John, I got the term “dilution” from Definition 1 in this paper by Luca Cardelli. As far as I know, he invented the term.

Luca Cardelli, “Strand algebras for DNA Computing”

Click to access Strand%20Algebras%20for%20DNA%20Computing%20(Natural%20Computing).pdf

]]>Thanks. There certainly needs to be a distinction, at least in technical work.

]]>No. -dilution is not a standard term. It’s just a term that Manoj and I have introduced in our paper. I just felt that when we talk about reaction networks in general, one should make a distinction between a dilution and a reaction.

]]>Thanks. Is ‘-dilution’ a standard terminology in some community? Part of my job is to learn all the jargon that everyone uses: we’ve got computer scientists studying Petri nets, chemists studying reaction networks, etc.

]]>Yes, this is precisely what I mean by -dilution. Thanks for the clarification.

]]>What’s a ‘-dilution’?

I don’t know that term, but I have a guess as to what you mean. If we have a transition

wolf + rabbit wolf + wolf

in our Petri net, we can also use this transition to turn

wolf + rabbit + rabbit

into

wolf + wolf + rabbit

(One rabbit stands by and watches the other get eaten.) Manoj actually gave an example of this sort when explaining the concept of :

By a complex I will mean a population vector: a snapshot of the number of tokens in each species. In the example above, (#rabbit, #wolf) is a complex. If are two complexes, then we write

if we can get from to by a single transition in our Petri net. For example, we just saw that

via the predation transition.

So, his use of does not correspond precisely to a transition, but rather a transition ‘with bystanders’. Is this what you’re calling ‘dilution’?

]]>Does your explanation extend to the question of why this is not true for all critical siphons?

My “translation” for conservation law should be made more precise by adding:

“conservation law”: *a group which forms an “island of stability” by having a net constant wealth via one but arbitrary way of exchange.*

That is in the X,Y example one would – if one would apply all two transitions and not just one, keep the numbers of tokens constant. Or in other words X+Y satisfies a kind of “total conservation law.” So the X,Y example is critical but would not be totally critical. I could imagine that eventually by demanding “total conservation” on a support one might get around the minimal condition, but this is just a rough guess.

]]>John, I believe such objects have been studied and are called “traps.” Take a look at Definition 5 of a “trap” in this book chapter. (I may be wrong, but it seems to me on a quick read right now that the authors may have inadvertently gotten the two definitions reversed.)

David Angeli, Patrick De Leenheer, Eduardo Sontag,

“A Petri Net Approach to Persistence Analysis in Chemical Reaction Networks.”

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.152.627&rep=rep1&type=pdf

The same authors also say,

“Traps for Petri-Nets enjoy the following invariance property: if a trap is non-empty at time zero (meaning that at least one of its places has tokens), then the trap is non-empty at all future times. In contrast, in a continuous set-up (when tokens are not integer quantities but may take any real value), satisfaction of the siphon-trap property does not prevent (in general) concentrations of species from decaying to zero asymptotically.”

in this paper:

A Petri net approach to the study of persistence in chemical reaction networks.

Click to access angeli_leenheer_sontag_math_biosciences_MBS-D-06-00188R1.pdf

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