Explicit estimates from capture-recapture data with both death and immigration-stochastic model

A brief review, with references, of the literature on capture-recapture theory is given in Jolly (1963). More recently, Cormack (1964) gives a solution, including asymptotic variances, for a specific situation involving the marking and release of a non-random sample of fulmar petrels. His model is stochastic and will be referred to in ? 4. Seber (1962, 1965) has also produced some interesting solutions, and these together with Darroch (1958, 1959) are most directly relevant to our present problem. Darroch (1958, 1959) shows that in a fully stochastic model with either immigration (often called dilution) or death (or emigration), the population parameters can be easily estimated by maximum likelihood. For the more general case when death and immigration are operating simultaneously he derives estimation equations by equating certain observations to their expectations, but does not give variances or covariances for the estimates. In a later paper, (Darroch, 1961), he considers estimation for a closed population consisting of different strata. Seber (1962) establishes a stochastic model for what he calls the multi-sample single recapture census in which an individual cannot be recaptured more than once. This situation arises, for example, when the recaptures are made in the course of hunting or fishing. He allows for both death and immigration in the population, provides explicit maximumlikelihood estimates of the parameters with variances, and suggests tests for certain of the assumptions. In a second paper, (Seber, 1965), he considers a multiple-recapture model differing only slightly from that of Darroch (1959), with both death and immigration. Again, he provides explicit maximum-likelihood estimates of the parameters with variances. A test is also given for equi-catchability in a closed population of individuals with different capture histories. The first purpose of the present paper is to derive a general probability distribution designed to fit the majority of capture-recapture problems involving a 'single' population. The word 'single' here denotes a population covering an area within whose boundaries the animals (or, in general, individuals or members) are free to move and to mix with others of their kind, but which is regarded as a single area in respect of which parameters are to be estimated. The type of situation which is thus excluded by this definition is one where the population is split into a number of defined areas, and separate population estimates are required for each area as well as for numbers of animals moving from one area to another. The single population, however, need not be homogeneous but may consist of different classes of animals behaving in different ways. The other assumptions underlying the model are stated with the notation in ? 2, and the generalized probability distribution is derived in ?3.