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The study of population dynamics in a country is based primarily on the knowledge of the rates of occurrence of two natural vital events of births and deaths in relation to the population size. A continuous appraisal of population necessitates availability of reliable birth and death data for understanding the factors influencing population growth and also for making projections for future on the basis of the prevalent and anticipated vital rates for use in drawing up future social and economic plans in accordance with the realistic requirements of the population. Measurements of fertility, mortality and population size are also required in planning and evaluation of health programmes and in assessing effectiveness of the population planning programme.

Let {XN(t)} be a sequence of continuous time Markov population processes on an n-dimensional integer lattice, such that XN has initial state Nx(0) and has a finite number of possible transitions J from any state X: let the transition X → X + J have rate NgJ(N–1X), and let gJ(x) and x (0) be fixed as N varies. The rate of convergence of √N(N–1XN(t) — ζ (t)) to a Gaussian diffusion is investigated, where ζ(t) is the deterministic approximation to N–1XN(t), and a method of deriving higher order asymptotic expansions for its distribution is justified. The methods are applied to two birth and death processes, and to the closed stochastic epidemic.

A simple birth-death model of particle fluctuations is studied where at each discrete time a birth and/or death may occur. We show that if the probability of a birth does not depend on the number of particles present and if births and deaths are independent, then the times between successive deaths are independent geometrically distributed random variables, which is false in the general case. Since the above properties of the times between successive neuron firings have been observed in nerve cells, the model proposed in [2] obtains added credence.

A detailed probabilistic treatment is given of a birth-and-death process proposed by Williams (1969) in which the elements of the process bear up to s inanimate marks. Equations for the second-order moments of the process, and approximate marginal univariate solutions, are derived. The exact bivariate solution is given for the case s = 1. For general s the variance of the mark population is also derived. MARKED BIRTH-AND-DEATH PROCESS; MULTIVARIATE LAGRANGE DIFFERENTIAL EQUATIONS; MULTITYPE BRANCHING PROCESS Introduction In a recent paper Williams (1969) has considered the following process. We have a population of elements developing as a linear birth-and-death process and each element bears a number of inanimate marks. When a marked element dies then it and its marks are removed from the population; when a marked element divides, its marks segregate independently and without loss, each with probability I into one of the two daughter elements. It is assumed that marking an element does not affect its birth and death rates (cf. Clifford and Sudbury (1972)). Williams' process was proposed as a model of an experiment conducted on bacteria (the elements) infected by phage (the inanimate marks), by Meynell (1959). (The bacteria are lysogenic with respect to the phage, which means that the phage do not interfere with the behaviour of the bacteria.) His object was to use the marking process in order to separate the birth and death rates of the bacteria; if we denote birth and death rates by A and p respectively, then the mean-value measures on the bacterial population (assumed to develop as a linear birth-and-death process) will estimate the difference (A p). As the marks are inanimate, however, they will only be affected by p and so mean-value measures on the mark population enable us to estimate p (see Williams (1969)). The papers of Meynell and Williams taken together provide a most interesting example of a virological experiment and the probability model of that experiment. Received in revised form 10 August 1973 Research sponsored by a Science Research Council grant and a World Health Organisation research traineeship. 423 This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms 424 BYRON J. T. MORGAN Williams considered the mean-value solution of his model; this was adequate for Meynell's experiment in which population sizes were large. Here we shall give a fuller probabilistic statement of the model and make some contributions to its solution. Two differences from Williams' work lie in the adoption of an upper bound, s, to the number of marks any element can have, and the assumption (except where stated) of birth and death rates independent of time. We 'shall assume general initial conditions which, in Meynell's experiment, correspond to the distribution of phage over bacteria once these have been initially exposed to a phage population. Williams took this distribution to be Poisson; the distribution of phage over bacteria has, however, been the subject of much probabilistic study (e.g., see Gani (1965) and Morgan (1971)). Finally, let us note that the value s = I, on which we shall particularly concentrate, probably gives the most realistic description of Meynell's experiment. 2. The probability model The model is a continuous-time Markov process with discrete states. Let N,(t) be a random variable denoting the number of elements with i marks at time t, i = 0, 1,2, ..., s. Let n = (n0,n1, l , n.., ), Iu = (Uo, u1, *,us), N(t) = (No(t), N(t), --,N,(t)), and suppose N,(O) = ai, i = 0, 1,2, --, s. Let P(n; t) = P(N(t) = n), then corresponding to death we have the transitions, (No, ..., Ni, ---, N,) (No, --, Ni 1, , N,) at probability rate pNi, i = 0, 1,2, ,s, while corresponding to birth we have the following transitions (No, N,) ,(No + 1, .., N,) at probability rate, 2 =1 N 2-' +,ANo, (No, , Ni ,N2i,..., X (No, Ni + 2, , N2iI, at probability rate, This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms On the distribution of inanimate marks over a linear birth-and-death process 425 where we use [is] to denote the integral part of is, and (No, " , Ni, * * *, Nk i Nk, "', Ns) -*(No, ..., Nj + 1, ..., Nk-i + 1, ..., Nj 1, ..., Nl) at probability rate 2N 2-k' i = 1,2,...,C[(k 1)], k = 3,4,...,s (smaller k give transitions already considered above). It is readily verified that the sum of the transition rates is (A + yp) X = oN,, as required in the overall birth-and-death process. Introducing the operator Di, which increases the argument ni of a function of n by 1, we then deduce that the Kolmogorov forward equations (for all possible n) are, d S S (1) +(n; t) P(n; t)(A+ t)1 (n2 + D,P(n;t)(n +1) dt ==0 1=0

We analyze a simple, deterministic model of the dynamics of population changes in a bisexual, reproductive system based on marriage. Our model is one of a general class, special cases of which have been previously discussed within the framework of population biology by D. G. Kendall, L. A. Goodman, J. H. Pollard, and others. Here, we extend and complete previous analyses of systems characterized by first-degree homogeneous, unbounded marriage functions, allowing for arbitrary birth and death rates. The dynamics of the model is determined by three coupled first-order, nonlinear differential equations, similar to those used in the description of chemical reactions and of radioactive decay chains. Solutions of the differential equation system are classified according to the associated patterns of birth and death rates of the two sexes, and growth and stability properties are discussed. This preliminary report gives conditions sufficient to insure the existence of a unique, exponential mode of population growth or decay, with a finite ratio of the sexes. We also exhibit other conditions which, in contrast to the standard, linear demographic analysis of Lotka, guarantee that the sex ratio asymptotically becomes infinite. The model manifests a delicate balance between the vital parameters that alerts one to the possibility of selfaggravating distortions of the sex ratio, once a monogamous society's fertility falls below the replacement value.

1. SUMMARY In this paper the main discussion is concerned with obtaining asymptotic results for sequences of birth and death processes which are similar to the central limit theorem for sequences of univariate random variables. The motivation is the need to obtain useful approximations to the distributions of sample paths of processes which arise as models for population growth, but for which Kolmogorov differential equations are intractable. In the first section, univariate processes are considered, and conditions are given for the weak convergence of Z N (t) = {X N (t) - aN}/N, where {X N (t), N = 1,2,…} is a sequence of ergodic birth and death processes, to those of an Ornstein-Uhlenbeck process N → ∞. A heuristic method is given which may help explain why this convergence holds, and some examples are given for purposes of illustration. The second part deals with multivariate processes, and three examples are considered in detail: a model for the growth of the sexes in a biological population, a multivariate Ehrenfest process, and a model for the growth and interreaction of two cities. The paper concludes with a discussion of various related results. It is shown that in certain special cases it is possible to obtain diffusions other than the Ornstein-Uhlenbeck process as limits. Finally, heavy traffic results are included for congestion situations originally considered in the special case of time-homogenous arrival rates by Kingman. Transient processes such as epidemics are also shown to exhibit a “central limit” behavior.

A large sample theory for birth and death queuing processes which are ergodic and metrically transitive is applied to make inferences about arrival and service rates. Likelihood ratio tests and maximum likelihood estimators are derived for simple models. Composite hypotheses, for example that the arrival rate does not vary with the number in the system, are considered. A numerical example illustrating these results, based on data generated by simulating a known model, is presented, which includes the testing of both true and false composite hypotheses.

In the applications one is given the matrix A and it is required to construct P(t) and to study the properties of the corresponding stochastic process. The existence, uniqueness, and the analytic properties of P(t) have been discussed in detail in [I]. The objective of this paper is to use the results of [1] to establish equivalences between properties of the stochastic process and properties of the sequences { Xn }, { n,u } and to evaluate, in terms of these sequences, some of the interesting probabilistic quantities associated with the process.

1. The development of populations of Daphnia magna was followed at two different constant temperatures. Sixteen populations were maintained at 18° and 21 at 25° C. The 50 cc. of pond water which served as medium were renewed every other day and always contained an excess quantity of the food-alga Chlorella.2. Population development at 25° proved oscillatory in nature, four peaks occurring in 234 days, with a maximum population size of 126 animals. In the 174 days of observation at 18° , one major peak was observed (maximum 241) followed by a decrease and virtual stabilization at a population density of about 135.3. Analysis of the oscillation disclosed that it is due to a delay in the expression of the effects of population density upon birth and death rates.4. The mechanism of oscillation at 25° is an alternation of fluctuations in numbers of births and numbers of deaths. The mechanism at 18° is the fluctuation in the number of births about a nearly constant number of deaths.5. Experiments with a series ...

The theory of fluctuations is extended to nonlinear systems far from equilibrium. Systems whose evolution involves two separate time scales, e.g., chemically reacting mixtures near a local equilibrium regime, are studied in detail. It is shown that the usual stochastic description of chemical kinetics based on a “birth and death” model is inadequate and has to be replaced by a more detailed phase-space description. This enables one to develop for such systems a plausible mechanism for the emergence of instabilities, in which the departure from the steady state is governed by large fluctuations of macroscopic size, while small thermal fluctuations are still described by a generalization of Einstein's equilibrium theory. On the other hand, far from a local equilibrium regime, infinitesimal fluctuations may increase and attain macroscopic values. In this case the system evolves to a state of “generalized turbulence”, in which the distinction between macroscopic averages and fluctuations becomes meaningless.

A population projection is a prediction of a random vector variable X T . which represents the size and age/sex distribution of the population in year T. The population is assumed to be closed and to develop according to fixed and known schedules of birth and death probabilities as a multitype branching process. The precision of the usual projection e T (= EX T) is studied by a family of prediction intervals of linear functions of the vector of deviations X T — e T , which has a preassigned probability level. This family is obtained by a multi-normal approximation and an argument similar to the one leading to Scheffe's method of multiple comparison. From the family of prediction intervals, an upper limit of the total absolute deviation Σ |X iT − e it | is obtained, and the ratio of this limit to the projected total population is proposed as a measure of the relative precision of the projection. For a numerical study, Norwegian population data is used.

SUMMARY In this paper we determine, for an individual of age x existing at some specified point in time, the probability p(x) that his line of descendants will eventually become extinct, for the age-dependent birth-and-death process studied by D. G. Kendall (1949) and Bartlett (1955) in which the time-scale and age-scale are continuous, and also for the analogous agedependent birth-and-death process earlier studied by Bartlett and by Pollard (1966) in which the time-scale and age-scale are discrete. The extinction probabilities are also determined for generalizations of these processes in which, in a unit time-interval in the discretetime, discrete-age process (or in a small time-interval in the continuous-time, continuous-age process), a given individual may produce more than one new individual (multiple births), and his chances of producing new individuals may depend, in ways other than those specified for the original processes, on whether he does or does not live through the time-interval. In addition, the extinction probabilities p(x) are determined for a multiphase birth-and-death process, which is a generalization of the multiphase birth process studied by Kendall (1948 a), in which an individual may move through a number of different phases during his lifetime, and the phase in which he is at a specified point in time may affect his chances of producing new individuals and his chances of dying at that time. For each of these processes, and for any value of x > 0, we present an explicit expression for the extinction probability p(x) in terms of p(O), and we determine p(O) from a single equation.

SUMMARY The processes discussed in this paper are models for populations of independent identical individuals reproducing by fission, with the alternative possibility of death. The probabilities for birth and death are considered to be age-dependent, so that under suitable assumptions the populations have a probability to become extinct, which is neither zero nor one. It is the purpose of the present paper to study conditional probabilities for such populations where the condition is that they do ultimately become extinct. This is done by working in terms of a space of family trees, and the results include conditional probabilities for the life-lengths of individuals and the number of their offspring. An example is given, and an application is made to a model of bacterial toxicity. For a Markov process possessing a set of transient states, and one or more sets of absorbing states, there is a simple method for deriving probabilities conditional on absorption in a given set, from unconditional probabilities for the same process (Breny, 1962; Kemeny & Snell, 1960, p. 64; Waugh, 1958). This method takes a particularly simple form for the Markovian binary-fission-or-death branching process when the condition is that of ultimate extinction, and it has been applied in this form to a model of the toxic effect of a bacterial invasion (Puri, 1966). In the present note, a related method is developed for nonMarkovian age-dependent processes. The results take rather simple forms when expressed in terms of the distributions, conditional and otherwise, of the life-lengths of individuals and of the number of their children. In the process discussed in this note the life-length distributions may take general forms, whereas in the Markovian case they are all negativeexponential. An example for Erlangian life-lengths is given, and the method is used to extend results of Jagers (1967) for the integral of the population size. Incidentally, Kendall (1966) has shown how to obtain conditional probabilities for Markovian birth processes that tend to infinity, where conditioning is on the random variable W = lim {Zt1f'(Z1)}, Zt being the population size at time t. The variable W is discussed

This paper presents the theory of a multivariate birth-and-death process and its representation as a branching process. The bivariate linear birth-and-death process may be used as a model for various epidemic situations involving two types of infective. Various properties of the transient process are discussed and the distribution of epidemic size is investigated. For the case of a disease spread solely by carriers when the two types of infective are carriers and clinical infectives the large population version of a model proposed by Downton (1968) is further developed and shown under appropriate circumstances to closely approximate Downton's model.

The random processes discussed here may be specified in the following way. A fixed population of N members is spilt into two distinct classes. Individuals move about randomly between the classes, and we are interested in the size of each class at any time, rather than in the behaviour of particular individuals. Let i(t) and N —i(t) be the numbers present in the repective classes at the time t. It is assumed that the process {i(t), t ≧ 0} is Markovian, and that transitions between the states j = 0, 1, … N, occur according to the conditional probabilities; and.

SummaryTwo cases of multiple linearly interconnected linear birth and death processes are considered. It is found that in general the solution of the Kolmogorov differential equations for the probability generating function (p.g.f) g of the random variables involved is not obtainable by standard methods, although one can obtain moments of the random variables from these equations. A method is considered for obtaining an approximate solution for g. This is based on the introduction of a sequence of stochastic processes such that the sequence {f(n)} of their p.g.f.'s tends to g as n → ∞ in an appropriate manner. The method is applied to the simple case of two birth and death processes with birth and death rates λi and μi, i = 1,2, interconnected linearly with transition rates v and δ (see Figure 2). For this case some limit theorems are established and the probability of ultimate extinction of both the processes is considered. In addition, for the special cases (i) λ1 = δ = 0, with the remaining rates time dependent and (ii) λ2 = δ = 0, with the remaining rates constant, explicit solutions for g are obtained and studied.

The relationship of the demographic parameters of a population to its ecological niche constitutes one of the central problems of population biology. A most interesting theoretical notion pertinent to this problem is r- and K-selection (MacArthur 1962; Cody 1966; MacArthur and Wilson 1967; Hairston, Tinkle, and Wilbur 1970; Roughgarden 1971). The central idea of r- and K-selection is that populations living in environments imposing high density-independent (D.I.) mortality (r-strategists) will be selectively favored to allocate a greater proportion of resources to reproductive activities at the cost of their capabilities to propagate under crowded conditions, and conversely, populations living in environments imposing high density-dependent (D.D.) regulation (K-strategists) will be selectively favored to allocate a greater proportion of resources to nonreproductive activities, at the cost of their capabilities to propagate under conditions of high D.I. mortality. From the argument just stated, it may be deduced that the birth rate of an r-strategist will be greater than that of a related K-strategist. However, increased birth rate under conditions of high D.I. mortality is not sufficient evidence for an r-strategy, because, as demonstrated later in this paper, any increase in D.I. mortality must by itself produce a new equilibrium of birth and death rates at higher values of both. The crucial evidence needed for r- and K-selection is whether an organism is allocating a greater proportion of its resources to reproductive activities (r-strategists) than another related one (K-strategist) under any and all D.D. and D.I. mortality conditions.

Two simulation models were used to predict species changes in three deciduous forest cover types in New England. A fixed—rate simulator, employing differences among species in birth/death rates, was used for stands approaching a steady—state condition. A density—related simulator, using birth and death rates related to both species and population size, was applied to stands in dynamic condition. Predicted and actual changes over a 25—year period were reasonably consistent. See full-text article at JSTOR

It is shown that the set of Markov random fields and Gibbs states with nearest neighbour potentials are the same for any finite graph. The set of Markov random fields is also shown to be the same as the equilibrium states of time-reversible birth/death processes with nearest neighbour interactions defined on the graph.

This subject was first considered statistically by the late Dr Farr. It is one of the brilliant attempts to extract the real meaning of figures so frequent in his work, but though this theory has not shared in the complete neglect that has been the lot of his attempt to put a quantitative measure to the course of epidemics, it has suffered as much from the kind of patronage with which it is usually discussed. On at least one of the great medical officers of health of his time, however,— the late Dr J. B. Russell—the theory exercised a strong fascination. My own copy of Fare's Vital Statistics came from Dr Russell's library, and the whole passage referring to the law is lined with his characteristic nervous pencil marks, while in much of his work on vital statistics the influence can easily be traced.

The mapping of ecological and selective events during the process of genetic change in natural populations is presented. The representation rests on a method for estimating the amount of population control arising from the birth and death effects at each age. The theoretical treatment is then applied to three current problems in evolutionary ecology to illustrate the difficulty and magnitude of such problems. The three problems treated are (1) the evolution of complex life cycles, (2) the evolution of colonizing ability, and (3) ecological convergence and divergence in the evolution of similar species coexisting in the same local species ensemble.

A simple model for a bivariate birth-death process is proposed. This model approximates to the host-vector epidemic situation. An investigation of the transient process is made and the mean behaviour over time is explicitly found. The probability of extinction and the behaviour of the process conditional upon extinction are examined and the probability distribution of the cumulative population size to extinction is found. Appropriate circumstances are suggested under which the model might possibly be applied to malaria. The host-vector model is classified within a general class of models which represent large population approximations to epidemics involving two types of infectives. BIVARIATE; BIRTH-DEATH PROCESS; CONDITIONED PROCESS; EXTINCTION; POPULATION; HOST-VECTOR; MALARIA; EPIDEMIC MODEL; CARRIER-BORNE DISEASE

To explain the growth of interacting populations, non-linear models need to be proposed and it is this non-linearity which proves to be most awkward in attempts at solving the resulting differential equations. A model with a particular nonlinear component, initially proposed by Weiss (1965) for the spread of a carrierborne epidemic, was solved completely by different methods by Dietz (1966) and Downton (1967). Immigration parameters were added to the model of Weiss and the resulting model was made the subject of a paper by Dietz and Downton (1968). It is the aim here to further generalize the model by introducing birth and death parameters so that the result is a linear birth and death process with immigration for each population plus the non-linear interaction component. Consider two populations which we refer to as type 1 and type 2. We suppose that at time t there are Xi(t) individuals of type i present in the habitat and X;(0) = min is the initial number. Further let the birth, death and immigration rates for population i be i, ~i and vi respectively. It is then supposed that the probability of increasing the type i population by one individual during the time

A majority of the papers appearing in statistical journals are concerned with development of statistical methods appropriate to particular theoretical models. Careful study of these publications is rewarded by a constantly growing repertory of well-founded techniques. However, for many practising statisticians, a recurrent problem is that of choosing a model appropriate to a given situation. It is important that the model shall be a sufficiently good approximation to reality, and yet be simple enough to yield practicable statistical techniques. Professor Cramer's paper provides an invaluable survey of possibilities open to a statistician faced by the need to choose a suitable model. The paper is based on the concept of 'stochastic processes' involving random variables in 'infinitely-dimensional space'. This does not mean, of course, that situations in which infinitely-many variables are measured are to be considered; rather, the values of random variables dependent on time, which latter variable is supposed capable of being given infinitely many values (in any time interval), are studied. It is possible to gain useful insight into the available models without a deep understanding of the nature of 'infinitelydimensional space' though this might be helpful. The following brief notes are intended as a guide to help gain a rapid appreciation of the useful results described in the paper. Sections 1-5 are generally of a historical nature. Section 6 contains important definitions: it should be noted that weak (second order) stationarity is defined in terms of moments only, not of distributions. The first part of Section 7 contains a lucid introduction to the nature of the Wiener process. The latter part of this Section may be found rather more difficult on first reading. Sections 8 and 9 on the Poisson process and generalizations thereof, are interesting reading. Quite probably the reader will recognize the 'risk process' of Section 9 in settings other than those mentioned by Professor Cram6r. Section 10 goes into little detail but is worth reference by those with problems likely to call for this kind of model. Section 11, and its specialized sequel, Section 12, contain a varied assortment of models. Here will be found the most intense concentration of potentially useful models in the paper, including the most common 'accident-proneness' model (part 3 of section 11) and the 'birth and death' and 'busy time' models of section 12. For most readers really pressed for time Sections 8-12 probably represent the best investment of their time. For those with more time, or with special interests in analysis of data of 'time-series type' the later sections of the paper, which are concerned with more formal 'stochastic process' theory, will well-repay reading. Particular mention may be made of the description of the Ornstein-Uhlenbeck process (section 14), and the integral representation ((23), section 16) of a stochastic process. N. L. Johnson

An important ecological goal is understanding patterns of coexistence and exclusion of species in the same geographic area. Because most multispecies ecological systems are exteremely complex, it is probably impossible to describe all the component populations in terms of their physiological status or demographic characteristics such as age distributions and birth and death rates. One current method of studying such ecological systems is to design a generalized model, based on a few premises or simplifying assumptions, which will generate testable predictions. If, upon being tested, these predictions are falsified, the simplifying assumptions are changed accordingly; but if the predictions are verified, the assumptions are often considered to be substantiated. This hypothetico-deductive approach has influenced ecologists to focus on more general questions rather than on specific factors which may have small effects and which, when applied to more general situations, lead to intractable mathematical equations. Here I emphasize, with Levins (1966) and Cohen (1968), that the use of generalized models can be misleading because of uncertainty as to whether a positive result depends on the reality of the model and the simplifying assumptions, or on entirely unrelated factors. I therefore urge that, whenever possible, simplifying assumptions themselves be tested as hypotheses and resultant models be based on a thorough understanding of the relevant natural history and, when possible, on experimental manipulation. Here I will test predictions generated by two models which might be used in the construction of a general competition-based theory of the structure of an intertidal community. In both cases, the predictions generated by these two models are verified, suggesting that their simplifying assumptions are based on parameters sufficient to describe the community process in question. In both cases, however, acceptance of the superficially plausible models is due to inadequate appreciation of natural history. Simple experimental manipulation shows that each model makes the right prediction for the wrong reason.

The development of a population over time can often be simulated by the behavior of a birth and death process, whose transition probability matrix P(t) = (Pij(t), where X(t) denotes the number of individuals at time t, satisfies the differential equations and the initial condition

A model proposed by Bailey (1968) for migratory individuals which reproduce according to a simple birth-death process is generalized to include time dependent birth and death rates. 1. The model Let I denote the set of all integers. We assume that an individual of the population can be at any point x e I. The individuals develop in numbers according to the generalized birth-death process of Kendall (1948), with birth rate A(t) and death rate #I(t), 0 < t < oo. They also undergo changes in position according to a simple random walk in continuous time, i.e., if an individual is at a position x e I at time t, then in (t, t + At) it may move to (x + 1) or to (x - 1), the probability of each shift being oAt + o(At). Here A(t), g(t) are functions of time only and a is a positive constant. The position of the newly born individual is, at birth, the same as that of its parent. The individuals act independently of each other and the probability of a multiple event in (t, t + At) is o(At). The above assumptions determine the stochastic process which is a multiplicative point process, (cf. Moyal (1962)). The individuals of the population are assumed to be indistinguishable. The development of the process can be described, as in Adke and Moyal (1963) or Adke (1964), in terms of the generating function: 00oo

SUMMARY The effect of migration between a finite number of colonies each of which undergoes a simple birth and death process is studied. The first two moments are obtained for the general process and deterministic solutions are developed for several special models including the finite linear model proposed by Bailey (1968). In a recent paper Bailey (1968) constructs a model for spatially distributed populations by considering a population to be composed of an infinite number of colonies situated at the integer points of a single co-ordinate axis represented by -xo < i < x. Each colony is assumed to be subject to a simple birth and death process with birth and death rates A and Iu respectively, and with migration rates to each of the two neighbouring colonies. He derives the mean number of individuals in each colony at time t together with the variances and covariances, and briefly considers the corresponding models where the population is extended to two and three dimensions. Clearly by assuming the existence of an infinite number of colonies he avoids the problem of 'edge effects' at the boundaries when the number of colonies is finite. We shall later examine this finite analogue of Bailey's model and show how his results for the infinite model follow as a special case. His paper has recently been extended by several authors. Adke (1969) generalizes Bailey's process to include time-dependent birth and death rates whilst Usher & Williamson (1970) use discrete time intervals to analyze the model when the number of colonies is finite. They consider the population to be split into migrants and nonmigrants, each group having different birth and death rates. Davis (1970) presents some results for a general Markov branching-diffusion process and then applies them to Bailey's model. Crump (1970) studies a general age-dependent branching process in which the population is distributed in N colonies with migration between them and he obtains asymptotic expressions for the first two moments in several special cases. The stochastic equations for the general parameter case of the simple birth-death-migration process are considered by Puri (1968).

We shall consider a many server (multiple channels in parallel) queueing process in which customers arrive at the queue according to a Poisson process. The service times are assumed to be independent and exponentially distributed. As usual, the service times are independent of the arrival process. We assume that no server is idle if there is a customer waiting, but that otherwise the service discipline is arbitrary. If there are n servers and we let Xn (t) denote the number of customers waiting or being served at time t, then it is well known that Xn (t) is a birth and death process with stationary transition probabilities. A very comprehensive analysis of this many server queue from the point of view of birth and death processes has been carried out by Karlin and McGregor [5].

A birth-and-death process taking immigration into account is considered. The complete solution of the equation governing the generalized birth-and-death process when the birth and death rates λ(t)and μ(t) and also the immigration factor ν(t) may be any specified functions of the time t is given. This solution can be applied to the fluctuation problem in electromagnetic showers. The distribution function of the number of particles at given depth t with given initial energy is derived. The results obtained are compared with recent experimental data, and the agreement is much better than in past theoretical works.

A technique is developed for finding the time dependent operating probabilities used by reliability systems designers for provisioning a system with N + k identical units, k of which are called spares and N called operating units, and s repair facilities. System failure occurs when less than N units are operational. Units fail with exponential interfailure times and are repaired with exponential service time. Idle spares fail due to deterioration at a rate possibly different from that of the operating units. Graphs are presented which show the minimum numbers of spares needed to achieve system reliabilities of 0.90 and 0.99 as a function of time. The technique is applicable for finding, numerically, the first passage time distribution for any system modeled by birth and death processes.

General mathematical problems arising in the scientific study of predation have been studied from a variety of viewpoints. Primary emphasis has been given to the descriptive aspects of the prey and predator populations under various assumptions concerning interactions among the different members of the populations during these processes and to birth and death processes. The major objective of this note is to show how the functional equation technique of a new mathematical discipline, dynamic programming, can be used in formulating and solving--both analytically and numerically--a variety of problems of optimal predation. We wish to determine optimal predation policies and are thus interested in the control, as opposed to the descriptive, aspects of predation processes.

This article reviews various analytic and computer simulation models of population growth with a special emphasis on biological factors. These models serve 3 main purposes: 1) to describe the past as simply and concisely as possible 2) to predict future populations and 3) to consider the possible effects on future population of alteration in influential characteristics. Over the past 50 years demography has included an increasing number of factors as relevant and necessary to achieve a realistic description of population processes. Growth curve models -- an early way to predict population growth -- were all based on the assumption that change in population size was a constant function of its current total size. Sharpe and Lotka and Leslie introduced age-specific birth and survival rates into their mathematical models. Lotka went on to develop the stable population theory inquiring into the effect of birth and death rates which remained constant over time. The next step in developing demographic models was to treat mortality and fertility rates as dependent variables and examine their determinants. Since fertility far more than mortality is the controlling factor in population size and age there has been an increased interest in the determinants of birth rates such as age at marriage and the spacing of children. Henry and Sheps showed that the birthrate approaches the inverse of the mean birth interval. These models are concerned with a small number of interacting parts of the reproductive process rather than with all parts of the demographic process. Yet they provide insights into the relationships between these factors and birthrates. They also provide an improved basis for certain types of prediction. At least 3 different types of extensions from this work on models have been developed 2 in the direction of more complex sets of assumptions about the biological factors. In the third mathematical demographers have incorporated simplified versions of the models for biological aspects of reproduction into analytic models that also consider other determinants of population growth such as marriage and mortality. The construction of more elaborate models would be a useless exercise unless further research provided crucial information on the aspects of reproduction and mortality in humans which affect population characteristics.

In recent years it has been found that in vitro many ferments, etc., lose their power of action at a rate corresponding to the monomolecular reaction, i.e. if the amount of the substance present at the beginning is denoted by unity, and, if after a certain period of time only one half of the substance remains active, then it may be predicted that at the end of an equal period of time, one half of one half, or only one quarter of the original amount, will still retain its power of a.ction. It is not necessary to give examples, they are sufficiently well known.

T HE 1JANDLING of data with high-speed electronic equipment suclh as computers with a "memory" core has come to be referred to as "automatic data processing." Such methods are well accepted in industry and the larger research institutions, but are not yet fullv exploited in public hea,lth and medical research. Vital statistics and other administrative health data are often required with sufficient frequency and detail to justify these newer methods even when the volume of data is small. Examples are birth and death statistics, hospital admissions and discharges, and rates, percentage distributions, life tables, and other computations based on these data. In addition, the efficiency and versatility of electronic computers enable the statistician to conduct many types of analyses of basic health data that would not have been practical before. One type of computer application is to case r egisters of chronic disease. By a chronic disease we mean "one which persists over a long period of time, sometimes even for most of the individual's life. . . . It is the changeability and variation, not stability, that is in fact the dominant characteristic of most long-lived conditions" (1). In a register we attempt to study the dynamic characteristics of a chronic disease by systematically following identified cases over long periods of time. This followupl, dynamic, or longitudinal aspect distinguishes the disease register from the disease index. The index is merely a file of diagnosed cases which have been identified at some time and do not reflect the current status of a disease. A variety of chronic disease registers have been established for cancer, tuberculosis, and rheumatic fever (2-8). Registers differ according to (a) principal purpose-medical service, health supervision, research; (b) location-in the clinic, hospital, health department, bjiometrics office; (c) type of initial and followup data collected and manner of collection; and (d) extent to which all diseased individuals in a community or defined population have been identified. We shall discuss certain common features of these registers and illustrate the use of the computer for a psychiatric case register.

~ (λt + [h)t + o(t) if j = i where Xt > 0 for i > 0, μt > 0 for i > 1 and μ0 > 0. We further assume that Pu(ί) satisfies the foward and backward equation in the usual form. In this paper we restrict attention to the case μ0 — 0 so that when the particle enters the state zero it remains there a random length of time according to an exponential distribution with parameter λ0 and then moves into state one etc. In order to avoid inessential difficulties we assume henceforth that the infinitesimal birth and death rates Xt and μi uniquely determine the process. This is equivalent to the condition Σ~-o (πn + l/λn7Γn) = °° where

In a large number of cases the curves which describe the rate at which ferments, antibodies, etc., disappear in the body accord in form with the equations of physical chemistry, but not so as to be readily interpretable. If one molecule, two molecules, etc., take part in a reaction then it is clear that the indices in the equations must bear certain relations to whole numbers. Failure to conform to these relations seems at first sight to imply that a fraction of a molecule takes part in a reaction, but this is not necessarily the case. Such a failure has been found to appear when the curves of many vital phenomena, e.g. the disappearance of agglutins in an organism, are fitted to physico-chemical equations. A possible meaning, however, can easily be seen.

The importance of stochastic processes in relation to problems of population growth was pointed out by W. Feller [1] in 1939. He considered among other examples the "birth-and-death" process in which the expected birth and death rates (per head of population per unit of time) were constants, $\lambda_o$ and $\mu_o$, say. In this paper, I shall give the complete solution of the equations governing the generalised birth-and-death process in which the birth and death rates $\lambda(t)$ and $\mu(t)$ may be any specified functions of the time. The mathematical method employed starts from M. S. Bartlett's idea of replacing the differential-difference equations for the distribution of the population size by a partial differential equation for its generating function. For an account of this technique,$^1$ reference may be made to Bartlett's North Carolina lectures [2]. The formulae obtained lead to an expression for the probability of the ultimate extinction of the population, and to the necessary and sufficient condition for a birth-and-death process to be of "transient" type. For transient processes the distribution of the cumulative population is also considered, but here in general it is not found possible to do more than evaluate its mean and variance as functions of $t$, although a complete solution (including the determination of the asymptotic form of the distribution as $t$ tends to infinity) is obtained for the simple process in which the birth and death rates are independent of the time. It is shown that a birth-and-death process can be constructed to give an expected population size $\bar n_t$ which is any desired function of the time $t$, and among the many possible solutions the unique one is determined which makes the fluctuation, Var$(n_t)$, a minimum for all. The general theory is illustrated with reference of two examples. The first of these is the $(\lambda_0, \mu_1t)$ process introduced by N. Arley [3] in his study of the cascade showers associated with cosmic radiation; here the birth rate is constant and the death rate is a constant multiple of the "age, $t$, of the process. The $\bar n_t$-curve is then Gaussian in form, and the process is always of transient type. The second example is provided by the family of "periodic" processes, in which the birth and death rates are periodic functions of the time $t$. These appear well adapted to describe the response of population growth (or epidemic spread) to the influence of the seasons.

Previous authors have used simple models to investigate the relative importance to population increase of variations in the total and age-specific reproductive rates. But while acknowledging that the latter were the product of the age specific birth and death rates, they have used their models only to investigate changes in total or age-specific birth rates and have not been concerned with variations in death rates. This paper extends the use of Lewontin's (1965) model, to a wide range of values of r, the exponential rate of population increase. It shows how the relative importance of changes in certain life-history features can change with r and be reversed when r is near to zero. It is also shown that variations in mortality rate are not necessarily best expressed in analogous terms to variations in birth rate. If more suitable terms are used it is seen that changes in mortality rate can be of varying importance depending on the existing mortality rate. They can be overwhelmingly important when the mortality rate is high.