Density independent population growth with random survival

A simplified model for the growth of a population is studied in which random effects arise because reproducing individuals have a certain probability of surviving until the next breeding season and hence contributing to the next generation. The resulting Markov chain is that of a branching process with a known generating function. For parameter values leading to non-extinction, an approximating diffusion process is obtained for the population size. Results are obtained for the number of offspring $r_h$ and the initial population size $N_0$ required to guarantee a given probabilty of survival. For large probabilities of survival, increasing the initial population size from $N_0=1$ to $N_0=2$ gives a very large decrease in required fecundity but further increases in $N_0$ lead to much smaller decreases in $r_h$. For small probabilities (< 0.2) of survival the decreases in required fecundity when $N_0$ changes from 1 to 2 are very small. The calculations have relevance to the survival of populations derived from colonizing individuals which could be any of a variety of organisms.