Bring the chapter together: find the equilibria of a nonlinear system, compute and read its Jacobian, decide stability where the linearisation can, recognise when it cannot, and interpret the predator–prey and competition models. No worked example sits above the answers.

A pilot’s checklist before landing: the same few checks in order, each quick, together catching almost everything.

Decide first what is asked. Equilibria: solve $f=0$ and $g=0$ together, factor by factor. The Jacobian: row one from $f$, row two from $g$, evaluated at the equilibrium. Its eigenvalues decide the local type unless they are purely imaginary; then look for a conserved quantity or a polar form. Nullclines and the rule that trajectories never cross join the local pictures into a portrait.

• Every eigenvalue with negative real part: asymptotically stable. One with positive real part: unstable.
• Purely imaginary eigenvalues: the Jacobian decides nothing.
• Predator–prey: closed orbits, from the conserved $V=dx-c\ln x+by-a\ln y$.
• Competition: coexistence is stable when $\alpha,\beta<1$ and a saddle when $\alpha,\beta>1$.