What's the transition probability according to this PDE? I'm trying to figure out how I can simulate markov chains based on an ODE:
dN/dt = alpha N (1 - N / K) - beta N

Thus N denotes total population, and I want to simulate through sampling for each present individual N(t) if they'd birth new ones alpha (1-N/k) or die due to death probability beta. I don't want to use exponential distribution for these.. But I don't know what to do when e.g. beta>1 or K>N, as then alpha(1-N/K) may not be a proper probability.

Based on additional details in the comments, you have a process with two types of transitions happening with certain rates per unit time (a birth-death process).  Given the current population size $N$ at time $t$, the transition rates are $\alpha N(1-N/K)$ and $\beta N$, respectively.  It follows that the waiting time $\tau$ to the next transition is exponentially distributed with rate parameter equal to the sum of the current transition rates.  At time $t+\tau$, the process then transitions to state $N+1$ or $N-1$ with probabilities proportional to the transition rates.  This simulation method is known as the Gillespie algorithm.  An implementation in R follows below.
gillespie <- function(alpha, beta, K, N0, tmax) {
  t <- 0
  N <- N0
  i <- 1
  repeat {
    rates <- c(alpha*N[i]*(1 - N[i]/K), beta*N[i])
    tau <- rexp(1, rate = sum(rates))
    if (t[i] + tau > tmax | N[i]==0)
      break()
    t[i + 1] <- t[i] + tau
    N[i + 1] <- N[i] + sample(c(1, -1), 1, prob = rates)
    i <- i + 1
  }
  data.frame(t,N)
}
set.seed(1)
realization <- gillespie(alpha = .5, beta = .1, K = 20, N0 = 5, tmax = 50)
plot(realization, type="s")


