Study of Lotka-volterra food chain chemostat with periodically varying dilution rate

In this paper, we introduce and study a model of Lotka-volterra chemostat food chain chemostat with periodically varying dilution rate, which contains with predator, prey, and substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey, and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, we numerically simulate a model with sinusoidal dilution rate, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the system experiences following process: periodic solution → periodic doubling cascade → chaos.     

Analysis of a Monod–Haldene type food chain chemostat with seasonally variably pulsed input and washout

In this paper, we introduce and study a model of a Monod–Haldene type food chain chemostat with seasonally variably pulsed input and washout. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, bifurcation diagrams have shown that there exists complexity for the pulsed system including periodic doubling cascade, periodic halving cascade and Pitchfork bifurcations and tangent bifurcations.      

Comparative energetics of glucose and xylose metabolism in recombinant Zymomonas mobilis

Recombinant Zymomonas mobilis CP4:pZB5 was grown with pH control in batch and continuous modes with either glucose or xylose as the sole carbon and energy source. In batch cultures in which the ratio of the final cell mass concentration to the amount of sugar in the medium was constant (i.e., under conditions that promote “coupled growth”), maximum specific rates of glucose and xylose consumption were 8.5 and 2.1 g/(g of cell…h), respectively; maximum specific rates of ethanol production for glucose and xylose were 4.1 and 1.0 g/(g of cell…h), respectively; and average growth yields from glucose and xylose were 0.055 and 0.034 g of dry cell mass (DCM)/g of sugar respectively. The corresponding value of Y_ATP for glucose and xylose was 9.9 and 5.1 g of DCM/mol of ATP, respectively. Y_ATP for the wild-type culture CP4 with glucose was 10.4g of DCM/mol of ATP. For single substratechem ostat cultures in which the growth rate was varied as the dilution rate (D), the maximum or “true” growth yield (max Y_a/s) was calculated from Pirt plots as the inverse of the slope of the best-fit linear regression for the specific sugar utilization rate as a function of D, and the “maintenance coefficient” (m) was determined as the y-axis intercept. For xylose, values of max Y _s/s and m were 0.0417g of DCM/g of xylose (Y_ATP=6.25) and 0.04g of, xylose/(g of cell…h), respectively. However, with glucose there was an observed deviation from linearity, and the data in the Pirt plot was best fit with a second-order polynomial in D. At D>0.1/h, Y_ATP=8.71 and m=2.05g of glu/(g of cell…h) whereas at D<0.1/h, Y_ATP=4.9g of DCM/mol of ATP and m=0.04g of glu/(g of cell…h). This observation provides evidence to question the validity of the unstructured growth model and the assumption that Pirt's maintenance coefficient is a constant that is in dependent of the growth rate. Collectively, these observations with individual sugars and the values assign ed to various growth and fermentation parameters will be useful in the development of models to predict the behavior of rec Zm in mixed substrate fermentations of the type associated with biomass-to-ethanol processes.     

Dynamics of a stochastic chemostat competition model with plasmid-bearing and plasmid-free organisms

In this paper, we consider a chemostat model of competition between plasmid-bearing and plasmid-free organisms, perturbed by white noise. Firstly, we prove the existence and uniqueness of the global positive solution. Then by constructing suitable Lyapunov functions, we establish sufficient conditions for the existence of a unique ergodic stationary distribution. Furthermore, conditions for extinction of plasmid-bearing organisms are obtained. Theoretical analysis indicates that large noise intensity $\sigma_{2}^{2}$ is detrimental to the survival of plasmid-bearing organisms and is not conducive to the commercial production of genetically altered organisms. Finally, numerical simulations are presented to illustrate the results.     

Bifurcation from a double eigenvalue in the unstirred chemostat

This article concerns a competition model in the unstirred chemostat. The bifurcation solution from a double eigenvalue is obtained. We see that this bifurcation solution connects the positive solution from the semitrivial solution (θ _a ,?0) with that from the other semitrivial solution (0,?θ _b ). Moreover, the asymptotic stability of the positive solution corresponding to this bifurcation is derived under certain conditions. The method we used here is based on spectral analysis, comparison principle, bifurcation theory and Lyapunov–Schmidt procedure.     

Dynamic Analysis of an Impulsive Chemostat Model with Microbial Competition and Nonlinear Perturbation

In this paper, we propose an impulsive chemostat model with microbial competition and nonlinear perturbation. First, thresholds for the extinction of both microoganisms are given. Second, we investigate the persistence in mean and boundedness of the chemostat system by constructing Lyapunov function. Moreover, we obtain the sufficient condition for the existence of an ergodic stationary distribution of the system. At last, numerical simulations are presented, and the results show that the competition between two species tends to make one species disappear from their common habitat, especially when the competition is concentrated in a single resource.