= Solution
For $\beta>0$ and positive initial populations, summing the coordinates of the attracting <equilibrium point> gives \b[the steady total population]
$$
\boxed{N_*(d)=\begin{cases}1-d,&0\leq d\leq\beta/(1+\beta),\\d/\beta,&\beta/(1+\beta)\leq d\leq\beta,\\1,&d\geq\beta.\end{cases}}
$$
Its minimum is $1/(1+\beta)$ at $d=\beta/(1+\beta)$. Small mortality leaves a fully infected population with only modest demographic loss. Raising mortality first reduces that population; after the threshold it suppresses infection sufficiently for the uninfected population to recover. Beyond the invasion threshold the disease cannot persist.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ii/paper-3-frog-population.png]
{title=Steady frog population against disease mortality, with transmission rate beta equal to two}
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