A population of organisms consists of both male and female members. Any particular male is likely to mate with any particular female, in any time interval of length $h$, with probability $\lambda h + o(h)$. Each mating immediately produces one offspring, equally likely to be male or female. There are no deaths. Let $N_1(t)$ and $N_2(t)$ denote the number of males and females in the population at time $t$. This can be modeled as a continuous time Markov chain.
(a) What is the state space?
(b) What are the parameters $v_i$ and $P_{ij}$ (rate to leave the state $i$ and the probability that the jump is to $j$ when this happens)?
(c) What do you think the limiting probabilities will be?
Here is my answer to the question but I am not really sure. Please let me know what you think.
(a) $S = \{(i,j)\}, i,j \in \mathbb{N_0}$ $i = N_1(t), j = N_2(t)$
(b) $v_i=\lambda$ only when $i\in \{(i,j)\}$, where both $i,j$ are positive integers. $v_i=0$ otherwise.
$P_{ij}=1$
(c) If the starting state is one of {(0,0), (0,1), (1,0)}, the limiting probability will be $P_{ii}=1$. Otherwise, limiting probability doesn't exists as $ N_1(t), N_2(t)$ can go to infinity.