mathematics//differential equations//interaction term

A term in a rate equation that is proportional to the **product** of two quantities, because the thing it describes only happens when the two meet.

A term in a rate equation that is proportional to the product of two quantities, because the thing it describes only happens when the two meet.

u˙=−βum,m˙=βum−γm.\dot u=-\beta u m,\qquad \dot m=\beta u m-\gamma m.u˙=−βum,m˙=βum−γm.

Read it as a transfer: what leaves uuu enters mmm, but only at the rate at which uuu and mmm encounter each other, hence the product umumum. A rumor needs one person who tells it and one who receives it; an infection needs a susceptible and an infected in the same place; a chemical reaction needs both reactants. If either quantity is zero the transfer stops, whatever the other one is.

The second equation also has a loss, −γm-\gamma m−γm: a decay that depends on mmm alone (exponential decay). Transfer and decay are different shapes of term: the transfer conserves u+mu+mu+m, the decay does not.

The product is what makes the system nonlinear. Doubling both populations quadruples the encounter rate, so the field is not the same pattern everywhere; it deforms with the state (linear field). The predator and prey model is built from the same term (autonomous system).

In the gains-minus-losses reading, an interaction term is the gain of one equation and the loss of another, which is why it produces a constraint when nothing else enters or leaves.