Remarks on invariance of population distributions for systems with equivariant internal dynamics

There has been recent interest, particularly in the systems biology literature, in the study of sym-metry invariances of responses of dynamical systems. The paper [1] obtained sucient characteriza-tions of symmetry invariance using a notion of equivariance, and this characterization was shown tobe necessary as well as sucient in [2]. Both [1] and [2] sketched how to extend the results to motilesystems that explore space, so long as the \motor dynamics" depends only on an invariant response.Speci cally, these results predicted that E. coli bacteria would produce scale-invariant searches,meaning that distributions of bacteria, even under non-uniform and time-varying chemoe ector elds, should be invariant under any rescaling of the input eld. This prediction was subsequentlyexperimentally veri ed in [3]. In this note, we remark that, for a velocity-jump Markov model, thePDE for the evolution of densities (or normalized concentrations) in time inherits the symmetry-invariance property from individual behaviors. Although not at all surprising, this provides furthertheoretical justi cation for passing from individual-based models to population predictions.