A model of myxomatosis based on hormonal control of rabbit-flea reproduction

A two-dimensional first-order nonlinear system of ordinary differential equations with constant coefficients is constructed to model the rabbit/flea dynamics of the European rabbit viral disease known as myxomatosis. It is proved that infected fleas induce stable oscillations of small amplitude for a range of coefficient values when the Rothschild coupling coefficient r, which models biochemical control of flea reproduction by the rabbit, attains a critical (Hopf) value rO. These oscillations may lead to rapid local extinction of rabbits depending on the virulence gamma 2 of myxoma. The coefficient gamma 1 = r gamma 2 measures the effect on the fleas. Since the four determining coefficients may change over evolutionary time-scales, the mathematical results together with a natural selection argument proves that virulence gamma 2 attenuates. This has been observed both in Australia and in Great Britain. However, the rabbit flea Spilopsyllus cuniculi is not an effective vector for myxoma in Australia, but is the principal vector in Britain, as verified by R. Muirhead-Thomson at the suggestion of M. Rothschild. This preliminary model assumes density-independent rabbit reproductivity pR, but the qualitative results hold for a wider class of density-dependent models. Yet, the former condition is basic for the technique of parameter reduction used to simplify statistical estimation of the more general density-dependent model.