  To examine the co-evolution of quantitative traits in hosts and parasites, we
present and study a co-evolutionary model of a social parasite-host system that
incorporates (1) ecological dynamics that feed back into their co-evolutionary
outcomes; (2) variation in whether the parasite is obligate or facultative; and
(3) Holling Type II functional responses between host and parasite, which are
particularly suitable for social parasites that face time costs for host
location and its social manipulation. We perform local and global analyses for
the co-evolutionary model and the corresponding ecological model. In the
absence of evolution, the facultative parasite system can have one, two, or
three interior equilibria, while the obligate parasite system can have either
one or three interior equilibria. Multiple interior equilibria result in rich
dynamics with multiple attractors. The ecological system, in particular, can
exhibit bi-stability between the facultative-parasite-only equilibrium and the
interior coexistence equilibrium when it has two interior equilibria. Our
findings suggest that: (a) The host and parasite can select different
strategies that may result in local extinction of one species. These strategies
can have convergence stability (CS), but may not be evolutionary stable
strategies (ESS); (b) The host and its facultative (or obligate) parasite can
have ESS that drive the host (or the obligate parasite) extinct locally; (c)
Trait functions play an important role in the CS of both boundary and interior
equilibria, as well as their ESS; and (d) A small variance in the trait
difference that measures parasitism efficiency can destabilize the
co-evolutionary system, and generate evolutionary arms-race dynamics with
different host-parasite fluctuating patterns.
