equation for $p(t)$, and that equation solved. Doing so is the next step, and the solution will also return an explicit formula for $K$ built from the four constants, turning the bare claim that a carrying capacity exists into an actual number.

The saturation equation

The linear birth and death rates describe how fast a population reproduces and dies, but they are not yet a model: on their own they say nothing about how the population moves. Turning them into an equation of motion for $p(t)$ is the short step this section takes — short, because most of the work was already done for Malthus.

Recall how the Malthus model was assembled, because the step here mirrors it. There the birth and death rates were counted per individual, so the population’s change over an interval had to be scaled up by the headcount $p(t)$ before it meant anything. The linear rates are bookkept differently, as noted when they were introduced: $\gamma(t)$ and $\delta(t)$ already count births and deaths for the whole population per unit of time. With the headcount thus baked in, building a difference equation and passing to the limit — the route already taken for Malthus — leaves the rate of change as births minus deaths outright, with no $p(t)$ factor anywhere:

Now substitute the linear forms. Taking $\gamma(t) = \gamma_0 - \gamma_1\, p(t)$ and $\delta(t) = \delta_0 + \delta_1\, p(t)$ and subtracting, the equation becomes

The right-hand side is a linear function of the population: a constant $\gamma_0 - \delta_0$, less a multiple of $p(t)$. This is already the model’s equation, but it reads better repackaged. Define two constants,

The first, $m$, is the coefficient of $p(t)$; the second, $K$, is built so that $m K$ equals the constant term $\gamma_0 - \delta_0$. With those choices the equation factors neatly, collapsing to a compact and very suggestive form:

Those two names are worth dwelling on, because neither is mere shorthand.

The constant $m = \gamma_1 + \delta_1$ is a sum of two positive numbers, so $m > 0$. It sets the pace of the model: it gathers how sharply the birth rate falls and the death rate climbs as the population grows into a single number, measuring how briskly the population responds to its own size.

The constant $K$ is the one promised earlier. The carrying capacity was introduced as a claim — a finite, positive saturation level the model would somehow have to produce. Here it is, produced:

It is finite because the denominator $\gamma_1 + \delta_1$ is nonzero, and strictly positive because the sign conditions on the rates make numerator and denominator both positive: $\gamma_0 > \delta_0$ gives $\gamma_0 - \delta_0 > 0$, and $\gamma_1, \delta_1 > 0$ gives $\gamma_1 + \delta_1 > 0$. So $0 < K < \infty$ is no longer an assumption — it is a consequence of the four constants the model began with. The bare claim has become a number.

The differential equation tells the population how to change from one instant to the next, but not where to begin. Populations that start at different sizes trace different curves, all of them obeying the same equation — so the equation on its own describes a whole family of curves, not a single one. To pick out the one curve a particular population follows, the model needs a second fact: the population’s size at the start, the initial condition $p(0) = p_0$. The equation and this initial condition together make the model complete, and — as with the Malthus model — its solution can be written in closed form.

With the solution in hand, the population’s entire future can be read straight off the formula $p(t) = K + (p_0 - K)\, e^{-m t}$. It has just two pieces: the fixed number $K$, and a time-varying term $(p_0 - K)\, e^{-m t}$. Because $m > 0$, the exponential $e^{-m t}$ equals $1$ at $t = 0$ and decays steadily toward zero, so that second term always shrinks away — whatever it does on the way, the population is bound for $K$.

What the starting population decides is how the approach happens, through the sign of $p_0 - K$:

• if $p_0 > K$, so the population starts above its carrying capacity, then $p_0 - K$ is positive: the second term is positive and shrinking, and $p(t)$ descends toward $K$ from above;
• if $p_0 < K$, so the population starts below capacity, then $p_0 - K$ is negative: the second term is negative and shrinking in magnitude, and $p(t)$ climbs toward $K$ from below;
• if $p_0 = K$ exactly, the second term is zero for all time, and the population holds at $K$ forever.

Let time run on without bound and the two cases merge into one. The exponential $e^{-m t}$ tends to zero, the time-varying term vanishes with it, and the population converges on its carrying capacity:

This is the saturation the refinement set out to deliver: starting high or starting low, every population the model describes is drawn to the same finite level $K$ and settles there — no unbounded boom, and no slide to extinction.

The figure puts that behavior in plain view. Each curve is one solution $p(t)$, started from a different population $p_0$; the dashed line is the carrying capacity $K$. The curves beginning above $K$ bend down toward it and those beginning below climb up to it, but every one of them flattens onto the dashed line — the limit $p(t) \to K$, seen directly. The approach is fast then slow: the exponential $e^{-m t}$ sheds most of the gap to $K$ early on, so each curve closes in quickly and then eases through the final stretch. Wherever a population starts, the model funnels it to the single level $K$.

Is saturation realistic?

The saturation model earns its keep. Its solution rises or falls to the carrying capacity $K$ and settles there — and because $K$ is the level the population approaches as time runs on without end, it also goes by the name $p_\infty$. That bounded behavior is the whole point of the refinement: where Malthus sent a population climbing forever toward infinity, the saturation model holds it to a finite ceiling. For any population living on limited resources — which is to say every real one — that is a genuine step toward realism.

And yet it is still not realistic, and the flaw shows the moment the clock starts. Picture a single pair of rabbits loosed into vast empty grassland — land that could in time feed millions of them. With only two rabbits and all that room, nothing whatsoever holds reproduction back, and the population ought to take off like a Malthusian exponential: slow at first, with just two rabbits to breed, then faster and faster as their numbers swell. The saturation model does the opposite. Its growth rate is set by the gap $K - p$ between the current population and the distant ceiling — and that gap is at its widest exactly when the population is smallest, at the very start. So the population grows fastest right at the start, and its growth slows a little more with every step after that, the whole way up to $K$. The rabbits behave as if they could feel a ceiling getting closer and were holding back for it — when in fact they have room for millions more.

What realism asks for is a slowdown that comes later. While the population is small and its surroundings empty, it should grow like the Malthus model — an accelerating, exponential take-off. Only as it starts to crowd its limit should it switch to the saturation model’s behavior, bending over and easing onto $K$. Both halves are already pictured on this page: the exponential climb in the Malthus figure, the saturating approach in the figure just above this section. A realistic history splices the two together, and spliced, it has the shape of an S — curving upward through the early boom, then curving over as it nears $K$, with a single turning point where it passes from one half to the other. The saturation model has only the second half: its curve bends toward the ceiling from the very first instant, with no exponential phase before it and no turning point at all. Supplying that missing first half, and the turn into the second, is the work of the second refinement.

A second refinement: logistic growth

The saturation model gave a population a finite ceiling but spent realism getting it: its brake is active from the very first instant, even when there is nothing yet to brake against. The second refinement keeps the ceiling but rebuilds the brake so its strength depends on the population itself — weak when the population is small, strong only as the population grows. Done right, this produces the S-shape the previous section asked for.

The way to build a brake that scales with the population is to write the rate of change as a product of two factors:

with positive constants $a$ and $b$, the first much larger than the second: $a \gg b > 0$. The second factor, $p(t)$, is just the current population — the “more makes more” multiplier already familiar from the Malthus model. The first factor, $a - b\, p(t)$, is the new piece: an effective per-population growth rate that starts at $a$ when the population is vanishing and shrinks linearly toward zero as the population grows. That first factor is the brake — a number that punishes size.

The product of the two reads naturally as exponential growth with a built-in brake, and the two regimes can be read straight off it.

• When $p$ is small, the brake term $b\, p$ is tiny next to $a$, so $a - b\, p \approx a$. The equation collapses to $\dot p \approx a\, p$ — the Malthus equation with growth rate $a$. The population takes off exponentially, exactly as it should in an empty habitat.
• As $p$ grows, $b\, p$ catches up to $a$, the first factor $a - b\, p$ shrinks, and with it the effective growth rate. Growth slows, the curve starts to bend, and the brake — invisible until now — takes over.

The condition $a \gg b$ is what cleanly separates those two regimes. Because $b$ is so small, the brake term $b\, p$ stays insignificant for a long stretch of population sizes, letting the Malthusian phase play out fully before saturation begins to bite. Take $b$ comparable to $a$ instead and the brake would press from the start, throwing the model back to the saturation behavior the refinement was meant to leave behind.

Distributing the product across $p(t)$ rewrites the equation in a second, equally useful form:

The two forms say the same thing arithmetically but emphasize different ideas. The product form $(a - b p)\, p$ makes the modelling intent visible — Malthus’s $p$ multiplied by a slowing factor — while the expanded form names the new ingredient explicitly: pure Malthus growth $a\, p$ corrected by a quadratic term $-b\, p^2$ that grows faster than the population and eventually overwhelms it. Both will be useful later.

A note on what is not in the derivation. For the Malthus model and the saturation model, the equation of motion was built up from a per-individual or per-population bookkeeping of births and deaths over a small step $\Delta t$, and passing to the limit gave the ODE. No such build-up happens here: the logistic ODE is posited directly, as a modelling choice for the shape of the per-population growth rate. The interpretation of $a$ and $b$ — and the case for the quadratic term over, say, a cubic one — is taken up later in this chapter; for now, the equation is the model.

Deriving that closed form is more involved than for the previous two models. The right-hand side is quadratic in $p$ rather than linear, so the change-of-variable trick that solved the saturation equation no longer applies. The full derivation pairs separation of variables with a partial-fraction decomposition; the latter doesn’t earn enough use elsewhere in this chapter to develop from scratch here, so we take the formula as given and turn next to what it actually does.

Reading the logistic curve

With the logistic growth model in hand, the same questions surface that we asked of the saturation model: where does the population end up, and what shape does the journey take?

The limit is read straight off the closed-form solution. As $t \to \infty$, the exponential $e^{-a t}$ vanishes (since $a > 0$), and the denominator $b\, p_0 + (a - b\, p_0)\, e^{-a t}$ collapses to just $b\, p_0$, leaving

The starting size $p_0$ cancels. Every solution ends up at the same level, set only by the model’s two constants — so saturation is built into the logistic curve just as it was into the Verhulst one. The level $a/b$ earns the same name, carrying capacity, only with a new formula:

What the starting point decides is how the population approaches $K$, and the comparison with the saturation model splits the cases.

• If $p_0 < a/b$, the population starts below carrying capacity. It rises toward $K$ with the S-shape the refinement was built to deliver: an accelerating early climb while the brake term $b\, p$ is still tiny, then a bend over the top as the brake catches up, easing onto $K$ in the final stretch. The turn happens at the inflection point $p = K/2$, where the curve switches from accelerating to decelerating, and the bound $p(t) < a/b$ holds for every $t$.
• If $p_0 > a/b$, the population starts above carrying capacity. It descends monotonically toward $K$ from above with no inflection point — the same monotone descent the saturation model produced for populations starting above their ceiling. The bound $p(t) > a/b$ holds throughout.
• If $p_0 = a/b$ exactly, the population is already at its carrying capacity and stays there for all time.

The S-curve in the first case is exactly the shape asked for earlier — a Malthusian boom in the empty-habitat phase and a saturating tail near the ceiling, joined by a single turning point. Earlier those were two regimes that would have had to be spliced by hand. Here they aren’t a splice at all: one equation produces both, and the inflection where the regimes meet emerges from the formula on its own.

The figure shows several solution curves started from different $p_0$, with the dashed line marking the carrying capacity $K = a/b$. Curves below $K$ trace the characteristic S — slow at first, then a fast climb through the middle, then a gentle bend onto the dashed line — while curves above $K$ slide down to it directly. All of them level off on the same line.

To put numbers to the formula, the model has been fitted to census figures for the population of the United States from 1790 to 1950, giving

The Malthusian rate $a \approx 3.1\%$ per year matches the early-American population boom, and the brake coefficient $b$ is small enough that the quadratic term $b\, p^2$ only seriously bites once the population reaches the hundreds of millions. Substituting gives a carrying capacity of $K = a/b \approx 2 \cdot 10^8$, roughly the U.S. population at the upper end of the fitting window.

With saturation built in and a realistic S-shape, the logistic model is decidedly more powerful than the two before it. What it still doesn’t capture is anything from outside the single species — no other species interacting with it, no external influences operating on it at all. Supplying those — first a framework for several species sharing an environment, then specific patterns like competition and predator–prey — is the work of the rest of this page.

Why quadratic, and not cubic?

The logistic ODE was posited directly as a modeling choice, with the quadratic brake $-b\, p^2$ taken on faith. We come back to that choice now. The exponent isn’t forced by the math: a cubic brake $-c\, p^3$, or some power in between, would also produce a bounded, saturating curve. Picking the power is a modeling decision. The choice of two, though, has a clean interpretation that makes it the natural pick.

The picture is one of pairwise interaction. In a population of size $p$, fix attention on a single individual and ask how much it is disturbed by the others — by competition for food, water, space, mates. The simplest assumption is that this disturbance grows in proportion to how many others are around: an individual surrounded by $100$ others is disturbed roughly $100$ times as much as one surrounded by just one. So the per-individual disturbance is proportional to $p$.

But the equation of motion tracks the whole population, not one individual. The community-wide disturbance is the per-individual amount summed across all $p$ individuals, each one contributing $\propto p$:

So the brake is quadratic because it is the product of two factors of $p$ — one for how many individuals feel the disturbance, another for how strongly each is disturbed by its neighbors. It is the slogan the whole population disturbs the whole population made precise. A cubic term would correspond to disturbance from triples of individuals interacting at once rather than pairs — admissible, but with no equally simple reading.

Read in that light, the two terms of $\dot p = a\, p - b\, p^2$ are opposite sides of the same crowd. The linear $a\, p$ is the population’s total Malthusian growth contribution, scaling one-for-one with the headcount — each individual contributes its share independently of the others. The quadratic $-b\, p^2$ is the disturbance working against that growth, scaling pair-by-pair. The brake is one power higher than the drive because the brake comes from interactions and the drive does not.

An aside: two generalizations

Logistic-like growth can also be obtained from two more general formulations of the population dynamics. Neither is common in practice — the standard $\dot p = (a - b\, p)\, p$ remains dominant — but they are worth naming, for completeness.

The first lets the growth rate depend on time:

with $\lambda(t)$ positive, monotonically decreasing, and continuous. The picture is a Malthusian model whose rate is no longer constant: the population still grows by a fixed fraction of itself per unit time, but that fraction fades as the years pass. Each condition does its own job: positive for the initial exponential climb, monotonically decreasing so that climb slows down over time, and continuous for mathematical reasons.

The second is more general still: the growth rate depends on the population itself,

with $f(p)$ positive, monotonically decreasing, and tending to zero as $t \to \infty$. This is a super-class that includes the logistic growth model as a special case — there $f(p) = a - b\, p$, linear in $p$, vanishing at $p = a/b = K$. Empirically this family is sufficient to describe closed populations — populations without external influences.

Another possibility: oscillations

The models so far all approach saturation smoothly. A natural question is whether a population might overshoot its limit, swing back below, and gradually settle in — the shape of a damped oscillation. There is a standard equation that produces exactly that:

a linear damped oscillator with $\mu > 0$, $4\omega^2 - \mu^2 \geq 0$, and limit $p_\infty$. With $p(0) = p_0$, its solution is

three pieces doing three jobs: the constant $p_\infty$ pins down the limit, the cosine produces the oscillation, and the decaying exponential $e^{-\mu t/2}$ shrinks that oscillation over time so the curve eventually settles.

What makes this qualitatively new is the second derivative $\ddot p$ — the equation now tracks not just how fast the population grows but how that growth rate itself is changing. The first derivative has a clean reading as net growth; the second derivative, the rate at which the rate changes, has no equally clean biological meaning, and without one there is little reason to bring it into a single-species model. Oscillations do show up naturally once two interacting species enter the picture — and that is where this page goes next.

More than one species

Every model so far has been a single species in isolation. The next step is coexistence — several species sharing an environment and influencing one another’s growth — and the simplest case is two species, $P$ and $Q$, tracked by their headcounts $p(t)$ and $q(t)$. Two species means two equations of motion, one per population, and the modeling choice is whether the two species feel each other or not.

If they ignore one another — separate habitats, no shared resources, no predation — the system decouples. Each population evolves on its own, governed only by its own headcount, and the whole arrangement is just the logistic growth model (or one of its predecessors) written twice. Nothing genuinely new happens.

The case that does call for new tools is the opposite one: each species’ growth rate depends on both populations. Rabbits and foxes are the standard picture — more rabbits feed more foxes, more foxes kill more rabbits — and neither population can be tracked without watching the other. Each equation keeps the now-familiar shape, a population multiplied by its growth rate, only the growth rate of each species is now a function of both populations.

The growth rates $f$ and $g$ generalize the constant $\lambda$ of the Malthus model and the linear factor $a - b\,p$ of the logistic growth model: the same “headcount times rate” structure as before, only with the rate now allowed to feel both populations at once.

The first question to ask of any such system is when it settles into a state that never changes: a pair $(\bar p, \bar q)$ at which neither species changes, $\dot p = \dot q = 0$ together, so the system would sit there forever if it ever arrived. Each equation is a product of two factors, the population and its growth rate, so each derivative vanishes the moment either factor does.

The name records what the system does at the pair, not how big the populations are: the state is stationary — no growth, no decay, for either species. Each derivative is a product — $\dot p = f(\bar p, \bar q)\, \bar p$ and $\dot q = g(\bar p, \bar q)\, \bar q$, a growth rate times a population — and a product is zero when either factor is zero. So a pair can be stationary in two ways. A population sitting at zero, $\bar p = 0$ or $\bar q = 0$, zeroes its own derivative no matter what its growth rate is doing: a degenerate stationary point, with one species simply extinct and any survivor balancing only against itself. The other way keeps both species present and instead drives both growth rates to zero, $f(\bar p, \bar q) = g(\bar p, \bar q) = 0$ — the genuinely interesting case, and the one with its own name.

This is the kind of stability the rest of the chapter chases: both species hanging on indefinitely, neither outgrowing the other, neither dying out. Whether a particular system has an equilibrium is one question, answered by solving $f = g = 0$ and checking the positivity of the resulting pair. Whether the system actually settles into the equilibrium from nearby starting points is another question entirely, and the next section takes it up.

Stability of equilibria

The first question about a two-species system is whether it has an equilibrium at all, answered by solving $f = g = 0$ and checking that the resulting pair has both components positive. The second question is whether the system cares: an equilibrium exists in the equations whether or not the dynamics ever go near it, and the system may simply pass the pair by, or actively flee from it, or be drawn straight in. Which of those happens depends on what the populations $p(t)$ and $q(t)$ do when the system is started near $(\bar p, \bar q)$. Geometrically, the populations trace a curve through the $(p, q)$-plane as time runs — the system’s trajectory — and the standard vocabulary classifies what nearby trajectories do into three cases.

The most demanding of the three is the one where the populations not only reach the equilibrium from a nearby start but stay there once they do.

The opposite extreme is the one where nothing nearby reaches the equilibrium at all.

In between is the mixed case, where some starting pairs feed in and others push out.

Telling the three cases apart from the equations themselves is the work of linear stability analysis: the Jacobian of the system, evaluated at the equilibrium, has eigenvalues whose real parts classify the equilibrium as attractive, repulsive, or a saddle. The linked page is the full lookup table — six rows covering every generic case — and a later section in this chapter applies the criterion to the two-species setting specifically. The immediate next step is to revisit the arms race of two superpowers, and watch all three behaviors emerge in a system that is mathematically a two-species model, with military budgets standing in for headcounts.

A worked example: the arms race

The model returns in essentially the form it had when first introduced. The picture is deliberately spare. Two countries are tracked, labeled $X$ and $Y$ (historically, the United States and the Soviet Union at the height of the Cold War). Each carries one number, its military expenditure at time $t$, written $x(t)$ and $y(t)$.

Read either equation term by term and the modeling intent reads off directly. Take the $X$ line:

• $-m\, x(t)$: the more $X$ already spends, the harder it gets to justify spending even more — domestic political and economic resistance scales with the existing budget. The minus sign makes this a brake on the rate.
• $+a\, y(t)$: the more the rival spends, the more pressure there is to keep up — rivalry scales with the enemy’s budget. The plus sign makes this a driver.
• $+c$: a constant baseline pulling toward more spending no matter what either side is doing.

The $Y$ equation has the same shape with the roles reversed: the brake $-n\, y$ scales with $Y$‘s own budget, and the driver $+b\, x$ scales with the rival’s. Whether the combined system runs away, settles to a balance, or oscillates depends on whether the armament coefficients $a$ and $b$ win against the disarmament coefficients $m$ and $n$ — the question the next sections take up.

Because every right-hand-side term is linear in $x$ and $y$, the system collapses into matrix form. Gathering the linear parts into a $2 \times 2$ matrix and the constants into a vector,

The diagonal entries carry each country’s own self-restraint, with their negative signs; the off-diagonal entries carry the cross-influence from the rival, with positive signs.

Equilibria of the arms race

The arms-race ODE is a linear system in $(x, y)$ — every right-hand-side term is linear in the budgets, plus a constant. Rewriting it in matrix form,