FIG. 3.

Predicted and actual location for Hopf bifurcation as a function of $ϵ$ and $\varphi$ for weak cross immunity in the case of no mortality $\left({\mu }_d=0\right)$. The full curve is the analytic approximation [zeros of Eq. (11)], while the dashed curve is the actual location of the Hopf bifurcation obtained numerically for the full system in the case of no mortality. The number of strains is $n=4$, and other parameters are as listed in the text.