  Many biological fluids are composed of suspended polymers immersed in a
viscous fluid. A prime example is mucus, where the polymers are also known to
form a network. While the presence of this microstructure is linked with an
overall non-Newtonian response of the fluid, swimming cells and microorganisms
similar in size to the network pores and polymer filaments instead experience
the heterogeneous nature of the environment, interacting directly with the
polymers as obstacles as they swim. To characterise and understand locomotion
in these heterogeneous environments, we simulate the motion of an undulatory
swimmer through suspensions and networks of elastic filaments, exploring the
effects of filament and link compliance and filament concentration up to 20\%
volume fraction. For compliant environments, the swimming speed increases with
filament concentration to values about 10\% higher than in a viscous fluid. In
stiffer environments, a non-monotonic dependence is observed, with an initial
increase in speed to values 5\% greater than in a viscous fluid, followed by a
dramatic reduction to speeds just a fraction of its value in a viscous fluid.
Velocity fluctuations are also more pronounced in stiffer environments. We
demonstrate that speed enhancements are linked to hydrodynamic interactions
with the microstructure, while reductions are due to the filaments restricting
the amplitude of the swimmer's propulsive wave. Unlike previous studies where
interactions with obstacles allowed for significant enhancements in swimming
speeds, the modest enhancements seen here are more comparable to those given by
models where the environment is treated as a continuous viscoelastic fluid.
