It is well known that a simple "rst-order di!erence equation can exhibit complex population dynamics, such as sustained oscillations and chaos. An interesting problem is whether such oscillatory dynamics are expected to occur in real populations. This paper assumes that the resident system is composed of 1-host and 1-parasitoid and that only the host is allowed to evolve, but not the parasitoid. Based on the invasibility of a host to host}parasitoid systems, we investigate the dynamics of the host}parasitoid system favored by natural selection. We consider two cases. In the "rst case, the host's evolution involving both the intrinsic growth rate and the sensitivity to density is considered. In the second case, the host's evolution involving both the intrinsic growth rate and the vulnerability to the parasitoid is considered. In both cases, we see that the dynamics with a stable equilibrium will not be favored by natural selection without the trade-o! between the host's traits which are allowed to evolve. The host}parasitoid system with a stable equilibrium will be eventually invaded by a host type that develops an unstable equilibrium with the parasitoid. If there is a trade-o! between the host's traits which are allowed to evolve, a host}parasitoid system with a stable equilibrium can be favored by natural selection. Academic Pres
