Plant SI model with logistic total population

The plant-disease modelhas total population $N=S+I$ and disease prevalence $\theta =I/N$ satisfyingThus infection does not alter total-population growth. The invasion threshold at the carrying capacity $N=1$ is $\beta =1$.

Removing susceptible and infected plants at the same per-capita rate $k$ changes the reduced equations toFor $0\le k<1$, the surviving population has carrying capacity $1-k$, and infection invades precisely when $\beta (1-k)>1$.