Neyman, Park and Scott (1956) describe an experiment which they performed to determine the spatial distribution of Tribolium confusum developing within a closed container. To explain the concentration of beetles at the boundary a birth–death–migration model is developed in which the beetles may migrate over a set of lattice points, and this is shown to produce a distribution of the required shape. Not only is this distribution independent of the number of lattice points, but it is also indistinguishable from the associated diffusion process.

For the n-dimensional birth, death and migration process with constant rates, Aksland (1975) found a necessary and sufficient condition that the extinction-probability be one. This condition is translated into the form of direct relations between the parameters of the model.

A finite number of colonies, each subject to a simple birth-death and immigration process is studied under the condition of migration between the colonies.

Kolmogorov's backward equations for the process are solved for some special cases, and a sequence of functions uniformly converging to the p.g.f. of the process is given for the general case. Further, a set of algebraic equations for the extinction probabilities are studied for the process without immigration, and a necessary and sufficient condition that the extinction probability be one is obtained.