Stochastic Equivalents of the linear and Lotka-Volterra Systems of Equations- a General Birth-and-Death Process Formulation WAYNE M. GETZ* National Research Institute for Mathematical Sciences, CSIR, P. 0. Box 395, Pretoria 0001, South Africa Communicated by Charles J. Mode ABSTRACT A general m-dimensional multivariate birth-and-death process is defined, and the corresponding Kolmogorov forward equations are formulated. The process is shown to be a denumerable Markov jump process. General birth and death parameters are defined. Special cases of the forms of these parameters are studied in depth. This investigation leads to the derivation of stochastic equivalents of a deterministic linear control system and two types of deterministic Lotka-Volterra systems, in the form of differential equa- tions describing the means variance and covariance trajectories of the individual popula- tions of the systems. A general appraisal is made of the derived systems of equations as models in population dynamics. 1. INTRODUCTION Extensive use has been made of the birth-and-death process to formulate the behavior of dynamic populations. There are many examples of uni- variate models of population growth, especially in biological applications. The pioneering work of Yule, Feller and Kendall in the field of univariate birth-and-death processes has been extended in a variety of applications including studies in epidemiology, population control and genetics [l-5]. General multivariate populations have been analyzed with less success, partly owing to the notational complexities involved. Expressions for the conditional and marginal distributions of an important class of bivariate [6] and multivariate [7] birth-and-death process models have been derived, but the joint bivariate and multivariate probability distributions have been *Partially supported by an A. E. & C. I. Postgraduate Research Fellowship. MATHEMATICAL BIOSCIENCES 29, 235-257 (1976) 8 American Elsevier Publishing Company, Inc., 1976 235
Transcript
Stochastic Equivalents of the linear
and Lotka-Volterra Systems of Equations-
a General Birth-and-Death Process Formulation
WAYNE M. GETZ*
National Research Institute for Mathematical Sciences, CSIR,
P. 0. Box 395, Pretoria 0001, South Africa
Communicated by Charles J. Mode
ABSTRACT
A general m-dimensional multivariate birth-and-death process is defined, and the
corresponding Kolmogorov forward equations are formulated. The process is shown to be
a denumerable Markov jump process. General birth and death parameters are defined.
Special cases of the forms of these parameters are studied in depth. This investigation
leads to the derivation of stochastic equivalents of a deterministic linear control system
and two types of deterministic Lotka-Volterra systems, in the form of differential equa-
tions describing the means variance and covariance trajectories of the individual popula-
tions of the systems. A general appraisal is made of the derived systems of equations as
models in population dynamics.
1. INTRODUCTION
Extensive use has been made of the birth-and-death process to formulate the behavior of dynamic populations. There are many examples of uni- variate models of population growth, especially in biological applications. The pioneering work of Yule, Feller and Kendall in the field of univariate birth-and-death processes has been extended in a variety of applications including studies in epidemiology, population control and genetics [l-5].
General multivariate populations have been analyzed with less success, partly owing to the notational complexities involved. Expressions for the conditional and marginal distributions of an important class of bivariate [6] and multivariate [7] birth-and-death process models have been derived, but the joint bivariate and multivariate probability distributions have been
*Partially supported by an A. E. & C. I. Postgraduate Research Fellowship.
MATHEMATICAL BIOSCIENCES 29, 235-257 (1976)
8 American Elsevier Publishing Company, Inc., 1976
235
236 WAYNE M. GETZ
derived only for a few special cases. Notable among these results are the distributions obtained for a class of multivariate linear birth-and-death processes under the initial conditions of zero members in each population [8], and migration processes between colonies of the same species where the birth, death and migration parameters are linearly dependent on population
size [9]. Usually it is sufficient for most modeling purposes if the means,
variances and covariances of the individual populations are the only param- eters of the joint multivariate distribution known. These results have been derived for the bivariate two-sex problem [lo, Chapter 161 and bivariate Lotka-Volterra predator-prey system [ 111.
A general procedure has been derived to generate the variance, covariance and higher-order terms for any Galton-Watson multitype pro- cess [12]. Pollard’s stochastic version of the Leslie model of population age structure is an important application of this theory. No methodology exists, however, for generating the desired statistics for general Markov jump processes. In addition, most analyses of systems in this area have been primarily concerned with the properties of the stochastic matrix of a particular Markov process. Inglehart [13], following Reuter’s method of analyzing the transition probability matrix of a Markov process, gives conditions of uniqueness recurrence and properties of absorption states of a class of multivariate Markov jump processes known as competition pro-
cesses. In this paper we shall derive a general method for generating the
statistics of Markov jump processes. We will primarily consider processes where only jumps of unit magnitude are allowed in a population at any instant in time, but the basic methodology can be extended to processes where jumps of several units of magnitude are allowed in a population at any instant in time. The general formulation will demonstrate the inclusion of functions in the system to facilitate control studies of the system, although further analysis in this direction is beyond the scope of this paper.
For a general survey of the application of stochastic processes, especially of the Markov type, the interested reader is referred to two volumes by Iosifescu and Tautu [ 14,151.
2. DEFINITIONS
In view of the lack of a standardized notation and the notational complexities involved when a multivariate system is analyzed, the following definitions are introduced, which will facilitate formulating the problem.
2.1. Let n, denote the ith element of a vector N E R”. Then
QL {NERm]fori=l , . . . , m each n, is a non-negative integer}.
A GENERAL BIRTH-AND-DEATH PROCESS FORMULATION 237
2.2. Let 0 be the null vector of W, 1 the vector with unit entries and e,, i=l , . . . ,m, the vector differing from the null vector only in its ith element, which is unity.
2.3. We define the I;” norm on the elements N, N* EO, i.e.,
(ii) IIN- N*ll= 2 Ini- $1. i-l
2.4. An m-species population process is an m-dimensional stochastic process X(t) defined on Q for t E[O, co) by the probability density function
P(N;t)=Pr{X(t)=N} forall NEG,
=o for all N $Z&?.
2.5. We define the transition probability distribution of the process in terms of any N, N* ~0 annd t,s E[O, co) as
P(N,N*;s,t)=Pr{X(t+s)=N*IX(s)=N}.
Also define
P(N,N*;s,O)=S(N,N*)= ; if;:,“: 1
so that Eq. (3.1) considered in the next section holds when At = 0. Further, since P (N, . ;s,t) is a probability distribution, we must have
P(N,N*;s,t)>O and 2 P(N,N*;,s,~)=L N’EB
In this paper we shall be primarily concerned with formulating, in detail, models of the following processes.
2.6. An m-dimensional multivariate birth-and-death process is a popula- tion process X(r) defined on D with a transition probability distribution satisfying the following conditions:
(i) P(N,N*;t,At)=o(At) for all N,N*ESl such that IIN-N*ll>2. (ii) P (N, N*; t, At) = 0 for all N, N* ED such that n, = 0 and n: > 0.
238 WAYNE M. GETZ
Condition (i) allows only single events to occur at any instant in time. This assumption is standard for most birth-and-death models, a notable excep- tion being models which provide for the occurrence of multiple births [l]. Condition (ii) states that no spontaneous generation (i.e., births) can take place in a species that has zero members. This assumption provides one of the essential differences between birth-and-death and queueing processes.
2.1. We define vectors U(t)E RP, V(t) E R4 for all t E[O, 00). Using
systems terminology, we shall then sometimes refer to X(t) as the state vector of the system (process), and U(t), V(t) as the external or control vectors of the system.
3. GENERAL FORMULATION
In this section we shall derive the forward Kolmogorov differential equation for P(N; t) to be solved under a given initial condition, P(N; 0) say, for the process defined in 2.6. This process will be shown to be a denumerable Markov jump process.
Since condition in 2.6 restricts state transitions to single events, we can write down the following equation at any instant in time, with probability
which holds for all N EQ. If the limits as defined in the following equation exist, we have, from Eq.
(3.1), the forward Kolmogorov differential equations of the process:
P’(N; t) = lim P(N,N;t,At)-1
At P (N; t) At-+0
+ lim
i
m P(N+e,,N;t,At) c P(N+ei;t) At-0 At
i-l
+ m P(N-e,,N;t,At) c At
P(N-e,;t)
! (3.2)
i-1
for all N EQ, which can be solved subject to a given initial condition P (N; 0).
A GENERAL BIRTH-AND-DEATH PROCESS FORMULATION 239
The existence and uniqueness of a solution P (N; t)t E [0, 00) will depend on the functional form of the limits as defined in the above equation. In addition, a limiting distribution
P(N)= lim P(N;t) *+m (3.3)
will exist for a time-homogeneous system under certain conditions. Since L? is denumerable [17], the process defined by 2.6 turns out to be a
denumerable Markov jump process under the following conditions [16, p. 1251:
We define
C (N; t) 4 lim I-P(N,N;t,At)
At-0 At
and
C (N*; t)II(N*, N; t) 9 dim, P(N*,N;t,At)
At ’
(3.4)
(3.5)
Then C(N; t),C(N*; t), II(N*,N; t) must be continuous in t for all pairs (N*,N), N*,N EO, with
C(N,t)aO, II(N,N;t)=O and
x II(N*,N; t) = 1 for all t ~(0, cm). NEB
C(N; t) is referred to as the “jump rate” from state N, and TI(N*, N; t) is the probability of a jump from N* to state N. The interested reader should refer to [16, Chapter 41 and [14, p. 2011 for an account of the theory on which the Kolmogorov forward and backward equations of a Markov jump process are based.
Suppose that the transitional probabilities of the process are functions of the state and external vectors of the process, as well as t explicitly. We define these probabilities as
P(N+e,,N;t,At) i 11g(N+e,,U(t),V(t),t)At+o(At), i=l,...,m.
(3.6)
240 WAYNE M. GETZ
Since P (N, N*; t, At) is a probability distribution satisfying the conditions of an m-dimensional multivariate birth-and-death process (definition 2.6) we have, using (3.6) above,
Further, II(N,N*; .)=O for all other N* E 3, including N* = N, so that clearly from (3.9) on (3.10)
c II(N,N*;r)=l N.EQ
for all t E [0, co). Therefore, under the stated conditions, if the expressions (3.9) and (3.10) are continuous in t for all N l s2, the process defined in 2.6 is a denumerable Markov jump process and Eq. (3.8) is the well-defined forward Kolmogorov equation of this process.
To sum up: If we choose A,(.;;;) and pi(.;;;) as continuous non- negative functions of time for all N EO, the continuity conditions are satisfied, the various probability distributions are well defined, and Eq. (3.8) is the forward Kolmogorov differential equation of the stochastic process X(t) defined in 2.6, which turns out to be a denumerable Markov jump process.
4. BUILDING SPECIFIC MODELS
In this section we shall choose specific forms for the functions p,(. , . , . , -) and &i(.;;, .) introduced in the previous section and state the physical interpretation that can be given to these forms. The systems considered turn out to be the stochastic analogues of the linear and Lotka-Volterra de- terministic systems that have been used widely in population dynamics [ 11,22,24], The building of controls into the system will also be considered, allowing the models to be used in optimal-control studies [25].
A. LINEAR CONTROL SYSTEM
For i= l,...,m we define
hi(N, u(t), k’(t),t) f 5 h,(t)nj+ 5 ai,(t)~,(t) for ni>O, j-1 r=l (4.1)
FO for ni=O,
P
/~(N,U(t),f’(t),t) A 5 pu(r)nj+ z &(t)u,(t) for n,>O, j=l r-1 (4.2)
GO for ~-0,
where all time-dependent functions are continuous in t, t E [0, co).
242 WAYNE M. GETZ
Using the above definitions, for all N EQ Eq. (3.8) becomes
The physical interpretation of (4.1) and (4.2) is that each ith species is influenced to increase and decrease at
(i) a linear time-dependent rate with respect to its own size and the size
of every other species; (ii) a linear time-dependent rate with respect to the size of the elements
of an external vector.
B. LOTKA-VOLTERRA COMPETITIVE SYSTEMS
For i= l,..., m and N EO we define
h,(N, U(r), V(r),r) b Ai( V(r),r)n,+ 5 hu(r)n,nj, j- 1
(4.4)
(4.5)
where all time-dependent functions are continuous in r, t E [0, 00). It is clear from definitions (4.4) and (4.5) that hi(. , . , . , a) and pi( 1,. , . , .)
are zero when ni = 0.
A GENERAL BIRTH-AND-DEATH PROCESS FORMULATION
Using the above definitions, for all N ~a Eq. (3.8) becomes
The physical interpretation of (4.4) and (4.5) is that each ith species is influenced to increase and decrease at
(i) a linear time-dependent rate with respect to its own size, where the rate may depend on an external vector I’(t);
(ii) a quadratic time-dependent rate with respect to its own size (the inclusion of this term is equivalent to building a Pearl-Verhulst-type limit term into the model [ 11, p. 2371);
(iii) a time-dependent rate directly proportional to the product of the sizes of the ith and jth species, for j = 1,. . . , m and j# i; this rate is usually termed the interaction rate between the ith andjth populations.
Note: Linear controls can be added to models described in B and C, as was done in A. More general models can be built using combinations of A and B as well as by introducing various non-linearities, e.g., bilinear terms in N and U(t) [26].
5. METHOD OF SOLUTION
We are now faced with the problem of solving equations of the type given by (4.3) and (4.6) for the probability density function P(N; t), which will then define our puocess X(t) (see definition 2.3), subject to a given initial condition P (N; 0). The method outlined in this section is versatile in that it can be applied to general population processes of the type discussed in Sec. 4, although only the statistics of the distribution P(N; t) can be
244 WAYNE M. GETZ
derived (e.g., means, variance and covariance terms) and not P (N; t) itself. We define a generating function
G(S,t)= 5 . .’ 5 P(N;t)S;!sp,...,S;m n,=O ?I,=0
4 2 P(N;t)SN. NEQ
(5.1.1)
The summation in (5.1.1) is well defined, since Q is denumerable, and uniformly convergent for lsi\< 1, i=l,...,m, as ENNEP(N;t)=l. For a well-behaved multivariate distribution P (N; t) (i.e., finite second and third order moments around the mean), we can derive the following relation- ships:
ac (S, t> ~ = NznniP(N;f)SNp”,
asi (5.1.2)
or multiplying (5.1.2) by si,
(5.1.3)
Clearly (5.1.2) can be written as
where we have changed the index of summation from running over the set
!J to running over the set Q2; e {N - e,lN l a}. It is easily seen that the only
element in fYl2; that is not in Ll is the element (0- e,), and that
(n,+ l)P(N+ei;t)SNI,_,=O,
since the ith element of the vector 0- e, is - 1, i.e., (ni + l)=O. Thus the specific index 0- e, can be dropped from the summation index
set 8;, leaving the index set G? itself-i.e.,
aG(S,t) -=Nza(ni+l)P(N+ei;t)SN.
asi (5.1.4)
A GENERAL BIRTH-AND-DEATH PROCESS FORMULATION 245
Similarly by changing the summation index set from D to QT L {N +
e,l N EL?} and multiplying by s,, (5.1.3) can be written as
s2 aG (s, l) - I as,
2 (ni- l)P(N-e,;t)SN. NE&+
Clearly S, N + q N E Q} = 8, and by definition 2.4, I’(@-- e,; t) = 0, since 0’-- e, EO, so that adding the vector @ to the index set 52: will not alter the summation, and the index set can be taken as 9, i.e.,
sZaG(SJ) - I asi
2 (n,-l)P(N-e,;t)SN. (5.15) IvErl
As in (5.1.2), we can easily show for i#j that
aZG(S,t)
asi asj = NzQninjP(N; t)SN-q-ei.
Using this identity, the identities (5.1.1) and (5.1.2), and where applicable the summation-index-set arguments discussed above, we can derive the following identities:
iG(S,r)= x P(N+ei;t)SN, (5.1.6) I N ESl-
siG(S,t)= x P(N-ei;t)SN, (5.1.7) NEO
zF= z njP(N+ei;t)SN, I J NE$
sisjF = 2 njP(N- ei;r)SN, J NE&?
azG(S,t)
fi asi asI = NzQ(ni+ l)njP(N+ei;t)SN,
a2G(s,t) ‘is/ asi asj = NznninjP(N; t)SN,
‘h 8 2G (S, t)
asi asj =Nzn(ni-l)njP(N-ei;t)SN.
(5.1.8)
(5.1.9)
(5.1.10)
(5.1.11)
(5.1.12)
246 WAYNE M. GETZ
Further, since
s;& &n;P(N:f)S”= x n’P(N;t), NECl
we have
4 a*G(S,t) + aG(S,t)
as,2 -= N~o(ni+l)2P(N+ei;t)S”, (5.1.13)
asi
(5.1.14)
Sf a*G(s,t) +s? aG(s>o
as; ~ = z (ni- l)*P(N-ei;t)SN. (5.1.15)
’ aS, NER
Note: Since (5.1.6) and (5.1.8) do not contain a term in ni + 1, they are the only expressions where summation cannot be taken over 3, as was the case in (5.1.4). We shall see that this problem arises in models of the type described in Sec. 4. A, but can be overcome because the birth-and-death parameters &(.;;;) and pi(.;;;) are defined to be zero when ni = 0.
Differentiating (5.1.1) with respect to t, we obtain
(5.2)
Consider Eq. (4.3). Multiplying it by S”’ and summing over all N EQ, we obtain
2 P’(N;t)SN NEO
m m
= =( h,(t) x njP(N-ei;t)SN-[Av(t)+pti(.,j(t)] x njP(N;t)SN
The process of taking the summation over N EQ inside the finite summa- tions over i, j and r is valid (as previously remarked) for well-behaved distributions which lead to uniformly convergent sums for Isi]< 1, i = ,...,m. 1
In (5.3), since the birth-and-death parameters of Sec. 4. A (defined in (4.1) and (4.2) are zero when the size of the ith species is zero, we can replace
am x n,P(N+ei;t)SN and ,f$(t)u,(t) 2 P(N+e,;t)S NEfl NESl
by the same expressions, except that the summation can be taken over Q2; instead of CL
Thus using expressions of the type (5.1.1) to (5.1.9) Eq. (5.3) can be written as
+ 2 f: ( ai,( +))(+- l)G(S,t)u,(t). (5.4) i-l r-1
Similarly, Eq. (4.6) can be multiplied by SN and summed over N EO, and using identities (5.1.1) to (5.1.15) we obtain
-= [~(v(r),t)Si-~(V(t),f)](Si-l)~ I
+[~j(t)sj-~i(r)l(si-l) a*G(s,t)
as:
+ aG(s,t)
asi II (5.5)
i-l j-1 j+i
Solving (5.4) and (5.5) for G(S,t) subject to the initial condition
G(S,O)= 2 P(N;O)SN, NEO
we can, from (5.1.1), find P(N; t) for each N EL? as the coefficient of SN in G (S, t). However, solving the above equations for G(S,t) proves to be extremely difficult, and even then it may be impossible to find a closed- form expression for the coefficients of SN in G(S,t) [5]. In most circum-
248 WAYNE M. GET2
stances it may be sufficient to know only the first few moments of the multivariate distribution-e.g., means and variance and covariance terms. These can be found as follows:
Put S=l (see definition 2.2) in the expressions (5.1.1) to (5.1.15) and (5.2). Then we have
G(l,r)= c P(N;t)=l, NEQ
since P (N; t) is a probability distribution.
Ep = ~p,P(N;r) 4 F(q),
a*G(l,t)
as; = 2 nJn,- l)P(N;t) 2 &(?I:)- G(n,),
NEP
a *G (1, t)
asi asj = 2 n,n,P(N; t) e &(ryj), izj.
NESl
(5.6.1)
(5.6.2)
(5.6.3)
(5.6.4)
Clearly, since G (S, t) is well behaved for well-defined probability distribu-
tions,
G (qn,) = & (n/z,),
a*G(l,t)
as,2 + EgL c nfP(N;t): &+I$.
NESl
Further, for a well-defined probability distribution,
so that
a aG(&t) a aG(Zt)
------‘at-, asi at asi
a aG(Lt) zi = $ &(n,).
(5.6.5)
(5.6.6)
Similarly, we can easily show that
asG(l,t)
asi asj as, = x n,njnkP(N; t) 2 &(ni,nj,n,). * *, i#j#k, (5.6.7)
N En
a 3G (1, t)
as; a+ = 2 ni(ni-l)njP(N;t) f &($nj)-&(n,nj), i#j, (5.6.8)
NEO
A GENERAL BIRTH-AND-DEATH PROCESS FORMULATION 249
a a*ww + aal, - asi I as’
~ = N;/2:(ni- l)P(N;f) asi
I
2 G($)- &($), (5.6.9)
a a'G(l,l) + aG(l,t) - asj
[ a.$
~ = 2 n,52,P(N;t) asi I NEfl
.I
s &(T+zj). (56.10)
If we denote the mean of the ith population by ei; its variance by uii; its
covariance with the jth population by ati; and the general third-order
moment of the ith, jth and kth populations around the mean, where
possibly j = i of k = j = i etc., by a+ then we have the following definitions
and relationships:
rii i G(Q), i= 1 ,...,m. (5.7.1)
Since oU b &(n,- fii)(nj- ri,), we have
& Cninj> =clu+iiiEj, ij= 1 ,..., m. (5.7.2)
Since ullk i & (ni - iii)(nj - iij)(nk - fiJ, we have
Clearly (5.7.3) is symmetrical with respect to the indices i, j and k. As in (5.6.6), we can derive the following expressions using (5.7.2):
aa:ca(slB:) = $ & ( ninj) 1 J
a a’G(l,t) + aG(l,t) z I a$ asi 1 = $&(nf)
(5.8.1)
da,, diii =dt+2qx. (5.8.2)
250 WAYNE M. GETZ
If we differentiate (5.4) or (5.5) with respect to si for some i E[ 1,. . . ,m] and then put S= 1, we obtain, using the expressions (5.6:) and (5.7:), an ordinary differential equation in t for ii,(t), which is then the mean of the distribution of the process corresponding to the particular set of equations that have been differentiated. Similarly differentiating (5.4) or (5.5) twice, first with respect to s, and then with respect to sj (where possibly j= i), using the expressions (5.6. .), (5.7. .) and (5.8. .), we obtain an ordinary differential in t for u,(t). Ordinary differential equations in t for higher-order moments can be similarly derived.
Proceeding as described above, we can derive from Eq. (5.4), the equa- tions describing the dynamic behavior of the means and the variance and covariance terms of the linear control system defined in Sec. 4. A. These equations turn out to be the following:
The equations (5.12) describe the dynamic behavior of the mean size of the m different species in a Lotka-Volterra competitive system as defined in
Sec. 4. B. The equations (5.13) and (5.14) describe the dynamic behavior of the variance and covariance terms of this system.
As previously mentioned, under certain conditions there will exist, for a time-homogeneous system, a limiting distribution [ 181. The generating func- tion for this distribution will, by (3.3) and (5.1.1), be
G(S)= 2 P(N)SN, NEQ
since the probabilities will no longer be a function of 1. Thus 8G (S)/ at = 0, so that replacing G (S,t) by G(S) in Eqs. (5.4) and (5.5) and setting the left-hand sides of these equations equal to zero, we can solve each of these two systems for G(S). Further, the equations for the means, variance and covariance of each system will satisfy dri,/dt=O, dukk/dt=O and dqk/dt =0 respectively and can be found by solving the corresponding set of algebraic equations that will arise by setting the left-hand sides of Eqs. (5.9) to (5.14) equal to zero.
An analysis of the system solution (especially of its stability properties, a knowledge of which is required to determine whether a limiting distribution exists) is, however, beyond the scope of this paper.
6. STOCHASTIC VERSUS DETERMINISTIC MODELS
In the previous section we derived differential equations for the means, and the variance and covariance terms of a population process for each of the three systems considered in this paper. The differential equations derived above for the set of means for each species of the stochastic system can be directly compared with the set of differential equations describing the magnitude of each species in the corresponding deterministic system. In the linear control system the set of differential equations for the means of the stochastic system and the deterministic system are mathematically identical, while for the Lotka-Volterra competitive system the two sets of
252 WAYNE M. GETZ
equations are identical except for one interesting modification in the stochastic system.
In this section we shall compare a stochastic approach to a particular problem with a deterministic approach, balancing realism with simplicity.
A. LINEAR SYSTEM
If we define
q(t)=&(r) - puLij(t), iJ= 1 ,...,m,
bi,(r)=cui,(f)-_i,(f), r=l ,...,p, i=l ,...> m
and
rn=(iT,(t) )..., Em(t)),
then the equations (5.9), k= 1,. . ., m, can be written in matrix form as
follows:
$ =A(t)P+B(t)U(t),
whereA(t)=(~~(t))ER’“~” and B(t)=(b,,(t))~R~~p. A deterministic model of this system would have the form
(6.1)
S$=Lt(t)N+B(I)U(r), (6.2)
where the parameters a,(t) and b,(t) would be considered as deterministic interaction and growth rates, as opposed to the more realistic interpretation given to these parameters in this paper, as rate measures of the probability of population changes due to interaction and growth.
Further, we see that the solution, to (6.1) and (6.2) must be identical when solved subject to the same initial conditions, i.e.,
2 n,P(N;O) NESl
N(O)=m(O)= :
A GENERAL BIRTH-AND-DEATH PROCESS FORMULATION 253
This solution, however, cannot be realistically interpreted in the context of the deterministic system, since we know from differential-equation theory that for A (t), B(t) and u(t) continuous in t, the solution N(t) will be continuous in t. Thus N(t) $?O for all t E [0, cc). However, m(t) is not expected to belong to 52, since m(t) is an average measure of a number of integers which need not itself be an integer to be realistically interpreted.
It is common practice to stochasticize a deterministic system by selecting a disturbance vector w(t) and suitably coupling it to the system through a matrix G(t) [19, p. 241, so that (6.2) becomes
(6.3)
To solve (6.3) for N(t), which will now represent a probable trajectory of the stochastic process X(t), we are required to know the statistics of the disturbance vector w(t) (which is often assumed to be Gaussian) and the matrix G(t). The statistics of the noise for the stochastic linear control system developed in this paper can, in contrast to the stochastic linear control system given by Eq. (6.3), be directly solved for, utilizing Eqs. (5.10) and (5.11) subject to given initial conditions-i.e., the stochasticity is implicitly defined in the system given by equations (5.9) (5.10) and (5.11). Further, if P(t) is a vector of elements ~~(2) listed in dictionary order, say, the equations (5.9) (5.10) and (5.11) form an extended linear system
(6.4)
where @ (t) and $8 (t) are appropriately defined in terms of the parameters of the three equations. Using the extended system (6.4) it is possible to formulate an optimal control policy that takes the cost of variance and covariance into account, by directly utilizing results of the well-analyzed
quadratic linear control problem [23,, Chapter 31, where T will now be ( 1
the system vector under consideration. A model of the type (6.4) avoids some of the mathematical intricacies
involved in analyzing continuous stochastic systems of the type (6.3) where, if w(t) is uncorrelated in time, the rigour of fto calculus is necessary. It is, however, simpler to work with the discrete form of Eq. (6.3) and well- known results exist for optimally controlling such systems [ 19,201.
The comparison of the stochastic model given by Eq. (6.3) with the stochastic model developed in this paper [i.e., Eqs. (5.9), (5.10) and (5.1 l)] applies only to the case where the stochasticity in the system arises out of the inherent uncertainty in the system parameters as being derivatives of
254 WAYNE M. GETZ
measures of the probability of birth, death and species interaction changing the size of the population. The comparison does not cover external noise interfering with the system, as this type of noise has clearly not been accounted for in building the model (5.9)-(5.11). More explicitly, we arrived at Eq. (5.9), (5.10) and (5.11) using a generalized stochastic birth-and-death formulation, and the resulting model represented by these three equations gererates the time-dependent means and variance and covariance terms of (in general) the non-stationary, multivariate probability distribution for this birth-and-death process. In contrast, the model described by Eq. (6.3) is based on the assumption that the population process has a deterministic growth rate. A random noise component is added, however, to compensate for this simplifying assumption.
B. LOTKA-VOLTERRA SYSTEMS
If we define
and
in equation (5.12), it can be written as
k= I,...,m. (6.5)
This system of differential equations is identical to the deterministic Lotka- Volterra system of differential equations,
dn, - =rk(t)nk+ 5 akj(t)n,nj, dt
k= I,...,m, j=l
except for the additional terms X7_ 1 akj(t)ukj, k=l ,...,m. Compared with the linear case, the behavior of the system (6.6) is more difficult to analyze, except possibly when all the system parameters are constant [21]. Furthermore, any results obtained for the system (6.6) cannot be directly extended to the system (6.5), owing to the additional terms in (6.5). The appearance of the covariance and variance terms in (6.5) is particularly interesting in the one-dimensional case, where the solutions to (6.5) and (6.6) can be easily compared. For a single population and constant parame-
A GENERAL BIRTH-AND-DEATH PROCESS FORMULATION 255
ter values (6.6) becomes the deterministic sigmoid growth equation
dn dt = rn - an=, r>O, a>O,
and
dii - =rE-a(H2+a2), dt
(6.7)
(6.8)
which is the stochastic equivalent of (6.7). Since u2, the population variance, is always non-negative, given
the population mean size in the stochastic model (6.8) will grow more slowly than the size of the population in the deterministic model (6.7) and in fact will always be the smaller of the two. Thus the deterministic sigmoidal growth model for a single population will, at any point in time, over estimate the mean population size as given by the equivalent stochastic model.
The system (6.5) cannot be solved for n(t), since the variance and covariance terms uU, iJ= 1,. . . , m, are now known. Furthermore, the system of equations {(5.12),(5.13),(5.14)} cannot be solved either, since the third- order moments arki, I, kj = 1,. . . , m, appear and are not known. It is possible to find differential equations describing the behavior of the third-order moments, but since these equations will include moments of the fourth order, and so on, it is impossible to get a finite closed system of equations by including the higher-order moments. Goel et al., discussing a two-species system [ll, p. 2621, suggest solving (6.5) (k = 1,2) by considering the variance and covariance terms as random driving forces. In making such an approximation no use would, however, have been made of the considerable knowledge available about the behavior of these second-order terms. A far better approximation would be to solve the system equations for the mean and variance and rather assume the form of the third-order moments urkj, for the following reasons:
(a) For reasonably large population sizes we can expect the probability distribution of the system to closely approximate a joint multivariate normal distribution for which u,~~ = 0 for all I, kj= 1,. . .,m, so that the assumption
ii, Ukj + Hj U,k + UCkj 25 ii,Ukj + ‘i/ U,k (6.9)
is a good one.
256 WAYNE M. GETZ
(b) If we either set urkl = 0 or replace urkl by an appropriate random noise term, we loose very little knowledge of the behavior of the second-order terms, and such an approximation would achieve a good estimate of the variance and covariance terms, which is always important information to have about a particular stochastic system.
7. CONCLUSION
Although only the stochastic-process equivalents of the linear and Lotka- Volterra deterministic systems have been derived in detail, it is possible to model any system embraced by the general formulation described in Sec. 3. The models chosen in Sec. 4 were used to demonstrate the general proce- dure for deriving the differential equations for the means and the variance and covariance terms; they were specifically chosen for their general appli- cability to many population processes and also because these models covered processes linear in state and control, quadratic in state and state- dependent on control.
The models formulated in this paper can be used to design optimal control policies to manipulate population processes. These models facilitate the inclusion of the cost of large variance and covariance terms in determin- ing the optimal control, as discussed in Sec. 6. A. Controls of the type F’(t) that directly affect the birth and death rates of an individual species are very important in population control, since these rates are susceptible to external influences.
If the process modeling a particular physical system is well formulated, the solution to the equations describing this process will exist and will be unique. A stability analysis of the system is important, however, as invari- ably the system stability will be parameter-dependent, and such an analysis would yield information on various parameter relationships and bounds.
No discussion has been included on the control and stability analysis of the models, since each system has its own peculiar features.
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