Answers · The mathematics underneath

What stays constant when a system just goes around in a loop forever?

In shortA fixed combination of the changing counts stays the same round the loop, and that is what keeps the system from settling or escaping.

The quantity that stays put is $V=cR-d\ln R+bF-a\ln F$ , where $R$ is the rabbit count, $F$ is the fox count, and $a,b,c,d$ are the four positive parameters in the rule. It is not the energy of anything; it is a combination forced by the algebra of the two rate equations.

V = cR - d\ln R + bF - a\ln F

Why a fixed number closes a loop

At every moment the state must satisfy one equation, $V=\text{constant}$ . One equation in two unknowns is a curve in the $(R,F)$ plane. Near the resting point these curves are closed loops, so fixing $V$ at its starting value confines the state to one loop. The state cannot stop on that loop because the only resting point is inside it, not on it, and it cannot leave the loop because leaving would change $V$ . The only remaining behaviour is to go round the loop forever.

The phrase closed loop is doing real work. A level curve of $V$ could in principle be an open arc, and then the state would run to an end of the arc or to infinity. Around the positive resting point of this rule the level curves are closed, so there is no end to run to. That is why a finite calculation about one equation answers a question about the infinite future.

\dot R = R(a-bF), \qquad \dot F = F(cR-d)

Divide the fox rate by the rabbit rate and the time step cancels, leaving a relation between the two counts with no clock in it. Accumulating that relation produces $V$ . The expression looks unmotivated until this is done: it is what remains after the clock has been eliminated.

That $V$ really is fixed follows by differentiating it and substituting the two rates. Writing the rates as $\dot R$ and $\dot F$ :

\dot V = c\dot R - d\frac{\dot R}{R} + b\dot F - a\frac{\dot F}{F} = cR(a-bF)-d(a-bF)+bF(cR-d)-a(cR-d) = caR-cbRF-da+dbF+bcRF-bdF-acR+ad = 0.

The cancellation is exact, not a small-oscillation approximation. It uses only $R>0$ and $F>0$ , so the logarithms are defined. No formula for $R(t)$ or $F(t)$ is needed. Whether the orbit repeats forever is settled by the level curve, not by solving the system. The numbers in the rule used here are $a=0.10$ , $b=0.004$ , $c=0.001$ , $d=0.08$ . The resting point is $R=d/c=0.08/0.001=80$ rabbits and $F=a/b=0.10/0.004=25$ foxes. Starting from $R=95$ , $F=22$ gives $V=0.001(95)-0.08\ln95+0.004(22)-0.10\ln22=-0.490414$ , and the derivative above says that the same number is still $-0.490414$ at every later time on that orbit. The counts change; this combination does not.

The guess that fails

The natural guess is that endless circling means nothing is conserved, or that if something is conserved it must be energy. Neither holds. The state is changing at every instant, sometimes violently, and what is conserved is a peculiar sum with one linear and one logarithmic term per population. A quantity can be conserved without being an energy: conservation here means the rule cannot change it, not that a physical law protects it. The logarithms are not decoration. They come from accumulating a rate proportional to the amount, so a term like $\dot R/R$ integrates to $\ln R$ . Looking for kinetic plus potential energy will not find this quantity, because there is no physical energy in the model to find.

Over one complete circuit the averages are not free to vary with the size of the loop. They are $\bar R=d/c=0.08/0.001=80$ and $\bar F=a/b=0.10/0.004=25$ , the same as the resting values. A small oscillation and a violent one have the same long-run average. That is a second constancy, and it is why spraying a predator and a pest together can raise the average pest count: the average pest count is $d/c$ , so increasing the predator death rate $d$ increases it.

Where it stops being true

All of this is a property of the continuous rule with no ceiling and no crowding. Add a carrying capacity for the rabbits, so that rabbit growth slows as $R$ approaches a limit, and the closed loop generally becomes a spiral that settles on a fixed point; $V$ is then no longer conserved. A numerical step also conserves $V$ only approximately, with an error that shrinks as the step is made smaller, and too large a step can make the computed orbit drift off the level curve or spiral outward. That is an artefact of the method, not a change in the mathematics. A system can also circle without having any such first integral: a damped pendulum spirals in, a driven one can spiral out, and many loops have no fixed combination at all. The constancy is a fact about this rule, not about looping in general.

Worth keeping

Going round forever does not mean no quantity is fixed: in a predator-prey loop a particular combination of the two counts is exactly conserved.
It is not energy, and the logarithms in it come from accumulating rates proportional to the amounts.
Adding crowding or a carrying capacity can turn the closed loop into a spiral, so the conservation belongs to the rule, not to cycles in general.

Where this goes next

This question is one step of Systems That Evolve, a course that works through its subject from the beginning. The page that teaches this part of it is Round and Round, and What Holds It There, and it reads the whole way through whether or not you have an account.

What an account changes is the practice: problems standing under every idea, marked with a reason rather than a tick, in a copy of the course that is yours.

Read that page The whole course