Modeling with Systems

Modeling with systems is simply writing down, for each quantity, how fast it changes because of everything else in the model. It's how you turn a story like "foxes eat rabbits" or "sick people infect healthy ones" into equations you can solve. The predator–prey equations $x' = ax - bxy$, $y' = -cy + dxy$ look like alphabet soup, but each term is one small idea: $ax$ (rabbits breed on their own), $-bxy$ (rabbits get eaten when they meet foxes), $-cy$ (foxes die off without food) and $dxy$ (foxes thrive by eating). The $xy$ terms count encounters, since meetings happen more often when there are more of both. It's just a few simple stories added up; bear with it.

Equilibria of nonlinear systems: set every rate to zero and solve together. That's where nothing changes, so it's the natural first question to ask of any model. For Lotka–Volterra the coexistence point is $\left(\frac cd, \frac ab\right)$, and populations cycle around it. You saw those loops in the direction field at the start of this unit.

Epidemics. In the SIR model, $S$, $I$ and $R$ are the fractions of people who are susceptible, infected and recovered. $R_0 = \frac\beta\gamma$ (with $S$ near 1) is the number of people each case infects at the start. Think about why: a case infects people at rate $\beta$ for an average time $\frac1\gamma$, so the total is $\frac\beta\gamma$. The epidemic grows when $R_0 > 1$; vaccinating a fraction above $1 - \frac{1}{R_0}$ prevents it, because then each case meets too few susceptible people to replace itself.

Linear compartments. Tanks connected by pipes give linear systems $\vec x' = A\vec x$ whose eigenvalues are decay rates. Each tank's salt changes by what flows in minus what flows out, and the outflow is the concentration (salt divided by volume) times the flow rate. Everything from the earlier lessons in this unit applies directly.

A common mistake. It's tempting to write a tank's outflow as just the flow rate, like $10$ g/min for $10$ L/min, because that's the number given in the problem. It isn't: what leaves is salt, not water, so the outflow is $\frac{x}{V}$ times the flow rate. For a $100$ L tank at $10$ L/min that's $0.1x$.

Public-health agencies forecast outbreaks with SIR-type systems, fisheries model predator–prey interactions, and pharmacologists track drugs through body compartments with linear systems.