  The May--Leonard model was introduced to examine the behavior of three
competing populations where rich dynamics, such as limit cycles and nonperiodic
cyclic solutions, arise. In this work, we perturb the system by adding the
capability of global mutations, allowing one species to evolve to the other two
in a linear manner. We find that for small mutation rates the perturbed system
not only retains some of the dynamics seen in the classical model, such as the
three-species equal-population equilibrium bifurcating to a limit cycle, but
also exhibits new behavior. For instance, we capture curves of fold
bifurcations where pairs of equilibria emerge and then coalesce. As a result,
we uncover parameter regimes with new types of stable fixed points that are
distinct from the single- and dual-population equilibria characteristic of the
original model. On the contrary, the linearly-perturbed system fails to
maintain heteroclinic connections that exist in the original system. In short,
a linear perturbation proves to be significant enough to substantially
influence the dynamics, even with small mutation rates.
