Chapter 3: Birth and Death processes

Transcription

1 Chapter 3: Birth and Death processes Thus far, we have ignored the random element of population behaviour. Of course, this prevents us from finding the relative likelihoods of various events that might be of interest, for example extinction. In this chapter, we focus on demographic stochasticity. This component arises from the intrinsically stochastic nature of birth and death processes. Even without external noise, one can not predict the future population numbers with certainty. Stochastic models often predict population behaviour that is significantly different from their deterministic equivalent. In these cases, the randomness itself is central to the population dynamic. This randomness can, under some nonlinearities have a systematic influence on the population. Such effects can not be captured by a deterministic model. 3.) Introduction to Birth and Death processes A birth and death process is defined as any continuous-time Markov chain whose state space is the set of all non-negative integers and whose transition rates from state i to state j, r,, are equal to zero whenever i j >. That is, a birth and death process i j that is currently in state i can only go to either state i or state i +. When the state increases by one, we say that a birth has occurred and when the process decreases by one, we say that a death has occurred. We thus have: B ( ) = r, +, D( ) = r, p ( t + t) = p ( t) B( ) t + p ( t) D( + ) t + + p ( t)[ B( ) t D( ) t] + o( t), for > where p ( t) is the probability that the population size at time t is 9 (3..) The Markov assumption states that only the current population size is of use in predicting future population behaviour. Thus, by definition, other possible transition rate predictors like the environmental condition are ignored. A population whose size at time t + t is could have only had one of four things occurring in the preceding interval [ t, t + t] : a member of the population could have given birth; a member of the population could have died; no births or deaths could have occurred and, finally, there could have been more than one event (whether birth or death or both) in the preceding interval. This means: (3..) The probabilities of the first three events are given by the first three terms on the RHS,

2 and o( t) is the (negligible) probability of more than one event occurring within the interval [ t, t + t], resulting in the final population size at time t + t being. We need to consider the case when = there is a death when =. In this case: By subtracting p p ( t + t) = D( + ) p ( t) t + p ( t) + o( t) p ( t) = p ( t) B( ) + p ( t) D( + ) p ( t)[ B( ) + D( )] for > + p ( t) = p ( t) D( ) for = written in matrix form as: p(t) = p(t)r where p( t) = p ( t) p ( t) p ( t), p( t) =.. R = r with r = ij i, j=,,,... ij separately, as such a circumstance will only arise if p ( t) p ( t) p ( t).. r i, i+ r i, i = B( i) = D( i) r = ( B( i) + D( i)) i, i r = for i j > i, j (3..3) ( t) from both sides of equations (3..) and (3..3), and dividing by t, we obtain the so-called Kolmogorov forward equations. As t : (3..4) The Kolmogorov equations are an infinite system of differential equations. They can be (3..5) The Kolmogorov equations are the primary means to define a time-homogeneous Markov process. By solving these equations, we can find the probability distribution of as a function of time. 3.) Solving the Kolmogorov equations The Kolmogorov equations can be solved for a linear birth-only model. Let denote the initial population size. For a birth-only model (i.e. a model that assumes that the members of the population cannot die), the Kolmogorov equations are as follows: p ( t) = B( ) p ( t) B( ) p ( t) for with boundary condition p The differential equation for p ( t) is: ( ) = p ( t) = B( ) p ( t) (3..) (3..)

3 The above equations can be solved directly. Suppose that B( ) = λ i.e. the birth rate is not density-dependent. Upon substituting this transitional form into (3..), we get: p ( t) = d ln p t ( ) = dt p ( t) ln p ( t) = tλ + C where C is a constant p ( t) e t λ λ Using the boundary condition that p ( ) =, we then have: p ( t) = e λ t This expression derived for p t ( ) can then be substituted into the Kolmogorov differential equation for p p λt λt ( t) = e e, p ( t) + λ( + ) p ( t) = λ + + p ( t) = e λ + = + ( t) thus allowing us to derive an expression for p ( + t ) : λt p ( t) p ( t) e + + λ + + = λ λ + t t p ( t) = e λ e + C where C is a constant + Using the boundary condition that p + ( ) =, we then have: λ ( + ) t λt λt λ( ) λ e e λt p ( t) = e e + Repeating the procedure with the boundary condition p + ( ) =, it can be shown that: p + t λt + λt + λt λt ( t) = e e The first three terms, suggest that the probability mass function for the population number is: λ e e t λt + t e e e dt (3..3) This is a negative binomial distribution with parameters and e λ t. If p ( t ) does indeed satisfy equation (3..3), then the differential equation for p + ( t) would be: (3..4) Equation (3..4) is a first order differential equation. We can thus apply a result given by Jaeger et al. (974) on equation (3..4), with the boundary condition p + ( ) = to get: (3..5)

4 We have thus shown that if p p ( t) = D( + ) p ( t) D( ) p ( t) for + with boundary condition p p ( t) satisfies equation (3..3) then p t ( ) = p ( t) = D( ) p ( t) for t e bt e bt ( ) =,,..., = + ( ) also satisfies equation (3..3). We also know that equation (3..3) is true when =. Hence, by the induction principle, we then know that equation (3..3) is always true. We have thus proved that the probability mass function for a linear birth-only process is a egative Binomial Distribution with parameters and e λ t. Similarly, one can directly derive the probability distribution function for a deathonly process. The Kolmogorov equations for the death process are: with the differential equation for p ( t) being: (3..6) (3..7) As before, we start of solving for p ( t) using the boundary condition that p ( ) =. After which we can solve for p ( t ), p ( t),, p ( t ) using the boundary condition that p ( ) = for. When D( ) = b (i.e. the death rate is proportional to the population size), the resulting probability mass function for the death only process has the form: (3..8) Unlike the births-only process, the deaths-only process has a finite set outcomes. Thus, for a linear deaths-only process, the population size is binomially distributed with parameters and e bt. 3.3) Alternatives to the Kolmogorov Equations The direct method of solving the differential equations is quite laborious in dealing with birth-only and death-only equations. This is due to the necessity of deriving the probabilities for the various possible population sizes (or at least the first few probabilities) separately before one can derive the general expression for p ( t ). This direct method is even less practical when dealing with a model that allows for both births and deaths. Due to the dependence of p ( t) on both p ( t) and p + ( t ) in such models, one must solve the differential equations simultaneously contrast this with the successive solution of the equations in the two earlier models. This makes the Kolmogorov equations unwieldy for large populations as, in such cases, obtaining even

5 a numerical solution is often difficult. The following alternatives to the Kolmogorov equations have proved useful. A. The Continuous Approximation By its very nature the population size,, is a discrete random variable. By treating as a continuous random variable and re-interpreting p p ( t) + + t B D p t ( ) ( ) ( ) B( ) D( ) p ( t) p ( ) = δ where δ E d = B( ) D( ) dt = + E d B( ) D( ) dt 3 ( t) as s probability density function (which we shall denote by p ( t) to avoid confusion), one can derive an approximate probability distribution for the population. isbet et al. (98) derived a single, approximate differential equation for p ( t). By performing a Taylor expansion on p ( t) and discarding terms that are of third order and higher, the authors showed that: is the Dirac Delta function (3.3.) A slightly modified version of the proof given by isbet and Gurney (98) is given in Appendix A (some of the elements of the proof have been re-ordered in an attempt to make the proof more comprehensible). is: The boundary condition for equation (3.3.) when the initial population size is (3.3.) Unfortunately, due to its non-linear form, equation (3.3.) is analytically intractable. Even a numerical solution to the differential equation cannot be found due to the discontinuous nature of the boundary condition given in (3.3.). This implies that the continuous approximation cannot be used to derive an approximate probability distribution for. However, equation (3.3.) can be used to generate an approximate, quasi-equilibrium distribution (covered in Section 3.6) which, in turn, can be used to derive an approximate analytical expression for the mean time to extinction. B. Stochastic Differential Equations Both the Kolmogorov equations and their continuous approximation model the population probability distribution through time. The following model is based on the population size itself and can thus be readily compared with the deterministic models covered in Chapter. Stochastic differential equations are often also used to model environmental stochasticity. It can be shown (see Appendix A) that:

6 Thus, provided dt is sufficiently small for terms of order dt we have: We can thus write: Var[ d ] = E[ d ] E[ d ] = B( ) + D( ) dt B( ) D( ) dt B( ) + D( ) dt provided dt is sufficiently small d = B( ) D( ) dt + η( t) B( ) + D( ) dt where η( t) is a random variable with zero mean and unit variance and higher to be ignored, (3.3.3) (3.3.4) Equation (3.3.4) is merely a cumbersome restatement of the Kolmogorov equations: η( t) has an unusual, discrete probability distribution to accommodate the fact that can only take on integer values. However a tractable approximation to this equation is possible. isbet et al. (98) stated, for all but the smallest populations, that any change in the population size which is large enough to affect the transition probabilities must be the result of a large number of statistically independent births and deaths. Thus, d can be taken over a relatively long time increment dt ; η( t) will then have an approximately normal probability distribution (by the Central Limit Theorem). regarding as a continuous variable (which implies that dt is small since dt ε By must be constant see proof of equation (3.3.) in Appendix A) and η( t) as being normally distributed (implying dt is large), we have: The Wiener process, ω( t) d = B( ) D( ) dt + B( ) + D( ) dω( t) where ω( t) is a Wiener process (3.3.5) is a continuous random process with independent increments and which is also time homogeneous (i.e. ω( t) and ω( t + s) ω( s) have the same distribution for s and ω( ) = ). In addition, the Wiener process, ω( t) is ormally distributed with mean and variance σ t (where σ > ). isbet et al. (98) did not offer a resolution of the requirement that dt is both small and large. However, from empirical evidence in Section 3.5, equation (3.3.5) does seem to provide a good approximation to the Kolmogorov equations. The stochastic differential equation can alternatively be written as: d dt dω( t) = B( ) D( ) + B( ) + D( ) γ ( t) where γ ( t) = white noise dt (3.3.6) 4

7 Equation (3.3.6) can be interpreted as the sum of deterministic and stochastic contributions to d dt. Care must be taken with this interpretation since white noise is not well behaved (e.g. E γ ( t) = ). Stochastic differential equations prove to be especially useful for deriving the gross fluctuation characteristics of a population. C. Generating Functions We now derive a singular differential equation for the population s moment generating function. The moment generating function of any random variable characterises that random variable s probability distribution. Hence such an equation implicitly models the population s probability distribution through time. Consider the random variables ( t) and ( t + t) ( t). The variables represent the population size at time t and the net change in the population size over the interval [ t, t + t] respectively. If we let f ) represent the continuous transition rate from population size to size P ( t + t) ( t) = j ( t) = j ( + j, then: equations (as shown in (3..5)): M( θ, t) jθ = ( e ) f j M( θ, t) t j θ (3.3.8) ote that the θ operator acts only on M ( θ, t). So, for example, if f ( ) = a b more than one unit in the interval t. Hence we have: M( θ, t) θ θ = ( e ) B M( θ, t) ( e ) D M( θ, t) t θ θ + f ( ) t + o( t) for j j j t f ( ) + o( t) for j = 5 j (3.3.7) For the birth and death model, f ( ) = B ( ), f ( ) D = ( ) and f ( ) = for j j,,. Also, f j ( ) = r, + j using the notation for the transition rates introduced in Section 3.. Bailey (964) showed that the following differential equation for the moment generating function, M ( θ, t), corresponds to the set of probability differential = then f a b θ θ θ equation (3.3.8) is given in Appendix B. = and thus f M t a M t ( θ, ) M t b ( θ, ) ( θ, ). A proof of θ θ θ The birth and death process assumes that the population size cannot change by (3.3.9) The advantage of the above equation is easy to see: instead of having a possibly infinite set of differential equations to solve simultaneously, we only need to solve a single

8 differential equation. The moment generating function characterises the probability distribution of so the solution of equation (3.3.8) helps to identify the correct probability distribution of. One can come up with corresponding differential equations for the probability generating function, P( ϕ, t) and the cumulant generating function, K( θ, t). For K( θ, t), we use the relationship K( θ, t) = log M ( θ, t) (equation (3.4.3) below gives the differential equation of K( θ, t) for a birth and death process). If we substitute e θ = ϕ and θ = ϕ ϕ in (3.3.9), we then have the following differential equation for the probability generating function, P( ϕ, t) : + P( ϕ, t) = ( ϕ ) B ϕ P( ϕ, t) ( ϕ ) D ϕ P( ϕ, t) t ϕ ϕ B( ) = a; D( ) = b where a and b are constants M( θ, t) θ θ M ( θ, t) = a e + b e t θ M( θ, t) = b e θ e a b t θ ae b a e θ e a b t θ ae b (3.3.) Consider the simple case where the transition rates are proportional to the population size: In this case, the differential equation for M ( θ, t) becomes: (3.3.) (3.3.) Equation (3.3.) is a linear differential equation. Hence an analytical solution for M ( θ, t) is easily obtainable. Bailey (964) showed that the solution of equation (3.3.) θ with boundary condition M ( θ, ) e = (i.e. the initial population size is ) is: (3.3.3) Since the birth and death rates were simply proportional to the population size, it was easy to derive this analytical solution for M ( θ, t). If the birth and death rates are nonlinear, differential equations (3.3.9) and (3.3.) can become intractable. In such cases, we unfortunately cannot derive the exact probability distribution of and thus we need to look at ways to approximate the true probability distribution or to solve the equation numerically. 6

9 3.4) Approximate solutions to the Kolmogorov equations The difficulties encountered in deriving an analytical solution to the Kolmogorov equations particularly when the birth and death rates are nonlinear forces one to look at various, more solvable, models that approximate a population s probability distribution. This section is based on the work of Matis et al. (). Consider transition rates with the following mathematical form: B( ) = f ( ) = a b D( ) = f ( ) = a + b for a b where a, b > ; i =, and B( ) = D( ) = i The transition rates shown above only hold for a b i M = θ + θ M θ θ M ( e ) a ( e ) a + ( e )( b ) + ( e ) b t θ θ (3.4.) This implies that the differential equation for the cumulant generating function, K( θ, t), K = θ θ K θ θ K K ( e ) a ( e ) a ( e )( b ) ( e ) b t θ θ θ i 3 θ κ i ( t) θ θ K( θ, t) = = θκ ( t) + κ ( t) + κ 3( t) +... where κ i is the i th cumulant i!! 3! i>. This suggests that a (3.4.) >> b. Consequently, we expect the per capita birth rate to dominate when is small and the term b to dominate when is large. b can thus be interpreted as the effect of crowding on the population. A similar interpretation also holds for the death rates. By applying equation (3.3.9) to the above transition rates, we obtain: is: (3.4.3) A derivation of equation (3.4.3) is given in Appendix E. either of the above two differential equations is analytically tractable. We thus are unable to find an exact analytical expression for the probability mass function. We thus need to look at ways to derive an approximate expression for the probability mass function as it evolves through time. One alternative to generate an approximate probability mass function would be to try to derive expressions for the first few cumulants of from equation (3.4.3). In deriving such expressions, we make use of the following relationship: Equation (3.4.4), implies that: (3.4.4) K = + θ t t t + θ θκ ( ) κ κ t! 3 ( )! ( ) (3.4.5)

10 Furthermore, we have: K θ = κ ( t) + θκ ( t) + κ 3( t) +... θ! K = κ + + ( t) θκ 3( t)... θ We also need to use the series expansion of e θ : e θ = + θ + θ + θ ! 3! Substituting equations (3.4.5) (3.4.8) into equation (3.4.3), we thus have: (3.4.6) (3.4.7) (3.4.8) 3 θ θ θκ ( t) + κ ( t) + κ ( t) +... = 3! 3! θ θ θ θ θ 3 3 θ θ... κ ( ) θκ ( ) κ ( )... 3! 3! a + + +! 3! a t + t + t + +! 3 3 θ θ θ θ θ b + θ b κ ( t! 3! ( )! 3! θ ) + ( ) ( ) + ( ) + ( ) +... ( )! θκ κ θκ κ t t t t 3 3 Expanding, we thus have: (3.4.9) 3 θ θ θκ ( t) + κ ( t) + κ ( t) +... = 3! 3! a a b + bκκ b + bκθ θ + 3a a 3b b 6b + bκκ 3b + bκ 3 4 a + a b b κ κ + a a b b 4 b + b κ κ b + b κ 3 a a b + bκκ + 3a + a b + b 6 b b κ 6 b + b κ κ +... It can be seen that the differential equation for the i -th cumulant function has terms up to the i +-th cumulant this is due to the non-linear birth and death rates. The presence of these higher-order cumulants prevents us from solving the differential 8 3! θ! (3.4.) By equating the coefficients for the various powers of θ in equation (3.4.), we obtain the following system of differential equations for the cumulants: + 3( a a ) 3( b b ) 6( b + b ) κ 3( b b ) κ + κ etc 3 4 κ ( t) = ( a a ) ( b + b ) κ κ ( b + b ) κ κ ( t) = ( a + a ) ( b b ) κ κ + ( a a ) ( b b ) 4( b + b ) κ κ ( b + b ) κ 3 κ ( t) = ( a a ) ( b + b ) κ κ + 3( a + a ) ( b + b ) 6( b b ) κ 6( b + b ) κ κ 3 (3.4.)

11 equations in (3.4.) directly. Matis et al. () proposed using a cumulant truncation procedure. Here, one approximates the first i cumulants by setting all the cumulants of order i + or higher to zero. If we set all the cumulants of order 4 and above to zero, we can then solve the resulting three differential equations in (3.4.) numerically to find values for κ ( t ), κ ( t) and κ 3 ( t) approximation derived by Renshaw (998): +. 5 / 4 3/ 3/ p ( t) 4π ψ exp / 6κ 3 κ 3κ ψ + ψ dn. 5 where ψ = κ + κ ( n κ ) at various values of t. The cumulant values obtained can then be used to create a saddle-point approximation of the probability distribution. The probability distribution of may be approximated by a saddle-point probability distribution. The saddle-point is a density function that will take as its parameters the values of s cumulants up to a specified order and force the values of the density function s cumulants to match those of. It is this matching of the cumulants that makes the saddle-point an approximation of the true distribution of : one would expect the approximation to be more accurate if more cumulants are being matched (however, in some cases, this is not true). The values for the first three cumulants (derived from equation (3.4.)) can be substituted into the following saddle-point 3 (3.4.) Matis et al. () have stated that the investigations which they have performed into the accuracy of the above approximation have yielded results that were very encouraging. 3.5) Example We again consider the transition rates introduced in Chapter : for < B( ) = otherwise (3.5.) D( ) =. +. By solving the equation B( ) D( ) =, we can see that the equilibrium population size is Since this was a stable equilibrium state, more births tend to occur when < 7. 5 and more deaths tend to occur when > Three simulations of the continuous-time Markov model were performed using the above transition rates so as to get a feel of what a typical population trajectory might look like. In order to execute the simulations, we needed to divide the timeline into 9

12 intervals of sufficiently small lengths (in this case, the interval length was set to.) so as to make the probability of more than one birth or death occurring within an interval negligible. For each interval, we thus know (using t =. ) that the probability that a birth occurs is B( ) t = The probability that a death occurs is D( ) t =. +. and the probability that nothing happens is B( ) + D( ) t = One can then simulate which of the three possible events occurs in each interval and hence replicate the population trajectory over the period of interest. In the simulations performed, the initial population size was set to. The three simulated population trajectories are shown below: Population Trajectories Time Figure 3. Simulated runs of the Population when = Similarly, one can simulate various trajectories for the stochastic differential equation (SDE) approximate representation of the birth and death process. The SDE for the transitions given in (3.5.) is: d = dt dω( t) where ω( t) is a Wiener process (3.5.) As with the earlier model, we assume that the initial population size is and we use time increments of size.. Thus, by simulating the values that the normal random variable dω( t) takes over each time increment, one can derive the population increments through time. By adding these increments to the initial population size, one can derive the population trajectory. 3

13 The diagram below shows 3 typical trajectories for (3.5.): Population Trajectories Time Figure 3. Simulated SDE runs of the Population when = In Figure 3., one can clearly see that the population size never moves by more than one unit in any instant (since dt is made sufficiently small to exclude the possibility of multiple births and deaths within any time increment). This serves to highlight that the birth-and-death process is a discrete-state process in continuous time. However in Figure 3. the population size can change to any value within an instant. This is to be expected as the SDE treats the population size as a continuous variable. Of course, if one were to repeat the simulations, one would, in all likelihood, obtain appreciably different population trajectories from the ones shown in Figures 3. and 3. (since the population movements depend on the occurrence of random events). The initial population size is. From Figure., we can see that the birth rates are higher than the death rates when = and hence we would expect an upward trend in the population numbers initially. Such a trend is clearly evident at the outset of all three population simulations in both Figures 3. and 3.. Once the equilibrium state is reached, one can see from the above figures that the population then tends to vacillate around this point. This is to be expected as this is a stable equilibrium state. A million simulations were then run for the birth-and-death process and these were compared with ten thousand simulations for the SDE. Both sets of simulations took roughly an hour to run in Microsoft Excel on a Pentium 4, 8 MHz, 5MB RAM: the SDE simulations are relatively more time-consuming as random numbers from a ormal distribution must be generated to carry out these simulations. This takes a considerably longer period of time to complete than the Uniform random number generation required for the birth-and-death process as the programming language used 3

14 to execute the simulations required one of Microsoft Excel s statistical functions to generate the ormal random numbers. For both types of simulation the initial population size is set to. The resulting population sizes after each simulation run was recorded at time. These results were then grouped under a frequency distribution (values for the SDE were rounded to the nearest integer) and consequently the probability distribution at time, p ( ) for both the birth-and-death process and the stochastic differential equation could be estimated as the number of times that a particular population value occurred, divided by the number of simulations undertaken. The estimated probabilities for the two models are shown below: Comparisonof the actual and SDEsimulated probabilitiesat time t=.5. Proba.5. Birth anddeathprobabilities SDEprobabilities PopulationSize Figure 3.3 Simulated Probability Distributions for the two processes One can clearly see that the probability distribution for both processes is negatively skew at time. This is due, in part, to the population starting below the equilibrium size. The probability distribution derived using the SDE is a good approximation to the true distribution obtained by simulating the Birth and Death process directly. This is despite the fact that the population size is quite small and so the normality assumption implicit in the SDE model is contentious. The above transition rates are nonlinear and so we need to use the cumulant truncation method in order to obtain approximate values for the cumulants. If we choose to truncate all cumulants of order four and higher, we will obtain the following system of differential equations (see equations in (3.4.)): κ ( t) =. 8. 6κ κ. 6κ κ ( t) =. 3. 4κ κ κ κ. 3κ 3 (3.5.3) κ ( t) =. 8. 6κ κ κ. 96κ κ κ κ 3 3 with boundary conditionsκ ( ) = = and κ ( ) = κ 3( ) = 3

15 The solution of the first three cumulants at time was obtained numerically. The following values were obtained: κ ( ) = 6. 49, κ ( ) = 353., κ ( ) = These estimates compare favourably with the first three cumulant values observed for the one million simulations of the birth-and-death process: κ ( ) = 6. 5, κ ( ) = 35., κ ( ) = The estimates of the cumulants can then be substituted into the saddle-point approximation given in (3.4.) so as to get an approximate probability mass function for. The diagram below compares the saddle-point probabilities with the simulated relative frequencies from the Birth and Death process: Accuracy of the Saddlepoint approximation Proba Simulated Probabilities Saddlepoint approximations Population Size Figure 3.4 Comparison of the Saddle-point and Simulated probabilities From the above figure, one can see that the saddle-point approximation deviates substantially from the true Birth and Death probabilities. The saddle-point approximation does not seem to be valid for the above three values obtained for the cumulants the probability density function will be a complex number when n > 7. 5 as ψ (as defined in (3.4.)) will be negative over this range. This means that the saddle-point approximation, p ( ), is not defined for > 7. Matis et al. () applied the saddlepoint approximation successfully to various other transitional forms. However, they did not apply the saddle-point approximation when considering the transitional rates given in (..). Further research needs to be done to ascertain the reason for the failure of the saddle-point approximation for the transition rates in (..). 33

16 To see whether the saddle-point approximation worked better after a longer time interval, the population size at time was also studied. A hundred thousand simulations were run. The values observed for the first three cumulants were: κ ( ) = 7. 9, κ ( ) =. 58, κ ( ) =. 3 3 Using the formula for ψ (given in (3.4.)), we now find that the density function is complex over the range: n > Compare the cumulant values at time with those observed at time 5: κ ( 5) = 7. 35, κ ( 5) =. 49, κ ( 5) =. 3 Here, the density function becomes complex over the range: n > There are minimal changes in the values of the cumulants. This seems to suggest that the population is close to equilibrium at time (the concept of a population being in equilibrium is considered in more detail in section 3.6). A process X ( t) is said to be ergodic if all its cumulants (e.g. µ = E X ( t) ) are equal to the matching time averages of the process, (e.g. X T lim T X ( t ) dt ). One of the T = conditions of ergodicity which is satisfied by all the birth-and-death models considered in this research report is that the population should forget its initial population size after a suitably long period of time (see isbet et al. (98) and section 3.7). Thus we expect the values of the cumulants by time to be independent of the starting population size. Thus, for the transition rates considered in this example, the saddlepoint approximation is inappropriate; irrespective of the initial value of the population. 3.6) Quasi-Equilibrium Distribution isbet et al. (98) stated that a population with a true equilibrium state, p has: B( ) p = D( + ) p for =,,... p + B( ) B( )... B( ) = D( ) D( )... D( ) B ( ) p, > (3.6.) Intuitively, equation (3.6.) signifies that a population at equilibrium has an equal probability of increasing from size to size + as it has of decreasing from size + to size. By repeatedly applying the above recurrence relationship, one can show that: (3.6.) 34

17 Since we are ignoring migration, B( ) is zero. This implies that p = >. Since p i = i=, this means that p =. This is to be expected since extinction is an absorbing state when migration is ignored. However this distribution is of limited interest. Consequently, the concept of the quasi-equilibrium distribution is considered instead. Quasi-equilibrium is defined as the equilibrium probability distribution that the population would ultimately be subject to if it were never to become extinct. We now look at two possible methods of deriving the quasi-equilibrium distribution. A. The modified Markov Process Matis et al. () modified the original birth and death process to create a new Markov process whose probability distribution did not degenerate at equilibrium to an extinction probability of one. The coefficient matrix for this new process R m m p ( t) = p ( t) R = ΠR was based on the coefficient matrix for the birth and death process R (see equation (3..5)). By deleting the first row and column of R (thus excluding the state = ) and by assuming that D( ) = (which removes the only transition to the state = ), one obtains the coefficient matrix for the new process, R. Let p let p m ( t) = p m ( t) m ( t ) be the probability that this modified process is equal to at time t. Also =,,... =,,... and p m ( t) = p m ( t). We then have: At equilibrium, we would expect p m ( t ) =. So if we let Π = p =,,... (3.6.3) be the equilibrium distribution for the modified process (and the quasi-equilibrium distribution for the birth and death process) we then have: (3.6.4) By solving equation (3.6.4) for Π, we get the quasi-equilibrium distribution. However, the algebra may become tedious when the population is large. ote that R singular matrix, otherwise Π =. B. Locally Linear Approximations must be a In Chapter, a locally linear approximation in the vicinity of the population s equilibrium state was used to derive an approximate population model. An approximation around the population s deterministic equilibrium state, can also be used to derive an 35

18 approximate quasi-equilibrium distribution. By definition, the deterministic equilibrium state satisfies the following relationship: B( ) = D( ) isbet et al. (98) defined the three functions, f ( ), g( ) and n( t) : f ( ) = B( ) D( ) g( ) = B( ) + D( ) n( t) = ( t) (3.6.5) (3.6.6) By performing a Taylor expansion of f ( ) around and retaining only the leading term in the expansion, we have: A similar procedure on g( ) gives: db dd f ( + n) λn where λ d = d (3.6.7) Equation (3.6.7) approximates g( + n) Q where Q B( ) + D( ) = t p t np t n n Qp t ( ) λ ( ) ( ) n n n approximate expression for the quasi-equilibrium distribution: p exp Q λ approximation, the population size has the following Q ormal (approximate) distribution:, (3.6.) λ ote that λ < for a stable equilibrium state (see equation (..6)) and thus the (3.6.8) f ( ) to the first order whilst equation (3.6.8) approximates g( ) to the zero th order. These expressions can then be substituted into the continuous approximation given by equation (3.3.) to obtain: By setting t p (3.6.9) t ( ) =, one can solve the resulting differential equation to derive an + n (3.6.) The function clearly is Gaussian in form. isbet et al. stated that, with this locally linear variance ( Q λ ) for is positive. 3.6.A) Example (continued) For the transition rates considered in Section 3.5, we have = Furthermore, we know that: λ = =. 8 Q = B( 7. 5) + D( 7. 5) = 35. Thus, we have that ormal( 7. 5, ). 36 = 7. 5

19 The corresponding modified transition matrix is obtained using the transition rates in (3.5.): By solving equation (3.6.4), one can calculate the quasi-equilibrium distribution. The quasi-equilibrium distribution and its approximation are independent of the initial population,. The diagram below shows the quasi-equilibrium distribution; the probability distribution at time (obtained by simulation); and the locally linear ormal approximation: The equilibriumprobability distribution Proba Quasi-Equilibrium Distribution Simulated Probability Distribution at Time t= Linear Approximation Population Size Figure 3.5 The population at equilibrium From the diagram, one can see that the locally linear approximation provides a relatively good fit to both the population s probability distribution at time and the quasiequilibrium distribution. Thus, after a long enough time interval, the population assumes an approximately normal distribution. 37

20 3.7) Gross Fluctuation Characteristics of a Population The population s probability distribution is usually a means to an end rather than the end itself since it is almost impossible to estimate for any natural population. This is due to both the unreliability of most ecological population data and the difficulty involved in setting up replicate populations so that various probabilities may be estimated. However, gross fluctuation characteristics of a population, such as the mean, the variance and the autocovariance function, can usually be observed over time for a population. Thus such characteristics prove to be invaluable in calibrating any population model. The gross fluctuation characteristics should describe the properties of a population at equilibrium. However, since extinction is an absorbing state, the equilibrium state of a population is extinction. The characteristics of such a state are not very interesting and so we would rather base the gross fluctuation characteristics on a population in quasiequilibrium. This section is based on the work of isbet et al. (98). As before we let p be the probability that a population in quasi-equilibrium has size. We then have: µ = p (3.7.) = σ = p µ = (3.7.) Unfortunately, the above equations cannot be used to calculate the mean and the variance as the quasi-equilibrium distribution of a population is seldom estimable. A good way to relate the gross fluctuation characteristics to a measurable quantity is to equate the above statistical expectations to the corresponding time averages of a single population. That is, we assume the population is ergodic (see Section 3.5 for a definition of ergodicity). In order for such a procedure to be valid, the following conditions for ergodicity must hold: i. After a suitably long period of time the population should forget its initial value. ii. A population starting from a particular value should, in principle, be able to reach any other value. isbet et al. stated that the above conditions are satisfied by all birth and death models. The time averages of a single population (which we denote by ) are defined as: T µ = tim lim T T ( t ) dt (3.7.3) T σ tim = µ tim lim µ tim dt T T (3.7.4) 38

21 The time averages, µ tim and σ tim should equal the mean and the variance of the quasiequilibrium distribution as the population should spend most of its time in the quasiequilibrium state. Thus, from equation (3.6.), we get: Q µ tim = ; σ tim = λ One can also define the autocovariance function, C( τ ) using time averages: C( τ ) ( t) ( t τ ) d = B( ) D( ) dt + B( ) + D( ) dω( t) (3.7.5) (3.7.6) The autocovariance function gives one an indication of the time it takes for a population to forget its initial value. An alternative method of deriving the gross fluctuation characteristics is to use the SDE formulation. The stochastic differential equation used to model the population (see equation (3.3.5)) was: (3.7.7) Retaining the usage of the functions f ( ) and g( ), as defined in section 3.6, we have: d = f ( ) dt + g( ) dω( t) (3.7.8) If, consequently, one were to approximate the functions f ( ) and g( ) around by the equations (3.6.7) and (3.6.8), we would obtain the following linear SDE: dn = λn + Q γ ( t) dt where n = dω( t) γ ( t) = = white noise dt One can easily derive the gross fluctuation characteristics for a linear SDE using T T ~ x ( ) x( t) e iωt ω = dt = iωx~ ω dx~ ω dt 39 (3.7.9) Fourier methods. Consequently, a brief description of Fourier analysis is given below. (It is also advisable to consult Appendix C as it gives proofs to some key Fourier theorems.) For any function x( t), its Fourier transform, ~ x ( ω) is defined over the interval T ; T as: For a linear process, x, t we have: It is this result in particular which makes a linear SDE amenable to Fourier analysis. (3.7.) (3.7.)

22 The spectral density, S x ( ω ), of the function x( t) is defined as: S x x~ ( ω ) ( ω) = lim T T dn~ ( ω ) = iωn~ ( ω ) = λn~ ( ω ) + Q ~ γ ( ω ) dt ~ Q n( ω ) = ~ ( ) iω λ γ ω Q (3.7.5) Sn( ω) = S γ ( ω) λ + ω The above relationships are useful as we know that white noise has the following E ~ ( ) ; E ~ γ ω = γ ( ω) = T ; S ( ω) = By applying some results proved in Appendix C to the population, we obtain: n ( t ) = T n ~ ( ) n( t) = Sn( ω) dω (3.7.8) π By substituting equation (3.7.4) into (3.7.7) and taking expectations, we have: Q E n( t) = T E ~ γ ( ) = λ ( t) = E ( t) = Q dω Q (3.7.) σ tim = σ = E n( t) = = π λ + ω λ The time averages given above agree with the gross fluctuation characteristics derived γ (3.7.) If the population s equilibrium state is stable, the transient initial condition-dependent term will decay to zero and the persisting term becomes dominant. Since equation (3.7.9) is linear, we know by Fourier transforming equation (3.7.9) (and applying equation (3.7.)) that: Rearranging the terms, we thus have: (3.7.3) (3.7.4) The spectral density of the population is upon substituting equation (3.7.4) in equation (3.7.) given by: properties: (3.7.6) (3.7.7) (3.7.9) In addition, by substituting (3.7.5) into (3.7.8) and applying the spectral property of white noise (given in (3.7.6)), we have: via the continuous approximation (see equation (3.7.5)). Unlike the continuous approximation however, the stochastic differential equation formulation allows us to 4

23 easily derive the autocovariance function, C( τ ), for the population. It can be proved (see Appendix C) that: C( τ ) = Sn ( ω) cos ωτ dω π Substituting equation (3.7.5) into (3.7.), we have: Q C ( τ ) e λτ = λ Thus, the autocorrelation function, ρ( τ ), is: (3.7.) (3.7.) ρ( τ ) = e λτ (3.7.3) The functional form of (3.7.3) implies that the population sizes at two distinct points in time can never be negatively correlated. 3.7.A) Example (continued) Using equation (3.7.3), we find that the autocorrelation function for the example in section 3.5 (where λ =. 8 ) is: Autocorrelation function Correlation o Time lag Figure 3.6 Correlation between two points a distance τ apart One can see that the population size in the near future is closely correlated to the population size now. This is to be expected as the time interval is too small to allow for any more than a few births and deaths to occur. One can see that the population at times a distance apart are virtually uncorrelated. Thus we would expect the initial population size of to have virtually no impact on the population size at time. 3.8) Extinction The extinction of a species is of particular interest in all population studies. Society is rarely indifferent to the prospect of extinction of a species (whether it be the rhino or smallpox!). Extinction is an absorbing state. The finality of the extinction state is, in large 4

24 part, the justification for any interest in this state. The probability of a population being extinct at time t is p ( t). Since extinction is an absorbing state, p ( t) is always an increasing function of time. Let T denote the time to extinction when the current population size is. Also let f ( t) be the density function and F ( t) the cumulative distribution function of T. We thus have: F ( t) = Pr ( t) = ( ) = = p ( t) given ( ) = By taking the derivatives of both sides, we have: E b t f ( t) = p ( t) with ( ) = E = tp ( t) dt = D( ) tp ( t) dt = e bt e bt dt = b i= i where E = S( t) dt = p ( t) dt S( t) = P[ T > t] where is the quasi equilibrium probability p ( t) p p ( t) p = dp( t) = D( ) p ( t) dt dp( t) = D( ) p p( t) dt The solution to equation (3.8.8) is: 4 p ( t) = exp D( ) p t (3.8.) (3.8.) Let E denote the mean of T. Using the functional form for p ( t) (shown in equation (3..4)), we then have: So for the deaths-only process in section 3. (where D( ) = b ), we have: (3.8.3) (3.8.4) For the example in Section 3.5, one cannot obtain an analytical expression for p t ( ) and hence we will only be able to derive the mean time to extinction numerically using the above method. Alternatively, one could try to derive an approximate analytical expression. An alternative expression for (3.8.3) is: (3.8.5) isbet et al. (98) showed that a when a population is close to its quasi-equilibrium state, then: The Kolmogorov equation when = (see equation (3..4)) is: Substituting equation (3.8.6) into (3.8.7), we have: (3.8.6) (3.8.7) (3.8.8) (3.8.9)

25 By substituting equation (3.8.9) into equation (3.8.5), one has: E = exp D( ) p t dt = D( ) p (3.8.) This expression is independent of the initial population size,. This is because we are assuming that the population has reached the quasi-equilibrium state, which is independent of the initial population size. For the example in Section 3.5, D( ) =. and p = 7. 8 (this probability was calculated in Section 3.6.A by solving equation (3.6.4)). Substituting these numbers into equation (3.8.), the model estimates the mean time to extinction to be 6. time units; for any initial population size. One can obtain an exact result for the example in Section 3.5 using the fact that we are modelling the population as a Markov process. Let R + be a modified coefficient matrix of R (see equation (3..5)), obtained by deleting the first row and the first column of R. ( R + is not quite the same as R which was used in equation (3.6.3) as we do not make D( ) =. As such, R + is the coefficient matrix of the Kolmogorov equations amongst the transient states.) Let M = m ij times. The element m ij be the matrix of so-called mean residence is defined as the expected value of the total elapsed time that a population, which starts at size ( ) = i, will be of size j prior to the population becoming extinct. Matis et al. () stated that: M = R + The mean time to extinction given that ( ) =, denoted by E, is: (3.8.) E = m, j j (3.8.) For the example in section 3.5, R is a matrix, as the population size cannot increase above. The matrix R + is thus invertible and consequently, one can derive the expected time to extinction using equation (3.8.) for any initial population size. The matrix R + is the same as the matrix shown in Section 3.6.A, except instead of having the top left entry of the matrix, r, =. 85, we have r, =. 36. By applying equations (3.8.) and (3.8.) in turn, we thus find: E = The mean time to extinction increases as the population size increases. This is fairly intuitive as the population has to suffer the loss of an additional member of the 43

26 population in order to become extinct. However, the mean time to extinction, E is effectively the same for initial population sizes of four and above. It thus seems that the fact that the approximate expression for E (equation (3.8.)) is independent of the initial population size in not all that unreasonable. Indeed, the estimated time to extinction using equation (3.8.) is not far from any of the true mean times to extinction. Unfortunately, equations (3.8.) and (3.8.) cannot be used when the population sizes are very large as the resulting matrices are also large and hence difficult to manipulate. This is when the approximated analytical expression derived by isbet et al. (98) becomes especially useful. The Birth-and-Death model is one of the most widely-used stochastic representations of a population. In addition to its flexibility, the Birth-and-Death model is also appealing due to the simplicity of its underlying principle: the population number can only change when a member of the population gives birth or dies. In the following Chapter, we look at an alternative method of modelling demographic stochasticity. 44