I am interested in growth dynamics and, in particular, how to derive difference/differential/stochastic equations from local behavior of a system.
For concreteness, let's imagine a simple predator/prey model with foxes and rabbits, described by the following local dynamics, with a discrete time assumption.
At each time step, a rabbit may be eaten by a fox; there should be some parameter $p$ which measures this, though I am not quite sure what the right interpretation for this parameter is. Suggestions welcome!
If a rabbit is eaten at time $t$, obviously it disappears; if not, it reproduces to create one new rabbit. If a fox eats a rabbit, it reproduces to create one new fox; if not, it disappears. Finally, there is a constraint that a fox eats no more than one rabbit for a given time step.
I think that we can use a recurrence relationship to describe the probability that $m$ rabbits are eaten at time $t$, given $f$ foxes and $r$ rabbits. Unfortunately, I'm not sure which recurrence relation.
Here are three options:
1) Look at things from the rabbit's perspective. Either he gets eaten (one rabbit eaten, one fox satisfied) or he doesn't (one rabbit safe, no foxes satisfied): $$P(m|f,r) = p\cdot P(m-1|f-1,r-1) + (1-p)P(m|f,r-1)$$
2) Look at things from the fox's perspective. Either he eats (one rabbit eaten, one fox satisfied) or he doesn't (no rabbits eaten, one foxes unsatisfied): $$P(m|f,r) = p\cdot P(m-1|f-1,r-1) + (1-p)P(m|f-1,r)$$
3) Try to balance the two views. Either eating occurs (one rabbit eaten, one fox satisfied) or it doesn't (one rabbit safe, one fox unsatisfied): $$P(m|f,r) = p\cdot P(m-1|f-1,r-1) + (1-p)P(m|f-1,r-1)$$
I think that 1 & 2 must be wrong, because the analysis is symmetric and, if both were correct, then $P(m|f-1,r)=P(m|f,r-1)$. But this must be false if, e.g., $m=f=1$ and $r=2$. Then the LHS is zero (no foxes means no catches) but the RHS should be $p$.
That leaves 3 as my best guess, though I would appreciate a good explanation of why.
Question 1) Which, if any, of the three descriptions above is the correct analysis?
Question 2) Where can I look for a good explanation of this sort of analysis? Is there a name for this sort of thing?
Question 3) Now suppose that there is an additional predator (snakes) that also eat rabbits, with parameter $q$. How can I express a recurrence relation for the probability that $m$ rabbits are caught by foxes and $n$ are caught by snakes: $$P(m,n|r,f,s)=?$$
Question 4) More generally, any (?) directed graph with parameters labelling edges should give a predator/prey model like the one in Q3. Is there a general theory of producing recurrence relations from graphs?
Thanks!