  Although ant--aphid interactions are the most typical example of mutualism
between insect species, some studies suggest that ant attendance is not always
advantageous for the aphids because they may pay a physiological cost. In this
study, we propose a new mathematical model of an ant--aphid system considering
the costs of ant attendance. It includes both mutualism and predation. In the
model, we incorporate not only the trade-off between the intrinsic growth rate
of aphids and the honeydew reward for ants, but also the facultative predation
of aphids by ants. The analysis and computer simulations of the two-dimensional
nonlinear dynamical system with functional response produces fixed points and
also novel and complex bifurcations. These results suggest that a higher degree
of dependence of the aphids on the ants does not always enhance the abundance
of the aphids. In contrast, the model without facultative predation gives a
simple prediction, that is, the higher the degree of dependence, the more
abundant the aphids are. The present study predicts two overall scenarios for
an ant--aphid system with mutualism and facultative predation: (1) aphids with
a lower intrinsic growth rate and many attending ants and (2) aphids with a
higher intrinsic growth rate and fewer attending ants. This seems to explain
why there are two lineages of aphids: one is associated with ants and the other
is not.
