For your first question, consider the following notation: At any given time, say $t$, the state of your system is the function $q_t : \Omega \mapsto \mathbb{R}^{+}$, where $q_t(x) \equiv m(x, t)$. How can we succinctly describe its time evolution? It is actually pretty simple:

$$ \frac{dq}{dt} = q \cdot (K q) $$

where $\cdot$ is ordinary multiplication and $K$ is the integral transform defined by the kernel $k : \Omega \times \Omega \mapsto \mathbb{R}^{+}$.

Note that the derivative $\frac{d}{dt}$ is now defined in an infinite dimensional function space. One can straightforwardly define this derivative using the traditional form

$$ \frac{dq}{dt} \equiv \lim_{\tau \rightarrow t} \frac{q_\tau - q_t}{\tau - t} , $$

where the limit is defined via a suitable norm in your function space. See these lecture notes for details.

For your first question, consider the following notation: At any given time, say $t$, the state of your system is the function $q_t : \Omega \mapsto \mathbb{R}^{+}$, where $q_t(x) \equiv m(x, t)$. How can we succinctly describe its time evolution? It is actually pretty simple:

$$ \frac{dq}{dt} = q \cdot (K q) $$

where $\cdot$ is ordinary multiplication and $K$ is the integral transform defined by the kernel $k : \Omega \times \Omega \mapsto \mathbb{R}$.

Note that the derivative $\frac{d}{dt}$ is now defined in an infinite dimensional function space. One can straightforwardly define this derivative using the traditional form

$$ \frac{dq}{dt} \equiv \lim_{\tau \rightarrow t} \frac{q_\tau - q_t}{\tau - t} , $$

where the limit is defined via a suitable norm in your function space. See these lecture notes for details.