Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 32A / b / i / Solution

Adding the two equations gives $\dot{x}+\dot{y}=b-x$, so a fixed point hasAt this point the Jacobian has determinant $a+b^2>0$ and traceThe fixed point is asymptotically stable exactly when $T<0$, orThe equality curve exists only for $a\le 1/8$ and isFor $0<a<1/8$ the fixed point is unstable between these two curves and stable outside them; for $a>1/8$ it is stable everywhere.