Bifurcations of a periodically forced microbial continuous culture model with restrained growth rate

Abstract

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.

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“…Recently, the differential equations theory (stability, bifurcation, chaos, etc.) has been used in several fields such as medicine, economics, life science, engineering, technology and sociology [13][14][15][16][17].…”

“…In order to support our theoretical analysis, we shall carry out the numerical simulations. Model (14) involves six parameters, including the delay τ which can be chosen γ = 0.9, a = 5, b = 5, d = 1, k = 0.1 and vary β ∈ [0, 1). Regarding to the semi-trivial equilibrium point E 1 (1, 1, 0), the characteristic Equation (18) can be written as…”

“…so according to Lemma 3, Equation (26) has two positive roots for β ∈ [0, 1). We simulate the critical time delays τ 1,j (blue curves) and τ 2,j (red dash curves) for system (14) with β by using Equations (25) and (27) in Figure 9, j = 0, 1, ..., 10. As seen in this figure, there are finite stability domains for E 2 .…”

“…As seen in Figure 10, delayed model (14) approaches stable fixed points for τ ∈ [0, τ 1,0 ) ∪ τ 2,j , τ 1,j+1 for j = 0, 1, ..., 5, for example τ = 0.9, 5, 13, 20, 27, 34.2 and 40.9. Moreover, the delayed model (14) has periodic solution when τ ∈ τ 1,j , τ 2,j ∪ (τ 1,6 , ∞) for j = 0, 1, ..., 5 as shown in Figure 11. Figure 11.…”

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“…As we know, the stable (unstable) fixed points of the kth iterate of the map correspond to the stable (unstable ) periodic solutions with period k of system (21). More details about the Poincaré map and notations in obtained bifurcation diagrams can be found in [18,22,23].…”