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Convergence of the mean and varlance of size for a stochastic population model
ELSEVIER
Convergence of the Mean and Variance of Size for a Stochastic Population Model SMALL MAHDI*
Department of Mathematics and Statistics, University of Montrea~ Montreal, Province of Quebec, H3C MT, Canada Received 28 August 1995; revised 8 March 1996
ABSTRACT The purpose of this article is to present convergence results for the mean and variance of fish size in a stochastic population model developed by Deriso and Parma. The proofs rely on the concavity property of the transformation of variance over successive generations. © Elsevier Science Inc., 1996
1. INTRODUCTION In the discrete stochastic model for the dynamics of size in a population of fish proposed by Deriso and Parma [1], the principal factors of evolution considered are fishing and growth. At the beginning of a year t, the probability density function ~b(x It) of the logarithm of length X(t) of the individuals of a cohort at the youngest age of recruitment into the fishery is normal with mean E(t) and variance V(t). The year is divided into two parts, with fishing taking place in one part and growth in the other. The size-selective risk of fishing mortality is separable into a product of full-recruitment fishing mortality rate F, and a size selectivity factor S(x). The probability that an individual of size x survives the fishery in year t is given by Pr(fish survives I x) = e x p [ - F t S ( x ) ] .
(1)
The probability density function ~b* of x after the fishing season can be written using Bayes theorem as ~b*( x I t) =
Pr (fish survives I x) ~ ( x I t)
(2)
f?ooPr(fish survivesl x)4~( x l t) dx
*Current address: 96A Normand Street, Ch~teauguay, PQ, J6J 2M8, Canada.
MATHEMATICAL BIOSCIENCES 138:23-29 (1996) © Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010
0025-5564/96/$15.00 PII S0025-5564(96)00096-X
24
S. MAHDI
The length frequency distribution for the cohort at the beginning of year t + 1 is obtained by applying to the log-length size after fishing X * ( t ) , the following growth model X ( t + 1) = ~ + / 3 X * ( t ) + ~.
(3)
Here ot > 0 and 0 0 are constant),
(4)
the normality of the distribution of the log-length prior to the fishing season is preserved after the harvest and application of the growth model (3). At the beginning of the year t + 1, the distribution of the log-length is normal with mean E ( t + 1) and variance V(t + 1) whose expressions are given in the next section. In this article, we study equilibrium of the dynamical system of the mean and the variance of the model and give analytical proofs of their global stability. This will show the existence of an asymptotic balance between the biological and environmental forces acting on size of individuals in the population. 2. DYNAMIC EQUATIONS In this section, we first recall the recurrence equations for the mean and the variance obtained by Deriso and Parma [1] in the case of constant full-recruitment fishing mortality risk. The dynamic equation for the mean is E(t+l)=a+/3[
[ E(t) - blFV(t ) ]
~
7
i
]
(5)
and for the variance is /3
v(0
v(t + 1) = 2bzFV(t) + 1 + II,.
(6)
Dynamical systems (5) and (6) admit a unique equilibrium. The equilibrium variance can be obtained by solving the following equation 2b:Fl )2 + (1-2bzFl/, - f l 2 ) I ) - V~= 0.
(7)
STOCHASTIC POPULATION MODEL
25
Since
(8)
A = ( 1 - 2 b 2 F V~ - / 3 2 ) 2 + 8 b 2 F V , > 0 and
-K
2b2F < 0,
(9)
it follows that the only admissible equilibrium point is 1~= (2b2FV~ + / 3 2 - 1)+ ~ ( 1 - 2 b E F V ~ -/32)2+8bEFV~ 4b2F
(10)
As a partial check of the above result, we put /3 = 0 and get 1~---V~, which agrees with hypotheses of the model. V > 0 since the normal growth random variable e has a constant variance II,. On the other hand, it is easy to ,prove that the recurrence for the mean admits a unique equilibrium E, which is given by
ff~ = (l + 2b2Fl~)°t - blFl~'fl
1-/3 + 2bEfl)
(11)
3. STABILITY OF EQUILIBRIUM VARIANCE AND MEAN The study of stability of equilibrium variance and mean is presented below. 3.1. CONVERGENCE OF THE VARIANCE SEQUENCE The dynamics of the variance is described by the following transformation: /32V ~b(V) = 2bEFV+ 1 +V~.
(12)
The proof of convergence of the variance sequence is based on the following basic convergence theorem recently established in Lessard and Mahdi [3]. Let us recall the following classical definition before presenting the convergence theorem for transformations of covariance.
26
S. MAHDI
DEFINITION 3.1 Let A and B be two positive semidefinite symmetric matrices. We say that A >! B or A > B if, respectively, A - B is positive semidefinite or positive definite. We are now ready to state the following theorem, proved in [3], on the iterates of transformations of covariance matrices, which are necessarily semidefinite symmetric matrices.
THEOREM3.1 Let T be a continuous transformation for positive semidefinite symmetric matrices which is concave, that is, for any positive semidefinite symmetric matrices C1 and C2, we have T( C1 +2C2 )>-" T(C,) +2T(C2) If T possesses a unique fixed point C, which is a positive definite symmetric matrix, that is, T(C)= C > O, then, for any positive definite symmetric matrix C, we have lim T(")(C) = C. n--~
The following result is a direct consequence of Theorem 3.1.
Result 3.1. The variance asymptotic body size lP of individuals in the population is globally stable. Proof In our case, the transformation ~0 is positive and continuous. According to Theorem 3.1, it suffices to prove the concavity property of for establishing the global stability of V. Taking the second derivatives of ~b with respect to V, we get d2~(V)
=
dv 2
_
4bz ~2F [2b2FV + 1]3 < O.
(13)
This proves the concavity property of the transformation qt. On the other hand, we have
-
=
(1 + 2b rv) (i + 2b Fe) "
(14)
STOCHASTIC POPULATION MODEL
27
Since I1 +2b2FVI >/1 and I1 + 2b2Fl~'l >/1, it follows that
lqJ(v)- Pl /321v-
(15)
Thus the sequence V(t) converges toward if" at a geometric rate (with factor/3 2). 3.2. CONVERGENCE OF THE MEAN SEQUENCE The dynamics of the mean can be written as
E ( t +1) = P ( t ) E ( t ) + ~ ( t ) ,
(16)
where /3 P(t) = l+2b2FV(t)
(17)
and
b 1/3FV(t) e(t) = a - l+2b2FV(t) .
(18)
Since V(t) converges, the sequences P(t) and e(t) also converge. Let /; and ~: the corresponding limits. For proving convergence of E(t), we have to demonstrate that the sequence W(t) = E ( t ) - E tends towards 0 as t tends to oo. The dynamics of W(t) is given by
W ( t + 1) = f i W ( t ) + S ( t ) ,
(19)
S ( t ) --- ( P ( t ) - t ~ ) E ( t ) + ( e ( t ) - ~).
(20)
where
PROPOSITION 3.1 The sequence S( t ) converges toward O. To prove Proposition 3.1, it suffices to prove that the sequene~ E(t) is bounded. Seeing that a(t) and P ( t ) are convergent, it follows that there exist t o > 0 and constants C > 0 and A such that Vt > to; we have le(t)[ < C and IP(t)] < A < 1. Iterating Eq. (16) n times from to, we get
S. MAHDI
28
for any arbitrary 7/> 0, IE(t0 +n)l~< A"lE(t0)l+C ~ Ak < ~
C
+n.
(21)
k=O
So for t sufficiently large, we have E(t)<~ C / ( 1 - )0+ ~/. The sequence E(t) is then bounded and there exist t I > 0 such that Yt > t~, we have IS(t)[ < r/. Iterating now (19) k times from t~, we obtain tl+k-1
W( t 1 + k) = f f k w ( tl) +
Y'~ lfit'+k-1-/S(j)
(22)
j= t I
and tl+k-1
i/~l,,+k-l-j< ii~lklW(tl)l÷
[W( t 1 + k)[ ~<[l~]klW( tl)l+ 71 j=t 1
n
1-1PI" (23)
As 7/is arbitrary, we have that lim W(t) = 0.
(24)
t --~ oo
Hence, we conclude the following:
Result 3.2.
The mean asymptotic body size /~ of individuals in the population is a unique fixed point, and it is globally stable.
Moreover, from Eq. (23), it follows that I2" is reached before /~ and, after completion of convergence of V(t), the sequence E(t) also converges toward E at a geometric rate with factor
Iel =1 l+2b2Fl~ ~<,8. 4.
DISCUSSION
We have proved in this article that the dynamical system of variability for the stochastic body size in a population, given in Deriso and Parma [1], possesses a unique and globally stable fixed point in the considered case. This result excludes the existence of limit cycles or chaos for the mean and variance asymptotic body size of individuals in
STOCHASTIC POPULATION MODEL
29
the population under the action of considered evolutionary forces. Moreover, numerical studies have shown rapid convergence to equilibrium, usually in fewer than 20 iterations. On the other hand, the study presented in this article can be extended to a general case of correlated characters using mathematical techniques developed in [4].
I express my gratitude to the anonymous referees for providing useful comments. REFERENCES 1 R. B. Deriso and A. M. Parma, Dynamics of age and size for a stochastic population model. Can~ J. Fish. Aquat. Sci. 45:1054-1068 (1988). 2 M. D. Cohen and G. S. Fishman, Modeling growth-time and weight-length relationships in a single year-class fishery with examples on North Carolina pink and brown shrimp. Can. J. Fish. Aquat. Sci. 37:1000-1011 (1980). 3 S. Lessard and S. Mahdi, Convergence of variability in Gaussian polygenic models. Genet. Sel. Evol. 27:395-421 (1995). 4 S. Mahdi, Convergence de la vadabilit6 de caract6res g6n6tiques quantitatifs, Ph.D. thesis, Department of Mathematics and Statistics, University of Montreal, Montreal (1994).
Convergence of the Mean and Variance of Size for a Stochastic Population Model SMALL MAHDI*
Department of Mathematics and Statistics, University of Montrea~ Montreal, Province of Quebec, H3C MT, Canada Received 28 August 1995; revised 8 March 1996
ABSTRACT The purpose of this article is to present convergence results for the mean and variance of fish size in a stochastic population model developed by Deriso and Parma. The proofs rely on the concavity property of the transformation of variance over successive generations. © Elsevier Science Inc., 1996
1. INTRODUCTION In the discrete stochastic model for the dynamics of size in a population of fish proposed by Deriso and Parma [1], the principal factors of evolution considered are fishing and growth. At the beginning of a year t, the probability density function ~b(x It) of the logarithm of length X(t) of the individuals of a cohort at the youngest age of recruitment into the fishery is normal with mean E(t) and variance V(t). The year is divided into two parts, with fishing taking place in one part and growth in the other. The size-selective risk of fishing mortality is separable into a product of full-recruitment fishing mortality rate F, and a size selectivity factor S(x). The probability that an individual of size x survives the fishery in year t is given by Pr(fish survives I x) = e x p [ - F t S ( x ) ] .
(1)
The probability density function ~b* of x after the fishing season can be written using Bayes theorem as ~b*( x I t) =
Pr (fish survives I x) ~ ( x I t)
(2)
f?ooPr(fish survivesl x)4~( x l t) dx
*Current address: 96A Normand Street, Ch~teauguay, PQ, J6J 2M8, Canada.
MATHEMATICAL BIOSCIENCES 138:23-29 (1996) © Elsevier Science Inc., 1996 655 Avenue of the Americas, New York, NY 10010
0025-5564/96/$15.00 PII S0025-5564(96)00096-X
24
S. MAHDI
The length frequency distribution for the cohort at the beginning of year t + 1 is obtained by applying to the log-length size after fishing X * ( t ) , the following growth model X ( t + 1) = ~ + / 3 X * ( t ) + ~.
(3)
Here ot > 0 and 0 0 are constant),
(4)
the normality of the distribution of the log-length prior to the fishing season is preserved after the harvest and application of the growth model (3). At the beginning of the year t + 1, the distribution of the log-length is normal with mean E ( t + 1) and variance V(t + 1) whose expressions are given in the next section. In this article, we study equilibrium of the dynamical system of the mean and the variance of the model and give analytical proofs of their global stability. This will show the existence of an asymptotic balance between the biological and environmental forces acting on size of individuals in the population. 2. DYNAMIC EQUATIONS In this section, we first recall the recurrence equations for the mean and the variance obtained by Deriso and Parma [1] in the case of constant full-recruitment fishing mortality risk. The dynamic equation for the mean is E(t+l)=a+/3[
[ E(t) - blFV(t ) ]
~
7
i
]
(5)
and for the variance is /3
v(0
v(t + 1) = 2bzFV(t) + 1 + II,.
(6)
Dynamical systems (5) and (6) admit a unique equilibrium. The equilibrium variance can be obtained by solving the following equation 2b:Fl )2 + (1-2bzFl/, - f l 2 ) I ) - V~= 0.
(7)
STOCHASTIC POPULATION MODEL
25
Since
(8)
A = ( 1 - 2 b 2 F V~ - / 3 2 ) 2 + 8 b 2 F V , > 0 and
-K
2b2F < 0,
(9)
it follows that the only admissible equilibrium point is 1~= (2b2FV~ + / 3 2 - 1)+ ~ ( 1 - 2 b E F V ~ -/32)2+8bEFV~ 4b2F
(10)
As a partial check of the above result, we put /3 = 0 and get 1~---V~, which agrees with hypotheses of the model. V > 0 since the normal growth random variable e has a constant variance II,. On the other hand, it is easy to ,prove that the recurrence for the mean admits a unique equilibrium E, which is given by
ff~ = (l + 2b2Fl~)°t - blFl~'fl
1-/3 + 2bEfl)
(11)
3. STABILITY OF EQUILIBRIUM VARIANCE AND MEAN The study of stability of equilibrium variance and mean is presented below. 3.1. CONVERGENCE OF THE VARIANCE SEQUENCE The dynamics of the variance is described by the following transformation: /32V ~b(V) = 2bEFV+ 1 +V~.
(12)
The proof of convergence of the variance sequence is based on the following basic convergence theorem recently established in Lessard and Mahdi [3]. Let us recall the following classical definition before presenting the convergence theorem for transformations of covariance.
26
S. MAHDI
DEFINITION 3.1 Let A and B be two positive semidefinite symmetric matrices. We say that A >! B or A > B if, respectively, A - B is positive semidefinite or positive definite. We are now ready to state the following theorem, proved in [3], on the iterates of transformations of covariance matrices, which are necessarily semidefinite symmetric matrices.
THEOREM3.1 Let T be a continuous transformation for positive semidefinite symmetric matrices which is concave, that is, for any positive semidefinite symmetric matrices C1 and C2, we have T( C1 +2C2 )>-" T(C,) +2T(C2) If T possesses a unique fixed point C, which is a positive definite symmetric matrix, that is, T(C)= C > O, then, for any positive definite symmetric matrix C, we have lim T(")(C) = C. n--~
The following result is a direct consequence of Theorem 3.1.
Result 3.1. The variance asymptotic body size lP of individuals in the population is globally stable. Proof In our case, the transformation ~0 is positive and continuous. According to Theorem 3.1, it suffices to prove the concavity property of for establishing the global stability of V. Taking the second derivatives of ~b with respect to V, we get d2~(V)
=
dv 2
_
4bz ~2F [2b2FV + 1]3 < O.
(13)
This proves the concavity property of the transformation qt. On the other hand, we have
-
=
(1 + 2b rv) (i + 2b Fe) "
(14)
STOCHASTIC POPULATION MODEL
27
Since I1 +2b2FVI >/1 and I1 + 2b2Fl~'l >/1, it follows that
lqJ(v)- Pl /321v-
(15)
Thus the sequence V(t) converges toward if" at a geometric rate (with factor/3 2). 3.2. CONVERGENCE OF THE MEAN SEQUENCE The dynamics of the mean can be written as
E ( t +1) = P ( t ) E ( t ) + ~ ( t ) ,
(16)
where /3 P(t) = l+2b2FV(t)
(17)
and
b 1/3FV(t) e(t) = a - l+2b2FV(t) .
(18)
Since V(t) converges, the sequences P(t) and e(t) also converge. Let /; and ~: the corresponding limits. For proving convergence of E(t), we have to demonstrate that the sequence W(t) = E ( t ) - E tends towards 0 as t tends to oo. The dynamics of W(t) is given by
W ( t + 1) = f i W ( t ) + S ( t ) ,
(19)
S ( t ) --- ( P ( t ) - t ~ ) E ( t ) + ( e ( t ) - ~).
(20)
where
PROPOSITION 3.1 The sequence S( t ) converges toward O. To prove Proposition 3.1, it suffices to prove that the sequene~ E(t) is bounded. Seeing that a(t) and P ( t ) are convergent, it follows that there exist t o > 0 and constants C > 0 and A such that Vt > to; we have le(t)[ < C and IP(t)] < A < 1. Iterating Eq. (16) n times from to, we get
S. MAHDI
28
for any arbitrary 7/> 0, IE(t0 +n)l~< A"lE(t0)l+C ~ Ak < ~
C
+n.
(21)
k=O
So for t sufficiently large, we have E(t)<~ C / ( 1 - )0+ ~/. The sequence E(t) is then bounded and there exist t I > 0 such that Yt > t~, we have IS(t)[ < r/. Iterating now (19) k times from t~, we obtain tl+k-1
W( t 1 + k) = f f k w ( tl) +
Y'~ lfit'+k-1-/S(j)
(22)
j= t I
and tl+k-1
i/~l,,+k-l-j< ii~lklW(tl)l÷
[W( t 1 + k)[ ~<[l~]klW( tl)l+ 71 j=t 1
n
1-1PI" (23)
As 7/is arbitrary, we have that lim W(t) = 0.
(24)
t --~ oo
Hence, we conclude the following:
Result 3.2.
The mean asymptotic body size /~ of individuals in the population is a unique fixed point, and it is globally stable.
Moreover, from Eq. (23), it follows that I2" is reached before /~ and, after completion of convergence of V(t), the sequence E(t) also converges toward E at a geometric rate with factor
Iel =1 l+2b2Fl~ ~<,8. 4.
DISCUSSION
We have proved in this article that the dynamical system of variability for the stochastic body size in a population, given in Deriso and Parma [1], possesses a unique and globally stable fixed point in the considered case. This result excludes the existence of limit cycles or chaos for the mean and variance asymptotic body size of individuals in
STOCHASTIC POPULATION MODEL
29
the population under the action of considered evolutionary forces. Moreover, numerical studies have shown rapid convergence to equilibrium, usually in fewer than 20 iterations. On the other hand, the study presented in this article can be extended to a general case of correlated characters using mathematical techniques developed in [4].
I express my gratitude to the anonymous referees for providing useful comments. REFERENCES 1 R. B. Deriso and A. M. Parma, Dynamics of age and size for a stochastic population model. Can~ J. Fish. Aquat. Sci. 45:1054-1068 (1988). 2 M. D. Cohen and G. S. Fishman, Modeling growth-time and weight-length relationships in a single year-class fishery with examples on North Carolina pink and brown shrimp. Can. J. Fish. Aquat. Sci. 37:1000-1011 (1980). 3 S. Lessard and S. Mahdi, Convergence of variability in Gaussian polygenic models. Genet. Sel. Evol. 27:395-421 (1995). 4 S. Mahdi, Convergence de la vadabilit6 de caract6res g6n6tiques quantitatifs, Ph.D. thesis, Department of Mathematics and Statistics, University of Montreal, Montreal (1994).