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I suggest we revive this classical Azimuth topic at the Forum -- Petri nets / reaction networks -- in light of the present day context. On the theoretical side, things have changed due to advances in the application of category theory to Petri nets. And in practice they have changed due to the pandemic, which brings matters of epidemiological modeling to a higher level of prominence on the social radar. Compartmental models are of signal importance to this subject. And the mathematics of Petri nets provides the stochastic and deterministic foundations for these models. This is math that matters for the social planet -- Azimuth math.

The subject of Petri nets is rich and expansive.

## Comments

Some topics within Petri nets:

Applied sciences

Abstract, categorical

Classical theoretical problems

Deterministic Petri net analysis

Stochastic Petri net analysis

Compartmental models in Epidemiology - SIR, SIRS, ...

Simulation

`Some topics within Petri nets: * Applied sciences * Chemical reaction networks - smaller number of molecules brings stochastic behavior to the fore * Biochemical reaction networks * Gene regulatory networks * Virological and immunological processes * Ecology - interacting species networks, e.g. predator-prey * Computer science applications, e.g., process modeling * R&D at Statebox * Oilfield depletion (Paul Pukite) * Epidemiology * Abstract, categorical * Open Petri nets (John Baez, Jade Master, Blake Pollard) * Classical theoretical problems * Reachability, termination, ... * Deterministic Petri net analysis * Rate equations; solutions and behavior * Equilibrium, periodicity, ... * Stochastic Petri net analysis * Qua Markov chains, Poisson processes * Formulation using techniques from quantum math (John Baez, Jacob Biamonte) * Compartmental models in Epidemiology - SIR, SIRS, ... * Deterministic form * Classical epidemic analysis; logistic equations, the sigmoidal curve * Stochastic form * Applications to infection processes in small populations, e.g., hospital wards, farms * Simulation * Algorithms - Gillespie * Software tools * Needed: concrete problems to be solved with software`

All told, I propose an Azimuth quest with the following focus:

`All told, I propose an Azimuth quest with the following focus: * Pursuit of applications of Petri nets to stochastic as well as deterministic epidemiology`

For study and development work at the forum, this can motivate a couple more branches of the quest:

The former taps into our interest in ACT at the forum. The latter taps into our interest in software development.

`For study and development work at the forum, this can motivate a couple more branches of the quest: * Understanding the contemporary research into the mathematical foundations of Petri nets * Development and application of simulation software The former taps into our interest in ACT at the forum. The latter taps into our interest in software development.`

To the extent that we can find interesting applications of

stochasticPetri nets to epidemiology, we are making connections between a vital application and a developing area of math that is already in Azimuth culture (thanks to John's leading efforts).Along these lines, I have a couple of ideas for avenues to explore, which I will post separately:

Here are a couple of references, for starters:

Priscilla E. Greenwood and Luis F. Gordillo, Stochastic epidemic modeling.

Gang Wang, Ying Zhang, Simon J. Shepherd, Clive B. Beggs, Nini Rao, Application of stochastic Petri nets for modelling the transmission of airborne Infection in Indoor Environments, Acta Medica Mediterranea, 2016, 32: 587.

`To the extent that we can find interesting applications of _stochastic_ Petri nets to epidemiology, we are making connections between a vital application and a developing area of math that is already in Azimuth culture (thanks to John's leading efforts). Along these lines, I have a couple of ideas for avenues to explore, which I will post separately: * [Research concept A](https://forum.azimuthproject.org/discussion/2512/petri-nets-research-concept-a) - composition of a global pandemic network from regional factor networks Here are a couple of references, for starters: * Priscilla E. Greenwood and Luis F. Gordillo, [Stochastic epidemic modeling](aimath.org/WWN/populationmodel/chapmar25.pdf). * Gang Wang, Ying Zhang, Simon J. Shepherd, Clive B. Beggs, Nini Rao, [Application of stochastic Petri nets for modelling the transmission of airborne Infection in Indoor Environments](https://pdfs.semanticscholar.org/c701/1a719988e589e0425ae7346b2202f93a78f6.pdf), Acta Medica Mediterranea, 2016, 32: 587.`

As for simulation and software, I'm all for it, with one main caveat. All the tools and technology will become interesting and come to life once we figure out some cool things to

dowith them. To get there, though, we need to learn enough of the theory and applications to start formulating specific interesting questions. It doesn't even have to be a grand research idea, it could even be some good and interesting exercises from a chemistry textbook, which can be fruitfully and interestingly approached via simulation.To avoid putting the cart before the horse, let's gather some real requirements before designing new systems. To formulate these requirements will work on our part -- fun work -- to learn enough about the application domain that we become the

usersof whatever software we may develop.`As for simulation and software, I'm all for it, with one main caveat. All the tools and technology will become interesting and come to life once we figure out some cool things to _do_ with them. To get there, though, we need to learn enough of the theory and applications to start formulating specific interesting questions. It doesn't even have to be a grand research idea, it could even be some good and interesting exercises from a chemistry textbook, which can be fruitfully and interestingly approached via simulation. To avoid putting the cart before the horse, let's gather some real requirements before designing new systems. To formulate these requirements will work on our part -- fun work -- to learn enough about the application domain that we become the _users_ of whatever software we may develop.`

To get there, we will need leadership by people who can bridge the gap between theory, meaningful "green" applications, and software engineering.

The more people who can work on this bridge, the better the bridge will be.

`To get there, we will need leadership by people who can bridge the gap between theory, meaningful "green" applications, and software engineering. The more people who can work on this bridge, the better the bridge will be.`

A special thanks goes to @DanielGeisler, who has made a

significantcommitment to working on this effort - and for the Azimuth Project in general.`A special thanks goes to @DanielGeisler, who has made a _significant_ commitment to working on this effort - and for the Azimuth Project in general.`

I want to thank @DavidTanzer for the quality and quantity of his work here. While I was committed to work somewhere, David's work showed me the Azimuth Project was where I wanted to invest my energy.

`I want to thank @DavidTanzer for the quality and quantity of his work here. While I was committed to work somewhere, David's work showed me the Azimuth Project was where I wanted to invest my energy.`

Some strategy ideas. Since much of what we are doing is software development, development methodologies may be of interest.

Scope

Analysis

Design

Implementation

Maintenance

I like the Scope part. @DavidTanzer has laid out an aggressive and necessary.

`Some strategy ideas. Since much of what we are doing is software development, development methodologies may be of interest. * Scope * Analysis * Design * Implementation * Maintenance I like the Scope part. @DavidTanzer has laid out an aggressive and necessary.`

I see a need to develop open source technology here and to peer-review it.

Scope - Do we want to impact schools, governments, scientific institutions or lay people? Maybe we can pretend that accepting the model is an epidemiology in it's own right. That suggests a second order process where the population has to "catch" social distancing. Basic graphics and animations that accepts parameters.

`I see a need to develop open source technology here and to peer-review it. Scope - Do we want to impact schools, governments, scientific institutions or lay people? Maybe we can pretend that accepting the model is an epidemiology in it's own right. That suggests a second order process where the population has to "catch" social distancing. Basic graphics and animations that accepts parameters.`

Fortunately I'm a "young" retiree, so I can think of long term projects, and more importantly scientific orientation and education. So my next step is to become a user to the system I want to construct. That means I need to become an epidemiologist and to learn how to think like one.

`Fortunately I'm a "young" retiree, so I can think of long term projects, and more importantly scientific orientation and education. So my next step is to become a user to the system I want to construct. That means I need to become an epidemiologist and to learn how to think like one.`

I'm taking a course

Epidemics, Pandemics and Outbreaksby University of Pittsburgh though Coursera.`I'm taking a course **Epidemics, Pandemics and Outbreaks** by University of Pittsburgh though Coursera.`

That great! Good choice. Could you start a discussion, and post some notes to it as you go along? Thanks again.

`That great! Good choice. Could you start a discussion, and post some notes to it as you go along? Thanks again.`

I haven't kept close track over the years, but simulation software for Petri nets has many roots in reliability analysis and that's where I think you may find usable tools. The first hit I find on Google Scholar for "stochastic petri net" is for a tool called SPNP, which I remember was one of the first packages available in the early 1990's. The one that has been around for probably as long is called GreatSPN and this one still is being maintained and they have a github account with recent commits, so this is far from dead -- https://github.com/greatspn/SOURCES

My own interest in stochastic Petri nets peaked back then as well, and a report I wrote for a NASA fault-tolerance analysis project is on ResearchGate, which has links to a long final report w/ lots of citations + source code snippets and also a link to a short conference paper: https://www.researchgate.net/publication/269227210_Intelligent_reliability_analysis_tool_for_fault-tolerant_system_design

`I haven't kept close track over the years, but simulation software for Petri nets has many roots in reliability analysis and that's where I think you may find usable tools. The first hit I find on Google Scholar for "stochastic petri net" is for a tool called SPNP, which I remember was one of the first packages available in the early 1990's. The one that has been around for probably as long is called [GreatSPN](https://books.google.com/books?id=us3tCwAAQBAJ&pg=PA250&lpg=PA250&dq=SPNP+GreatSPN&source=bl&ots=ZoRrY4UkKo&sig=ACfU3U2IMD8BbH3xJW2BzcNTjaNYeaV-4A&hl=en&sa=X&ved=2ahUKEwjc4rrXvb3pAhWRLs0KHdBlAbQQ6AEwAHoECAkQAQ#v=onepage&q=SPNP%20GreatSPN&f=false) and this one still is being maintained and they have a github account with recent commits, so this is far from dead -- https://github.com/greatspn/SOURCES My own interest in stochastic Petri nets peaked back then as well, and a report I wrote for a NASA fault-tolerance analysis project is on ResearchGate, which has links to a long final report w/ lots of citations + source code snippets and also a link to a short conference paper: https://www.researchgate.net/publication/269227210_Intelligent_reliability_analysis_tool_for_fault-tolerant_system_design`

Thanks for the information @WebHubTel, I'm reading your report right now. - Daniel

`Thanks for the information @WebHubTel, I'm reading your report right now. - Daniel`

@WebHubTel Hi Paul, For the benefit of people who haven't read up on Petri nets / reaction networks, could you say a few words about Lotka-Volterra, and why it is of interest? Do you come across it in your work?

Also, what specific classes of questions are well addressed through simulation. This is a wide-net question, which covers both stochastic and deterministic nets. One clear example is finding equilibrium states. What else?

I'm looking for some case studies which could be explained to developers, as motivating examples for developing simulation software for Petri nets.

Thanks

`@WebHubTel Hi Paul, For the benefit of people who haven't read up on Petri nets / reaction networks, could you say a few words about Lotka-Volterra, and why it is of interest? Do you come across it in your work? Also, what specific classes of questions are well addressed through simulation. This is a wide-net question, which covers both stochastic and deterministic nets. One clear example is finding equilibrium states. What else? I'm looking for some case studies which could be explained to developers, as motivating examples for developing simulation software for Petri nets. Thanks`

I wrote something recently about L-V in an applied context here : https://geoenergymath.com/2020/03/29/lemming-fox-dynamics-not-lotka-volterra/

For real-time software development everything is done through simulation. Every software thread corresponds to a Petri net token stepping through the source code. Can even simulate logic designs with Petri nets as we described in this paper: https://dl.acm.org/doi/pdf/10.1145/1315580.1315592

`> "could you say a few words about Lotka-Volterra" I wrote something recently about L-V in an applied context here : https://geoenergymath.com/2020/03/29/lemming-fox-dynamics-not-lotka-volterra/ > "Also, what specific classes of questions are well addressed through simulation." For real-time software development everything is done through simulation. Every software thread corresponds to a Petri net token stepping through the source code. Can even simulate logic designs with Petri nets as we described in this paper: https://dl.acm.org/doi/pdf/10.1145/1315580.1315592`

Thanks Paul. I have a pedagogical challenge. Suppose one was explaining L-V and simulation to, say a functional programmer or software engineer, who didn't have any background in the natural sciences. We're at a party with them, and are trying to convey to them at a high level the spirit of these topics.

For now let's start with the deterministic case, which is a big subject in itself. How would the "mini talk" go?

I would start by saying something like L-V is a system of ODEs, which arise as the rate equations for a predator/prey population model based on the following premises XYZ. For simulating these kinds of deterministic systems the goal boils down to understanding the structure of the solutions, which are all the paths through the vector field defined by the equation. Stable equilibrium points show up as attractor points that solution paths converge to. Unstable equilibria are repelling points which the paths lead out of. Oscillatory solutions are bounded paths that never fall into an attractor points.

Now how can simulators help us to understand this structure of solution paths? For starters, they can identify the equilibrium and attractor points. Next step is that they can be used to make a map of the basins of attractions for the attractor points. It is can be useful to make a "color chart" where the color shows ABC. Regions of oscillatory paths may be conjectured if they remain bounded for a "long time".

The structure of the L-V solutions has the following general topology... To explore aspect ABC of these solutions in greater detail, simulation type DEF is useful.

An important question in population dynamics is what are the regions that lead to the extinction of some subset of the populations...

`Thanks Paul. I have a pedagogical challenge. Suppose one was explaining L-V and simulation to, say a functional programmer or software engineer, who didn't have any background in the natural sciences. We're at a party with them, and are trying to convey to them at a high level the spirit of these topics. For now let's start with the deterministic case, which is a big subject in itself. How would the "mini talk" go? I would start by saying something like L-V is a system of ODEs, which arise as the rate equations for a predator/prey population model based on the following premises XYZ. For simulating these kinds of deterministic systems the goal boils down to understanding the structure of the solutions, which are all the paths through the vector field defined by the equation. Stable equilibrium points show up as attractor points that solution paths converge to. Unstable equilibria are repelling points which the paths lead out of. Oscillatory solutions are bounded paths that never fall into an attractor points. Now how can simulators help us to understand this structure of solution paths? For starters, they can identify the equilibrium and attractor points. Next step is that they can be used to make a map of the basins of attractions for the attractor points. It is can be useful to make a "color chart" where the color shows ABC. Regions of oscillatory paths may be conjectured if they remain bounded for a "long time". The structure of the L-V solutions has the following general topology... To explore aspect ABC of these solutions in greater detail, simulation type DEF is useful. An important question in population dynamics is what are the regions that lead to the extinction of some subset of the populations...`

Now there I was shooting from the hip, having never actually done any of the above things. (But getting interested in it, as our focus is turning this way.)

Paul could you elaborate / amend the above, to make it more concrete, still maintaining the pedagogical tone which is directed who is not experienced with natural science or reading science papers.

`Now there I was shooting from the hip, having never actually done any of the above things. (But getting interested in it, as our focus is turning this way.) Paul could you elaborate / amend the above, to make it more concrete, still maintaining the pedagogical tone which is directed who is not experienced with natural science or reading science papers.`

Sidenote: of course, a lot of the general discussion above will carry over to the solution space for e.g. SIRS models, and we can ask the same questions about the topology of the solution path space, the attractors, etc. So, beyond its specific interest, L-V functions here as a teaching example for more general ideas.

`Sidenote: of course, a lot of the general discussion above will carry over to the solution space for e.g. SIRS models, and we can ask the same questions about the topology of the solution path space, the attractors, etc. So, beyond its specific interest, L-V functions here as a teaching example for more general ideas.`

Autonomous versus non-autonomous equations. Lotka-Volterra belongs to the former category, a time-invariant system -- is that a stretch for describing real systems where populations depend on the environment? What good will that behavioral description do when the prey species is susceptible to e.g. drought cycles?

So it then becomes a forced system and the focus on an attractor orbit goes out the window. I actually get annoyed by how much people cling to the notion that internal eigenvalues have to be the solution to everything. In many practical cases, it's the forced response and not the natural response that governs the evolution. For this predator-prey system, it appears that ENSO climate cycles and not the internal L-V dynamics drive the cycles. Moreover, even ENSO isn't an internally natural response system, as that is obviously forced by external tidal cycles. Perhaps that's why the scientists are all mystified by this, as they may be deeply attached to the mathematical idealism of eigenvalue-based solutions. But then even this is odd, because climate change and AGW is well-agreed to be a forced response system, driven by adding CO2 to the atmosphere. So I can't generalize either.

I can discuss this aspect of natural vs forced response all day.

`Autonomous versus non-autonomous equations. Lotka-Volterra belongs to the former category, a time-invariant system -- is that a stretch for describing real systems where populations depend on the environment? What good will that behavioral description do when the prey species is susceptible to e.g. drought cycles? So it then becomes a forced system and the focus on an attractor orbit goes out the window. I actually get annoyed by how much people cling to the notion that internal eigenvalues have to be the solution to everything. In many practical cases, it's the forced response and not the natural response that governs the evolution. For [this predator-prey system](https://geoenergymath.com/2020/03/29/lemming-fox-dynamics-not-lotka-volterra/), it appears that ENSO climate cycles and not the internal L-V dynamics drive the cycles. Moreover, even ENSO isn't an internally natural response system, as that is obviously forced by external tidal cycles. Perhaps that's why the scientists are all mystified by this, as they may be deeply attached to the mathematical idealism of eigenvalue-based solutions. But then even this is odd, because climate change and AGW is well-agreed to be a forced response system, driven by adding CO2 to the atmosphere. So I can't generalize either. I can discuss this aspect of natural vs forced response all day.`

This are very interesting point you are making! And I am interested to further discuss it. But it is veering away from the topic of this discussion, which is how we can get started working on Petri net education, simulations, etc. at Azimuth. Rather than pull a discussion in the direction of a theme that is on your mind, why not step right up and start a new discussion with a title that directly states your theme.

`> I can discuss this aspect of natural vs forced response all day. This are very interesting point you are making! And I am interested to further discuss it. But it is veering away from the topic of this discussion, which is how we can get started working on Petri net education, simulations, etc. at Azimuth. Rather than pull a discussion in the direction of a theme that is on your mind, why not step right up and start a new discussion with a title that directly states your theme.`

I propose a lasting accord -- call it the treaty of Azimuth -- between those of us who are mathematicians at heart and those who are empirical scientists. :)

`> Perhaps that's why the scientists are all mystified by this, as they may be deeply attached to the mathematical idealism of eigenvalue-based solutions. I propose a lasting accord -- call it the treaty of Azimuth -- between those of us who are mathematicians at heart and those who are empirical scientists. :)`

I do get your point, though, which can be boiled down to saying: what's the point of studying equilibrium solutions if the system is being driven by external forces, so there is no equilibrium.

`I do get your point, though, which can be boiled down to saying: what's the point of studying equilibrium solutions if the system is being driven by external forces, so there is no equilibrium.`

But the charge against those who are "deeply attached to the mathematical idealism of eigenvalue-based solutions" has an edge to it which could easily be taken out of context.

Your point is that the 'first order' L-V model, which doesn't take into account forcing, won't give accurate predictions in real-world scenarios. Fine.

But eigenvalue-based solutions and equilibria are absolutely foundational concepts, which apply almost perfectly in cases like chemical reactions, which are not dominated by external forces. So let's not get discouraged from opening up the topic just because it's empirical applicability needs to be evaluated on a case-by-case basis.

I can also imagine that even where it doesn't literally apply, the theory of equilibrium may provide a framework for understanding the disruption of equilibrium by slight to moderate amounts of forcing. For example, with slight forcing, there will no longer be points of equilibrium, but perhaps small fuzzy regions of "quasi-equilibrium" around the theoretical equilibrium points.

`But the charge against those who are "deeply attached to the mathematical idealism of eigenvalue-based solutions" has an edge to it which could easily be taken out of context. Your point is that the 'first order' L-V model, which doesn't take into account forcing, won't give accurate predictions in real-world scenarios. Fine. But eigenvalue-based solutions and equilibria are absolutely foundational concepts, which apply almost perfectly in cases like chemical reactions, which are not dominated by external forces. So let's not get discouraged from opening up the topic just because it's empirical applicability needs to be evaluated on a case-by-case basis. I can also imagine that even where it doesn't literally apply, the theory of equilibrium may provide a framework for understanding the disruption of equilibrium by slight to moderate amounts of forcing. For example, with slight forcing, there will no longer be points of equilibrium, but perhaps small fuzzy regions of "quasi-equilibrium" around the theoretical equilibrium points.`

Distinction perhaps between equilibrium and steady-state. Tidal cycles for a given geographical location will not deviate from predictions over decades. Even though this has the properties of a stationary equilibrium, we are taught that this should be referred to as steady-state behavior since forcing is supplied externally. The only equilibrium behavior observable for this system would be to remove the moon and the sun, and only then will the sea surface achieve a stable "equilibrium" level.

Isn't that wild? The equilibrium calculation is trivially uninteresting. This happens in electrical circuit theory as well:

The "broken circuit" is essentially the one where someone clipped a connection and the circuit doesn't do anything. Totally uninteresting, yet that's the definition for equilibrium.

But then if one reads further across disciplines:

In terms of tidal forces only, the Earth/Moon/Sun (+other planets) system is closed to other flows. So perhaps by definition, the observed tidal cycles actually are the equilibrium solution. Or that whatever caused the initial orbital motions long ago, and in the context of extremely low dissipation of energy, the tides are still part of a natural response to the original (solar-system formation) stimulus.

`> "I do get your point, though, which can be boiled down to saying: what's the point of studying equilibrium solutions if the system is being driven by external forces, so there is no equilibrium." Distinction perhaps between equilibrium and steady-state. Tidal cycles for a given geographical location will not deviate from predictions over decades. Even though this has the properties of a stationary equilibrium, we are taught that this should be referred to as steady-state behavior since forcing is supplied externally. The only equilibrium behavior observable for this system would be to remove the moon and the sun, and only then will the sea surface achieve a stable "equilibrium" level. Isn't that wild? The equilibrium calculation is trivially uninteresting. This happens in electrical circuit theory as well: > ["The simplest example is a broken circuit vs a closed circuit consisting of a battery and a wire. The first is in equilibrium, the second is in a steady state."](https://forum.allaboutcircuits.com/threads/difference-between-steady-state-and-equilibrium.70546/) The "broken circuit" is essentially the one where someone clipped a connection and the circuit doesn't do anything. Totally uninteresting, yet that's the definition for equilibrium. But then if [one reads further across disciplines](http://www.projects.bucknell.edu/LearnThermo/pages/Equilibrium%20and%20Steady%20State/equilibrium-and-steady-state.html): > "Equilibrium and Steady State. A state of chemical equilibrium is reached when the concentration of reactants and product are constant over time (Wikipedia). ... In contrast, steady state is when the state variables are constant over time while there is a flow through the system (Wikipedia)." In terms of tidal forces only, the Earth/Moon/Sun (+other planets) system is closed to other flows. So perhaps by definition, the observed tidal cycles actually are the equilibrium solution. Or that whatever caused the initial orbital motions long ago, and in the context of extremely low dissipation of energy, the tides are still part of a natural response to the original (solar-system formation) stimulus.`

Can we put a stop here please. You clearly know a lot more about equilibrium than I do. In this thread I am trying to achieve some progress towards getting a Petri net thing going for us at Azimuth. I started with a basic question, perhaps I didn't phrase it well, but somehow we've veered into ruminative discussions about equilibrium, tidal forces, etc. I know I contributed to this direction by delving into equilibrium, but it's not productive for the stated purpose of this discussion.

In an attempt to reset, I will rephrase my question. Thanks.

`Can we put a stop here please. You clearly know a lot more about equilibrium than I do. In this thread I am trying to achieve some progress towards getting a Petri net thing going for us at Azimuth. I started with a basic question, perhaps I didn't phrase it well, but somehow we've veered into ruminative discussions about equilibrium, tidal forces, etc. I know I contributed to this direction by delving into equilibrium, but it's not productive for the stated purpose of this discussion. In an attempt to reset, I will rephrase my question. Thanks.`

Another indicator to look at regarding the scope of comments is the category of a discussion. This discussion is "Petri nets at Azimuth in a new context," in category Strategy.

Aided and abetted by each other, we have led the discussion rather out of the realm of strategy. Strategy discussions give a public window into the directions and planning of Azimuth, and when they veer away into technical areas it has a diluting and disorganizing effect. To remedy this I have copied the substance of our discussion to a new thread - let's continue there, I am interested in those ideas. Also, unless you object, I'm going to trim down the contents here as they've been copied to a new context. Thanks.

`Another indicator to look at regarding the scope of comments is the category of a discussion. This discussion is "Petri nets at Azimuth in a new context," in category Strategy. Aided and abetted by each other, we have led the discussion rather out of the realm of strategy. Strategy discussions give a public window into the directions and planning of Azimuth, and when they veer away into technical areas it has a diluting and disorganizing effect. To remedy this I have copied the substance of our discussion to a new thread - let's continue there, I am interested in those ideas. Also, unless you object, I'm going to trim down the contents here as they've been copied to a new context. Thanks.`

Being a veteran of threaded discussions, it's always useful to go to the source. David, In this thread you directed this question to me :

I responded that I knew about L-V and provided a link to a blog post I wrote in March called Lemming/Fox Dynamics not Lotka-Volterra. Then you asked me to elaborate, and so I summarized my opinion that L-V does not explain the cycles because of the evidence that an external forcing is involved in controlling the population levels. I mentioned this to imply that I was wary about the general utility of L-V for modeling predator-prey dynamics.

`Being a veteran of threaded discussions, it's always useful to go to the source. David, In this thread [you directed this question to me](https://forum.azimuthproject.org/discussion/comment/22202/#Comment_22202) : > @WebHubTel Hi Paul, For the benefit of people who haven't read up on Petri nets / reaction networks, could you say a few words about Lotka-Volterra, and why it is of interest? Do you come across it in your work? I responded that I knew about L-V and provided a link to a blog post I wrote in March called [Lemming/Fox Dynamics not Lotka-Volterra]([https://geoenergymath.com/2020/03/29/lemming-fox-dynamics-not-lotka-volterra/). Then you asked me to elaborate, and so I summarized my opinion that L-V does not explain the cycles because of the evidence that an external forcing is involved in controlling the population levels. I mentioned this to imply that I was wary about the general utility of L-V for modeling predator-prey dynamics.`

That's a fair point. I see now that my wording of the question didn't really express what I was trying to get at. I tried again with the parable of the software engineer at the party, but still that wasn't clear enough. That's on me, sorry for the confusion. I will regroup and start afresh. Best Regards!

`That's a fair point. I see now that my wording of the question didn't really express what I was trying to get at. I tried again with the parable of the software engineer at the party, but still that wasn't clear enough. That's on me, sorry for the confusion. I will regroup and start afresh. Best Regards!`

Diagnosis: since I was fishing for something not yet clearly formulated - and which opens a big topic - to keep things organized I should have asked my technical question in a separate discussion off the strategy thread. So yes the bug was on my end. Well, live and learn. That aside, this has been an informative discussion. Thanks Paul.

`Diagnosis: since I was fishing for something not yet clearly formulated - and which opens a big topic - to keep things organized I should have asked my technical question in a separate discussion off the strategy thread. So yes the bug was on my end. Well, live and learn. That aside, this has been an informative discussion. Thanks Paul.`

I see three major divisions for a Petri net initiative: math, science and computation.

Starting from their definition, Petri nets can be treated as purely mathematical entities. On the one hand, there is the classical mathematics which studies them a a special kind of Markov process, looks at the rate equations and the structure of the solutions, etc. And then there are more modern, abstract extensions of this math into category theory.

In the scientific branch, we study the application of these models to actual empirical systems. As part of this, we there investigate the extent to which the models capture the actual phenomena.

The computational branch can be partitioned into algorithms and development.

`I see three major divisions for a Petri net initiative: math, science and computation. Starting from their definition, Petri nets can be treated as purely mathematical entities. On the one hand, there is the classical mathematics which studies them a a special kind of Markov process, looks at the rate equations and the structure of the solutions, etc. And then there are more modern, abstract extensions of this math into category theory. In the scientific branch, we study the application of these models to actual empirical systems. As part of this, we there investigate the extent to which the models capture the actual phenomena. The computational branch can be partitioned into algorithms and development.`

I think those are appropriate divisions:

The first two are closely linked and are most easily distinguished by how much linearity is imposed. I have written one book that is almost exclusively on Markov models and another book that is more geared to compartmental models. The third is essentially development and engineering as you said, and my experience in this is related here: https://forum.azimuthproject.org/discussion/comment/22259/#Comment_22259

`I think those are appropriate divisions: 1. Markov process modeling - purely stochastic : appropriate for formal classical mathematical analysis 2. Compartmental modeling - could be stochastic but also could be mean-value deterministic : more empirical, c.f. epidemiology, chemistry 3. Computation realization - deterministic : appropriate for testable and verifiable systems The first two are closely linked and are most easily distinguished by how much linearity is imposed. I have written one book that is almost exclusively on Markov models and another book that is more geared to compartmental models. The third is essentially development and engineering as you said, and my experience in this is related here: https://forum.azimuthproject.org/discussion/comment/22259/#Comment_22259`