First-order dynamics

In brief

A colony of bacteria in a dish with unlimited food does something almost embarrassingly simple: the more bacteria there are, the faster new ones appear. There is no memory of how the colony grew yesterday, no dependence on how fast it was growing a moment ago, only a dependence on how many bacteria are present right now. Systems like this, where the instantaneous rate of change is a function of the current state alone, are called first-order dynamics. The restriction sounds modest, but it is powerful enough to force the long-term behavior of the system into one of only three possible shapes: growing without bound, decaying toward zero, or settling at a fixed equilibrium. Recognizing that a system is first order is often enough, by itself, to know its entire qualitative future before solving anything.

The full treatment

First look: a colony with unlimited food

Let y(t) be the number of bacteria at time t. The rule "more bacteria means faster growth, in direct proportion" is written as dy/dt = k*y, where k is a positive constant, the growth rate per bacterium. This equation is first order because only the first derivative of y appears, and the right-hand side depends only on y itself, never on t directly and never on a second derivative. Compare this to a rule where the rate depended on how the population was changing a week ago; that would need a memory of the past beyond the current count, which first-order dynamics explicitly excludes.

Solving the rule: separating the pieces

The equation dy/dt = ky can be solved directly by isolating the y-terms on one side. Divide both sides by y (assuming y is never zero): (1/y) dy/dt = k. The left side is exactly the rate of change of the natural logarithm of y, since the derivative of ln(y) with respect to t, by the chain rule, is (1/y) times dy/dt. So the equation says: the rate of change of ln(y) is the constant k. A quantity whose rate of change is a constant must be that constant times t, plus whatever it started at, by the same reasoning used to recover a state from a constant rate. So ln(y(t)) = kt + ln(y0), where y0 is the population at time zero. Exponentiating both sides gives y(t) = y0 * e^(k*t). The entire future of the colony, for every future time, is captured in one formula built from only two numbers: the starting population and the growth rate.

Fixed points and which way they pull

Many first-order systems are not pure growth or decay but a pull toward a resting value. Write the rule as dy/dt = k*(y minus y_eq), where y_eq is some target value and k is a constant. When y equals y_eq exactly, the rate of change is zero: the state does not move. This is called a fixed point. What happens near it depends on the sign of k. If k is negative, then whenever y is above y_eq the rate is negative (pulling y down toward y_eq), and whenever y is below y_eq the rate is positive (pushing y up toward y_eq); the fixed point attracts nearby states, and the system settles at equilibrium. Newton's law of cooling has exactly this form, with y_eq as room temperature. If k is positive instead, the fixed point repels: any small nudge away from y_eq grows rather than shrinks, and the state races away toward unbounded growth or decay. Whether an equilibrium is stable or unstable is decided entirely by one sign.

When there is no clean formula: stepping forward numerically

Not every first-order rule can be solved by the separating trick above; many realistic rate functions mix y and t in ways that resist an explicit formula. Even then, the local nature of first-order dynamics gives a way forward: knowing the rate at the current state lets you estimate the state a short time later, by moving in the direction of that rate for a small step, then recomputing the rate at the new state and repeating. This step-by-step approximation, developed by Leonhard Euler, produces a sequence of estimated values that traces out the trajectory even when no closed-form solution exists. The accuracy of the approximation depends on how small the steps are and raises exactly the question of whether repeated refinement of the step size makes the sequence of estimates settle toward a true answer.

Only three long-term fates

Because a first-order equation's rate depends only on the current value, its long-term behavior cannot do anything more elaborate than three things: race off to infinity, decay to zero, or converge to a stable fixed point. It cannot oscillate, since oscillation requires the state to reverse direction periodically, which needs the rate itself to depend on more than just the current position, typically on a velocity as well. That richer behavior needs a second layer of change altogether.

Lineage

Thomas Malthus's 1798 population essay is the most famous early statement of unconstrained proportional growth, though the mathematics of exponential growth and decay predates him in compound interest calculations and in Newton's own treatment of cooling. Radioactive decay, discovered to follow exactly this pattern by Ernest Rutherford and Frederick Soddy around 1902, gave the equation a second, unrelated physical home a century later, underscoring that the mathematics does not care which physical mechanism produces the proportional rate. Leonhard Euler's eighteenth-century work on step-by-step approximation supplied the numerical companion to the analytic solution, needed whenever the closed-form trick fails.

The strongest case for it

The same equation, with different constants and different interpretations of y, describes population growth without resource limits, radioactive decay, cooling toward ambient temperature, the discharge of a capacitor through a resistor, and the clearance of a drug from the bloodstream. Wherever the rate of change is proportional to the current amount or to the gap from a resting value, the exact solution is already known and requires no further derivation, only fitting two numbers to data.

The strongest case against it

The proportional-rate assumption is an idealization that typically holds only over a limited range. Real populations run into finite food and space, so unconstrained exponential growth always eventually breaks down and needs a correction term, such as the one used in the logistic model. Radioactive decay is a rare case where the constant-rate assumption is extremely well supported at the level of individual atoms, but this should not be generalized: many everyday "constant rate" claims about decay or growth do not hold up under scrutiny and need direct verification. A common misconception is treating first-order dynamics as though it could produce oscillation or overshoot; a single first-order equation, by construction, moves monotonically toward its fate and never crosses back over an equilibrium once it approaches from one side.

Where it stands now

The mathematics of linear first-order dynamics is completely solved and has been for over two centuries; nothing about the closed-form solution or the classification into three long-term fates is in dispute. What remains an open, case-by-case question is whether a given real system genuinely obeys the proportional-rate assumption over the range being modeled, which is an empirical matter separate from the mathematics.

Test yourself

A savings account earns continuous interest at a fixed proportional rate, but the account holder also withdraws money at a constant fixed amount per year, regardless of the balance. Write the differential equation governing the balance, being careful to note that this rule is not purely proportional to the balance. Find the equilibrium balance at which withdrawals and interest exactly cancel, and determine whether that equilibrium is stable or unstable using the reasoning about signs developed above. Finally, explain what happens to an account that starts below that equilibrium value, and state, in plain terms, why the answer would be different if the growth were not first order but depended on the account's recent rate of change as well.