= Solution
Uniform per-capita culling subtracts $kS$ and $kI$ from the two equations:
$$
\boxed{
\begin{aligned}
\dot S&=(S+I)-(S+I)S-\beta IS-kS,\\
\dot I&=-(S+I)I+\beta IS-kI.
\end{aligned}}
$$
Adding and again differentiating $\theta=I/N$ gives the <Uniform per-capita culling in the plant SI model> equations
$$
\boxed{
\dot N=N(1-k-N),
\qquad
\dot\theta=\theta\{\beta N(1-\theta)-1\}.}
$$
The direct culling terms cancel from the prevalence equation because both classes are culled equally, but culling lowers the eventual total population. For $0\leq k<1$, $N(t)\to1-k$, and a rare infection grows precisely when
$$
\beta(1-k)>1.
$$
Therefore, when $\beta>1$, disease is eliminated while a positive plant population remains exactly for
$$
\boxed{1-\frac1\beta<k<1.}
$$
For $\beta\leq1$ the disease already dies out without culling. If $k\geq1$, culling drives the entire plant population to extinction, which removes the disease only by removing its host.
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