signal-flow graph. Between then and now I’ve thought more about those graphs…. and in fact I want to talk about them now in my network theory series, though it may take a while for me to get there!

I really do need to understand WHY this notation is evidently not obvious to those outside of a few fields.

It’s not that the notation was inherently hard, it’s that I was busy trying to understand Petri net notation, and you were translating it into some other notation, which looks slightly similar (networks of boxes and edges) but works differently. It was too much for me to process! The fact that you called a ‘state’ an ‘integrator’ was the straw that broke the camel’s back. But it’s just like how we get the charge on a capacitor by integrating the current through it.

]]>The Le Chatelier Principle I was talking about, is indeed about reversible reactions, and the principle is concerned with the situation at equilibrium, i.e. when the forward and backward rates of reactions are equal. ]]>

What is the logic behind writing the rate as the product of the powers of the inputs?

Is it that we are counting the number of possible encounters between the interacting species?- eg.: if a wolf eats 2 and only 2 rabbits at once, and there are rabbits and wolves resp., the number of encounters would be about .

Right, that’s the logic.

This is true only if are very large, so that non-allowed encounters are negligible (a wolf cannot eat the same rabbit twice).

Right. The **rate equation** is an approximation that’s only good in a certain limit where there are many things of each kind. In later sections we discuss the **master equation**, which is more accurate. Here you’ll see that the number replaces the number Also, the master equation treats the numbers of things as integers rather than real numbers. Also, it treats the processes using

probability theory.

Also, if the order in which the species interact to complete one encounter is immaterial, the extra factors can be adjusted into the rate constant.

Right.

So, in general shouldn’t the products of a transition also play a role in determining the rate as the products might start interfering?

Chemists use the rate equation and master equation that I’m discussing here. However, in chemical reactions for every transition there is always a *reverse* transition that converts the products back into the inputs. I bet what gives the effect you’re mentioning. Maybe you can look up the Le Chatelier principle you’re remembering, and try to derive it from a suitable master equation.

Also, we have considered transitions like death which have only inputs. Why not source like transitions?

Those are allowed too: in mathematics whatever is not explicitly forbidden, is allowed. I consider a transition like this in Part 5.

Are we also going to consider time varying rates later?

No. We could, but there is too much to do and not enough time to do it all!

]]>Is it that we are counting the number of possible encounters between the interacting species?- eg.: if a wolf eats 2 and only 2 rabbits at once, and there are rabbits and wolves resp., the number of encounters would be about . This is true only if are very large, so that non-allowed encounters are negligible (a wolf cannot eat the same rabbit twice). Also, if the order in which the species interact to complete one encounter is immaterial, the extra factors can be adjusted into the rate constant.

Is this understanding okay?

Also, I remember a bit about Le Chatelier’s principle for equilibrium reactions which had to do something with the rate being proportional to the product of powers of the inputs divided by the product of the powers of the outputs. So, in general shouldn’t the products of a transition also play a role in determining the rate as the products might start interfering. Either we could get the products to appear in the denominators or create a separate transition to allow obstruction(which we are doing here) – why is the second approach better?

Also, we have considered transitions like death which have only inputs. Why not source like transitions? (E.g. a human planting seeds at a constant rate, irrespective of the network the seed is involved in.) Are we also going to consider time varying rates later?

]]> Big fan posting.

I was fooling around with the diagrammatic calculus (Coecke and Penrose) and discovered that catalysis is the trace in the symmetric monoidal category. Here’ s the blog posting:

http://whyilovephysics.blogspot.com/

It also shows an autocatalytic reaction as trace. Maybe this has something to do with your idea of green math.

]]>The arrows in the diagram must be interpreted as a signal propagating in the arrow direction, or, also, as an assignment of a variable from the departure to the destination, but they do not represent a flow, in which a balance equation mandates that stuff must disappear from one place in order to get somewhere else.

This is the reason why negative feedback loops have to be implemented separately to make stuff disappear (the -1 gains in the transition blocks).

]]>… Isn’t that reasonable?

Yes!

At a minimum, it looks like this would require models with some built-in mechanism for defining when (or by how much) the system is departing from the model’s range of applicability.

Does anyone know of examples of models like that?

]]>I don’t know, what exactly is not clear to you ?

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