American Institute of Mathematical Sciences

Advanced
March  2008, 9(2): 321-351. doi: 10.3934/dcdsb.2008.9.321

Feedback-mediated coexistence and oscillations in the chemostat

Willard S. Keeran 1, , Patrick D. Leenheer 1, and  Sergei S. Pilyugin 1,

1. 

Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, United States, United States, United States

Received  July 2007 Revised  August 2007 Published  December 2007

We consider a mathematical model that describes the competition of three species for a single nutrient in a chemostat in which the dilution rate is assumed to be controllable by means of state dependent feedback. We consider feedback schedules that are affine functions of the species concentrations. In case of two species, we show that the system may undergo a Hopf bifurcation and oscillatory behavior may be induced by appropriately choosing the coefficients of the feedback function. When the growth of the species obeys Michaelis-Menten kinetics, we show that the Hopf bifurcation is supercritical in the relevant parameter region, and the bifurcating periodic solutions for two species are always stable. Finally, we show that by adding a third species to the system, the two-species stable periodic solutions may bifurcate into the coexistence region via a transcritical bifurcation. We give conditions under which the bifurcating orbit is locally asymptotically stable.
Keywords: state-dependent feedback, oscillatory coexistence, supercritical Hopf bifurcation, Chemostat, transcritical bifurcation of limit cycles..
Mathematics Subject Classification: 34C23, 34H05, 37G10, 92D25, 93C1.
Citation: Willard S. Keeran, Patrick D. Leenheer, Sergei S. Pilyugin. Feedback-mediated coexistence and oscillations in the chemostat. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 321-351. doi: 10.3934/dcdsb.2008.9.321
[1]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[2]

Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197
[3]

Maoan Han, Tonghua Zhang. Some bifurcation methods of finding limit cycles. Mathematical Biosciences & Engineering, 2006, 3 (1) : 67-77. doi: 10.3934/mbe.2006.3.67
[4]

Qingwen Hu, Huan Zhang. Stabilization of turning processes using spindle feedback with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4329-4360. doi: 10.3934/dcdsb.2018167
[5]

Salomón Rebollo-Perdomo, Claudio Vidal. Bifurcation of limit cycles for a family of perturbed Kukles differential systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4189-4202. doi: 10.3934/dcds.2018182
[6]

Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489
[7]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[8]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[9]

Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038
[10]

Kie Van Ivanky Saputra, Lennaert van Veen, Gilles Reinout Willem Quispel. The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 233-250. doi: 10.3934/dcdsb.2010.14.233
[11]

Russell Johnson, Francesca Mantellini. A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 209-224. doi: 10.3934/dcds.2003.9.209
[12]

Benjamin B. Kennedy. A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1633-1650. doi: 10.3934/dcdsb.2013.18.1633
[13]

Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 47-66. doi: 10.3934/dcdss.2020003
[14]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045
[15]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152
[16]

Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065
[17]

Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71
[18]

R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147
[19]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247
[20]

Hans-Otto Walther. On Poisson's state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences