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We analyze a mathematical model of a simple food web consisting of one predator and two prey populations in a chemostat. Monod's model is employed for the dependence of the specific growth rates of the two prey populations on the concentration of the rate-limiting substrate and a generalization of Monod's model for the dependence of the specific growth rate of the predator on the concentrations of the prey populations. We use numerical bifurcation techniques to determine the effect of the operating conditions of the chemostat on the dynamics of the system and construct its operating diagram. Chaotic behavior resulting from successive period doublings is observed. Multistability phenomena of coexistence of steady and periodic states at the same operating conditions are also found.

We study a model of three microbial populations competing for three complementary nutrients in a single chemostat. By using methods of numerical bifurcation theory we analyze the model equations and determine the effect of the model parameters on the dynamics of the system. The main question to be answered is whether there exist conditions under which the three populations can coexist in a stable state in the chemostat. The analysis shows that coexistence can be obtained as a stable steady state but also as a stable periodic state for a wide range of operating conditions of the chemostat.

We examine the question of coexistence of three microbial populations competing for two complementary rate-limiting substrates in configurations of interconnected chemostats. It is known that coexistence of two populations competing for two rate-limiting substrates is possible in a single chemostat, but coexistence is not possible when three populations are involved. We examine whether coexistence of three populations becomes possible by considering configurations of two or three interconnected chemostats, thus allowing for effects of spatial heterogeneity. Computational analysis of the model equations indicates that in the case of two chemostats coexistence is possible only for specific discrete parameter values where the system is structurally unstable and therefore the coexistence state is not practically attainable, whereas in the case of three chemostats coexistence is possible for a whole range of parameter values where the system is structurally stable and therefore the coexistence state can be realized in practice.

It is known that, when two microbial populations competing for a single rate-limiting nutrient are grown in a spatially uniform environment, such as a single chemostat, with competition being the only interaction between them, they cannot coexist, but eventually one of the two populations prevails and the other becomes extinct. Spatial heterogeneity has been suggested as a means of obtaining coexistence of the two populations. A configuration of two interconnected chemostats is a simple model of a spatially heterogeneous environment. It has been shown that, when Monod's model is used for the specific growth rates of the two populations, steady-state coexistence can be obtained in such systems for wide ranges of operating conditions. In the present work, we study a model of microbial competition in configurations of interconnected chemostats and we show that, if a substrate inhibition model is used for the specific growth rates of the two populations, coexistence in a periodic state is also possible. The analysis of the model is done by numerical bifurcation theory methods.

Coexistence of three microbial populations engaged in pure and simple competition is not possible in a chemostat with time-invariant operating conditions under any circumstances. It is shown that by periodic variation of the chemostat dilution rate it is possible to obtain a stable coexistence state of all three populations in the chemostat. This is accomplished by performing a numerical bifurcation analysis of a mathematical model of the system and by determining its dynamic behavior with respect to its operating parameters. The coexistence state obtained in the periodically operated chemostat is usually periodic, but cases of quasi-periodic and chaotic behavior are also observed.

Predator-prey systems in continuously operated chemostats exhibit sustained oscillations over a wide range of operating conditions. When two such chemostats interact through flow exchange, the interplay of the oscillation frequencies gives rise to a wealth of dynamic behavior patterns. Using numerical bifurcation techniques, we perform a detailed computational study of these patterns and the transitions between them as the coupling strength and relative frequencies of the two chemostats vary. We concentrate on certain strong resonance phenomena between the two frequencies as well as their mutual extinction and provide a representative sampling of possible phase portraits for our model system. Our observations corroborate recent mathematical results and case studies of coupled nonlinear chemical oscillators in which regions of mutual extinction as well as the Arnol'd structure for two-parameter families of maps of the plane have been observed. We highlight certain unexpected features of the operating diagram discovered through our computational study and discuss their implication for the dynamic response of the chemostat system.

It is well known that when two microbial populations competing for a single rate-limiting nutrient are grown in a chemostat with time-invariant inputs, with competition being the only interaction between them, they cannot coexist, but eventually one of the two populations prevails and the other becomes extinct. It has been suggested that periodic variation of one of the chemostat's operating parameters can stabilize the coexistence state of the two microbial populations. A systematic numerical study of the model equations describing microbial competition in a chemostat with periodically varying dilution rate is performed, and it is shown that coexistence of the competing microbial populations is obtained for a wide range of operating conditions. The coexistence state is usually in the form of limit cycle oscillations. However, cases of chaotic behavior resulting from successive period doublings and quasi-periodicity are also observed.

Predator-prey systems in continuously operated chemostats exhibit sustained oscillations over a wide range of operating conditions. When the chemostat is operated periodically, the interaction of the natural oscillation frequency with the external forcing gives rise to a wealth of dynamic behavior patterns. Using numerical bifurcation techniques, we perform a detailed computational study of these patterns and the transitions (local and especially global) between them as the amplitude and frequency of the forcing vary. The transition from low-forcing-amplitude quasiperiodicity to entrainment of the chemostat behavior by strong forcing (involving the concerted closing of resonance horns) is analyzed. We concentrate on certain strong resonance phenomena between the two frequencies and provide an extensive atlas of computed phase portraits for our model system. Our observations corroborate recent mathematical results and case studies of periodically forced chemical oscillators. In particular, the existence and relative succession of several distinct types of global bifurcations resulting in chaotic transients and multistability are studied in detail. The location in the operating diagram of several key codimension 2 local bifurcations of periodic solutions is computed, and their interaction with an interesting feature we name "real-eigenvalues horns" is examined.