Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 14A / a / Solution

For a positive homogeneous equilibrium,sorequiring $\alpha >\beta$. The reaction Jacobian there isIts determinant is ${\beta }^2/(\alpha -\beta )>0$, while its trace is $\beta (1-\beta )/(\alpha -\beta )$. Stability therefore requiresThis is the region above the line $\alpha =\beta$ and to the right of $\beta =1$ in the positive quadrant.