Plant SI model with logistic total population
= Plant SI model with logistic total population
{c}
The plant-disease model
$$
\dot S=(S+I)(1-S)-\beta IS,
\qquad
\dot I=-(S+I)I+\beta IS
$$
has total population $N=S+I$ and <disease prevalence> $\theta=I/N$ satisfying
$$
\dot N=N(1-N),
\qquad
\dot\theta=\theta\{\beta N(1-\theta)-1\}.
$$
Thus infection does not alter total-population growth. The invasion threshold at the carrying capacity $N=1$ is $\beta=1$.