Michael Malisoff’s Research on Chemostats

Chemostat models provide the foundation for much current research in bioengineering, ecology, and population biology [8]. In engineering, the chemostat is known as the continuous-stirred tank reactor. It is used to model the dynamics of interacting organisms in waste-water treatment plants, lakes and oceans, and in biomedical contexts. The basic mathematical model of the chemostat is a system of nonlinear differential equations whose state variables are the species levels and the levels of the limiting nutrients. Well-mixed chemostats lead to ordinary differential equations. Our chemostat-related research provides mechanisms for guaranteeing robust asymptotically stable coexistence of two species in well-mixed chemostats with only one limiting nutrient. Much of this work uses Lyapunov-based nonlinear control theory or Poincaré-Bendixson theory to design the dilution rate to be a feedback controller [3, 6]. When the dilution rate is a positive constant and the uptake functions are strictly increasing, the competitive exclusion principle implies that generically, all trajectories converge to an equilibrium where only one species survives [8]. Therefore, to guarantee coexistence, we choose a nonconstant dilution rate feedback controller depending on the species levels. However, the actual species levels may be uncertain, because of photometric methods that only calculate the sum of the species levels rather than the individual species levels, or because the measurements may take time, so only old measurements may be available. To allow for this uncertainty, we design output feedback controllers that depend only on the sum of the species levels, and that also yield coexistence under unknown measurement delays in the feedback controller [2]. The designs are based on Lyapunov-Krasovskii functionals, which are natural analogs of Lyapunov functions that apply to time-delayed systems and that depend on the recent history of the states. The uptake functions help explain how the species metabolize the nutrients. In much of the literature, the uptake functions are Monod [8], and so are concave strictly increasing functions of the nutrient level. To model cases where high levels of the nutrient impede species growth, it is important to allow nonmonotone uptake functions. In addition to covering Monod uptake functions, we cover cases where the uptake functions are Haldane functions, which increase over small values of the nutrient level but then decrease towards zero [2]. Other results lead to methods for tracking prescribed oscillatory trajectories for the species levels [5, 6]. We also obtained coexistence results under relaxed assumptions on the relative sizes of the growth yield constants without assuming concavity of the uptake functions [3], including extensions that give feedback stabilizers that are robust to uncertainty in the uptake functions [4]. Much chemostat literature is based on nonstrict Lyapunov functions, which are positive definite proper functions that are nonincreasing along all trajectories of the dynamics. By LaSalle invariance or Barbalat’s Lemma, one can often use nonstrict Lyapunov functions to prove asymptotic stability properties. By contrast, much of our work uses global strict Lyapunov functions, which are proper positive definite functions that strictly decrease along all trajectories outside the equilibrium [1]. Using this additional strictness property, one can often do an input-to-state stability robustness analysis that quantifies the effects of actuator errors [5, 7]. Actuator errors are measurable essentially bounded functions that are added to the dilution rate controller in the equations to model uncertainty in the control mechanism.