## Abstract

The notion that the nature of a measurement is critical to its outcome is usually associated with quantum phenomena. In this paper, we show that the observed statistical properties are also a function of the measurement technique in the case of simple classical populations. In particular, the measured and intrinsic statistics of a single population may be different, while correlation and transfer of individuals between two populations may be hidden from the observer.

## 1. Introduction

In most scientific disciplines, processes underlying the behaviour of both natural and man-made complex systems are inherently discrete. Such processes range from the interaction of particles of matter and energy in physics and chemistry to that of individual organisms in biology. A similar situation exists in fields of human activity such as finance (Scnerb *et al*. 2000) and perhaps most obviously in the behaviour of human populations. Although a range of discrete stochastic models are available for modelling the underlying processes (Shepherd 1981; Matthews *et al*. 2003; Jakeman *et al*. 2003), macroscopic phenomena are often a consequence of the interaction of many individuals. It is then found that continuum or mean field approximations, in which noise or randomness is introduced as an external perturbation, provide a more tractable mathematical approach and these theories have successfully explained the observed behaviour of many large systems. However, such an approach is not necessarily suitable for describing the interaction of small numbers of individuals and indeed may overlook new effects that can arise or even make erroneous predictions. Situations of this kind have been encountered in quantum optics. For example, it has long been known that single-photon effects provide the ultimate limitation on optical measurement accuracy. More recently, recognition of the discrete nature of light has led to the development of novel light sources and applications such as quantum cryptography. It is self-evident that the discrete nature of fluctuations in a small population becomes an important consideration in the evolution and ultimate survival of interacting chemical or biological species. In this context, the intrinsic noise of discrete models will often provide a more realistic description than the more familiar additive noise introduced in continuum approximations. Furthermore, measurements on small populations may be sensitive to the method of observation. For example, if a measurement involves the deliberate removal of individuals, as for example in photon absorption, the evolution of a small population will be disrupted. Alternatively, if the individuals leaving a population in a natural way are monitored, then the information so obtained may not accurately reflect the internal evolution of the parent population.

With the above considerations in mind, this paper will explore the predictions of some simple stochastic population models that illustrate how the discrete nature of small populations is manifest in the way they evolve and in the process by which they are measured. The dynamics of both the population and measurement processes can be obtained by straightforward analytical methods for these models.

Most of our deliberations will be concerned with measurements on a system comprising two populations between which individuals may transfer. The simplest case considers that of a conservative system where an individual leaves one population and enters the other, but in the more general case individuals may also leave either population for the outside world or transfer in from it. Two kinds of measurement on the resulting populations will be investigated. In the first, ‘internal’ sampling (counting) is carried out within the population at a prescribed rate. For short sample times this measurement accurately reflects the fluctuation characteristics of the population, while for long sample times the fluctuations are averaged during the sampling period. In the second, ‘external’ measurement, the train of events caused by individuals leaving the population is sampled. It is found in this case that the measurement may not be sensitive to the fluctuation characteristics of the population nor reveal the existence of the transfer process which is therefore effectively hidden from this measurement technique.

Section 2 reproduces the basic theory of the death–immigration process and shows how information obtained by the two observation methods differs. The analysis is straightforward and the monitoring results have indeed appeared in various forms in previous publications (Jakeman & Shepherd 1984). Nevertheless, this section serves to introduce the formalism and provides a starting point for the calculations reported in the following sections. Section 3 presents an analysis of the two-population exchange model where there is no interaction with the outside world, while §4 solves the more general model of two death–immigration populations that are coupled through the transfer of individuals between them. The results obtained, and discussions thereon, are contained in §5. Mathematical details are assigned to an appendix of the electronic supplementary material.

## 2. Death–immigration model

This paper considers a number of different population models and for each of these there is defined a primary and secondary process. The primary process relates to the population itself and describes how the population *size* changes according to the rate (or Kolmogorov) equations (Cox & Miller 1995). These processes represent a view of the population's evolution that could be sensed only from the privileged vantage point afforded to a ‘virtual’ observer who does not interact with the population in any way. The secondary process accounts for the additional sampling or monitoring of this primary population and must necessarily be susceptible to the fluctuations that occur within it. The monitoring occurs over an interval of time, the sample time, and it is in this way that a time series of events is formed from the primary process. The secondary process therefore gives rise to a sequence of events in time, being driven by the primary process. The different processes by which these observations are made will be referred to as internal or external monitoring.

External monitoring describes a sampling process whereby an individual is removed from the population once it has been counted, and can therefore be regarded as a movement of individuals and information from inside the population to the outside world. External monitoring models are relevant to the detection of photons in quantum optics (Shepherd 1981), where it is more commonly referred to as ‘destructive counting’ since photons are annihilated through their detection.

By contrast, internally monitored individuals remain within the population there being a transfer of information only. Internal monitoring is more representative of the type of sampling used in biological systems and censuses where individuals are recorded before being ‘released’ back into the population.

It should be noted that either method of monitoring results in a counted population that increases in integer increments, and it is this counted population that is used to infer the properties of the primary process.

### (a) Primary process

One of the simplest discrete population models is the death–immigration model, in which deaths occur at the rate *μN* and immigrations at the rate *ν*, where *N* is the instantaneous number in the population. This model has been studied extensively (e.g. Shepherd 1981; Cox & Miller 1995), and its rate equation is(2.1)where *P*_{N}(*t*) is the probability that there are *N* individuals in the population at time *t*. The problem is solved in terms of a generating function given by(2.2)which transforms the rate equation into the partial differential equationwith solution(2.3)where is the generating function of the initial state. The population reaches an equilibrium for any *μ*>0, with the generating function for the equilibrium distribution given by(2.4)where is the equilibrium mean. The corresponding equilibrium probability distribution is the Poisson distribution,(2.5)The correlation function in this case is of the standard form for a death–immigration process (Shepherd 1981), namely(2.6)It can be seen that as *τ*→∞ the correlation function tends to unity indicating that the process is uncorrelated in this limit.

### (b) External monitoring

The external monitoring process involves counting the number of individuals within the population over time intervals of duration *T* at a rate that is proportional to the number present. Individuals are removed from the population once counted, introducing an additional death process of rate *ηN* into the primary process. In order to compute the fluctuations of the number of counts, it is necessary to introduce the joint distribution *P*_{N,m}(*t*), being the probability that there are *N* individuals in the population at the commencement of the monitoring interval [0,*T*) during which *m* are counted. For a simple Markov model such as that considered here, the rate equation for *P*_{N,m}(*t*) is the same as that for the primary population but with the addition of two terms involving the count rate *η*,(2.7)This is solved using the joint generating function(2.8)which transforms (2.7) into(2.9)subject to the boundary conditions and , where *Q*_{∞}(*s*) is given by equation (2.4) with *μ* replaced by *μ*+*η* to account for the augmented death rate. The solution of equation (2.9) is(2.10)which is a product of two Poisson distribution generating functions: one in the variable *s* parametrized by the mean of the primary process (where the superscript indicates that this mean has been calculated with the augmented death rate), and the other in the variable *z* with mean number of counts(2.11)The mean count number 〈*m*〉^{ext} scales linearly with the counting interval *T* and the normalized factorial moments of the counts are unity. The correlations between the counts occurring in two disjoint intervals of duration *T* separated by a delay *τ* can be calculated to be (see electronic supplementary material, appendix (b))(2.12)Counts are therefore uncorrelated and form a classic Poisson train of events in the sense defined by Cramér & Leadbetter (1967). This is despite the primary process being correlated as given by equation (2.6), and external monitoring fails to register correlations within the population.

### (c) Internal monitoring

For internal monitoring, it is assumed that counted individuals continue to participate in the evolution of the population after they have been counted, and there is consequently no augmentation of the death rate in the primary population. The rate equation for *P*_{N,m}(*t*) in this case is(2.13)which, using a joint generating function of the form (2.8), is equivalent to(2.14)with the same boundary conditions as for the external monitoring case. The solution to this problem is(2.15)and the mean number of counts is given by(2.16)which again scales linearly with the counting interval *T*.

The correlation function is calculated to be(2.17)which in the short sample time limit, *μT*≪1, becomes(2.18)and so for sufficiently short sample times *T*, the correlation properties of the primary process are shared by the internal monitoring count statistics.

The origin of the difference between the two monitoring schemes is captured by the following simplifications. Suppose that the counting intervals are sufficiently short that only a single event may occur in each of them (i.e. *T*≪*η*^{−1}, *μ*^{−1} and *ν*^{−1}), then(2.19)where *P*(*m*, *m*′) is the probability that there are *m* counts in the first interval and *m*′ in the second. If the counting intervals are assumed to be contiguous, then there is no opportunity for the primary population to evolve between them and(2.20)where *N* is the primary population size. The effect of external monitoring is to reduce the primary population size by one for every count, and so(2.21)while for internal monitoring, where the counting process does not diminish the population size,(2.22)Thus, external monitoring yields a value for the correlation function that is less than that for the internal monitoring by an amount (*ηT*)^{2}〈*N*〉. This deficit translates as the difference of between equations (2.12) and (2.18) when *τ*=0. Since the term proportional to contains the time dependence in equation (2.18), the absence of this term accounts for the independence on the delay time *τ* in equation (2.12). Furthermore, since the only process under consideration here is the monitoring, it is the external monitoring process itself that suppresses the term.

Internal monitoring therefore retains more information about the primary process than external monitoring, despite the fact that the counted mean values share an identical dependence upon the mean of the primary process. This is particularly important for the death–immigration model since the fluctuation characteristics of the primary population are lost by external monitoring in this case. However, it shall be seen that this result also has implications for other models to be considered.

## 3. Exchange model

### (a) Primary process

A simple model that illustrates further the perturbative effects of monitoring a system is the exchange model, in which individuals are transferred between two populations, labelled 1 and 2. This process is considered first so that its unusual properties can be distilled and appreciated in isolation of other effects to be included in §4, but whose vestiges remain nevertheless. An individual leaves population 1 (2) at a rate *μαN*_{1} (*μβN*_{2}) proportional to the instantaneous size of the population of origin, and instantly enters the other population. The exchange model corresponds to the schematic shown in figure 1 with *μ*_{1}=*ν*_{1}=*μ*_{2}=*ν*_{2}=0 and has the rate equation(3.1)whose solution is facilitated by the joint generating function(3.2)which satisfies(3.3)The solution to the above (and indeed all subsequent models) is obtained by the method of characteristics outlined in Richards (1959) and is given in full in the electronic supplementary material, appendix (a). At equilibrium(3.4)where *Q*_{0}(*s*_{1}, *s*_{2}) is the generating function corresponding to the initial distribution of individuals within the system. *Q*_{∞}(*s*_{1}, *s*_{2}) is therefore a function of the single variable , a weighted average of *s*_{1} and *s*_{2} that can be interpreted as the sum of the rates at which members from population 1 are converted into those of population 2 and vice versa. Note that although there is mixing of the populations, the form of the generating function is necessarily preserved by virtue of the conservative nature of the process: no individual is gained or lost from the coupled system. This property ensures that the system retains the memory of its initial state at all times despite the shuffling of its constituents.

Taking the initial number in populations 1 and 2 to be and , respectively, the initial generating function is , and the mean of each population is(3.5)At equilibrium and exchange between populations occur at precisely the same rate, therefore having no net effect.

Since there are two distinct populations, there are now several normalized correlation functions(3.6)(3.7)(3.8)and all of these are governed by the same time scale [*μ*(*α*+*β*)]^{−1}, characterizing the circulation of an individual in the system. It can be seen from equations (3.6) and (3.8) that the auto-correlation functions are greater than or equal to unity, indicating that a population is correlated with itself, and that this correlation decays with the delay time *τ* on the time scale of the exchange process. Conversely, the correlation functions (3.7) are always less than or equal to unity, so the number in population 1 is anti-correlated with that in population 2 and vice versa. This must necessarily be so since the total population size is constant, meaning that when one population is large the other must be correspondingly small, and that a decrease in one implies an increase in the other.

For two populations that are uncoupled by an exchange process and are therefore independent of one another, the cross-correlation function (3.7) is unity, as expected. The auto-correlations (3.6) and (3.8) exceed unity due to the exchange process alone, whereas the auto-correlations seen earlier for the death–immigration model were governed by the external death rate, which is zero for the exchange model under consideration here.

The correlation function for the total population, , is(3.9)and displays no temporal variation as expected for a system that conserves the total number of individuals.

### (b) External monitoring

When external monitoring is performed upon the exchange model, the system empties since the exchange process conserves the total number and there is no process to replace the counted individuals that leave the system. Consequently, the equilibrium state is of a completely empty system, and so this process is not considered further.

### (c) Internal monitoring

Internal monitoring is now performed at the rate *η* upon both of the constituent populations of the system separately, yielding counts *m*_{1} and *m*_{2}. This is treated in the same way as in §2 by introducing into the rate equation (3.1) the terms(3.10)and the generating function satisfies(3.11)whose solution is again obtained using the methods outlined in electronic supplementary material, appendix (a). The characteristic time scales are governed by the rates *λ*_{+} and *λ*_{−}, which are the eigenvalues of the system of ordinary differential equations used in solving equation (3.11), given by(3.12)Their interpretation can be gleaned by examining the small count rate regime *η*≪*μ*, *α*, *β*, for which(3.13)(3.14)linearization in *η* representing the approximation that an individual is counted once only. Subsequent terms in the expansion of the eigenvalues for small *η* correspond to additional counts. The five terms in the above expressions for the eigenvalues each have an interpretation in terms of time scales governing the passage of an individual through the system. The *μ*(*α*+*β*) term is independent of *η* and therefore can be ascribed to an individual that is not counted but rather circulates inside the system. This interpretation is supported through noting that *μ*(*α*+*β*) is the total rate of flux between the two populations, as was noted when considering the primary process. The total rate at which counts are made from population 1 is *ηz*_{1} (cf. the *ηz* term in the simple Poisson model earlier), which comprises two contributions. The first term in equation (3.14) carries a weighting of *β*/(*α*+*β*) corresponding to the ratio of the equilibrium mean in population 1 to the total population size, and can be interpreted as the rate at which counted individuals of type 1 emerge directly from population 1. The second term in equation (3.13) is weighted by *α*/(*α*+*β*), corresponding to the ratio of the mean size of population 2 to the total, and this is the rate at which individuals that originated in population 2 at the start of the counting interval are counted from population 1. The two terms in equations (3.13) and (3.14) involving *ηz*_{2} possess a similar interpretation.

When *z*_{1}=*z*_{2}=*z*, the population types are not distinguished and the eigenvalues are(3.15)Following the considerations above, the *ηz* term in the expression for *λ*_{+} above relates to the rate at which individuals are counted from a population different from the one in which they began the counting interval, while the corresponding term in the expression for *λ*_{−} describes the rate at which individuals are counted from their initial population. The external counting eigenvalues can be obtained from those of the internal counting eigenvalues by setting *z*_{1}=*z*_{2}=1 and can be interpreted in the same way, noting that the *ηz* terms now become simply *η*, indicating the destructive nature of the counting. The eigenvalues for external monitoring are linear in *η* since an individual may be counted only once under that scheme.

Since the internal counting scheme does not introduce a death rate from the perspective of the primary population, the population is not depleted by the counting process as in the case of external monitoring, and the total number within the system remains conserved. This is manifest in the mean number of counts(3.16)The four normalized correlation functions, obtained by assuming deterministic initial conditions, are(3.17)(3.18)(3.19)where *Θ*(*τ*, *T*) is given by(3.20)In the small *T* limit, the above correlation functions reduce to those of the primary population, as given by equations (3.6)–(3.8). Internal monitoring again reproduces the correlation properties of the primary population. The deviations from unity in the correlation functions above arise from the exchange of individuals alone, as was the case for the primary process.

Another statistical measure is the probability *w*(*τ*)d*τ* that the time between successive count events lies between *τ* and *τ*+d*τ*. For two populations, the three inter-event times that may be considered are between counts from population 1, counts from population 2 and counts of any type. The densities for these inter-event times are denoted *w*_{1}(*τ*), *w*_{2}(*τ*) and *w*_{t}(*τ*), respectively, and are given by (Lowen & Teich 2005)(3.21)(3.22)(3.23)The last of these is(3.24)and the density for the time between all counts has the single time scale , which is the reciprocal of the total count rate. Since the total number in the system is constant, only this count rate affects the inter-event time, and the *total* number of counts forms a Poisson process.

The forms of *w*_{1}(*τ*) and *w*_{2}(*τ*) are more complex since they are contingent on the fluctuations of the individual population sizes—fluctuations that are absent in the total number. Both *w*_{1}(*τ*) and *w*_{2}(*τ*) have the form(3.25)where *i*=1, 2 and are the eigenvalues (3.12) evaluated at *z*_{1}=1, *z*_{2}=0 for *w*_{1}(*τ*) and *z*_{1}=0, *z*_{2}=1 for *w*_{2}(*τ*). The and are constants that depend only on the parameters of the model: the process rates *μ*, *η*, *α* and *β* and the initial occupation levels and . It can therefore be seen that in contrast to the single time scale observed in *w*_{t}(*τ*), there are time scales in each of *w*_{1}(*τ*) and *w*_{2}(*τ*)(3.26)This multitude of time scales reflects the complexity of the process under consideration. Each member of the population has two time scales, and , associated with it and each of the individuals within the system, as they move from one population to the other, will contribute to the distribution of the inter-event times *w*_{1}(*τ*) and *w*_{2}(*τ*). Since , in the limit , it follows that , and the asymptotic probability densities have exponential tails, mirroring the form of *w*_{t}(*τ*), albeit with different rates.

These considerations are illustrated in figure 2 that shows *w*_{1}(*τ*) (dot-dashed line), *w*_{2}(*τ*) (dashed line) and *w*_{t}(*τ*) (solid line) plotted on a log-linear scale against *τ* for *α*=2, and *η*=0.1. Also shown (as dotted lines) are the appropriate asymptotes given by for *i*=1, 2 from which there is significant deviation for shorter times. This deviation is a manifestation of the many time scales that come into play away from the tail and is indicative of a richness in structure that is lacking from *w*_{t}(*τ*), which has only the single time scale.

## 4. Coupled death–immigration model

### (a) Primary process

In addition to the exchange process, death and immigration processes are introduced to each of the constituent populations of the system to form the coupled death–immigration model. This is shown in figure 1 and has rate equation(4.1)where *μ*_{i} and *ν*_{i} are the death and immigration rates, respectively, for population *i*, and the rate of exchange from population 1 to 2 is *μα*, while that from 2 to 1 is *μβ*. The generating function (3.2) satisfies(4.2)and the solution of this is a product of two independent Poisson generating functions of the type seen in equation (2.4) with mean values given by(4.3)the equilibrium total number being given by the sum of the above. These are independent of and , indicating that the memory preserving property of the exchange process identified in §3 is destroyed by the additional death and immigration processes.

An interesting comparison occurs between two uncoupled death–immigration models where the immigration and death rates are identical with the coupled model, but the exchange rates are zero (*α*=*β*=0). The equilibrium mean values of the uncoupled system are and for populations 1 and 2, respectively, and so(4.4)(4.5)(4.6)where *A* is a positive factor that depends upon the system parameters. With appropriate choices of parameters, the difference between the coupled and uncoupled total means can be negative or positive, whereas the differences between the coupled and uncoupled means for populations 1 and 2 separately always differ in sign. This indicates that the modification of the total arises as a result of manipulating the relative size in the two constituent populations. For example, if population 1 has a high immigration rate and low death rate compared with population 2, by making *α*<*β* it is possible to make individuals more likely to be in population 1 where the conditions are more favourable for a large mean. This necessarily causes the corresponding low equilibrium occupancy in population 2.

The exchange process itself conserves the total number of individuals in the system, but in conjunction with other processes it can lead to a modification of this same quantity. This effect is evocative of constructive and destructive interference of waves, and the exchange process represents the ‘interference’ between one population and the other.

The correlation properties of the coupled system are now considered, and following the approach outlined in electronic supplementary material, appendix (b), the four normalized correlation functions are found to be(4.7)where *i*, *j*=1, 2 and the *A*_{ij} and *B*_{ij} are constants defined in terms of the constant parameters of the system. These are plotted in figure 3 for *ν*_{1}=*μ*_{1}=*μ*=*β*=1, *ν*_{2}=*α*=2. Both auto-correlation functions, given by the solid and by the dashed lines for populations 1 and 2, respectively, decay exponentially away from the value at *τ*=0 towards unity. This accords with the single death–immigration model, although the decay rate is now governed by the additional time scales of the system.

More noteworthy is the cross-correlation function, shown in figure 3 as the dot-dashed line, which peaks at a non-zero value of *τ* indicating the effect of the exchange process. This peak occurs at a non-zero lag time since there is a finite time scale associated with the exchange process. It is also worth noting that although for the parameters used in figure 3, this equality need not necessarily be the case, and the coupled death–immigration model differs in this regard from the simple exchange model of §3. It is the interaction of the exchange process with the dynamics of each population that gives rise to this distinction, in a similar way to the modification of the mean values observed earlier.

### (b) External monitoring

For external monitoring, the count generating function satisfies(4.8)which, subject to the initial condition , has solution(4.9)where and are obtained from equation (4.3) with and . The count distribution for external monitoring is therefore Poisson, with mean values given by and for populations 1 and 2, respectively. This was also seen to be the case for external monitoring of the single death–immigration population in §2, and will also apply to a pair of uncoupled death–immigration populations. Since the number of counts is Poisson distributed for both coupled and uncoupled death–immigration populations, there is no way of distinguishing these using external monitoring. External monitoring of either population, coupled or not, will yield a Poisson distribution of counts parametrized by a single quantity: the mean. This statistic alone cannot tell the observer what type of system is under consideration because it can be derived from both cases with appropriate parameter choices. For example, a coupled population with has mean values , precisely the same as would be obtained from an uncoupled system with *ν*_{1}=*ν*_{2}=1 and *μ*_{1}=*μ*_{2}=1. Since the count distribution is Poisson, all the normalized factorial moments are unity and no further insight can be gleaned from their consideration.

The exchange process couples the two populations and so the correlations might appear to provide a means of distinguishing between the coupled and uncoupled systems. However, both the auto- and cross-correlations are unity, and this is the standard result for a Poisson process such as the one described by equation (4.9). External monitoring eliminates the time dependence of the correlation function of the primary process, and with it the possibility of using the correlations to detect the presence of coupling. Further, since the process is Poisson, the inter-event times are exponentially distributed with a single time scale, and this time scale depends solely upon the mean values that have already been shown incapable of identifying the presence of the exchange process. The inter-event times are therefore unable to differentiate between populations that share the same mean, whether they are coupled or not. The exchange process is therefore invisible to the external monitoring scheme, and can be regarded as a *hidden process* with respect to this counting process.

### (c) Internal monitoring

For internal monitoring, the generating function satisfies(4.10)whose solution is given in electronic supplementary material, appendix (a). Similar to the internal counting case for the single death–immigration population, but unlike the external monitoring case considered above, the resulting generating function does not factorize into a product of Poisson generating functions (one for each of the variables *s*_{1}, *s*_{2}, *z*_{1} and *z*_{2}). The mean values of counts remain of the form and , where and are given by equation (4.3) but the second factorial moments and correlations are no longer Poissonian in nature.

The expressions for the correlation functions are too cumbersome to be informative but are plotted in figure 4 for *T*=0.1, 0.2, 0.3, 0.4 and 0.5 with *ν*_{1}=0.2, *ν*_{2}=0.1, *α*=2 and . It is clear that as *T* becomes smaller, the count correlation function converges upon that of the primary process (shown as the dashed line). As was seen for the single death–immigration process, the internal monitoring scheme is capable of reproducing the correlations of the primary system under examination.

It is apparent that the correlation function is decreasing as the count interval *T* is increased: becoming both lower at its peak and increasing in width. This attenuation represents the fact that as the count interval increases, the correlation function is effectively averaged over a longer interval. In the large-*T* limit, the completely flat form of the Poisson correlation function (shown as the dotted line in figure 3) will be retrieved as this averaging takes place across all time, destroying the correlation.

Since the correlation functions *f*_{12}(*τ*) and *f*_{21}(*τ*) are unity in the absence of coupling, their internal monitoring values *can* be used to identify the presence of the exchange process. The coupling process is visible when monitoring is performed under the internal scheme.

## 5. Discussion and conclusions

The death–immigration process discussed in §1 of this paper has found applications in photon statistics, the photon number distribution for the laser well above threshold being Poisson (Sargent *et al*. 1974; Bulsara & Schieve 1978). In Shepherd (1981), the monitoring problem is tackled in the case of the laser both above and below threshold (where it can be modelled as a birth–death–immigration process), and the equivalence of the classical treatment with the normal ordering of operators within the quantum formalism is established, this work being extended in Jakeman & Shepherd (1984), Srinivasan (1988) and Jakeman & Loudon (1991). This model serves to illustrate a number of general points, the first of which is the importance of the discrete nature of the population under consideration. This quality is manifest in, among other things, the term proportional to in the correlation function. For large populations, characterized by , the correlation function tends to unity, the result obtained in the continuum limit by mean field treatments that neglect fluctuations. The discrete approach is a necessary requirement of any system for which the instantaneous population size becomes small. Such a scenario is likely to arise when considering systems approaching extinction, either in biological populations or chemical reactions. It has been shown (Scnerb *et al*. 2000) that the discrete approach predicts that a population can survive despite the continuum approach predicting its extinction. In addition to simple biological populations, such arguments will bear upon other inherently discrete systems pertaining to finance and genetics among others.

The models considered in this paper are all linear, but the results derived from these take on added significance where nonlinear models are to be considered. Internal fluctuations of the type shown to arise within small populations have the potential to be amplified by the nonlinearity and have profound effects upon the population at a larger scale (Leonard & Reichl 1994). This could potentially invalidate quantitative predictions obtained from the continuum approach even for larger-sized populations.

The impact of monitoring processes is the central theme of this paper. The effect of external monitoring is to suppress the correlations of the primary process by eliminating the term, along with the associated time dependence. This implies that external monitoring forms a Poisson process and the counts registered in two distinct intervals are independent random variables separated by exponentially distributed intervals. This is the result that would be obtained in the continuum limit, and yet the primary process *is* correlated. There is a difference, therefore, between the statistics of the primary process and those of its external monitoring. The internal monitoring is capable of reproducing the statistics of the primary process and therefore also differs from the external scheme.

This discrepancy is most pronounced for small populations and originates from the removal of the counted individual from the primary population during external monitoring, which represents a perturbation to the unmonitored population. In nonlinear systems such as that described in Leonard & Reichl (1994), the fluctuations introduced by the monitoring process have the potential to be amplified by feedback mechanisms and thereby affect the evolution of the primary process.

The count statistics represent the information gleaned from a process by performing measurements upon it. External monitoring of the single death–immigration process yields uncorrelated count statistics. This is precisely the result obtained in the continuum limit and might ostensibly suggest that the continuum limit is valid *regardless of population size*. This is not the case, however, and care needs to be taken when monitoring systems via the external scheme to ensure that erroneous conclusions are not drawn about the primary process from the counting statistics.

For the internal monitoring scheme, the correlation properties of the primary process are retained by the count statistics, which therefore differ from the continuum results, except in the limit . Internal monitoring statistics, when compared with those of external monitoring, contain more information about the primary processes considered in this paper.

The exchange process considered in §3 had the property that the general form of the generating function was unchanged over time despite the mixing of constituents. As a result, memory of the initial conditions is retained in equilibrium. There are also correlations associated with the exchange process: in addition to the auto-correlations that were seen in the death–immigration process, cross-correlations arise between the populations due to the exchange of individuals.

The exchange process can be used to model simple chemical reactions where one reactant changes into another and vice versa. Models of this type are considered in Gómez-Uribe & Verghese (2007) and statistics for the primary process are calculated up to the second moments. The coupled death–immigration model is found in biological systems such as that considered by Brookfield & Johnson (2004). Other instances in which the dynamics of two-population systems are of relevance include stem cell growth where the change from one cell phenotype to another during culturing is of critical importance (Pollock *et al*. 2006; Lu *et al*. 2007). In none of these works is the effect of measurement considered.

When the exchange process is used to couple a pair of death–immigration models, the coupling results in a modification of the total number in the two populations, despite the exchange process itself conserving that number. This effect arises through the interplay of the exchange process with the death and immigration processes and is analogous to interference, a phenomenon usually associated with wave behaviour. It is possible that two sets of parameter values can give the same mean population values, even if one of the sets corresponds to an uncoupled process. The correlation properties of the coupled death–immigration models are such that the cross-correlation functions can display a peak for a non-zero delay time.

The results of §4 are summarized in table 1. It can be seen from the comparison of the fourth and seventh columns that in the case where the mean values of the primary process are equal, a pair of uncoupled death–immigration processes is indistinguishable from a coupled pair using external monitoring and the effect of the exchange process is hidden.

This reasoning can be extended: if external monitoring of a process yields Poisson count statistics, there is no way of telling whether these statistics arise from a single, isolated death–immigration process or a coupled system. Hence in such a case, there is no way of identifying whether a second *population* is present, this population being hidden from the external monitoring scheme. Furthermore, it is not unreasonable to assume that any number of death–immigration processes that are coupled in the manner described in this paper will give rise to Poisson count statistics for each population, and that under external monitoring a single death–immigration model is therefore indistinguishable from an arbitrary network of such populations.

The internal monitoring is sensitive to the correlations and can therefore successfully identify the presence of the coupling. The counting process used in monitoring a system is therefore crucial to interpreting the data that are obtained, and careful consideration is required before conclusions are drawn from the results of these data with regard to the nature of the underlying process.

## Acknowledgments

This work was funded by the Engineering and Physical Sciences Research Council in conjunction with QinetiQ, Malvern.

## Footnotes

- Received March 13, 2008.
- Accepted May 19, 2008.

- © 2008 The Royal Society