Master Equation for a Savana Population I was reading the following post Master equation for reproduction and mutual annihilation process and I began to wonder - What If the population also suffers of lethal competition?
We would have the following process:
$$A + A \xrightarrow{\text{$\lambda$}} A$$
How would be transition rate in this case? Would be something like $\omega_{n, n-1} = \frac{\lambda}{2} \left( n(n-1) \right)?$
 A: I suppose that for an arbitrary process
$$mA \xrightarrow{\lambda} pA$$
we could write
$$
\omega_{n\rightarrow n -m+p}=\lambda {n\choose m}.$$
Obviously, this assumes that the individuals meet randomly and encounter any other with equal probability.
More detailed explanation
Suppose that we have events where $m$ individuals out of the population can meet and engage in a battle or an orgy, which transforms them into $p$ individuals (i.e., if some individuals are killed we have $p<m$, whereas if some are born, we have $p>m$). Such an event changes the number of individuals in the population from $n$ to $n-m+p$, so we are talking about transition $\omega_{n,n-m+p}$.
There are $n$ possibilities to choose one individual from the population, $n-1$ possibilities to choose the second, and so on till $n-m+1$ possibilities to choose the $m$-th. In the same time the order in which we select the individuals does not matter, so we need to divide it by the number of all possible orderings of $m$ individuals, i.e., $m!$. Thus, our rate is
$$
\omega_{n,n-m+p}\propto \frac{n(n-1)(n-2)...(n-m+1)}{m!}=\frac{n!}{m!(n-m)!}={n\choose m},$$
where I used the standard notation for a binomial coefficient.
Given that the rate of such events is given by $\lambda$, we have our final expression as
$$
\omega_{n, n -m+p}=\lambda {n\choose m}.$$
In particular, if we take the example from the OP, $$A+A\rightarrow A,$$ where two individuals are converted to one, we have $m=2$ and $p=1$, so that the rate according to the above formula is
$$
\omega_{n, n -1}=\lambda {n\choose 2}=\lambda\frac{n(n-1)}{2}.$$
