Explicit solution of a Lotka-Sharpe-McKendrick system involving neutral delay differential equations using the r-Lambert W function

Abstract

Structured population models, which account for the state of individuals given features such as age, gender, and size, are widely used in the fields of ecology and biology. In this paper, we consider an age-structured population model describing the population of adults and juveniles. The model consists of a system of ordinary and neutral delay differential equations. We present an explicit solution to the model using a generalization of the Lambert W function called the r-Lambert W function. Numerical simulations with varying parameters and initial conditions are done to illustrate the obtained solution. The proposed method is also applied to an insect population model with long larval and short adult phases.

Full text

http://www.aimspress.com/journal/MBE
MBE, 17(5): 5686–5708.
DOI: 10.3934/mbe.2020306
Received: 25 June 2020
Accepted: 03 August 2020
Published: 28 August 2020
Research article
Explicit solution of a Lotka-Sharpe-McKendrick system involving neutral
delay differential equations using the r-Lambert Wfunction
Cristeta U. Jamilla∗, Renier G. Mendoza, and Victoria May P. Mendoza
Institute of Mathematics, University of the Philippines Diliman, Quezon City, Philippines
*Correspondence: Email: cujamilla@upd.edu.ph.
Abstract: Structured population models, which account for the state of individuals given features
such as age, gender, and size, are widely used in the fields of ecology and biology. In this paper,
we consider an age-structured population model describing the population of adults and juveniles.
The model consists of a system of ordinary and neutral delay differential equations. We present an
explicit solution to the model using a generalization of the Lambert Wfunction called the r-Lambert
Wfunction. Numerical simulations with varying parameters and initial conditions are done to illustrate
the obtained solution. The proposed method is also applied to an insect population model with long
larval and short adult phases.
Keywords: age-structured population model; r-Lambert Wfunction; neutral delay differential
equation; explicit solution; Lotka-Sharpe-McKendrick system; insect population model
1. Introduction
Delay differential equation (DDE) is a type of differential equation (DE) where the time derivatives
at the present time are dependent on the solution and its derivatives at a previous time. A k-th order
DDE takes the form
y(k)(t)=f(t,y(t),...,y(k−1)(t),y(d0),...,y(k)(dk)),(1.1)
where dj=dj(t,y(t)) is called the delay satisfying dj≤tfor all times ton the interval [t0,t1], j=
0,...,k.A DDE is said to have discrete delays if the delays are constant. DDEs can have a single delay
or multiple delays. A DDE is of retarded type (RDDE) when there is no time-delay in the derivative
terms of the DDE. Its simplest form is
˙y(t)=f(t,y(t),y(d0)).

5687
A DDE is of neutral type (NDDE) if there are delays in the derivatives or in the solution of the
DDE. Its first-order form can be written as
˙y(t)=f(t,y(t),y(d0),˙y(d1)).
Recall that the unique solution to an ordinary differential equation (ODE) is dependent on an initial
condition. Analogously, the unique solution to a DDE is dependent on some initial function φ(t)
defined on a previous time, say [−h,0], for some delay h∈R+. The function φ(t) is called the preshape
or history function.
Asl and Ulsoy showed an approach for obtaining analytical solutions to systems of DDEs,
particularly, first-order, linear, constant-coefficient RDDEs with constant delays using the Lambert W
function [1]. This was then extended by Yi to solve nonhomogeneous systems [2–4]. These results on
DDEs are notable since there are few studies on explicit solutions to these types of differential
equations.
The Lambert Wfunction, also known as the omega function, is defined to be the function W(a)
satisfying
W(a)eW(a)−a=0.(1.2)
If ais a nonzero real number, then there are at most two real values of W(a) satisfying (1.2). If
ais a complex number, then there is an infinite number of complex values of W(a) satisfying (1.2).
An in-depth discussion on the Lambert Wfunction, as well as the branch cut, asymptotes, and other
properties, are found in [5].
A generalization of the Lambert Wfunction, introduced by Mez˝
o [6–8], is defined as a multi-valued
inverse of the transcendental function
e−x=a(x−t1)(x−t2)...(x−tn)
(x−s1)(x−s2)...(x−sm)
at a, where a∈C,a,0 and n,m∈N. We consider a simple form of the generalization,
e−x=a(x−t1)
(x−s1),(1.3)
with nand mequal to 1. Equation (1.3) can be transformed into the form as xex+rx =a, where r∈R,
and hence, is coined as the r-Lambert Wfunction. Therefore, the r-Lambert Wfunction is defined to
be the function Wr(a) satisfying the equation
Wr(a)eWr(a)+rWr(a)−a=0.(1.4)
In (1.4), we note the presence of the term rWr(a), which does not appear in the definition of the
Lambert Wfunction in (1.2). If ais a nonzero real number, then depending on the value of r, there
can be one, two, or three real solutions of (1.4) with respect to Wr(a). This means that there are at
most three branches of the r-Lambert Wfunction, denoted W(r,0)(x),W(r,−1)(x), and W(r,−2)(x), having
nonempty intersection with the real line. Extending to the complex plane, if ais a complex number,
then (1.4) has infinitely many complex solutions Wr(a). This is similar to the solutions of (1.2) in
the complex plane. However, Mez ˝
o suggests that there are different structures of the branches of the
r-Lambert Wfunction depending on the value of r[8]. For a discussion on the solutions of (1.4), the
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5688
branch structures, and other properties of the r-Lambert Wfunction, we refer the readers to [7, 8]. The
generalized Lambert Wfunction appears in the study of the solution to the Bose-Fermi mixtures [6] and
on the asymptotic estimation of the Bell and generalized Bell numbers [9]. In [10], it was shown how
the generalized Lambert Wfunction can be used to obtain a closed-form of the inverse of Langevin and
Brillouin functions. These functions are used in several fields of physics. In [11], it was illustrated how
the r-Lambert Wfunction can be used to characterize the equilibrium strategy in the symmetric two-
player Hirshleifer contest. In [12], the r-Lambert function was used to show the asymptotic equivalence
of the proximity operator of the logistic function and a composition of polynomial and exponential
functions. For other applications of the generalized Lambert Wfunction in engineering, physics, and
biology, we refer the readers to [13–28].
In [29], Jamilla et. al showed an approach for obtaining an analytical solution to first-order, linear,
constant-coefficient NDDEs with constant delays using the r-Lambert Wfunction. The solution takes
the form of a series whose terms are dependent on the parameters and the preshape function of the
NDDE. In this paper, we adopt an age-structured population model described by a system of ODEs
and NDDEs. We utilize the r-Lambert Wfunction to provide an explicit solution to the model using
the approach proposed in [29]. One advantage of this method is that its form is similar to the form of
the solution of ODEs [1]. Moreover, as stated above, the terms in the obtained solution are dependent
on the parameters of the NDDE, and hence, one can analyze the effect on the solution of varying
parameter values.
The outline of this paper is as follows. We discuss in section 2 the Lotka-Sharpe-McKendrick age-
structured population model presented in [30]. We show how the model is reduced to a system of ODEs
and NDDEs, along with the necessary conditions and relationships of the parameters of the reduced
and original model. In section 3, we solve the reduced system using the r-Lambert Wfunction. An
approach to numerically approximate the solution obtained in section 3 is detailed in section 4. Because
the true solution of the model is not available, we compare our method to the MATLAB built-in solvers.
We test our method by varying essential parameters and initial conditions, which we illustrate in three
examples. The summary, conclusion, and future works are in section 5.
2. Age-structured model
Bocharov and Hadeler in [30] considered a linear Lotka-Sharpe-McKendrick system of an age-
structured population given by
nt+na+µ(a)n=0,(2.1)
n(t,0) =Z∞
0
b(a)n(t,a)da,(2.2)
n(0,a)=n0(a),(2.3)
where n(t,a) is a non-normalized age distribution of the population at time t,ais the chronological
age, µ(a) is the mortality rate, and b(a) is the fertility rate. Both µ(a) and b(a) are functions dependent
on age, and are assumed to be sums of step functions and delta peaks of the following forms:
µ(a)=µ0+(µ1−µ0)Hτ(a)+µ2δτ(a),(2.4)
b(a)=b0+(b1−b0)Hτ(a)+b2δτ(a),(2.5)
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5689
where
Hτ(a)=






0 for a< τ
1 for a≥τ,
δτ(a) is the Dirac delta function, and µkand bk,k=0,1,2, represent death and birth rates, respectively.
We assume that µkand bkare nonnegative. The coefficients µkand bkare chosen so that they mimic the
qualitative behavior of real populations such as non-constant mortality, reproductive window, etc. [30].
In Figure 1, we see three coefficients b0,b1,b2that constitute the fertility rate b(a). When a< τ, then
b(a)=b0, while if a> τ,b(a)=b1. Notice that b(a) has a peak at a=τ. Similarly, the coefficients
µ0and µ1appear in µ(a), and the graph of µ(a) has a jump at a=τ. The functions for the mortality
and fertility rates are chosen such that the rates are different when a≤τand when a> τ. Based on
this, the population is divided into two classes, the juveniles (a≤τ) and the adults (a> τ), each with
corresponding mortality and fertility rates.
Figure 1. Sample graphs of the birth rates (left) with peaks and jump and the death rates
(right) with jump considered in (2.6)–(2.7). The blue lines represent the graph of the
functions µ(a) and b(a), while the red dotted lines show a continuous approximation of the
functions.
The following new variables are then introduced:
U(t)=Zτ
0
n(t,a)da,
V(t)=Z∞
τ
n(t,a)da,
where U(t) is the population of the juveniles and V(t) is the population of the adults at time t. We
assume that the juveniles do not reproduce (b0=0). For example, the nymph or larval stage of most
insects lacks functional reproductive organs [31]. We also assume that there is no sudden increase in
the death rate at the age τ(µ2=0). Using the functions U(t) and V(t), the partial differential Eq (2.1)
is transformed into a system of the form
˙
U(t)=






−µ0U(t)+b1V(t)+(b2−1)n0(τ−t)e−µ0t,0≤t≤τ,
c0U(t)+c1V(t)+c2V(t−τ)+c3˙
V(t−τ),t> τ, (2.6)
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5690
˙
V(t)=






n0(τ−t)e−µ0t−µ1V(t),0≤t≤τ,
a1V(t−τ)+a2˙
V(t−τ)−a3V(t),t> τ, (2.7)
with initial conditions
U(0) =Zτ
0
n(0,a)da =Zτ
0
n0(a)da,(2.8)
V(0) =Z∞
τ
n(0,a)da =Z∞
τ
n0(a)da,(2.9)
dependent on n0(a).
The parameters aiand cjare related to the parameters µkand bkin the following manner:
c0=−µ0,
c1=b1,
c2=(b2−1)(b1+b2µ1)e−µ0τ,
c3=(b2−1)b2e−µ0τ,
a1=(b1+b2µ1)e−µ0τ,
a2=b2e−µ0τ,
a3=µ1,
(2.10)
where i=1,2,3 and j=0,1,2,3. Moreover, from [30], it is necessary that ai≥0 for all iand that
a1≥a2a3.
We see that the partial differential equation in (2.1) is reduced to a system involving ODEs and
NDDEs. Throughout this paper, we consider the age-structured population model (2.6)–(2.9). We
present an explicit solution to this model and illustrate numerically its solution using the r-Lambert W
function.
3. Solution to the model
In this section, we consider system (2.6)–(2.7), along with the initial conditions (2.8) and (2.9).
First, we determine the solutions U(t) and V(t) to the model for t∈[0, τ]. The solutions on [0, τ]are
the preshape functions necessary to find the solutions U(t) and V(t) for t> τ.
3.1. Solutions for t ∈[0, τ]
We begin by solving for U(t) and V(t), t∈[0, τ], from the ODEs in (2.6) and (2.7). Note that V(t) is
independent of U(t), and thus, can be solved separately. Applying techniques in solving ODEs, we get
V(t)="Zn0(τ−t)e(µ1−µ0)tdt +C1#e−µ1t,(3.1)
where C1is an arbitrary constant. Next, we solve for C1using the initial condition V(0). We let
X(t)=Zn0(τ−t)e(µ1−µ0)tdt.(3.2)
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Substituting (3.2) to (3.1) and solving for C1at t=0, we get
C1=V(0) −X(0).
Therefore,
V(t)=[X(t)+V(0) −X(0)]e−µ1tfor t∈[0, τ].(3.3)
Next, we solve for U(t) for t∈[0, τ]. We plug-in the expression for V(t) from (3.3) to the first
equation in (2.6). The resulting equation is a non-homogeneous ODE, and its solution is given by
U(t)="Z(eµ0tb1V(t)+(b2−1)n0(τ−t))dt +C2#e−µ0t,(3.4)
where C2is an arbitrary constant. We then solve for C2by letting
Y(t)=b1ZV(t)eµ0tdt,
and
Z(t)=(b2−1) Zn0(τ−t)dt.
Using the initial condition U(0), we get
C2=U(0) −Y(0) −Z(0).
Thus,
U(t)=[Y(t)+Z(t)+U(0) −Y(0) −Z(0)]e−µ0tfor t∈[0, τ].(3.5)
3.2. Solutions for t > τ
We now solve system (2.6)–(2.9) for t> τ. Since V(t) is independent of U(t), we solve V(t) first
following the methodology in [29]. Suppose that the solution V(t) to the second equation in (2.7) is of
the form eλt. Plugging in V(t)=eλtinto the NDDE, we get
λeλt+a3eλt−a1eλ(t−τ)−a2λeλ(t−τ)=0.
Multiplying both sides of the equation by τeτ(λ+a3)−λtyields
τ(λ+a3)eτ(λ+a3)−a1τeτa3−a2τλeτa3=0.
Now, adding a constant term −a2τa3eτa3to both sides of the equation and rearranging terms, we
obtain
τ(λ+a3)eτ(λ+a3)−a2eτa3τ(λ+a3)=a1τeτa3−a2τa3eτa3,(3.6)
which is of the form xex+rx =a, with x=τ(λ+a3),a=a1τeτa3−a2τa3eτa3, and r=−a2eτa3. Using
the r-Lambert Wfunction, we can write λas
λ=1
τWr(a1τeτa3+ra3τ)−a3,
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5692
where Wris defined to be solution of the r-Lambert Wfunction. Noting that there is an infinite number
of solutions to the r-Lambert Wfunction, we express V(t) as
V(t)=∞
X
k=−∞
Ckeλkt,(3.7)
where
λk=1
τW(r,k)(a1τeτa3+ra3τ)−a3,(3.8)
W(r,k)is defined to be the kth branch of the r-Lambert Wfunction with r=−a2eτa3, and Ckis dependent
on V(t) for t∈[0, τ].
Next, we solve for U(t) for t> τ. From (3.7), we obtain
V(t−τ)=∞
X
k=−∞
Ckeλk(t−τ),
˙
V(t)=∞
X
k=−∞
λkCkeλkt,
and
˙
V(t−τ)=∞
X
k=−∞
λkCkeλk(t−τ).
Substituting these expressions to (2.6), we get
˙
U(t)=c0U(t)+c1
∞
X
k=−∞
Ckeλkt+c2
∞
X
k=−∞
Ckeλk(t−τ)+c3
∞
X
k=−∞
λkCkeλk(t−τ).(3.9)
Suppose that a solution to the associated homogeneous DE ˙
U(t)=c0U(t) is of the form U(t)=eλt.
Then ˙
U(t)=λeλt, and hence, λ=c0. Thus, the complementary function for (3.9) is
Ug(t)=Aec0t,(3.10)
for some constant A. To find a particular solution Up(t) to (3.9), suppose that
Up(t)=∞
X
k=−∞
αkeβkt.
Then
˙
Up(t)=∞
X
k=−∞
αkβkeβkt.
From (3.9), we have
∞
X
k=−∞
αkβkeβkt=c0
∞
X
k=−∞
αkeβkt+c1
∞
X
k=−∞
Ckeλkt+c2
∞
X
k=−∞
Ckeλk(t−τ)+c3
∞
X
k=−∞
λkCkeλk(t−τ).
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5693
Rearranging the terms, we get
∞
X
k=−∞
(βk−c0)αkeβkt=∞
X
k=−∞
γkCkeλkt,(3.11)
where
γk=c1+c2e−λkτ+c3λke−λkτ.
Comparing coefficients on both sides of (3.11), it follows that for each k,
βk=λk
and
(βk−c0)αk=γkCk=(c1+c2e−λkτ+c3λke−λkτ)Ck.
Then
(λk−c0)αk=(c1+c2e−λkτ+c3λke−λkτ)Ck
αk=(c1+c2e−λkτ+c3λke−λkτ)Ck
λk−c0
,
and
U(t)=Ug(t)+Up(t) for t> τ. (3.12)
Now, we solve for Ain (3.10). Since we are solving for U(t) for t> τ, then it follows from (3.12)
that the initial condition is given by
U(τ)=Aec0τ+∞
X
k=−∞
αkeλkτ.
Hence,
A=





U(τ)−∞
X
k=−∞
αkeλkτ





e−c0τ.
Finally, the solution to system (2.6)–(2.9) is given by
U(t)=












(Y(t)+Z(t)+U(0) −Y(0) −Z(0))e−µ0t,0<t≤τ,






U(τ)−∞
X
k=−∞
αkeλkτ





ec0(t−τ)+∞
X
k=−∞
αkeλkt,t> τ, (3.13)
V(t)=












(V(0) −X(0) +X(t))e−µ1t,0<t≤τ,
∞
X
k=−∞
Ckeλkt,t> τ, (3.14)
where
X(t)=Zn0(τ−t)e(µ1−µ0)tdt,
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5694
Y(t)=b1ZV(t)eµ0tdt,
Z(t)=(b2−1) Zn0(τ−t)dt,
αk=(c1+c2e−λkτ+c3λke−λkτ)Ck
λk−c0
,
λk=1
τW(r,k)(a3τr+a1τea3τ)−a3,
r=−a2eτa3,
and Ckis dependent on V(t) on [0, τ].
4. Numerical illustration
Note that the explicit solution computed in the previous section for t> τ involves an infinite series,
and thus, cannot be exactly solved. We can only approximate the solution by limiting the summations,
i.e., limiting the number of solutions to the r-Lambert Wfunction being considered. For N∈N
sufficiently large, we set



















U(t)≈





U(τ)−
N
X
k=−N
αkeλkτ





ec0(t−τ)+
N
X
k=−N
αkeλkt,
V(t)≈
N
X
k=−N
Ckeλkt,
(4.1)
on the interval [τ, T], where T> τ. Also, note that the solution V(t) on (τ, ∞) is dependent on a history
function φ(t), which is V(t) on the interval [0, τ]. To solve for V(t) numerically on MATLAB, we need
an approximate for the coefficient vector Ck. For this, we use the procedure discussed in [29]. The
history function φ(t) can be approximated as follows:
φ(t)≈
N
X
k=−N
Ckζk(t),(4.2)
where ζk(t)=eλkt, with the same λkas in (3.8). Partitioning the interval [0, τ] to get 2N+1 points,
equation (4.2) becomes

























φ(−h)
φ−h+h
2N
φ−h+2h
2N
.
.
.
φ(0)

























=
























ζ−N(−h)··· ζN(−h)
ζ−N−h+h
2N··· ζN−h+h
2N
ζ−N−h+2h
2N··· ζN−h+2h
2N
.
.
..
.
.
ζ−N(0) ··· ζN(0)















































C−N
C−N+1
C−N+2
.
.
.
CN























.
Solving the system above gives the coefficients Ck,k∈ {−N,...,N}in (4.1).
We illustrate in the following three examples the solution obtained in section 3 using the
approximation (4.1). In the first two examples, we vary the initial age distribution n0(a). In the third
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5695
example, we investigate under certain conditions the long-term behavior of the solution. The
parameter trepresents time in months. Because we do not know the true solution to system
(2.6)–(2.9), we compare our obtained solution with the solution using the MATLAB built-in functions
ode23 and ddensd.
In the next two examples, we use the parameter values shown in Table 1.
Table 1. Parameter values for Examples 1 and 2 obtained from [32, 33].
parameter b1b2µ0µ1τ
value 0.0133 0.045 0.0011 0.00056 36
We set T=144 and compute for the following parameters:
c0=−µ0=−0.0011,
c1=b1=0.0133,
c2=(b2−1)(b1+b2µ1)e−µ0τ
=(0.045 −1)(0.0133 +0.045(0.00056))e−0.0011(36) =−0.0122,
c3=(b2−1)b2e−µ0τ=(0.045 −1)0.045e−0.0011(36) =−0.0413,
a1=(b1+b2µ1)e−µ0τ
=(0.0133 +0.045(0.00056))e−0.0011(36) =0.0128,
a2=b2e−µ0τ=0.045e−0.0011(36) =0.0433,
a3=µ1=0.00056.
Example 1. We set the initial age distribution to be uniform, that is,
n0(a)=1,0<a≤amax =50,
where amax is the maximum age in the population. Thus, the initial conditions are
U(0) =Zτ
0
n(0,a)da =36,
V(0) =Z∞
τ
n(0,a)da =14.
For t∈[0,36],the solutions V(t) and U(t) are given by
V(t)=[V(0) −X(0) +X(t)]e−µ1t
=1865.9e−0.00056t−1851.9e−0.0011t,(4.3)
U(t)=[Y(t)+Z(t)+U(0) −Y(0) −Z(0)]e−µ0t
=45956e−0.00056t−(25.5853t+45920)e−0.0011t,(4.4)
with
X(t)=Zn0(τ−t)e(µ1−µ0)tdt =−1851.9e−0.00054t,
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5696
X(0) =−1851.9e−0.00054·0=−1851.9,
Y(t)=b1ZV(t)eµ0tdt =45956e0.00054t−24.6303t,
Y(0) =45956e0.00054·0−24.6303 ·0=45956,
Z(t)=(b2−1) Zn0(τ−t)dt =−0.955t,
Z(0) =−0.955 ·0=0.
Moreover, the approximate solutions U(t) and V(t) on the interval [36,144] are given by
U(t)=









U(36) −
N
X
k=−N
αkeλkτ









ec0(t−τ)+
N
X
k=−N
αkeλkt
=









16.40079 −
N
X
k=−N
αke36λk









e−0.0011(t−36) +
N
X
k=−N
αkeλkt,(4.5)
and
V(t)=
N
X
k=−N
Ckeλkt,(4.6)
where
αk=[c1+c2e−λkτ+c3λke−λkτ]Ck
λk−c0
=[0.0133 −(0.0122 +0.0413λk)e−36λk]Ck
λk+0.0011 ,
λk=1
τW(r,k)(a3τr+a1τea3τ)−a3=0.0277W(r,k)(0.4693) −0.00056,
r=−a2ea3τ=−0.0442,
and Ckis a column vector dependent on φ(t)=V(t) for t∈[0,36].
We evaluate the explicit solutions U(t) and V(t) on the interval [0,144].Note that we approximate
the solutions U(t) and V(t) on the interval [36,144] using N=50. We also compute for the solution
using the built-in functionsode23 and ddensd from MATLAB with stepsize s=0.01. We then plot
our proposed solutions, U(t) and V(t), together with the solutions obtained using the MATLAB built-in
functions, denoted by UM(t) and VM(t), for 30 equally spaced points on [0,144]. Figures 2a and 2b
show the plots of these results.
Mathematical Biosciences and Engineering Volume 17, Issue 5, 5686–5708.

5697
]
(a) The plot of U(t) vs UM(t)(b) The plot of V(t) vs VM(t)
Figure 2. Comparison of our proposed solutions (solid blue), U(t) and V(t), with solutions
using the MATLAB built-in functions ode23 and ddensd (red asterisks), UM(t) and VM(t),
for Example 1.
Notice that there is a decline in the juvenile population for the first 36 months, compared to the
steady increase in the adult population. Moreover, there is an almost steady increase for both
populations for t>36.Also, at around t=76 months, the total population is doubled, and at t=144
months, the total population is quadrupled.
Comparing our solutions U(t) and V(t), with the numerical solutions UM(t) and VM(t) using the
built-in functions at 30 different points, we see that the two solvers obtain similar results. The relative
differences in approximation are
RU=
30
X
i=1
kUM(ti)−U(ti)k2
kUM(ti)k2=9.20418 ×10−10 and RV=
30
X
i=1
kVM(ti)−V(ti)k2
kVM(ti)k2=9.30018 ×10−10.
Going back to system (2.6)–(2.9) and the parameter values (2.10), notice that the absence of the
parameter b2will transform the NDDEs into RDDEs. Thus, we consider b2, together with the delay τ,
as the two most significant parameters. To show that our method is robust, we implement our method
by varying the parameter values of b2and τ. We then compare our results with the MATLAB built-in
functions using the following measure of relative difference
R(b2, τ)=
30
X
i=1
kUM(b2, τ, ti)−U(b2, τ, ti)k2
kUM(b2, τ, ti)k2+kVM(b2, τ, ti)−V(b2, τ, ti)k2
kVM(b2, τ, ti)k2,(4.7)
where ti,∈[0,144],i=1, ..., 30. We use the same values in Table 1 for the parameters b1,µ0, and µ1,
and choose b2and τsuch that b2∈[0.0013,0.0045] and τ∈[24,48]. In particular, we evaluate R(b2, τ)
at 50 different values of b2and τ, and create a heat map to compare the obtained results. Figures 3a
and 3b show the values of Rin (4.7) for different values of the parameters b2and τwhen N=10 and
N=50, respectively.
Mathematical Biosciences and Engineering Volume 17, Issue 5, 5686–5708.

5698
(a) N=10 (b) N=50
Figure 3. Contour plot of Rin (4.7) for varying parameters τand b2, for Example 1.
Notice that the relative difference when N=50 is smaller than when N=10, for all b2and τ. Also,
the minimum value of Ris 4.75501 ×10−8when N=10, while the maximum value is 1.83404 ×10−5.
When N=50, the maximum value of Ris 6.96986 ×10−8, and the minimum value is 3.75354 ×10−10.
This shows that when b2varies between [0.0013,0.0045] and τvaries between [24,48], our proposed
method can give a good approximate solution.
Example 2. We also consider a normal distribution as the initial age distribution, given by
n0(a)=25
σ√2π
e−
(a−25)2
2σ2,0≤a≤amax =50,
where µand σare the mean and standard deviation, respectively. The values of µand σare set to 25
and 5, respectively. This distribution follows a bell-shape structure and indicates that the age awith
the highest number of initial population is 25. The plot of n0(a) is illustrated in Figure 4.
Figure 4. Plot of n0(a) in Example 2.
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5699
Again, we set the maximum age to be 50. Solving for the initial conditions, we have
U(0) =Zτ
0
n(0,a)da =Zτ
0
n0(a)da =Z36
0
25
5√2π
e−(a−25)2
2·52da,(4.8)
V(0) =Z∞
τ
n(0,a)da =Z∞
τ
n0(a)da =Z50
36
25
5√2π
e−(a−25)2
2·52da.(4.9)
We let t=a−25
5√2. Then we have dt =da
5√2.Transforming (4.8) and (4.9), we have
U(0) =25
√πZ36−25
5√2
0−25
5√2
e−t2dt =25
2





−2
√πZ−25
5√2
0
e−t2dt +2
√πZ11
5√2
0
e−t2dt





,
and
V(0) =25
√πZ50−25
5√2
36−25
5√2
e−t2dt =25
2






2
√πZ25
5√2
0
e−t2dt −2
√πZ11
5√2
0
e−t2dt





.
Note that the addends above for U(0) and V(0) are similar to the form of the error function defined
as
erf(x)=2
√πZx
0
e−t2dt.
The error function arises when integrating the normal distribution. Substituting erf(x) to U(0) and
V(0), we get
U(0) =25
2 erf 5
√2!+erf 11
5√2!!,
and
V(0) =25
2 erf 5
√2!−erf 11
5√2!!.
For t∈[0,36],the solutions V(t) and U(t) are given by
V(t)=[V(0) −X(0) +X(t)]e−µ1t
=




12.42601erf 





√2(t−10.9865)
10 




+12.07809




e−0.00056t,(4.10)
U(t)=[Y(t)+Z(t)+U(0) −Y(0) −Z(0)]e−µ0t
=




306.04815erf 





√2(t−10.9865)
10 




+306.03984




e−0.00056t
−




319.80787erf 





√2(t−11)
10 




+294.82334




e−0.0011t,(4.11)
with
X(t)=Zn0(τ−t)e(µ1−µ0)tdt
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5700
=25
2·e−0.0059363 ·erf 





√2(−10.9865 +t)
10 




,
X(0) =25
2·e−0.0059363 ·erf 





√2(−10.9865 +0)
10 




=−12.07809,
Y(t)=b1ZV(t)eµ0tdt
=25 




12.24192erf 





√2(t−10.9865)
10 




+12.24159




e0.00054t
−25 ·12.31481 ·erf 





√2(t−11)
10 




,
Y(0) =25 




12.24192erf 





√2(0 −10.9865)
10 




+12.24159




e0.00054·0
−25 ·12.31481 ·erf 





√2(0−11)
10 




=307.87019,
and
Z(t)=(b2−1) Zn0(τ−t)dt
=−0.955 ·25
2erf t−11
5√2!=−11.9375erf 





√2(t−11)
10 




,
Z(0) =−11.9375erf 





√2(0 −11)
10 




=11.60555.
Approximate solutions U(t) and V(t) on the interval [36,144] are given by
U(t)=









U(36) −
N
X
k=−N
αkeλkτ









ec0(t−τ)+
N
X
k=−N
αkeλkt
=









9.10443 −
N
X
k=−N
αke36λk









e−0.0011(t−36) +
N
X
k=−N
αkeλkt,(4.12)
and
V(t)=
N
X
k=−N
Ckeλkt,(4.13)
where
αk=[c1+c2e−λkτ+c3λke−λkτ]Ck
λk−c0
=[0.0133 −(0.0122 +0.0413λk)e−36λk]Ck
λk+0.0011 ,
λk=1
τW(r,k)(a3τr+a1τea3τ)−a3=0.0277W(r,k)(0.4693) −0.00056,
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5701
r=−a2ea3τ=−0.0442,
and Ckis a column vector dependent on φ(t)=V(t) for t∈[0,36].
(a) The plot of U(t) vs UM(t)(b) The plot of V(t) vs VM(t)
Figure 5. Comparison of our proposed solutions (solid blue), U(t) and V(t), with the
solutions using MATLAB built-in functions ode23 and ddensd (red asterisks), UM(t) and
VM(t), for Example 2.
Again, we evaluate the solutions U(t) and V(t) on the interval [0,144] using the parameter values in
Table 1. We approximate the solutions on the interval [36,144] with N=50. We also compute for the
solutions using the MATLAB built-in functions ode23 and ddensd with stepsize s=0.01. We then
compare our solutions U(t) and V(t), with the numerical solutions UM(t) and VM(t) obtained using the
built-in functions, for 30 equally spaced points in [0,144]. We see in Figures 5a and 5b that the graphs
of U(t) and V(t) are close to the graphs of UM(t) and VM(t), respectively. The relative differences in
approximation are
RU=
30
X
i=1
kUM(ti)−U(ti)k2
kUM(ti)k2=1.7498 ×10−09 and RV=
30
X
i=1
kVM(ti)−V(ti)k2
kVM(ti)k2=3.8372 ×10−09.
Similar to the previous example, there is a decline in the population of the juveniles during the first
36 months, while there is an increase in the population of the adults. We can also see that for t>36,
both populations increase. In fact, at t=95, the total population is twice the initial total population. It
takes more time to double the population in this example, compared to Example 1.
We also compare the difference in approximation of our proposed method and the MATLAB built-
in functions. We calculate R(b2, τ) in (4.7) at 50 different values of b2and τ, and plot the results when
N=10 and N=50.Figures 6a and 6b show the results. As expected, the values are smaller when
N=50 than when N=10. When N=10, the maximum value of Ris 4.51858 ×10−7, while the
minimum value is 1.79839 ×10−9.When N=50, the maximum value of Ris 6.96986 ×10−8and the
minimum value is 3.75354×10−10.The results show that our proposed explicit formulation can be used
to numerically solve (2.6)–(2.9).
Mathematical Biosciences and Engineering Volume 17, Issue 5, 5686–5708.

5702
(a) N=10 (b) N=50
Figure 6. Contour plot of Rin (4.7) for varying parameters τand b2, for Example 2.
Example 3. In this example, we consider the model studied by Gourley and Kuang in [34]. The
model is a system of ODEs and NDDEs given by
˙ui(t)=












(b2−1)u0(τ−t)e−µt+b0ui(t)+b1um(t)−µui(t),t≤τ,
(b2−1){b2˙um(t−τ)+b2d(um(t−τ)) +b0ui(t−τ)+b1um(t−τ)}e−µτ
+b0ui(t)+b1um(t)−µui(t),t> τ,
(4.14)
˙um(t)=












u0(τ−t)e−µt−d(um(t)),t≤τ,
{b2˙um(t−τ)+b2d(um(t−τ)) +b0ui(t−τ)+b1um(t−τ)}e−µτ
−d(um(t)),t> τ,
(4.15)
where uiand umrepresent the immature (juvenile) and mature (adult) members of the population,
respectively, and u0(a) represents the initial age distribution of the corresponding PDE system, similar
to system (2.1). Moreover, they considered a similar birth rate function (2.5) and set µas the constant
linear death rate for the juveniles and d(um(t)) as the adult mortality function satisfying
d(0) =0 and d(um) strictly increasing in um.(4.16)
System (4.14)–(4.15) can be used to describe the population of insects with a very long larval stage
and a short adult stage allotted for mating [34]. One example of such insects is the 17-year periodical
cicadas, particularly the species Magicicada septendecim,M. cassini, and M. septendecula, which
spend years of their life as nymphs, feeding underground on plant root xylems and resurfacing as
adults only to mate for a few weeks [35]. Another example is the marine midges of genera Pontomyia
and Clunio, which spend a month as benthic larvae and a few hours as adults [36]. Mayflies of order
Ephemeroptera are another example, since they generally live from 3–4 weeks to more than 2 years as
nymphs, and from a few hours to a few days as adults [37]. Note that the juveniles of these insects are
not capable of reproduction and hence, we set the juvenile reproduction to be 0 (b0=0).
Mathematical Biosciences and Engineering Volume 17, Issue 5, 5686–5708.

5703
Assuming that the adult mortality function d(um) is linear, i.e.,
d(um(t)) =kum(t),
for some k>0, system (4.14)-(4.15) is reduced to
˙ui(t)=












−µui(t)+b1um(t)+(b2−1)u0(τ−t)e−µt,t≤τ,
−µui(t)+b1um(t)+(b2−1)(b1+b2k)e−µτum(t−τ)
+(b2−1)b2e−µτ ˙um(t−τ),t> τ,
(4.17)
˙um(t)=






u0(τ−t)e−µt−kum(t),t≤τ,
(b1+b2k)e−µτum(t−τ)+b2e−µτ ˙um(t−τ)−kum(t),t> τ. (4.18)
A theorem on the existence of a solution to system (4.14)–(4.15) is proven in [38], and this also
holds for the reduced system. Note that system (4.17)–(4.18) is composed of linear ODEs and NDDEs
and thus, using our method, a solution to this system is given by
ui(t)=












(y(t)+z(t)+ui(0) −y(0) −z(0))e−µt,t≤τ,








U(τ)−∞
X
j=−∞
αjeλjτ







e−µ(t−τ)+∞
X
j=−∞
αjeλjt,t> τ, (4.19)
um(t)=












(um(0) −x(0) +x(t))e−kt,t≤τ,
∞
X
j=−∞
Ckeλjt,t> τ, (4.20)
where
x(t)=Zu0(τ−t)e(k−µ)tdt,
y(t)=b1Zum(t)eµtdt,
z(t)=(b2−1) Zu0(τ−t)dt,
αj={b1+(b2−1)(b1+b2k)e−(µ+λj)τ+(b2−1)b2λje−(µ+λj)τ}Cj
λj+µ,
λj=1
τW(r,j)(kτr+(b1+b2k)τe(k−µ)τ)−k,
r=−b2e(k−µ)τ,
and Cjis dependent on um(t) on [0, τ]. Table 2 lists the parameter values used in the simulations.
Furthermore, we set k=0.5 and the initial age distribution to be
u0(a)=10,0≤a≤amax =0.6.(4.21)
The chosen parameter values for b0,b1and b2in Table 2 imply that reproduction occurs at a very
short period when the insect reaches adulthood. Figures 7a and 7b show the plots of the immature and
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5704
mature populations (labeled uiand um, respectively) using (4.19)–(4.20) on the interval [0,30] with
N=50, and using the MATLAB built-in functions ode23 and ddensd with s=0.01 (labeled u∗
iand
u∗
m, respectively).
Table 2. Parameter values for Example 3 obtained from [34].
parameter b0b1b2µ τ
value 0 0 1.2 0.7 0.5
(a) The plot of ui(t) vs u∗
i(t)(b) The plot of um(t) vs u∗
m(t)
Figure 7. Comparison of our proposed solutions (solid blue), ui(t) and um(t), with the
solutions using MATLAB built-in functions ode23 and ddensd (red asterisks), u∗
i(t) and u∗
m(t),
using parameter values from Table 2.
The relative differences in approximation of the solutions for 30 equally spaced points in [0,30] are
Rui=3.7641 ×10−07 and Rum=3.7869 ×10−06, which imply that ui(t) and um(t) are close to u∗
i(t)
and u∗
m(t), respectively. Moreover, we observe in Figure 7b that as t→ ∞,um(t) converges to 0. Also,
since the juvenile population is dependent on the adult population, ui(t) also converges to 0 (Figure 7a).
This convergence agrees with Theorem 2 of [34], which states that if b0=b1=0 and b2eµt<1,and
d(um(t)) is a continuous strictly monotonic increasing function of umsatisfying d(0) =0, then um→0
as t→ ∞.
Next, we consider the case when b1,0. We verify the asymptotic behavior of the resulting system
using Theorem 3 in [34], which states that if b0=0,b1>0,
b1ume−µτ <d(um)(1 −b2e−µτ)∀um>0,
and d(um) is a continuous strictly monotonic increasing function of umsatisfying d(0) =0,then um→0
as t→ ∞.Using our assumption on d(um) and simplifying the inequality above, we infer that if we set
k>1−b2e−µτ
b1e−µτ ,
then um→0.Consequently, using the parameter values for b0,b2, µ and τin Table 2 and setting b1=2,
then um→0 if k>9.1296. Figures 8a and 8b show our solutions ui(t) and um(t) with k=13 and
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5705
N=200, in comparison with the solutions u∗
i(t) and u∗
m(t) using the MATLAB built-in functions with
s=0.01.
(a) The plot of ui(t) vs u∗
i(t)(b) The plot of um(t) vs u∗
m(t)
Figure 8. Comparison of our proposed solutions (solid blue), ui(t) and um(t), with the
solutions using MATLAB built-in functions ode23 and ddensd (red asterisks), u∗
i(t) and
u∗
m(t)
We see in Figures 8a and 8b that ui→0 and um→0 as t→ ∞.Moreover, the relative differences of
the solutions for 30 equally spaced points on [0,50] are Rui=8.0737 ×10−07 and Rum=1.1265 ×10−05.
This implies that our solution is close to the solution obtained using the MATLAB built-in solvers.
In this example, we have shown how our proposed numerical method can be used to investigate the
asymptotic behavior of the solution.
5. Conclusion and future works
We have presented an explicit solution to an age-structured model. By assuming that the juveniles
do not reproduce and that there is no peak in the death rate when juveniles become adults, the age-
structured model is reduced to a system of ODEs and NDDEs. We have shown that the arising NDDE
can be solved using a generalization of the Lambert Wfunction. An explicit solution in the form of an
infinite series was obtained. The terms of this series depend on the parameters of the NDDE. Hence,
one can identify the effects of changing parameter values in the solution. We tested our approach
numerically and compared our computed solution with that of the MATLAB built-in solvers.
For future work, one may study the solution of the problem when there is no assumption on the
non-reproduction of juveniles and no peak in the death rate when juveniles become adults. In Example
3, we have shown that our proposed numerical approach can be used to study the long-term behavior
of the solution. However, an in-depth theoretical analysis of the convergence of the solution of the
model using our proposed explicit solution is not trivial. This would require a thorough analysis of the
r-Lambert Wfunction and would demand a separate study. The explicit solution derived in this work
relies heavily on the structure of the solutions of (1.4). To deal with a nonlinear model, one needs to
study the structure of the solutions of (1.4) that is not only dependent on the constant abut also on
time t. This is an exciting direction for future research.
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Acknowledgments
The authors acknowledge the Office of the Chancellor of the University of the Philippines, through
the Office of Vice Chancellor for Research and Development, for funding support through the Outright
Research Grant.
Conflict of interest
All authors declare no conflicts of interest in this paper.
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