A three dimensional microbial continuous culture model with a restrained microbial growth rate is studied in this paper. Two types of dilution rates are considered to investigate the dynamic behaviors of the model. For the unforced system, fold bifurcation and Hopf bifurcation are detected, and numerical simulations reveal that the system undergoes degenerate Hopf bifurcation. When the system is periodically forced, bifurcation diagrams for periodic solutions of period-one and period-two are given by researching the Poincar e map, corresponding to different bifurcation cases in the unforced system. Stable and unstable quasiperiodic solutions are obtained by NeimarkSacker bifurcation with different parameter values. Periodic solutions of various periods can occur or disappear and even change their stability, when the Poincar e map of the forced system undergoes Neimark-Sacker bifurcation, flip bifurcation, and fold bifurcation. Chaotic attractors generated by a cascade of period doublings and some phase portraits are given at last. Published by AIP Publishing.[http://dx.doi.org/10.1063/1.5000152]The microbiological fermentation technique is widely applied in many fields for its economic importance. It is also investigated due to the complex behaviours observed during the process. We study bifurcations of the microbial continuous culture model with two types of dilution rates: the steady dilution rate and periodically forced dilution rate. For the steady dilution rate, we prove that the model undergoes fold bifurcation and Hopf bifurcation. When the dilution rate is periodically forced, we find that the bifurcations of the equilibria of the unforced system can be extended to the forced system as bifurcations of periodic solutions. Furthermore, periodic perturbation can give rise to complex dynamics, such as quasiperiodic solutions, periodic solutions of various periods, and chaos. In addition, the various periodic solutions can well explain the oscillation phenomena observed in laboratory experiments.
We study the effect of interest rate on phenomenon of business cycle in a Kaldor-Kalecki model. From the information of the People's Bank of China and the Federal Reserve System, we know the interest rate is not a constant but with remarkable periodic volatility. Therefore, we consider periodically forced interest rate in the model and study its dynamics. It is found that, both limit cycle through Hopf bifurcation in unforced system and periodic solutions generated by period doubling bifurcation or resonance in periodically forced system, can lead to cyclical economic fluctuations. Our analysis reveals that the cyclical fluctuation of interest rate is one of a key formation mechanism of business cycle, which agrees well with the pure monetary theory on business cycle. Moreover, this fluctuation can cause chaos in a business cycle system.
<p style='text-indent:20px;'>This manuscript examines the dynamics of a predator-prey model of the Beddington-DeAngelis type with strong Allee effect on prey growth function. Conditions for the existence and equilibria types are established. By taking Allee effect, predation rate of the prey and growth rate of the predator as bifurcation parameters, different potential bifurcations are explored, including codimension one bifurcations: fold bifurcation, transcritical bifurcation, Hopf bifurcation, and codimension two bifurcations: cusp bifurcation, Bogdanov-Takens bifurcation, and Bautin bifurcation. In addition, to confirm the dynamic behavior of the system, bifurcation diagrams are given in different parameter spaces and phase portraits are also presented to provide corresponding interpretation. The findings indicate that the dynamics of our system is much richer than the system with no strong Allee effect.</p>
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