Multistability, oscillations and bifurcations in feedback loops

Previous Article
Discrete host-parasitoid models with Allee effects and age structure in the host
MBE Home
This Issue
Next Article
Modeling of mosquitoes with dominant or recessive Transgenes and Allee effects

2010, 7(1): 83-97. doi: 10.3934/mbe.2010.7.83

Multistability, oscillations and bifurcations in feedback loops

Maria Conceição A. Leite ^1, and Yunjiao Wang ^2,

1.	Department of Mathematics, University of Oklahoma, Norman, OK 73019-0315, United States
2.	School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom

Received May 2009 Revised October 2009 Published January 2010

Abstract
Related Papers

Feedback loops are found to be important network structures in regulatory networks of biological signaling systems because they are responsible for maintaining normal cellular activity. Recently, the functions of feedback loops have received extensive attention. The existing results in the literature mainly focus on verifying that negative feedback loops are responsible for oscillations, positive feedback loops for multistability, and coupled feedback loops for the combined dynamics observed in their individual loops. In this work, we develop a general framework for studying systematically functions of feedback loops networks. We investigate the general dynamics of all networks with one to three nodes and one to two feedback loops. Interestingly, our results are consistent with Thomas' conjectures although we assume each node in the network undergoes a decay, which corresponds to a negative loop in Thomas' setting. Besides studying how network structures influence dynamics at the linear level, we explore the possibility of network structures having impact on the nonlinear dynamical behavior by using Lyapunov-Schmidt reduction and singularity theory.

Keywords: oscillations, coupled feedback loops, transcriptional regulatory networks, bifurcation., multistability.

Mathematics Subject Classification: Primary: 37N25, 37G10; Secondary: 34C23, 34A3.

Citation: Maria Conceição A. Leite, Yunjiao Wang. Multistability, oscillations and bifurcations in feedback loops. Mathematical Biosciences & Engineering, 2010, 7 (1) : 83-97. doi: 10.3934/mbe.2010.7.83

[1]	Qingqing Li, Tianshou Zhou. Interlocked multi-node positive and negative feedback loops facilitate oscillations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3139-3155. doi: 10.3934/dcdsb.2018304
[2]	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
[3]	Freddy Dumortier, Robert Roussarie. Bifurcation of relaxation oscillations in dimension two. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 631-674. doi: 10.3934/dcds.2007.19.631
[4]	Sanling Yuan, Yongli Song, Junhui Li. Oscillations in a plasmid turbidostat model with delayed feedback control. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 893-914. doi: 10.3934/dcdsb.2011.15.893
[5]	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
[6]	Guy Katriel. Stability of synchronized oscillations in networks of phase-oscillators. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 353-364. doi: 10.3934/dcdsb.2005.5.353
[7]	Orit Lavi, Doron Ginsberg, Yoram Louzoun. Regulation of modular Cyclin and CDK feedback loops by an E2F transcription oscillator in the mammalian cell cycle. Mathematical Biosciences & Engineering, 2011, 8 (2) : 445-461. doi: 10.3934/mbe.2011.8.445
[8]	Bernold Fiedler, Isabelle Schneider. Stabilized rapid oscillations in a delay equation: Feedback control by a small resonant delay. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-41. doi: 10.3934/dcdss.2020068
[9]	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
[10]	Rachel Kuske, Peter Borowski. Survival of subthreshold oscillations: The interplay of noise, bifurcation structure, and return mechanism. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 873-895. doi: 10.3934/dcdss.2009.2.873
[11]	Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523
[12]	Kaïs Ammari, Mohamed Jellouli, Michel Mehrenberger. Feedback stabilization of a coupled string-beam system. Networks & Heterogeneous Media, 2009, 4 (1) : 19-34. doi: 10.3934/nhm.2009.4.19
[13]	Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933
[14]	Yong Zhao, Qishao Lu. Periodic oscillations in a class of fuzzy neural networks under impulsive control. Conference Publications, 2011, 2011 (Special) : 1457-1466. doi: 10.3934/proc.2011.2011.1457
[15]	Julien Coatléven, Claudio Altafini. A kinetic mechanism inducing oscillations in simple chemical reactions networks. Mathematical Biosciences & Engineering, 2010, 7 (2) : 301-312. doi: 10.3934/mbe.2010.7.301
[16]	Oskar Weinberger, Peter Ashwin. From coupled networks of systems to networks of states in phase space. Discrete & Continuous Dynamical Systems - B, 2018, 23 (5) : 2021-2041. doi: 10.3934/dcdsb.2018193
[17]	Eleonora Catsigeras. Dynamics of large cooperative pulsed-coupled networks. Journal of Dynamics & Games, 2014, 1 (2) : 255-281. doi: 10.3934/jdg.2014.1.255
[18]	Xiwei Liu, Tianping Chen, Wenlian Lu. Cluster synchronization for linearly coupled complex networks. Journal of Industrial & Management Optimization, 2011, 7 (1) : 87-101. doi: 10.3934/jimo.2011.7.87
[19]	Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
[20]	Jui-Pin Tseng. Global asymptotic dynamics of a class of nonlinearly coupled neural networks with delays. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4693-4729. doi: 10.3934/dcds.2013.33.4693

2018 Impact Factor: 1.313

American Institute of Mathematical Sciences