This work surveys techniques of Grasman and Veling
[1973], Vasil'eva and Belyanin [1988] and Shih [1996] for computing
the relaxation oscillation period of singularly perturbed Lotka-Volterra systems.
Grasman and Veling [1973] used an implicit function theorem to
derive an asymptotic formula for the period;
Vasil'eva and Belyanin [1988] employed a method of matched
asymptotic expansions to obtain an approximation to the period;
Shih [1996] obtained two (exact)
integral representations for the period in terms of
two inverse functions
W(-
k,
x) of
.
These results are compared numerically and asymptotically. In particular,
the integral representation of the period in Shih [1996] is computed
numerically using a Gauss-Tschebyscheff
integration rule of the first kind, and is further investigated
asymptotically by virtue of the asymptotics of
W(-
k,
x),
Laplace's method, and a method of consequent representation.
Computational results indicate that the Gauss-Tschebyscheff approximation of
the period in Shih [1996] is uniformly accurate for a wide range of
the singular parameter (
in the paper).