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....
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