  In \cite{BeEr}, Bergweiler and Eremenko computed the number of critical
points of the Green's function on a torus by investigating the dynamics of a
certain family of antiholomorphic meromorphic functions on tori. They also
observed that hyperbolic maps are dense in this family of meromorphic functions
in a rather trivial way. In this paper, we study the parameter space of this
family of meromorphic functions, which can be written as antiholomorphic
perturbations of Weierstrass Zeta functions. On the one hand, we give a
complete topological description of the hyperbolic components and their
boundaries, and on the other hand, we show that these sets admit natural
parametrizations by associated dynamical invariants. This settles a conjecture,
made in \cite{LW}, on the topology of the regions in the upper half plane
$\mathbb{H}$ where the number of critical points of the Green's function
remains constant.
