mathematics//differential equations//interaction term

An interaction term is 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.

An interaction term is 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.

peak of I0.41 S left at the end0.02 S + I + R1.00 at every instant What leaves S enters I at the rate βSI, which needs both; what leaves I enters R at the rate γI. The total never changes.

Play it and watch the band, the transfer βum\beta umβum: it is near zero at the start, when almost nobody knows, and again at the end, when nobody is left to tell; press Decay off and u+mu+mu+m stays flat.

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. That is what sets it apart within coupling, any dependence of one rate on another variable: a linear coupling kx2kx_2kx2​ keeps acting when x1x_1x1​ is zero, a product term does not.

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.