scholarly journals Equilibrium switching in nonlinear biological interaction networks with concurrent antagonism

2014 ◽  
Author(s):  
Jomar Rabajante ◽  
Cherryl O. Talaue
Keyword(s):  
Steady State ◽  
Basin Of Attraction ◽  
Gene Interaction ◽  
Oscillatory Behavior ◽  
Interaction Networks ◽  
Dominant Component ◽  
Maximal Growth Rate ◽  
Decision Making Model ◽  
Parameter Values ◽  
The Basin Of Attraction

Concurrent decision-making model (CDM) of interaction networks involving more than two antagonistic components can represent various biological systems, such as gene interaction, species competition and mental perception. The model assumes sigmoid kinetics where every component stimulates itself but concurrently represses the others. Here we prove general dynamical properties of the CDM (e.g., location and stability of steady states) for any dimension of the state space even if the reciprocal antagonism between two components is asymmetric. Significant modifications in parameter values serve as biological regulators for inducing steady state switching by leading a temporal state to escape an undesired equilibrium. Increasing the maximal growth rate and decreasing the decay rate expand the basin of attraction of a steady state with the desired dominant component. Perpetually adding an external stimulus can shut down multi-stability of the system that increases the robustness of the system against stochastic noise. We further show that asymmetric interaction that forms a repressilator-type network generates oscillatory behavior.

scholarly journals Equilibrium switching and mathematical properties of nonlinear interaction networks with concurrent antagonism and self-stimulation

2015 ◽  
Author(s):  
Jomar Rabajante ◽  
Cherryl O. Talaue
Keyword(s):  
Steady State ◽  
Basin Of Attraction ◽  
Gene Interaction ◽  
Oscillatory Behavior ◽  
Interaction Networks ◽  
Dominant Component ◽  
Mathematical Properties ◽  
Maximal Growth Rate ◽  
Decision Making Model ◽  
Parameter Values

Concurrent decision-making model (CDM) of interaction networks with more than two antagonistic components represents various biological systems, such as gene interaction, species competition and mental cognition. The CDM model assumes sigmoid kinetics where every component stimulates itself but concurrently represses the others. Here we prove generic mathematical properties (e.g., location and stability of steady states) of n-dimensional CDM with either symmetric or asymmetric reciprocal antagonism between components. Significant modifications in parameter values serve as biological regulators for inducing steady state switching by driving a temporal state to escape an undesirable equilibrium. Increasing the maximal growth rate and decreasing the decay rate can expand the basin of attraction of a steady state that contains the desired dominant component. Perpetually adding an external stimulus could shut down multi-stability of the system which increases the robustness of the system against stochastic noise. We further show that asymmetric interaction forming a repressilator-type network generates oscillatory behavior.

scholarly journals Equilibrium switching and mathematical properties of nonlinear interaction networks with concurrent antagonism and self-stimulation

2015 ◽  
Author(s):  
Jomar Rabajante ◽  
Cherryl O. Talaue
Keyword(s):  
Steady State ◽  
Basin Of Attraction ◽  
Gene Interaction ◽  
Oscillatory Behavior ◽  
Interaction Networks ◽  
Dominant Component ◽  
Mathematical Properties ◽  
Maximal Growth Rate ◽  
Decision Making Model ◽  
Parameter Values

Concurrent decision-making model (CDM) of interaction networks with more than two antagonistic components represents various biological systems, such as gene interaction, species competition and mental cognition. The CDM model assumes sigmoid kinetics where every component stimulates itself but concurrently represses the others. Here we prove generic mathematical properties (e.g., location and stability of steady states) of n-dimensional CDM with either symmetric or asymmetric reciprocal antagonism between components. Significant modifications in parameter values serve as biological regulators for inducing steady state switching by driving a temporal state to escape an undesirable equilibrium. Increasing the maximal growth rate and decreasing the decay rate can expand the basin of attraction of a steady state that contains the desired dominant component. Perpetually adding an external stimulus could shut down multi-stability of the system which increases the robustness of the system against stochastic noise. We further show that asymmetric interaction forming a repressilator-type network generates oscillatory behavior.

scholarly journals Equilibrium switching in nonlinear interaction networks with concurrent antagonism

2014 ◽  
Author(s):  
Jomar Rabajante
Keyword(s):  
Steady State ◽  
Basin Of Attraction ◽  
Oscillatory Behavior ◽  
Interaction Networks ◽  
Stochastic Noise ◽  
Maximal Growth ◽  
State Switching ◽  
Maximal Growth Rate ◽  
Decision Making Model ◽  
The Basin Of Attraction

In this paper, we examine a nonlinear concurrent decision-making model (CDM) of interaction networks that involve more than two antagonistic components (e.g., proteins, species, communities, mental choices). The model assumes sigmoid kinetics where every component stimulates itself but represses all others. We are able to prove general dynamical properties of the CDM (e.g., location and stability of steady states) for any dimension of the state space even if the reciprocal antagonism between two components is asymmetric. There are cases where asymmetric interaction generates oscillatory behavior. Some parameters can serve as biological regulators for inducing steady state switching by leading a temporal state to escape an undesired equilibrium. Increasing the maximal growth rate and decreasing the decay rate can expand the basin of attraction of a steady state with the desired component having the dominant value. We further show that perpetually adding an external stimulus can shutdown multi-stability of the system that increases the robustness of the system against stochastic noise.

THE BASIN OF ATTRACTION OF LOZI MAPPINGS

2009 ◽  
Vol 19 (03) ◽  
pp. 1043-1049 ◽  
Author(s):  
DIOGO BAPTISTA ◽  
RICARDO SEVERINO ◽  
SANDRA VINAGRE
Keyword(s):  
Strange Attractor ◽  
Basin Of Attraction ◽  
Parameter Values ◽  
The Basin Of Attraction

For parameter values which assure its existence, we characterize the basin of attraction for the Lozi map strange attractor.

THE EFFECT OF DAMPING ON THE STEADY STATE AND BASIN BIFURCATION PATTERNS OF A NONLINEAR MECHANICAL OSCILLATOR

1992 ◽  
Vol 02 (01) ◽  
pp. 81-91 ◽  
Author(s):  
MOHAMED S. SOLIMAN ◽  
J.M.T. THOMPSON
Keyword(s):  
Steady State ◽  
Basin Of Attraction ◽  
Period Doubling ◽  
Qualitative And Quantitative ◽  
Periodically Driven ◽  
Resonance Response ◽  
Nonlinear Mechanical ◽  
Damped Oscillator ◽  
The Basin Of Attraction

This paper examines the role of damping on both the steady state and basin behavior of a periodically driven damped oscillator with the ability to escape from a potential well. We examine the effect of damping on both the qualitative and quantitative resonance response of the system. Particular attention is paid to how the damping scales the main steady state bifurcations; saddle-nodes, period-doubling flips, cascades to chaos, boundary crises, etc. We also investigate how the damping level effects the main homoclinic and heteroclinic basin bifurcations that may result in a rapid erosion and stratification of the basin of attraction and hence a loss of engineering integrity of the system.

scholarly journals Capture Zones of the Family of Functions λzmexp(z)

2003 ◽  
Vol 13 (09) ◽  
pp. 2623-2640 ◽  
Author(s):  
Núria Fagella ◽  
Antonio Garijo
Keyword(s):  
Fixed Point ◽  
Critical Point ◽  
Basin Of Attraction ◽  
Capture Zone ◽  
Connected Component ◽  
Dynamical Plane ◽  
Capture Zones ◽  
The Family ◽  
Parameter Values ◽  
The Basin Of Attraction

We consider the family of entire transcendental maps given by Fλ,m(z)=λzm exp (z) where m≥2. All functions Fλ,m have a superattracting fixed point at z=0, and a critical point at z = -m. In the dynamical plane we study the topology of the basin of attraction of z=0. In the parameter plane we focus on the capture behavior, i.e. λ values such that the critical point belongs to the basin of attraction of z=0. In particular, we find a capture zone for which this basin has a unique connected component, whose boundary is then nonlocally connected. However, there are parameter values for which the boundary of the immediate basin of z=0 is a quasicircle.

Intermingled basins for the triangle map

1996 ◽  
Vol 16 (4) ◽  
pp. 651-662 ◽  
Author(s):  
James C. Alexander ◽  
Brian R. Hunt ◽  
Ittai Kan ◽  
James A. Yorke
Keyword(s):  
Lebesgue Measure ◽  
Basin Of Attraction ◽  
The Other ◽  
Positive Lebesgue Measure ◽  
Quadratic Maps ◽  
Open Set ◽  
Parameter Values ◽  
The Basin Of Attraction

AbstractA family of quadratic maps of the plane has been found numerically for certain parameter values to have three attractors, in a triangular pattern, with ‘intermingled’ basins. This means that for every open set S, if the basin of attraction of one of the attractors intersects S in a set of positive Lebesgue measure, then so do the other two basins. In this paper we mathematically verify this observation for a particular parameter, and prove that our results hold for a set of parameters with positive Lebesgue measure.

scholarly journals Reversible Environmental Catastrophes with Disconnected Generations

De Economist ◽  
2020 ◽  
Author(s):  
Ben J. Heijdra ◽  
Pim Heijnen
Keyword(s):  
Steady State ◽  
Environmental Quality ◽  
Basin Of Attraction ◽  
Environmental Taxes ◽  
Social Optimum ◽  
Stable Steady State ◽  
The Government ◽  
Random Shocks ◽  
The Basin Of Attraction

Abstract We study environmental policy in a stylized economy–ecology model featuring multiple deterministic stable steady-state ecological equilibria. The economy–ecology does not settle in either of the deterministic steady states as the environmental system is hit by random shocks. Individuals live for two periods and derive utility from the (stochastic) quality of the environment. They feature warm-glow preferences and engage in private abatement in order to weakly influence the stochastic process governing environmental quality. The government may also conduct abatement activities or introduce environmental taxes. We solve for the market equilibrium abstracting from public abatement and taxes and show that the ecological process may get stuck for extended periods of time fluctuating around the heavily polluted (low quality) deterministic steady state. These epochs are called environmental catastrophes. They are not irreversible, however, as the system typically switches back to the basin of attraction associated with the good (high quality) deterministic steady state. The paper also compares the stationary distributions for environmental quality and individuals’ welfare arising under the unmanaged economy and in the first-best social optimum.

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