parameter identification in multistage population dynamics model
TRANSCRIPT
Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328
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Nonlinear Analysis: Real World Applications
journal homepage: www.elsevier.com/locate/nonrwa
Parameter identification in multistage population dynamics modelDelphine Picart a,∗, Bedr’eddine Ainseba b
a Arizona State University, School of Mathematical and Statistical Sciences, Tempe, AZ 85287-1804, United Statesb Bordeaux University, IMB UMR CNRS 5251, Anubis Team INRIA Bordeaux Sud Ouest, 3 Place de la Victoire, 33076 Bordeaux Cedex, France
a r t i c l e i n f o
Article history:Received 28 September 2010Accepted 28 May 2011
Keywords:Population dynamicsAge-structured modelParameter estimationExperimental dataLagrangianQuasi-Newton algorithm
a b s t r a c t
We present a numerical analysis to solve a parameter identification problem. We identifythe demographical parameters of a multistage population dynamics model (Ainseba et al.,2011 [12]). Our nonlinear optimization problem with constraints is solved by a Quasi-Newtonmethod. The convergence proof of this numerical method is performed here. Somenumerical applications of it are also given at the end of the paper.
© 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Some classical models used in population dynamics are the linear von Foerster–McKendrick model [1], its homologousGurtin–MacCamy model [2] that considers nonlinearities in vital rates, or the Sinko–Streifer model [3], for example. Themodeling idea is to follow the growth of the population split into several groups by computing their sizes as time passesaccording to the individual gain or loss. In the nonlinear cases, variations of the population size are taken into account by thevital rates. These models consist of partial differential equations and one can find a rich literature as much in mathematicalanalysis [4–6] as numerical analysis [7–10] about them.
For the study of a specific population, we have to define the vital rates, i.e. the mortality and the growth rates. Theestimation of these parameters is possible from experimental data as shown by Banks [4] for the Sinko–Streifer model, or byAckleh [11] for the nonlinear size-structuredmodel. The input data used are either the change in total number of individualsper unit of time for each individual group or the total number of individuals per group in a fixed time. The general methodused for parameter identification is least-squares estimation.
In this paper, we are interested in a model developed for studying the population dynamics of a wine pest, the Europeangrapevine moth [12,13]. The model is a system of four equations each related to the Sinko–Streifer model and is written as
∂
∂tue(t, a) +
∂
∂a
ve(t, a)ue(t, a)
= −me(t, a)ue(t, a) − βe(t, a)ue(t, a),
∂
∂tul(t, a) +
∂
∂a
vl(t, a)ul(t, a)
= −ml(P l(t), t, a)ul(t, a) − β l(t, a)ul(t, a),
∂
∂tuf (t, a) +
∂
∂a
vf (t, a)uf (t, a)
= −mf (t, a)uf (t, a),
∂
∂tum(t, a) +
∂
∂a
vm(t, a)um(t, a)
= −mm(t, a)um(t, a),
(1.1)
∗ Corresponding author. Tel.: +1 480 727 7575.E-mail addresses: [email protected], [email protected] (D. Picart), [email protected] (B. Ainseba).
1468-1218/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2011.05.030
3316 D. Picart, B. Ainseba / Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328
where t is the time variable, a the age variable and ue, ul, um, and uf are the age density distributions of, respectively, egg,larva,malemoth and femalemoth populations at time t . The individual groups are structuredwith respect to their individualages. The total population for stage k is defined by
Pk(t) =
∫ Lk
0uk(t, a)da, t ≥ 0.
As in the classical models, the system (1.1) includes the mortality, mk, k = e, l, f ,m, and growth, vk, k = e, l, p, functionsbut here they are defined for each stage. To model the food competition between larvae, the larval stage mortality functionml depends on the total population of larvae.
Because of the specific biological cycle of our insect, the model (1.1) also contains two other functions that are thetransition functions βe and β l. The first one models the biological transformation of eggs into larvae whereas the secondone models the passage from the larval to the moth stage.
The boundary conditions of Eqs. (1.1) are defined by
Be(t) : ve(t, 0)ue(t, 0) =
∫ Lf
0β f (P f (t), Pm(t), t, s)uf (t, s)ds,
Bl(t) : vl(t, 0)ul(t, 0) =
∫ Le
0βe(t, s)ue(t, s)ds,
Bf (t) : vf (t, 0)uf (t, 0) =
∫ Ll
0τβ l(t, s)ul(t, s)ds,
Bm(t) : vm(t, 0)um(t, 0) =
∫ Ll
0(1 − τ)β l(t, s)ul(t, s)ds,
(1.2)
where τ is the sex-ratio, β f corresponds to the well know age-specific per capita birth rate and is dependent on the totalpopulation of moths. The initial conditions are
uk(0, a) = uk0(a), a ∈ [0, Lk], k = e, l, f ,m. (1.3)
Our objective is to estimate all the parameters that constitute the model (1.1)–(1.2)–(1.3). To reach this goal we need touse experimental data. There exist four different types of measures for these populations that are the hatching, flying, andegg-laying temporal dynamics, and larval age distribution. The three first give the total number of new individuals in eachage group in time, whereas the last data gives the total number of individuals per age group for a fixed time. Let ze,n, z l,n, andz f ,n, for n from 1 to Nt , be the temporal data and zi, for i from 1 to Na, be the age data. We use the set of these experimentaldata to estimate the model parameters.
Throughout this paper, we are interested in the following parameter identification problem
[P]
minq∈Q
J(q) =
−k=e,l,f
Nt−n=0
∫ tn+1
tn
Bk(t; q) − zk,n
2dt
+
Na−i=1
∫Ci
ul(T , a) − zi
2da,
where Bk(t; q), ul(T , a) are given by (1.1)–(1.2)–(1.3) and ze,n, z l,n, z f ,n, zi are the data. The parameter q is a vector givenby
q = (vk,mk, β k), k = e, l, f ,m, k = e, l, f ,
and is defined in the space Q which is a compact subset of K = (C(Ω))11 where
C(Ω) = f (t, a) ∈ L∞(Ω); 0 < f (t, a) ≤ f ,
with Ω set to [0, T ] × [0, Lmax] and Lmax
= maxLe, Ll, Lf , Lm. The constants 1t and 1a are the time and age mesh sizes.The grid points are defined by tn = n1t, 0 ≤ n ≤ Nt and ai+ 1
2=i + 1
2
1a, 0 ≤ i ≤ Na − 1, and they form the elements
[tn, tn+1[×Ci with Ci = [ai− 1
2, ai+ 1
2[.
The study of [P] is organized as follow: In Section 2,we are interested in themodel (1.1)–(1.2)–(1.3) and give convergenceand continuity results with respect to the set of parameters. In Section 3, we construct the parameter identification schemeand prove the convergence of its solution to the continuous solution. Sections 4 and 5 are, respectively, devoted to presentingsome numerical examples and conclusions.
2. Direct problem analysis
The first step in solving the optimization problem [P] is the approximation of (1.1)–(1.2)–(1.3). As said in the introduction,there exists an exhaustive literature on the numerical study of the structured equations. For example, Douglas andMilner [7]developed a finite difference scheme for the Gurtin–MacCamy model. This scheme was improved by Know and Cho [14] to
D. Picart, B. Ainseba / Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328 3317
get better precision, second order in time. Sulsky [9] compared it with other schemes for the Sinko–Streifer model. Sheproved that the finite volume scheme (FV), has a precision of second order in time and better approximates the solutionof the model. Angulo and Lopez-Marcos [15,16] and Kostova [8] also developed schemes on the nonlinear model but theybased them on the solution of the model along the characteristics. Their precision is of the second order for Angulo andLopez-Marcos, and third order for Kostova. In our case, we adopt a FV scheme to approximate the equations defined in(1.1)–(1.2)–(1.3).
Let uk∆ be the approximated solution of the density function related to the k-stage equation. The initial condition is
approximated by its mean value on each interval Ci so that
uk,0i =
11a
∫Ciuk0(x)dx, k = e, l, f ,m.
Let vk∆, βk
∆ andmk∆ be the approximated functions of, respectively, the functions vk, βk andmk. All these functions belong to
the spaceK∆ = f (t, a) = f ni , (t, a) ∈ [tn, tn+1
[×Ci, 0 ≤ f ni ≤ f , i = 1, . . . ,Na, n = 0, . . . ,Nt
that is an approximating finite dimensional compact set of K . The k-stage equation of model (1.1)–(1.2)–(1.3) isapproximated by
uk,n+1i =
uk,ni
1 −
1t1av
k,ni
+
1t1av
k,ni−1u
k,ni−1
1 + 1tmk,n+1i + 1tβk,n+1
i
, i = 1,Na,
vk,n+10 uk,n+1
0 = 1aNa−j=1
βp,n+1j up,n+1
j ,
(2.1)
for n from 0 to Nt and p is related to the stage equation preceding the k stage. The CFL condition is
0 <1t1a
maxk,n
‖vk,n∆ ‖∞ ≤ 1, (2.2)
for n = 0,Nt and k = e, l, f ,m. For any fixed n ≥ 0, the complete scheme isDe,n+1 0 0 0
0 Dl,n+1 0 00 0 E f ,n+1 00 0 0 Em,n+1
Ue,n+1
U l,n+1
U f ,n+1
Um,n+1
=
Ae,n 0 0 00 Al,n 0 00 0 Af ,n 00 0 Am,n 0
Ue,n
U l,n
U f ,n
Um,n
+
Be,n
Bl,n
Bf ,n
Bm,n
, (2.3)
where Uk,n is the vector (uk,n1 , uk,n
2 , . . . , uk,nNa
)T , and the k stage matrix Dk,n+1, Ek,n+1, and Ak,n are
Dk,n+1=
1 + 1t(mk,n+1
1 + βk,n+11 ) 0 · · ·
0 0
0. . .
......
...
0 · · · 1 + 1t(mk,n+1Na
+ βk,n+1Na
)
,
for k = e, l,
Ek,n+1=
(1 + 1tmf ,n+1
1 ) 0 · · ·
0 0
0. . .
......
...
0 · · · (1 + 1tmf ,n+1Na
)
,
for k = f ,m,
Ak,n=
1 −
1t1a
vk,n1
0 · · · · · · 0
1t1a
vk,n1
1 −
1t1a
vk,n2
0 0
0. . .
. . ....
......
0 · · · · · ·1t1a
vk,nNa−1
1 −
1t1a
vk,nNa
,
3318 D. Picart, B. Ainseba / Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328
for k = e, l, f ,m, and the last vector in (2.3) is defined byBe,n
Bl,n
Bf ,n
Bm,n
= 1a
0 0 F f ,n 0
F e,n 0 0 00 τF l,n 0 00 (1 − τ)F l,n 0 0
Ue,n
U l,n
U f ,n
Um,n
, (2.4)
with
F k,n=
β
k,n1 · · · β
k,nNa
0 · · · 0... · · ·
...0 · · · 0
for k = e, l, f . The matrices Dk,n, Ek,n are diagonal with positive coefficients, whereas Ak,n is lower bidiagonal with positivecoefficients. Under the CFL condition, we easily conclude that the complete scheme (2.3) is well posed.
We now set,
‖Uk,n‖1 = 1a
Na−i=1
|uk,ni |, ‖Uk,n
‖∞ = sup1≤i≤Na
|uk,ni |,
Uk,n∈ BV([0, Lmax
]) ⇔
Na−1−i=0
|uk,ni+1 − uk,n
i | ≤ c, c > 0,
and we impose the following conditions on the parameter set.
H 1. The growth functions vk∆, for k = e, l, f ,m, are bounded, non-negative, continuously differentiable with respect to the
age variable,
0 < ‖vk,n∆ ‖∞ < vk, n = 0,Nt ,
and belong to the class C1([0, Lk]); let Cvk be a positive constant such that
vk,ni−1 − v
k,ni
1a≤ Cvk , i = 1, . . . ,Na, n = 0, . . . ,Nt .
H 2. The hatching function βe∆, the flying function β l
∆ and the birth function βf∆ are non-negative, l1, l∞ bounded, and of
bounded variation with respect to the age variable,
0 < ‖βk,n∆ ‖∞ < βk, ‖β
k,n∆ ‖1 < βk
1, βk,n∆ ∈ BV([0, Lmax
]), n = 0, . . . ,Nt .
H 3. The mortality functions mk(E(t), a), for k = e, l, f ,m, are non-negative, l∞ bounded, and of bounded variation withrespect to the age variable,
0 < ‖mk,n‖∞ < mk, mk,n
∈ BV([0, Lmax]), n = 0,Nt .
H 4. The initial functions uk0, for k = e, l, f ,m are non-negative, L1 and L∞ bounded, and of bounded variation with respect
to the age variable,
0 < ‖uk0‖∞ < u0
k, ‖uk0‖1 < uk
0, uk0 ∈ BV([0, Lmax
]).
To prove the convergence of the discrete solution to the continuous solution, the scheme (2.3) has to satisfy some propertiesthat are recalled in the following lemma.
Lemma 1. Suppose that the hypotheses (4) and (2.2) hold, then the scheme (2.3) defines positive approximations and satisfiesthese four inequalities:
‖Uk,n‖∞ ≤ Mk
∞, ‖Uk,n
‖1 ≤ Mk1, k = e, l, f ,m,
maxk=e,l,f ,m
Uk,n
∈ BV([0, Lmax]),
maxk=e,l,f ,m
‖Uk,n+1− Uk,n
‖1 ≤ Nk∞
,
for n = 0,Nt . The constant Mk∞
,Mk1 and Nk
∞are positive and time independent.
D. Picart, B. Ainseba / Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328 3319
The methods used in [17] to prove positivity, stability, bounded variation and time bounded variation in l1 norm of theapproximated scheme are adaptable on our scheme (2.3). But for an exact proof, you can refer on [13]. The convergenceresult is then summarize in the following theorem.
Theorem 1. Suppose that hypotheses (H 1)– (H 4) and CFL condition hold. Let uk∆, for k equals to e, l, f ,m, be the solution of (2.3).
Since 1t and 1a converge to zero, there exists a subsequence, extracted from uk∆, that converges to uk in L1([0, Lmax
] × [0, T ])space. Moreover, this limit is the weak solution of (1.1)–(1.2)–(1.3).
The reader can refer on [17,7,13] for the proof. The next lemma is an important result required for the convergence proofstudied in Section 3. It gives a continuous property of the discrete state functions uk
∆, for k equals to e, l, f ,m, with respectto the set of parameters q∆.
Lemma 2. Under the condition (2.2), there exist positive constants Cke , Ckl , Ckf , Ckm , k = e, l,m, f , non-dependent of Na andNt such that for all n
‖Uk,n+1− Uk,n+1
‖∞ ≤
−p=e,l,f
Ckp · ‖qp∆ − qp∆‖∞ + Ckm · ‖qm∆ − qm∆‖∞,
where qp∆ = (vp∆,mp
∆, βp∆), qp∆ = (v
p∆, mp
∆, βk∆), qm∆ = (vm
∆,mm∆) and qm∆ = (vm
∆, mm∆) are defined in Q∆, Uk,n+1 is the solution
of (2.3) with (qp∆, qm∆).
Proof. We proceed by a recursive reasoning to proof the lemma. The first step consists to the estimate in l∞ norm of thedifference between uk,n
i and uk,ni for all n and k. Let
ξ k,n= sup
i=1,Na
|ξk,ni | = sup
i=1,Na
|uk,ni − uk,n
i |, for all n and k,
and the functionshe,ni = me,n
i + βe,ni ,
hl,ni = ml,n
i + βl,ni ,
hf ,ni = mf ,n
i ,
h,ni = mm,n
i .
From Eqs. (2.1), we compute the variable ξk,ni for i = 1,
ξk,n+1i = ξ
k,ni
1 −
1t1a
vk,ni
−
1t1a
(vk,ni − v
k,ni )uk,n
i +1t1a
ξk,ni−1v
k,ni−1 +
1t1a
(vk,ni−1 − v
k,ni−1)u
k,ni−1
− 1thk,n+1i ξ
k,n+1i − 1tuk,n+1
i (hk,n+1i − hk,n+1
i ).
Assuming that the Lemma 1 is satisfied, then the maximum norm of this last equation is
sup2≤i≤Na
ξk,n+1i (1 − 1thk
∞) ≤ ξ k,n
+ 1tMk∞
2‖vk
∆ − vk∆‖∞ + ‖hk
∆ − hk∆‖∞
,
where hk∞
= supi,n |hk,ni |. In the same way, the absolute value of the difference between uk,n
i and uk,ni for i = 1 is
|ξk,n+11 |(1 − 1thk
∞) ≤ ξ k,n
+ 1tβ k1ξ
k,n+ 1tM k
1‖βk∆ − β k
∆‖∞ + 1tMk∞
‖vk
∆ − vk∆‖∞ + ‖hk
∆ − hk∆‖∞
,
where k = f , e, l, l. We then deduce the l∞ norm of ξ k,ni that is
ξ k,n+1(1 − 1thk∞
) ≤ ξ k,n+ 1tβ k
1ξk,n
+ 1tM k1‖β
k∆ − β k
∆‖∞ + 1tMk∞
2‖vk
∆ − vk∆‖∞ + ‖hk
∆ − hk∆‖∞
,
or equivalently by making the recurrence on n
ξ k,n+1≤ 1tβ k
1
n−i=0
eThk∞ξ k,n−i
+ TeThk∞
2Mk
∞‖vk
∆ − vk∆‖∞ + Mk
∞‖hk
∆ − hk∆‖∞ + M k
1‖βk∆ − β k
∆‖∞
, (2.5)
where eThk∞ is the approximation of (1 − 1thk
∞)−Nt . For the first row, n = 0, the inequality (2.5) is written by
ξ k,1≤ TeTh
k∞
2Mk
∞‖vk
∆ − vk∆‖∞ + Mk
∞‖hk
∆ − hk∆‖∞ + M k
1‖βk∆ − β k
∆‖∞
.
3320 D. Picart, B. Ainseba / Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328
Setting ck0 = TeThk∞Mk
∞(2, 1, 1), and c k0 = TeTh
k∞M k
1(0, 0, 1) then Lemma 2 is proved for the row n = 0. Now, we assumethat the two following inequalities are truth for n
ξ k,n≤
−p=e,l,f
dp · ‖qp∆ − qp∆‖∞ + dm · ‖qm∆ − qm∆‖∞,
ξ k,n≤
−p=e,l,f
dp · ‖qp∆ − qp∆‖∞ + dm · ‖qm∆ − qm∆‖∞,
and we prove that they are satisfied for n = n + 1, we get from (2.5)
ξ k,n+1≤ Tβ k
1eThk∞
−p=e,l,f
dp · ‖qp∆ − qp∆‖∞ + dm · ‖qm∆ − qm∆‖∞ + ck0 · ‖qk∆ − qk∆‖∞ + c k0 · ‖qk∆ − qk∆‖∞.
The lemma is proved by settingCkp = TeTh
k∞β k
1 dp+ ck0, p = k,
Ckk = TeThk∞β k
1 dk+ c k0.
3. The discrete parameter identification problem
In this section, we define the parameter identification scheme and present the convergence proof of this scheme.The approximate problem of [P] is defined by
[P∆] =
J∆(q∆) = 1t−
k=e,l,f
Nt−n=0
Bk,n(q∆) − zk,n
2+ 1a
Na−i=1
ul,Nti (q∆) − zi
2,
where Bk,n(q∆) and ul,Nti (q∆) are computed by (2.3)–(2.4).
The discrete vector q∆ ∈ Q∆ is the set of unknown parameters defined by
q∆ = (vk∆,mk
∆, β k∆), k = e, l, f ,m, k = e, l, f ,
and Q∆ is the compact subset of K 11∆ of the admissible solutions. We now are looking for the existence solution of [P∆].
Theorem 2. The problem [P∆] admits at least one optimum.
Proof. Let d be the finiteminimum born of the cost function J∆(q∆). Let qh∆h, where h is a nonzero integer, be aminimizingsequence of the space Q∆, such that
d < J∆(qh∆) ≤ d +1h.
The value of the cost function at the point qh∆ is given by
J∆(qh∆) = 1t−
k=e,l,f
Nt−n=0
Bk,n(qh∆) − zk,n
2+ 1a
Na−i=1
ul,Nti (qh∆) − zi
2,
where Bk,n(qh∆) and ul,Nti (qh∆) are the solution of (2.3)–(2.4) with qh∆ for all i from 1 to Na and for all n from 0 to Nt .
The sequence qh∆h is bounded we can extract, according to the theorem of B.W, a sequence, termed qhn∆ hn , thatconverges through q∗
∆ of the space Q∆. The first brace of the cost function is written by
Nt−n=0
Bk,n(qh∆) − zk,n
2=
Nt−n=0
Bk,n(qh∆) − Bk,n(q∗
∆) + Bk,n(q∗
∆) − zk,n2
,
where
Bk,n(qh∆) − Bk,n(q∗
∆) = 1aNa−i=1
up,ni (qh∆)
(β
p,ni )h − (β
p,ni )∗
+ (β
p,ni )∗
up,ni (qh∆) − up,n
i (q∗
∆),
converges to 0 when h → +∞, according to the Lemma 2.With the same argument, the second bracket of the cost functionis written by
ul,Nti (qh∆) − zi −−−−→
h→+∞
ul,Nti (q∗
∆) − zi.
D. Picart, B. Ainseba / Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328 3321
We then get the conclusion,
J∆(qh∆) −−−−→h→+∞
1t−
k=e,l,f
Nt−n=0
Bk,n(q∗
∆) − zn2
+ 1aNa−i=1
ul,Nti (q∗
∆) − zi2
= J∆(q∗
∆) = d.
In the following, we consider the Lagrangian formulation of [P∆] that is
L(S) = J∆(q∆) +
−k=e,l
Nt−n=0
Na−i=1
λk,ni
uk,n+1i −
uk,ni
1 −
1t1av
k,ni
+
1t1av
k,ni−1u
k,ni−1
1 + 1tmk,n+1i + 1tβk,n+1
i
+
−k=f ,m
Nt−n=0
Na−i=1
λk,ni
uk,n+1i −
uk,ni
1 −
1t1av
k,ni
+
1t1av
k,ni−1u
k,ni−1
1 + 1tmk,n+1i
, (3.1)
where the vector S is (q∆, u∆, λ∆), the variable λ∆ = (λe∆, λl
∆, λf∆, λm
∆) corresponds to the dual variables.To numerically solve the problem [P∆], we use the Quasi-Newton algorithm (QN), see [18]. According to Yabe et al. in [1],
the algorithm converges if the second derivative of the Lagrangian with respect to q∆ is twice differentiable and Lipschitz.The first derivatives of the Lagrangian with respect to q∆ are
∂L∂v
e,n0i0
(S) =1t1a
λe,n0i0
ue,n0+1i0
1 + 1tme,n0+1i0
+ 1tβe,n0+1i0
−1t1a
λe,n0i0+1u
e,n0i0
1 + 1tme,n0+1i0+1 + 1tβe,n0+1
i0+1
,
∂L∂me,n0
i0
(S) = 1tλe,n0−1i0
ue,n0−1i0
1 −
1t1av
e,n0−1i0
+1t1av
e,n0−1i0−1 ue,n0−1
i0−1
(1 + 1tme,n0
i0+ 1tβe,n0
i0)2
,
∂L∂β
e,n0i0
(S) =∂L
∂me,n0i0
(S) − 1tλl,n01 ue,n0
i0
1 + 1tml,n0+11 + 1tβ l,n0+1
1
+ 21t1aue,n0i0
(Be,n(q∆) − ze,n0),
(3.2)
∂L
∂vl,n0i0
(S) =1t1a
λl,n0i0
ul,n0+1i0
1 + 1tml,n0+1i0
+ 1tβ l,n0+1i0
−1t1a
λl,n0i0+1u
l,n0i0
1 + 1tml,n0+1i0+1 + 1tβ l,n0+1
i0+1
,
∂L
∂ml,n0i0
(S) = 1tλl,n0−1i0
ul,n0−1i0
1 −
1t1av
l,n0−1i0
+1t1av
l,n0−1i0−1 ul,n0−1
i0−1
(1 + 1tml,n0
i0+ 1tβ l,n0
i0)2
,
∂L
∂βl,n0i0
(S) =∂L
∂ml,n0i0
(S) − 1tτλf ,n01 ul,n0
i0
1 + 1tmf ,n0+11
− 1t(1 − τ)λm,n01 ul,n0
i0
1 + 1tmm,n0+11
+ 21t1aul,n0i0
(Bl,n(q∆) − z l,n0),
(3.3)
for i0 from 1 to Na and n0 from 1 to Nt , and
∂L
∂vk,n0i0
(S) =1t1a
λk,n0i0
uk,n0+1i0
1 + 1tmk,n0+1i0
−1t1a
λk,n0i0+1u
k,n0i0
1 + 1tmk,n0+1i0+1
,
∂L
∂mk,n0i0
(S) = 1tλk,n0−1i0
uk,n0−1i0
1 −
1t1av
k,n0−1i0
+1t1av
k,n0−1i0−1 uk,n0−1
i0−1
(1 + 1tmk,n0
i0)2
,
∂L
∂βf ,n0i0
(S) = −1tλe,n01 ul,n0
i0
1 + 1tme,n0+11 + 1tβe,n0+1
1
+ 21t1auf ,n0i0
(Bf ,n(q∆) − z f ,n0),
(3.4)
where k = f ,m. The equations of systems (3.2)–(3.4) are dependent on the density functions uk∆, the dual variables λk
∆
and the set of unknown parameters q∆. To prove the second properties of the Lagrangian, we need the continuous propertyof the density functions and the dual variables with respect to the unknown parameters. The first condition is filled with
3322 D. Picart, B. Ainseba / Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328
Lemma 2, whereas the second one, by considering the following equations to compute the dual variables
λe,n0−1i0
=1t1a
λe,n0i0+1v
e,n0i0
1 + 1tme,n0+1i0+1 + 1tβe,n0+1
i0+1
+ λe,n0i0
1 −
1t1av
e,n0i0
1 + 1tme,n0+1
i0+ 1tβe,n0+1
i0
+ 1tλl,n01 β
e,n0i0
1 + 1tml,n0+11 + 1tβ l,n0+1
1
− 21t1aβe,n0i0
(Be,n(q∆) − ze,n),
λl,n0−1i0
=1t1a
λl,n0i0+1v
l,n0i0
1 + 1tml,n0+1i0+1 + 1tβ l,n0+1
i0+1
+ λl,n0i0
1 −
1t1av
l,n0i0
1 + 1tml,n0+1
i0+ 1tβ l,n0+1
i0
+ 1tτλf ,n01 β
l,n0i0
1 + 1tmf ,n0+11
+ 1t(1 − τ)λm,n01 β
l,n0i0
1 + 1tmm,n0+11
− 21t1aβ l,n0i0
(Bl,n(q∆) − z l,n),
λf ,n0−1i0
=1t1a
λf ,n0i0+1v
f ,n0i0
1 + 1tmf ,n0+1i0+1
+ λf ,n0i0
1 −
1t1av
f ,n0i0
1 + 1tmf ,n0+1
i0
+ 1tλe,n01 β
f ,n0i0
1 + 1tme,n0+11 + 1tβe,n0+1
1
− 21t1aβ l,n0i0
(Bf ,n(q∆) − z f ,n),
λm,n0−1i0
=1t1a
λm,n0i0+1v
m,n0i0
1 + 1tmm,n0+1i0+1
+ λm,n0i0
1 −
1t1av
m,n0i0
1 + 1tmm,n0+1
i0
(3.5)
for i0 from 1 to Na and n0 from 1 to Nt , is presented in the next lemmas. First, we prove the l∞ norm stability.
Lemma 3. Under the condition (2.2), there exist positive constants Nk > 0 for k = e, l, f ,m such that for all n
‖λk,n‖∞ ≤ Nk.
Proof. The method used to prove this lemma is the one of the Lemma 1. The estimate of |λk,ni | is given by
‖λe,n−1‖∞ ≤ ‖λe,Nt ‖∞ + 1tβe
Nt−j=n
‖λl,j‖∞,
‖λl,n−1‖∞ ≤ ‖λl,Nt ‖∞ + 1tτ β l
Nt−j=n
‖λf ,j‖∞ + 1t(1 − τ)β l
Nt−j=n
‖λm,j‖∞, +2T1aβ l Lβ lM l
∞+ Q
,
‖λf ,n−1‖∞ ≤ ‖λf ,Nt ‖∞ + 1tβ f
Nt−j=n
‖λe,j‖∞,
‖λm,n−1‖∞ ≤ ‖λm,Nt ‖∞,
where Q is the maximum bound of the vector zn. The recurrence on n starts with Nt and stops with n = 1. Assuming thatthe vector λk,Nt is uniformly bounded by the constant F , then we conclude the proof.
Lemma 4. Under the condition (2.2), there exist positive constants ξ ke , ξ kl , ξ kf , ξ km , k = e, l, f ,m, non-dependent of Na andNt such that for all n
‖λk,n− λk,n
‖∞ ≤
−p=e,l,f
ξ kp · ‖qp∆ − qp∆‖∞ + ξ km · ‖qm∆ − qm∆‖∞,
where qp∆ = (vp∆,mp
∆, βp∆), qp∆ = (v
p∆, mp
∆, βk∆) and qm∆ = (vm
∆,mm∆) are defined in Q∆, λk,n is the solution of (3.5) with q∆ and
k = e, l, f ,m.
Proof. Since the results of the Lemmas 1 and 3, we apply the method used to prove Lemma 1 with the following estimates
‖λk,n−1− λk,n−1
‖∞ ≤ ‖λk,Nt − λk,Nt ‖∞ + 1tβk∞
Nt−i=n
‖λk,i− λk,i
‖∞
+ 1tηkk · ‖qk,n∆ − qk,n∆ ‖∞ + ηkk · ‖qk,n∆ − qk,n∆ ‖∞
,
D. Picart, B. Ainseba / Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328 3323
where ηkk = (2Nk, 3Nk, 3Nk+N k), ηkk = N kβk
∞(0, 1, 1) and k = l, e for the dual variables associated to the egg and female
stages, and the estimate
‖λl,n−1− λl,n−1
‖∞ ≤ ‖λl,Nt − λl,Nt ‖∞ + 1tβ lτ
Nt−i=n
‖λf ,i− λf ,i
‖∞ + 1tβ l(1 − τ)
Nt−i=n
‖λm,i− λm,i
‖∞
+ 1t
−p=e,l,f
ηlp · ‖qp,n∆ − qp,n∆ ‖∞ + ηlm · ‖qm,n∆ − qm,n
∆ ‖∞
,
whereηle = 2L(β l
∞)2C le ,
ηll = (2N l, 3N l, 3N l+ τN f
+ (1 − τ)Nm+ 4Lβ l
∞M l
∞+ 2Q ) + 2L(β l
∞)2C ll ,
ηlf = τN f β l∞
(0, 1, 1) + 2L(β l∞
)2C lf ,
ηlm = (1 − τ)Nmβ l∞
(0, 1) + 2L(β l∞
)2C lm ,
for the dual variable associated to the larval stage and the estimate
‖λm,n−1− λm,n−1
‖∞ ≤ ‖λm,Nt − λm,Nt ‖∞ + 1tηmm · ‖qm,n
∆ − qm,n∆ ‖∞
,
where
ηmm = Nm(2, 3)
for the dual variable associated to the male stage. The constants C lk are given by the Lemma 1.
Now, let A be a matrix, given in ℜ11N×11N , such that its infinity norm is
‖A‖∞ = max1≤k≤11
max1≤ik≤N
11−p=1
N−jp=1
|aikjp |,
and Dϵ be the set,
Dϵ = q∆ ∈ ℜ11N , ‖q∆ − q∗
∆‖∞ < ϵ ∈ Q∆.
From the derivatives of the gradient of the Lagrangian with respect to q∆, given in Annex, and the above lemmas, we get
Lemma 5. Under the condition (2.2), The Lagrangian defined in (3.1) is twice continuously differentiable with respect to q∆ onDϵ and ∂2L
∂q2∆is locally Lipschitz in q∗
∆ ∈ Q∆.
Proof. As the CFL condition (2.2) holds, we prove that the derivatives are bounded in all points.Let S and S be respectively the vectors (q∆, u∆, λ∆) and (q∆, u∆, λ∆). We have to find positive constants Ls, s = e, l, f ,m,
such that ∂2L∂q2∆
(S) −∂2L∂q2∆
(S)
∞
≤
−s=e,l,f ,m
Ls‖qs∆ − qs∆‖∞,
where qs∆, qs∆ ∈ Q∆, and ∂2L∂q2∆
(S) −∂2L∂q2∆
(S)
∞
= max1≤k≤11
max1≤ik≤N
11−p=1
N−jp=1
∂2L∂qkik∂q
pjp
(S) −∂2L
∂qkik∂qpjp
(S)
.From the equations given in Appendix and the preceding Lemmas 2–4, we easily estimate the above absolute value for eachk and ik using the same method than the one developed in Lemma 2.
This last result is helpful to prove the next theorem that gives the convergence result of the QN algorithm.
Theorem 3. Let J∆(q∆) be the cost functional, Hk be the matrix updated with the BFGS formula and qk∆k the sequence computedwith the QN algorithm. Then, for υ ∈ (0, 1), and under the hypothesis (2.2), there exists two positive constants ϵ and σ suchthat if
q0∆ ∈ Dϵ
3324 D. Picart, B. Ainseba / Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328
and H0 −∂2L
(∂q∆)2(q∗
∆)
∞
≤ σ
the sequence qk∆k is well defined and converges linearly through q∗∆ with
‖qk+1∆ − q∗
∆‖∞ ≤ υ‖qk∆ − q∗
∆‖∞. (3.6)
The proof of this theorem can be find in [1]. We finally conclude the section by the convergence result of the parameteridentification scheme.
Theorem 4. Let q0∆ ∈ Q∆ be the initial data of the QN scheme and J∆ be the cost function defined in [P∆]. Since 1a and 1tconverge through 0, then the parameter identification scheme converges through the continuous cost function of [P] at theoptimum,
|J∆(qk∆) − J(q∗)| −−−→k→∞
0.
Proof. First, we prove, with Theorem 3 and Lemma 2 that
J∆(qk∆) −−−−→k→+∞
J∆(q∗
∆),
using the same method than the one presented in the proof of Theorem 2. And then, by applying methods of [11] weget
J∆(q∗
∆) = 1t−
k=e,l,f
Nt−n=0
∫ tn+1
tn
11t
Bk(t; q∗
∆) − zk,n2
dt
+ 1a
Na−i=1
∫Ci
11a
ul(T , a) − zi
2da
1a,1t→0−−−−−→
−k=e,l,f
Nt−n=0
∫ tn+1
tn
Bk(t; q∗
∆) − zk,n2
dt
+
Na−i=1
∫Ci
ul(T , a) − zi
2da
= J(q∗),
and q∗∈ Q .
4. Numerical examples
With few numerical examples, we test the parameter identification scheme developed in this article. We consider thefollowing simplified version of the model
∂
∂tue(t, a) +
∂
∂aue(t, a) = −βe(t, a)ue(t, a),
∂
∂tul(t, a) +
∂
∂aul(t, a) = −β l(t, a)ul(t, a),
∂
∂tuf (t, a) +
∂
∂auf (t, a) = 0,
∂
∂tum(t, a) +
∂
∂aum(t, a) = 0,
(4.1)
with
Be(t) : ue(t, 0) =
∫ Lf
0β f (t, s)uf (t, s)ds,
Bl(t) : ul(t, 0) =
∫ Le
0βe(t, s)ue(t, s)ds,
Bf (t) : uf (t, 0) =
∫ Ll
0τβ l(t, s)ul(t, s)ds,
Bm(t) : um(t, 0) =
∫ Ll
0(1 − τ)β l(t, s)ul(t, s)ds,
D. Picart, B. Ainseba / Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328 3325
Table 1Values of the cost function at the QN solution and estimation error between theexperimental and the approximated flying dynamics.
Figures 1 2(a) 2(b)
J∆(qQN) 5.32 × 10−7 1.51 × 10−7 7.84 × 10−9
‖z l,n(q∗) − z l,n(qQN)‖∞ 1.07 × 10−3 3.87 × 10−4 1.01 × 10−4
as the boundary conditions and
uk(0, a) = uk0(a), a ∈ [0, Lk], k = e, l, f ,m
as the initial conditions, to model the population dynamic in laboratory conditions. The model is then composed of threeparameters that are the hatching function βe, the flying function β l and the birth function β f . In these numerical examples,we identify the two first functions with respect to the age, q = (βe, β l), using experimental flying dynamic.
The maximal time T is set to 10 days, and the maximum age Lmax to 5 days. The initial population is composed of eggs ofthe same age such that
ue0(a) = e−
a−10,25
2, ul
0(a) = uf0(a) = um
0 (a) = 0, a ∈ [0, Lmax].
The experimental flying dynamic z l,n is computed with the discrete equation Bl(t; q∗) where the parameters are given by
βe∗(a) = e−
a−1,50,25
2, β l
∗(a) = e−
a−3,50,5
2, a ∈ [0, Lmax
].
To limit oscillations, we work with a penalized form of the cost function that is
Jp(q) = J∆(q) +
−k=e,l
ηk‖βk‖22 +
−k=e,l
µk‖∂aβk‖22.
The BCONG routine of IMSL library is used to compute theQNalgorithmon Jp(q). Once the solution obtained qQN, we computethe flying dynamic z l,n(qQN) with the discrete equation given in (2.4). The time and age steps are set to 0, 1. Values of thecost function at the QN solution and estimation error between the experimental and the approximated flying dynamics aregiven in Table 1.
The first example (Fig. 1) is the superposition between the exact solution q∗= (βe
∗, β l
∗) and the QN solution qQN with
respectively ηk and µk equal to 10−8 and 10−10. This result proves that there exist at least two couples of functions forwhich the simulations of the discrete model coincide with the experimental flying dynamic. However, if we use this data toestimate one of these two functions, we have a unique numerical solution (Fig. 2).
5. Conclusion
In this paper we propose a numerical method to estimate the demographic parameters of a multistage populationdynamics model. The method resolves an optimization problem with constraints with a nonlinear quadratic cost function.As a first step we describe the numerical method used for the approximation of the solution of the population model, afinite volumemethod. Then we introduce the parameter identification problem in the form of a least-squares problem for avector of approximate grid values of the coefficient functions (the parameters of the model) comparing observed densitiesand stage transitions at a given time with those predicted by the model.
The method and the convergence proof are different than Ackleh’s [11] because of the form of our cost function. Indeed,we have to study properties of the cost function and its derivatives. Moreover, our resolution of this problem is based on aQuasi-Newton algorithm, though other methods such as the Levenberg–Marquardt algorithm can be adapted.
The numerical applications provided above show that themethod converges nomatter how long the vector of estimatedparameters; however, the estimated parameters do not necessarily lead to the observed solution. In fact, using the samedata, the QN algorithm converges to a unique solution or does not, depending on the parameter that we want estimate. Thisquestion of uniqueness will be the topic of a future paper. Nonetheless, this method was applied in [13] to estimate thetransition functions from each stage to the next, as well as the birth function, and it allowed us to accurately describe thepopulation dynamics of the European grapevine moth in laboratory conditions.
Acknowledgments
The authors would like to thank Pr. F. Milner for his comments and suggestions about this paper. D. Picart was fundedduring her thesis by a grant from Region Aquitaine and the French National Institute for research in computer science andcontrol.
3326 D. Picart, B. Ainseba / Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
age (days)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig. 1. Exact and estimated functions βe (left side curves) and β l (right side curves) with respect to the age. The dotted curves are the minimizers of Jpwith ηk and µk equal to 10−8 .
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.0
0.2
0.4
0.6
0.8
1.0
age (days)0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.0
0.2
0.4
0.6
0.8
1.0
age (days)
a b
Fig. 2. Exact and estimated functions βe and β l with respect to the age. The two curves on the left side of the graphs are the exact and estimated functionsβe whereas the two curves on the right side are the exact and estimated functions β l .
Appendix
The above equations are the second derivatives of the Lagrangian with respect to the unknown parameters q∆,
∂2L
∂vk,n0i0
∂mk,n0+1i0
(S) = −1t2
1a
λk,n0i0
uk,n0+1i0
(1 + 1tmk,n0+1i0
+ 1tβk,n0+1i0
)2,
∂2L
∂vk,n0i0
∂mk,n0+1i0+1
(S) =1t2
1a
λk,n0i0+1u
k,n0i0
(1 + 1tmk,n0+1i0+1 + 1tβk,n0+1
i0+1 )2,
∂2L
∂vk,n0i0
∂βk,n0+1i0
(S) =∂2L
∂vk,n0i0
∂mk,n0+1i0
(S),
∂2L
∂vk,n0i0
∂βk,n0+1i0+1
(S) =∂2L
∂vk,n0i0
∂mk,n0+1i0+1
(S),
(A.1)
D. Picart, B. Ainseba / Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328 3327
where k = e, l, i0 from 1 to Na, and n0 from 1 to Nt ,
∂2L
∂mk,n0i0
∂vk,n0−1i0
(S) = −1t2
1a
λk,n0−1i0
uk,n0−1i0
(1 + 1tmk,n0i0
+ 1tβk,n0i0
)2,
∂2L
∂mk,n0i0
∂vk,n0−1i0−1
(S) =1t2
1a
λk,n0−1i0
uk,n0−1i0−1
(1 + 1tmk,n0i0
+ 1tβk,n0i0
)2,
∂2L
(∂mk,n0i0
)2(S) = −2λk,n0−1
i01t2
uk,n0−1i0
1 −
1t1a
vk,n0−1i0
+1t1av
k,n0−1i0−1 uk,n0−1
i0−1
(1 + 1tmk,n0i0
+ 1tβk,n0i0
)3,
∂2L
∂mk,n0i0
∂βk,n0i0
(S) =∂2L
(∂mk,n0i0
)2(S),
(A.2)
where k = e, l, i0 from 1 to Na, and n0 from 1 to Nt ,
∂2L
∂βk,n0i0
∂vk,n0−1i0
(S) =∂2L
∂mk,n0i0
∂vk,n0−1i0
(S),
∂2L
∂βk,n0i0
∂vk,n0−1i0−1
(S) =∂2L
∂mk,n0i0
∂vk,n0−1i0−1
(S),
∂2L
∂βk,n0i0
∂mk,n0i0
(S) =∂2L
(∂mk,n0i0
)2(S),
∂2L∂(β
e,n0i0
)2(S) =
∂2L(∂me,n0
i0)2
(S),
∂2L
∂(βl,n0i0
)2(S) = 21a21t(ul,n0
i0)2 +
∂2L
(∂ml,n0i0
)2(S),
∂2L
∂βl,n0i0
∂βl,m0i0
(S) = 21a21tul,n0i0
ul,m0j0
,
∂2L
∂βl,n0i0
∂mf ,n0+1i0
(S) =1t2τλ
f ,n01 ul,n0
i0
1 + 1tmf ,n0+11
,
∂2L
∂βl,n0i0
∂mm,n0+1i0
(S) =1t2(1 − τ)λ
m,n01 ul,n0
i0
1 + 1tmm,n0+11
,
where k = e, l, i0 from 1 to Na, and n0 from 1 to Nt ,∂2L
∂vk,n0i0
∂mk,n0+1i0
(S) = −1t2
1a
λk,n0i0
uk,n0+1i0
(1 + 1tmk,n0+1i0
)2,
∂2L
∂vk,n0i0
∂mk,n0+1i0+1
(S) =1t2
1a
λk,n0i0+1u
k,n0i0
(1 + 1tmk,n0+1i0+1 )2
,
where k = f ,m, i0 from 1 to Na, and n0 from 1 to Nt ,
∂2L
∂mk,n0i0
∂vk,n0−1i0
(S) = −1t2
1a
λk,n0−1i0
uk,n0−1i0
(1 + 1tmk,n0i0
)2,
∂2L
∂mk,n0i0
∂vk,n0−1i0−1
(S) =1t2
1a
λk,n0−1i0
uk,n0−1i0−1
(1 + 1tmk,n0i0
)2,
∂2L
(∂mk,n0i0
)2(S) = −2λk,n0−1
i01t2
uk,n0−1i0
1 −
1t1a
vk,n0−1i0
+1t1av
k,n0−1i0−1 uk,n0−1
i0−1
(1 + 1tmk,n0i0
)3,
3328 D. Picart, B. Ainseba / Nonlinear Analysis: Real World Applications 12 (2011) 3315–3328
where k = f ,m, i0 from 1 to Na, and n0 from 1 to Nt and,∂2L
∂βf ,n0i0
∂me,n0+11
(S) =1t2λe,n0
1 uf ,n0i0
1 + 1tme,n0+11 + 1tβe,n0+1
1
,
∂2L
∂βl,n0i0
∂βe,n0+11
(S) =∂2L
∂βf ,n0i0
∂me,n0+11
(S),
where i0 from 1 to Na, and n0 from 1 to Nt .
References
[1] H. von Foerster, Some remarks on changing populations, Kinetics of Cell Proliferation (1959) 382–407.[2] M.E. Gurtin, R.C. MacCamy, Non-linear age-dependent population dynamics, Archive for Rational Mechanics and Analysis 54 (1974) 281–300.[3] J.W. Sinko, W. Streifer, A new model for age-size structure of a population, Ecology 48 (1967) 910–918.[4] H.T. Banks, J.E. Banks, L.K. Dick, J.D. Stark, Estimation of dynamics rate parameters in insect populations undergoing sublethal exposure to pesticides,
Bulletin of Mathematical Biology 69 (2007) 2139–2180.[5] A. Calsina, J. Saldana, A model of physiologically structured population dynamics with a nonlinear individual growth rate, Journal of Mathematical
Biology 33 (1995) 335–364.[6] G.F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.[7] J. Douglas, F.A. Milner, Numerical methods for a model of population dynamics, Calcolo 24 (1987) 247–254.[8] T. Kostova, An explicit third-order numerical method for size-structured population equations, Methods for Partial Differential Equations 19 (2003)
1–21.[9] D. Sulsky, Numerical solution of structured population models. II Mass structure, Journal of Mathematical Biology 32 (1994) 491–514.
[10] C.G. Broyden, J.E. Dennis, J.J. Moré, On the local and super linear convergence of Quasi–Newton methods, Journal of the Institute of Mathematics andits Applications 12 (1973) 223–245.
[11] A.S. Ackleh, Parameter identification in size-structured population models with nonlinear individual rates, Mathematical and Computer Modelling30 (1999) 81–92.
[12] B. Ainseba, D. Picart, D. Thiéry, An innovating multistage, physiologically structured, population model to understand the European grapevine mothdynamics, Journal of Mathematical Analysis and Applications 382 (2011) 34–46.
[13] D. Picart, Modélisation et estimation des paramètres liés au succès reproducteur d’un ravageur de la vigne (Lobesia botrana DEN. & SCHIFF.), Ph.D.Thesis, No. 3772, University of Bordeaux, 2009.
[14] Y. Kwon, C.K. Cho, Second-order accurate difference methods for a one-sex model of population dynamics, SIAM Journal on Numerical Analysis 30 (5)(1993) 1385–1399.
[15] O. Angulo, J.C. Lopez-Marcos, Numerical integration of nonlinear size-structured population equations, Ecological Modelling 133 (2000) 3–14.[16] O. Angulo, J.C. Lopez-Marcos, Numerical integration of fully nonlinear size-structured population models, Applied Numerical Mathematics 50 (2004)
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Further reading
[1] H. Yabe, H. Ogasawara, M. Yoshino, Local and superlinear convergence of Quasi–Newton methods based on modified secant conditions, Journal ofComputational and Applied Mathematics 205 (2007) 617–632.
[2] M. Gyllenberg, A. Osipov, L. Päivärinta, The inverse problem of linear age-structured population dynamics, Journal of Evolution Equations 2 (2002)223–239.
[3] M. Pilant, W. Rundell, Determining a coefficient in a first-order hyperbolic equation, SIAM Journal on Applied Mathematics 51 (1991) 494–506.[4] M. Pilant, W. Rundell, Determining the initial age distribution for an age structured population, Mathematical Population Studies 3 (1991) 3–20.[5] W. Rundell, Determining the birth function for age structured population, Mathematical Population Studies 1 (4) (1989) 377–395.[6] W. Rundell, Determining the death rate for an age-structured population from census data, SIAM Journal of AppliedMathematics 53 (1993) 1731–1746.