  The journey of mammalian spermatozoa in nature is well-known to be reliant on
their individual motility. Often swimming in crowded microenvironments, the
progress of any single swimmer is likely dependent on their interactions with
other nearby swimmers. Whilst the complex dynamics of lone spermatozoa have
been well-studied, the detailed effects of hydrodynamic interactions between
neighbors remain unclear, with inherent nonlinearity in the governing
hydrodynamics and potential dependence on the details of swimmer morphology. In
this study we will attempt to elucidate the pairwise swimming behaviors of
virtual spermatozoa, forming a computational representation of an unbound
swimming pair and evaluating the details of their interactions via a
high-accuracy boundary element method. We have explored extensive regions of
parameter space to determine the pairwise interactions of synchronized
spermatozoa, with synchronized swimmers often being noted in experimental
observations, and have found that two-dimensional reduced autonomous dynamical
systems capture the anisotropic nature of the swimming speed and stability
arising from near-field hydrodynamic interactions. Focusing on two
configurations of spermatozoa, namely those with swimmers located side-by-side
or above and below one another, we have found that side-by-side cells attract
each other, and the trajectories in the phase plane are well captured by a
recently-proposed coarse-graining method of microswimmer dynamics via
superposed regularised Stokeslets. In contrast, the above-below pair exhibit a
remarkable stable pairwise swimming behavior, corresponding to a stable
configuration of the plane autonomous system with swimmers lying approximately
parallel to one another....
