Equilibria Exist in Compact Convex Forward-Invariant Sets

18 June 2011

Theorem. Consider a continuous map f : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} and suppose that the autonomous dynamical system \dot{x} = f(x) has a semiflow \varphi : {\mathbb{R}}_{\geq{0}} \times {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}. Let K \subseteq {\mathbb{R}}^{n}. If K is nonempty, compact, convex and forward-invariant, then K contains an equilibrium of the dynamical system, i.e. a zero of the map f.

According to a reliable source, the above theorem is a standard result everyone uses in dynamical systems without proof. I propose a proof in Equilibria Exist in Compact Convex Forward-Invariant Sets. I am interested in comments on this proof, in references to this or other proofs in the literature, and in new/better proofs. Please contribute here or on MathOverflow at Equilibria Exist in Compact Convex Forward-Invariant Sets.

Map Transformation to Force Convergence to Unique Fixed Point

19 December 2010

I posted a question on MathOverflow at http://mathoverflow.net/questions/49202/map-transformation-to-force-convergence-to-unique-fixed-point. The question has to do with forcing convergence to a fixed point which is known to exist and be unique. The posted question refers to this document which contains and elaborates on the question: http://math.gillesgnacadja.info/files/FixedPointAlgo_OPEN.html.

Introducing Vacuous Persistence

23 January 2010

The previous blog post “Persistence via reachability vs. via Petri nets” mentioned vacuous persistence. This post is devoted to this concept. 

\omega-limit point (Definition)
Consider a trajectory c in a closed subset P \subseteq {\mathbb{R}}^{n}, i.e. a map c : {\mathbb{R}}_{\geqslant{0}} \rightarrow P. An \omega-limit point of c is any point in P that c approaches in discrete time. More precisely, an \omega-limit point of c is any point x \in P such that x = \lim_{k \rightarrow \infty}c( t_{k}) for some sequence {(t_{k})}_{k \geqslant 0} in \mathbb{R}_{\geqslant{0}} such that \lim_{k \rightarrow \infty} t_{k} = \infty.

Persistent trajectory (Definition)
A trajectory in {\mathbb{R}}_{\geqslant{0}}^{n} is persistent if it has no omega-limit points on the boundary \partial\mathbb{R}_{\geqslant{0}}^{n} = \mathbb{R}_{\geqslant{0}}^{n}\!\setminus\!\mathbb{R}_{>0}^{n}.

If c is a trajectory in {\mathbb{R}}_{\geqslant{0}}^{n} whose components c_{1}, \ldots, c_{n} quantify n particular entities, to say that c is persistent is to say that these entities do not experience recurring, worse-trending episodes of near extinction, and in particular do not approach extinction. 

Positive dynamical system (Definition)
A dynamical system {\dot{x}} = f(x), where f is a vector field {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n}, is a positive dynamical system if the nonnegative orthant {\mathbb{R}}_{\geqslant{0}}^{n} is forward-invariant, i.e. if all solution trajectories originating in \mathbb{R}_{\geqslant{0}}^{n} range in \mathbb{R}_{\geqslant{0}}^{n}.

Examples of positive dynamical systems include the dynamical systems that describe the kinetics of mass-action reaction networks. 

Persistent dynamical system (Definition)
A positive dynamical system in {\mathbb{R}}^{n} is persistent if all solution trajectories originating in {\mathbb{R}}_{>0}^{n} are persistent.

I will state an equivalent formulation of persistence which applies to reaction networks and provides a convenient transition to vacuous persistence. I will use the language of Chemical Reaction Network Theory. For readers unfamiliar with this area, two classic references are the lecture notes of Martin Feinberg at http://www.che.eng.ohio-state.edu/~Feinberg/LecturesOnReactionNetworks/ and the tutorial of Jeremy Gunawardena at http://www.jeremy-gunawardena.com/papers/crnt.pdf

If P is a (nonnegative) stoichiometric compatibility class of a reaction network, I shall denote P_{>0} and P_{\ngtr{0}} the (relative) interior and the (relative) boundary of P respectively. These notations are chosen to highlight the fact that the interior P_{>0} consists of the points of P whose components are all positive, and the boundary P_{\ngtr{0}} consists of the points of P whose components are not all positive. We have P = P_{>0} \sqcup P_{\ngtr{0}}. I shall say that the class P is degenerate if P = P_{\ngtr{0}}, or equivalently if P_{>0} = \varnothing. Also, a set \mathcal{Z} of species is P-admissible if there exists in P  a point of which \mathcal{Z} is the support.

Persistence for mass-action reaction networks (Reformulated definition)
For every nondegenerate stoichiometric compatibility class P, there are no omega-limit points on the boundary P_{\ngtr{0}} for trajectories in P originating in the interior P_{>0}.

Vacuous persistence for mass-action reaction networks (Definition)
For every nondegenerate stoichiometric compatibility class P, there are no omega-limit points on the boundary P_{\ngtr{0}} for trajectories in P.

Ordinary persistence allows opportunities for non-persistence, in the sense that there can be boundary points that are omega-limit points, as long as they are so for trajectories originating on (and confined to) the boundary. The attribute “vacuous” is to indicate that such opportunities for non-persistence do not exist in vacuous persistence. One can see that vacuous persistence is equivalent to persistence together with the non-existence of trajectories entirely contained in the boundary of nondegenerate stoichiometric compatibility classes. It has been suggested to me (on 5 January 2010) that vacuous persistence might be equivalent to persistence together with the non-existence of equilibria on the boundary of nondegenerate stoichiometric compatibility classes. This condition is obviously necessary, but it is not yet fully established that it is sufficient.

Persistence has long been an area of interest in mathematical modeling, in particular in population modeling. To my knowledge, mathematical models of population dynamics assume that all living species (plants, animals, etc) are present at the chosen initial time. This may explain why vacuous persistence has as far as I can tell not been considered before. But vacuous persistence is relevant in chemistry because chemical species can appear after initial time as a result of chemical reactions, e.g. association (i.e. binding), dissociations (i.e. unbinding), isomerization, etc. The box below shows an if-and-only-if condition for the vacuous persistence of mass-action networks with bounded trajectories. The result is from my paper Reachability, Persistence, and Constructive Chemical Reaction Networks.

Characterization of vacuous persistence (Theorem)
Consider a mass-action reaction network for which all concentration trajectories are bounded. The following are equivalent:

  • The reaction network is vacuously persistent.
  • Among the subsets of the set of all species, only the full set is both reach-closed and P-admissible for every nondegenerate stoichiometric compatibility class P.

What it means to be reach-closed is discussed in the paper. I might in the future have a blog post dedicated to reachability. The fact that there is this nice structural (i.e. independent of rate constants) characterization (i.e. necessary and sufficient condition) makes me think that vacuous persistence is a natural concept of non-extinction for mass-action networks with bounded trajectories. This is of course a very subjective view, as evidenced by disputable attributes such as “nice” and “natural”.

Persistence via reachability vs. via Petri nets

3 January 2010

I have received and responded to an email question regarding my paper Reachability, Persistence, and Constructive Chemical Reaction Networks. The question and the response could be beneficial to others. I am posting them here with the kind permission of the person who asked the question.

QUESTION (4 December 2009):
Can you give a couple of interesting examples that can be proved to be persistent (or not to be) using your theorem, but not using the Petri net methods developed by David Angeli, Patrick De Leenheer and Eduardo Sontag?  That will help appreciate the impact of the work.

RESPONSE (14 December 2009):

I do not have examples in response to this question, but I have some relevant comments.

I actually prove “vacuous persistence”. (I would welcome suggestions for better terminology.) This is important because boundary initial states are a real concern. One can prove vacuous persistence by proving persistence for all “closed” subnetworks. But it’s a lot of extra work. I actually had to do essentially that in the paper [1] on the monotonicity of IL-1 receptor-ligand binding equilibrium. This was in response to a request from a referee. I felt at the time that this made the paper unnecessarily longer but I did not have solid arguments for that hunch.

I wanted to be able to tell whether a reaction network is vacuously persistent just by “looking at it” and not having to compute siphons. This is what I achieved with the result that the absence of isomerism among the elementary species in an explicitly-reversibly constructive network implies persistence, and the result that futile cascaded binary enzymatic networks are persistent.

I suspect that the “each siphon contains the support of a P-semiflow” condition is in fact equivalent to vacuous persistence for conservative networks. Other than just intuition, I observe that the networks that were shown in [2] to be persistent through this condition are in fact vacuously persistent. Furthermore, the examples (I know of) for which proving persistence required more effort are not vacuously persistent. I am referring to the following networks from [3], one of which is also mentioned in [2]:
(a) 2A+B -> A+2B , B -> A ;
(b) 2A+B -> C -> A+2B -> D -> 2A+B ;
(c) A -> B , B+C -> 2A , 2B -> 2C .
I also observe in these three examples that elementary species are isomeric to each other and that explicit or implicit isomerizations occur without catalytic (e.g. enzymatic) help.

[1] http://dx.doi.org/10.1016/j.jtbi.2006.07.023
[2] http://dx.doi.org/10.1016/j.mbs.2007.07.003
[3] http://dx.doi.org/10.1007/978-3-540-71988-5_9

Reachability, Persistence, and Constructive Chemical Reaction Networks

6 December 2009

I have just completed a paper titled Reachability, Persistence, and Constructive Chemical Reaction Networks. The paper is available at http://math.gillesgnacadja.info/files/ConstructiveCRNT.html.

This post represents my first experimentation with blogging. The plan is to blog on mathematics and systems biology topics that I find very intriguing and like cogitating about. As with most hobbies however, there are time constraints. I do not have much time for such cogitation and I expect to have even less time for blogging. I welcome and intend to respond to all comments, but I will often be unable to respond in a timely fashion.