Crooke et al (1980) showed that simple chemostat models can not have periodic solutions for any choice of the specific growth function μ(S) if the yield is constant. However as biological system have an ubiquity for periodicity this result motivated an interest in deriving simple extensions of the standard model which do exhibit periodicity. Numerical solutions presented in Crooke et al (1980) showed that limit cycles can be observed under certain circumstances if the yield coefficient is a function of the substrate concentration. (The use of a variable yield coefficient had been suggested earlier by Essajee & Tanner, 1979). These findings motivated the initial investigations into systems with a variable yield carried out by Crooke & Tanner (1982) and Agrawal et al (1982).
Crooke & Tanner (1982)
showed that when Monod growth rate
kinetics are assumed
μ(S) = &mum/(K+S)
Hopf bifurcations can occur when the yield coefficient (Y(S)) increases linearly with substrate concentration, i.e.
Y(S) = α +βS.
In the same year Agrawal et al (1982) established conditions for Hopf bifurcations to occur in a model with a general growth rate function, μ(S), and a general yield function, Y\left(S\right). They showed that a Hopf bifurcation will occur provided that the yield coefficient increases `sufficiently fast' as a function of the substrate concentration. (Their result provides another proof of the result established by Crooke et al (1980) that a Hopf bifurcation can not occur for systems with a constant yield). Since these pioneering studies, subsequent research has often split into those investigations using specific growth rate and yield functions and those using general growth rate and yield functions.
Agrawal et al (1982) considered two expressions for the specific growth rate: Monod kinetics and a substrate inhibition model. For both models they established parameter regions in which the steady-state diagram contains two Hopf bifurcation points.
Other investigators have considered the two-component system with Monod kinetics and a linear yield (Balakrishnan & Yang, 2002; Nelson & Sidhu, 2005; Nelson & Sidhu, 2006; Yang & Su, 1993). Yang & Su (1993) investigated the productivity of a cascade of two reactors. They fixed the total residence time of the cascade and varied the residence time in the first reactor. By choosing appropriate parameters it is possible to ensure that the stable attractor in the first reactor is a limit cycle. These oscillations then force the second reactor. Their primary interest was whether such forcing could increase the productivity of the cascade. They showed that in some circumstances an ``enormous improvement'' in performance could be obtained, compared to a single reactor of the same residence time. Nelson & Sidhu (2006) re-investigated the model considered by Yang & Su (1993), arguing that Yang & Su had over-estimated the increase in performance that can be achieved by a cascade because they had not compared ``like with like'': the performance of the optimal cascade had not been compared against the optimal performance of a single reactor. Balakrishnan & Yang (2002) investigated numerically the productivity of a single reactor as a function of the residence time. Nelson & Sidhu (2005) re-investigated the behaviour of a single reactor, identifying the residence time which maximised the biomass concentration in the effluent.
Wu & Chang (2007) considered the two-component system with Monod kinetics and a variable yield of the form Y(S) = (α+βS)γ. Their aim was to derive a control scheme which could be used to eliminate self-generated oscillations from the system. Their scheme was shown to be successful in one simulation.
It is natural to consider a general class of two-component model in which the growth rate law, μ(S) is assumed to be monotonic, subject to μ(0)=0, and in which the yield coefficient, Y(S), is strictly positive, subject to Y(0)=1 (Agrawal et al, 1982; Huang, 1990; Pilyugin & Waltman, 2003; Sun & Chen, 2008; Zhu & Huang, 2005; Zhu & Huang, 2006).
Agrawal et al (1982) showed that such systems have a unique non-washout solution and established conditions for the non-washout solution to lose stability at a Hopf bifurcation. Huang (1990) used the Poincaré-Bendixson theorem to show that when the no-washout solution is unstable there exists at least one limit cycle. This general results was applied to a system governed by Monod kinetics with a linearly increasing yield coefficient. Pilyugin & Waltman (2003) investigated the global stability of the steady-state solutions. They establish conditions showing that when the no-washout solution branch loses stability at a sub-critical Hopf bifurcation that there is a range of parameter values over which there are at least two limit cycles. It was shown that a subcritical bifurcation can not occur for a system with Monod kinetics if the yield varies linearly with the substrate concentration but it can if the yield takes the form Y(S) = 1 +cS2. The finding of two limit cycles was extended by Zhu & Huang (2006), who found conditions that guarantee that there exist at least three limit cycles. Zhu & Huang (2005) constructed an annular region with the property that all limit cycles of the system must be contained within it. Finally, Sun & Chen (2008) investigated the dynamics of a system in which the substrate concentration is subject to a periodically impulsive perturbation. They found conditions under which the boundary periodic solution is globally asymptotically stable. As the size of the perturbation is varied the system exhibits a range of complex dynamics.
A number of authors have investigated models in which two species compete for the same substrate (Pilyugin & Waltman, 2003; Huang & Zhu, 2005; Huang et al, 2006; Huang et al, 2007; Huang & Zhu, 2007). Of interest in these investigations is whether the two species can co-exist. These papers can be classified into two types, depending upon if one of the species produces a substance that is toxic to the other species or not. We first consider those papers where no toxin is produced (Pilyugin & Waltman, 2003; Huang & Zhu, 2005; Huang et al, 2007). In (Pilyugin & Waltman, 2003; Huang & Zhu, 2005) Monod kinetic are used for both species whilst in (Huang et al, 2007) general growth rate expressions μ(S) are used.
Pilyugin & Waltman (2003) showed numerically that both species can co-exist periodically when one of them has a constant yield coefficient and the other has a variable yield coefficient (Y(S)=1+50S3). Huang & Zhu (2005) studied the system with quadratic yields (A+BS2 and C+DS2). It is assumed that one of the species has a natural death-rate whilst the other does not. The stability of the steady state solutions was discussed. The Hopf bifurcation theorem is applied to the two subsystems that arise when one of the species becomes extinct. When this happens, the model reduces to one covered by the results in (Huang, 1990). Conditions were proved that guarantees a parameter region over which two limit cycles coexist surround an asymptotically stable steady-state solution. Zhu (2007) has shown that for systems with Monod kinetics the assumption of a non-constant yield coefficient is not necessary for periodicity as a Hopf bifurcation can occur when both yield coefficients are constant.
Huang et al (2007) considered a model with general growth rates and general variable yields. o They investigated the stability of the steady-state solutions. When one of the species is driven to extinction the model reduces to a special case of the system considered in (Huang, 1990) and results on the existence of a limit cycle follow immediately. They established the condition under which a Hopf bifurcation occurs when one of the species is driven to extinction. They did not investigate whether the resulting periodic solutions can be continued into a parameter region in which it can support both species, as demonstrated numerically by Pilyugin & Waltman (2003).
The system in which one species produces a toxin has been investigated in (Huang et al, 2006; Huang & Zhu, 2007; Zhu et al, 2007b). In Huang & Zhu (2007)) Monod growth rates are assumed for both species and the yields are given by A1+B1Sm and A2+B2Sb An equation for the concentration of the toxin is not included. Huang & Zhu (2007b) discussed the stability of steady-state solutions and showed the existence of limit cycles through use of the Hopf bifurcation theorem.
In (Huang et al, 2006) Monod growth rates are assumed for both species and the yields are quadratic (Ai +BiS +CiS2). The system has four components as an equation is included for the concentration of the `toxin'. (In this approach the `toxin' is considered to be an inhibitor.). This system has a special structure which enables it to reduced to a three-variable system. The asymptotic behaviour of the three-variable system is analyzed and it is shown that the steady-state solution in which the two species co-exist is always unstable. When one of the species is driven to extinction the model reduces to a special case of the two-dimensional system considered in (Huang, 1990) and results on the existence of a limit cycle follow immediately. The conditions for a three dimensional Hopf bifurcation to occur are derived. (Huang et al, 2006b) have shown that the assumption of non-constant yields is not required for periodicity. A three-dimensional Hopf bifurcation can occur in this model when the yields are constant). Zhu et al (2007b) have extended the results presented in (Huang et al, 2006) to the case of general yield functions, subject to the restrictions that Yi(0)=0 and Yii >= 0. The theorems in this paper are valid in the limiting case when no toxin is produced and therefore generalize many of the results noted in this section up to this point.
We now consider papers where the underlying kinetic model is
given by the Andrews inhibition law
μ(S) = &mumS/(K+S+S2/K_i).
Suzuki et al (1985) investigated the dynamics when a single species grows on a substrate with a linear yield. They obtained the regions in parameter space in which different dynamic behaviour can be observed. Subsequently Shimizu & Matsubara (1985) investigated the effect of P-control and PI-control upon the dynamics of this system. S.C. Wu et al (2007) have investigate a double-substrate interactive model in which a micro-organism grows in the presence of two limiting substrates. The yield factor for one of the substrates was constant whilst the other could either be constant or a linear function of limiting substrate. A Hopf bifurcation can not occur when both yields are constant.
The investigations detailed in the preceding discussion assumed
that the growth rate law was of the form
Recently Nelson and co-workers
(Nelson & Sidhu, 2007);
Nelson et al, 2008a)
the dynamics of a system with Contois growth kinetics
μ(S,X) = &mumS/(KX+S),
and a linear yield. These investigations were motivated by experimental studies which have shown that the rate determining step in the cleaning of wastewaters and slurries from a variety of agricultural processes is governed by the Contois expression (Nelson et al, 2008b). In (Nelson & Sidhu, 2007) a well-stirred reactor was considered whereas in (Nelson et al, 2008a) a well-stirred membrane reactor was considered. A common feature of interest in (Nelson & Sidhu, 2007); Nelson et al, 2008a) is the considerable decrease in effluent concentration that can be achieved by using a cascade of two reactors rather than a single reactor. In these systems Hopf bifurcations are undesirable because, compared against the effluent concentration at the unstable steady-state solution, they increase the average effluent concentration leaving the reactor.