 Solution

Uniform per-capita culling subtracts $kS$ and $kI$ from the two equations:Adding and again differentiating $\theta =I/N$ gives the Uniform per-capita culling in the plant SI model equationsThe direct culling terms cancel from the prevalence equation because both classes are culled equally, but culling lowers the eventual total population. For $0\le k<1$, $N(t)\to 1-k$, and a rare infection grows precisely whenTherefore, when $\beta >1$, disease is eliminated while a positive plant population remains exactly forFor $\beta \le 1$ the disease already dies out without culling. If $k\ge 1$, culling drives the entire plant population to extinction, which removes the disease only by removing its host.