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) = &mu _{m}/(K+S)`

Hopf bifurcations can occur when the yield coefficient (

In the same year Agrawal

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 +cS ^{2}`.
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+50S ^{3}`).
Huang & Zhu (2005)
studied the system with quadratic
yields (

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 `A _{1}+B_{1}S^{m}` and

In (Huang `et al`, 2006)
Monod growth rates are assumed for both species
and the yields are quadratic
(`A _{i} +B_{i}S +C_{i}S^{2}`).
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

We now consider papers where the underlying kinetic model is
given by the Andrews inhibition law
(Andrews, 1968)

` μ(S) = &mu _{m}S/(K+S+S^{2}/K_i).`

Suzuki

The investigations detailed in the preceding discussion assumed
that the growth rate law was of the form
`μ(S)`.
Recently Nelson and co-workers
(Nelson & Sidhu, 2007);
Nelson `et al`, 2008a)
have investigated
the dynamics of a system with Contois growth kinetics

`μ(S,X) = &mu _{m}S/(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

- P. Agrawal, C. Lee, H.C. Lim, and D. Ramkrishna.
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A mathematical model for the continuous culture of microorganisms
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Complex dynamics of a chemostat with variable yield and periodically
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