If there are two strains spreading concurrently, the deterministic rate of change of immunity, y, and the second strain x2, is given by the following set of differential equations:

This can be seen by forming dxz / dy as before, and after substituting either (p2 —> (p2 c or 02 —> az/c, one sees that each transformation results in the same rescaled differential equation (Section 3 of 81 Text):

Our analytical approach involves using a set of deterministic differential equations to ascertain pathogen spread in a stochastic birth-death process, where an infection (or immune cell) can only either die or produce 1 offspring.

By dividing Equation 1 by Equation 2, we obtain a differential equation for x1 as a function of y:

This differential equation is straightforward to solve (Section 1 of 81 Text), and yields the following function for x1(y):

In-depth mathematical analyses of the differential equations used, and how to derive the emergence probability if affected by immune growth (Equation 10 in the main text).

This framework allows us to formulate the essential dynamical ingredients of the genetic circuit of a single cyanobacterium into a set of differential equations describing the time evolution of the concentrations of the relevant molecular products.

(16) is a set of stochastic differential equations (SDE) so its numerical integration requires generating a statistical representative trajectory for a discrete set of time-values.

In this section we translate the genetic circuit previously described into a set of differential equations , for Which we follow the derivation in [46—48].

Finally we introduce the differential equations governing cyanobacterial reaction to nitrogen deprivation.

We used a Runge-Kutta method for the numerical integration of stochastic differential equations