Abstract

Polynomial dynamical systems (i.e. dynamical systems with polynomial right hand side) are ubiquitous in applications, especially as models of reaction networks and interaction networks. The properties of general polynomial dynamical systems can be very difficult to analyze, due to nonlinearity, bifurcations, and the possibility for chaotic dynamics. On the other hand, toric dynamical systems are polynomial dynamical systems that appear naturally as models of reaction networks, and have very robust and stable properties. A disguised toric dynamical system is a polynomial dynamical system generated by a reaction network N and some choice of positive parameters, such that (even though it may not be toric with respect to N) it has a toric realization with respect to some network N′. Disguised toric dynamical systems enjoy all the robust stability properties of toric dynamical systems. In this paper, we study a larger set of dynamical systems where the rate constants are allowed to take both positive and negative values. More precisely, we analyze the R-disguised toric locus of a reaction network N, i.e., the subset in the space rate constants (positive or negative) of N for which the corresponding polynomial dynamical system is disguised toric. We focus especially on finding a lower bound on the dimension of the R-disguised toric locus.

Abstract

We show that there exists endotactic and strongly endotactic dynamical systems that are not weakly reversible and possess infinitely many steady states. We provide a few examples in two dimensions and an example in three dimensions that satisfy this property. In addition, we prove for some of these systems that there exist no weakly reversible mass-action systems that are dynamically equivalent to mass-action systems generated by these networks.

Abstract

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Abstract

Reaction networks can display a wide array of dynamics. However, it is possible for different reaction networks to display the same dynamics. This phenomenon is called dynamical equivalence and makes network identification a hard problem. We show that to find a strongly endotactic/endotactic/consistent/conservative realization (when it exists) that is dynamically equivalent to the mass-action system generated by a given network it suffices to consider only the source vertices of the given network. In addition, we show that weakly reversible deficiency one realizations are not unique. We also present a characterization of the dynamical relationships that exist between several types of weakly reversible deficiency one networks.

Abstract

Systems of differential equations with polynomial right-hand sides are very common in applications. In particular, when restricted to the positive orthant, they appear naturally (according to the law of mass-action kinetics) in ecology, population dynamics, as models of biochemical interaction networks, and models of the spread of infectious diseases. On the other hand, their mathematical analysis is very challenging in general; in particular, it is very difficult to answer questions about the long-term dynamics of the variables (species) in the model, such as questions about persistence and extinction. Even if we restrict our attention to mass-action systems, these questions still remain challenging. On the other hand, if a polynomial dynamical system has a weakly reversible single linkage class (WR1 ) realization, then its long-term dynamics is known to be remarkably robust: all the variables are persistent (i.e., no species goes extinct), irrespective of the values of the parameters in the model. Here we describe an algorithm for finding WR1 realizations of polynomial dynamical systems, whenever such realizations exist.

Abstract

The structure of invariant regions and globally attracting regions is fundamental to understanding the dynamical properties of reaction network models. We describe an explicit construction of the minimal invariant regions and minimal globally attracting regions for dynamical systems consisting of two reversible reactions, where the rate constants are allowed to vary in time within a bounded interval.

Abstract

Homeostasis represents the idea that a feature may remain invariant despite changes in some external parameters. We establish a connection between homeostasis and injectivity for reaction network models. In particular, we show that a reaction network cannot exhibit homeostasis if a modified version of the network (which we call homeostasis-associated network) is injective. We provide examples of reaction networks which can or cannot exhibit homeostasis by analyzing the injectivity of their homeostasis-associated networks.

Abstract

Autocatalytic systems called hypercycles are very often incorporated in "origin of life" models. We investigate the dynamics of certain related models called bimolecular autocatalytic systems. In particular, we consider the dynamics corresponding to the relative populations in these networks, and show that it can be analyzed using well-chosen autonomous polynomial dynamical systems. Moreover, we use results from reaction network theory to prove persistence and permanence of several families of bimolecular autocatalytic systems called autocatalytic recombination systems. Keywords: Autocatalytic recombination networks; Origin of life; Permanence; Persistence. Copyright © 2022 Elsevier Inc. All rights reserved.

Abstract

Toric differential inclusions occur as key dynamical systems in the context of the Global Attractor Conjecture. We introduce the notions of minimal invariant regions and minimal globally attracting regions for toric differential inclusions. We describe a procedure for explicitly constructing the minimal invariant and minimal globally attracting regions for two-dimensional toric differential inclusions. In particular, we obtain invariant regions and globally attracting regions for two-dimensional weakly reversible or endotactic dynamical systems (even if they have time-dependent parameters).