Consider the two-dimensional system of ODEs arising in the study of transmission of parasitic infections: $$ (s)\quad w'=-\mu\sb 1w-\mu\sb 1T\sb 1y,\quad y'=-\mu\sb 2y+\mu\sb 2T\sb 2\phi (w)(1-y), $$ with initial conditions w(0)$\ge 0$, and $0\le y(0)\le 1$. Here $\mu\sb 1$, $\mu\sb 2$, $T\sb 1$, $T\sb 2$ are positive constants and the function $\phi$, which represents the expected number of ovipositing worms per vertebrate host, depends on the sexuality of the parasite.
The authors regard $\phi$ as an arbitrary function subject only to mild regularity conditions and study solutions of the initial value problem for (s). They discovered three new methods of establishing asymptotic behavior.
Contents: 1. Introduction; 2. The problem of global existence; 3. The method of the energy integral; 4. A generalized l’Hospital rule and the method of fluctuations, 5. The Sturm comparison method.