Posted Monday January 29 2024.

Let N be the Petri net (equivalently chemical reaction network) with a single species X and two transitions, b: ∅ → X and d: X → ∅. If you assign each of these transitions a *reaction rate*, say \(r_b\) and \(r_d\) respectively, then this defines a birth-death process where (using Wikipedia’s notation) \(\lambda_n = r_b\) and \(\mu_n=nr_d\). The system has a steady state (in the probablistic sense) when \(\lambda_n=\mu_n\), i.e. when \(n = \frac{r_b}{r_d}\), the ratio of the two rates and the “carrying capacity” of the model.

Extreme technial bit incoming. Let C be the commutative monoidal category associated to this Petri net (see Proposition 3.3 in arXiv:2101.04238), and let S be any dynamical system, regarded as a category, where objects are states and morphisms are transitions. Define a “birth-death system” to be any functor S → C.

I wonder if this has a name? It seems to extend the notion of birth-death process beacuse it not only encodes the probabilities of transition between different n but also *tracks each individual*, since any time-evolution in the dynamical system S is mapped to a morphism in C, and every such morphism is generated by (consists of sequential and parallel composition of) b and d.