Parameter-robustness analysis for a biochemical oscillator model describing the social-behavior transition phase of myxobacteria

We develop a tool based on bifurcation analysis for parameter-robustness analysis for a class of oscillators and in particular, examine a biochemical oscillator that describes the transition phase between social behaviors of myxobacteria. Myxobacteria are a particular group of soil bacteria that have two dogmatically different types of social behavior: when food is abundant they live fairly isolated forming swarms, but when food is scarce, they aggregate into a multicellular organism. In the transition between the two types of behaviors, spatial wave patterns are produced, which is generally believed to be regulated by a certain biochemical clock that controls the direction of myxobacteria’s motion. We provide
a detailed analysis of such a clock and show that, for the proposed model, there exists some interval in parameter space where the behavior is robust, i.e., the system behaves similarly for all parameter values. In more mathematical terms, we show the existence and convergence of trajectories to a limit cycle, and provide estimates of the parameter under which such a behavior occurs. In addition, we show that the reported convergence result is robust, in the sense that any small change in the parameters leads to the same qualitative behavior of the solution.