A class of piecewise linear differential equations arising in biological models

101

Cite this article as: Vlastimil Křivan, On the gause predator-prey model with a refuge: a fresh look at the history, Journal of Theoretical Biology, doi:10.1016Biology, doi:10. /j.jtbi.2011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.On the Gause predator-prey model with a refuge: A fresh look at the history AbstractThis article re-analyses a prey-predator model with a refuge introduced by one of the founders of population ecology G. F. Gause and his co-workers to explain discrepancies between their observations and predictions of the Lotka-Volterra prey-predator model. They replaced the linear functional response used by Lotka and Volterra by a saturating functional response with a discontinuity at a critical prey density. At concentrations below this critical density prey were effectively in a refuge while at a (1) trajectories converge to a limit cycle, as predicted by F. G. Gause, (2) trajectories converge to an equilibrium, or (3) the prey population escapes predator control and grows to infinity.

Section: Discussionmentioning

confidence: 99%

On the Gause predator–prey model with a refuge: A fresh look at the history

Křivan

2011

101

“…The first three cases, shown in Figure 14 stability of equilibria in switching domains, was presented in [5], where the work of Gouze et al [24] and de Jong et al [10] was extended using the framework of differential inclusions and Filippov solutions. Similarly to the SNM, the SPWLM does not predict oscillations for the activation-inhibition network.…”

mentioning

confidence: 99%

Comparing different ODE modelling approaches for gene regulatory networks

Polynikis

Hogan

Bernardo

2009

156

103

A fundamental step in synthetic biology and systems biology is to derive appropriate mathematical models for the purposes of analysis and design. For example, to synthesize a gene regulatory network, the derivation of a mathematical model is important in order to carry out in silico investigations of the network dynamics and to investigate parameter variations and robustness issues. Different mathematical frameworks have been proposed to derive such models. In particular, the use of sets of nonlinear ordinary differential equations (ODEs) has been proposed to model the dynamics of the concentrations of mRNAs and proteins. These models are usually characterized by the presence of highly nonlinear Hill function terms. A typical simplification is to reduce the number of equations by means of a quasi-steady-state assumption on the mRNA concentrations. This yields a class of simplified ODE models. A radically different approach is to replace the Hill functions by piecewise-linear approximations [5]. A further modelling approach is the use of discretetime maps [7] where the evolution of the system is modelled in discrete, rather than continuous, time. The aim of this paper is to discuss and compare these different modelling approaches, using a representative gene regulatory network. We will show that different models often lead to conflicting conclusions concerning the existence and stability of equilibria and stable oscillatory behaviours. Moreover, we shall discuss, where possible, the viability of making certain modelling approximations (e.g. quasi-steady-state mRNA dynamics or piecewise-linear approximations of Hill functions) and their effects on the overall system dynamics.

“…The proof follows by checking that the vector fields for x ∈ L s are compatible with Filippov's construction, see [14]. If there is sliding motion in L s , then there exists ∆ > 0 such thaṫ…”

Section: Part (B) Analogous To Part (A)mentioning

confidence: 99%

“…The use of PA models for biochemical systems was pioneered by Glass and Kaufmman [12] and has lead to a number of extensions [13,14,15,16] and the development of dedicated simulation tools [17]. They provide a convenient way of encoding switch-like regulation with a small parameter set (i.e.…”

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confidence: 99%

See 1 more Smart Citation

Multistability and oscillations in genetic control of metabolism

Oyarzún

Chaves²,

Hoff-Hoffmeyer-Zlotnik³

2012

Genetic control of enzyme activity drives metabolic adaptations to environmental changes, and therefore the feedback interaction between gene expression and metabolism is essential to cell fitness. In this paper we develop a new formalism to detect the equilibrium regimes of an unbranched metabolic network under transcriptional feedback from one metabolite. Our results indicate that one-to-all transcriptional feedback can induce a wide range of metabolic phenotypes, including mono-, multistability and oscillatory behavior. The analysis is based on the use of switch-like models for transcriptional control and the exploitation of the time scale separation between metabolic and genetic dynamics. For any combination of activation and repression feedback loops, we derive conditions for the emergence of a specific phenotype in terms of genetic parameters such as enzyme expression rates and regulatory thresholds. We find that metabolic oscillations can emerge under uniform thresholds and, in the case of operoncontrolled networks, the analysis reveals how nutrient-induced bistability and oscillations can emerge as a consequence of the transcriptional feedback.