DTIC ADA453957: A Finite Difference Approximation for a Coupled System of Nonlinear Size-Structured Populations

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A FINITE DIFFERENCE APPROXIMATION FOR A COUPLED SYSTEM
OF NONLINEAR SIZE-STRUCTURED POPULATIONS
(cid:3) y (cid:3)
A.S. Ackleh , H.T. Banks and K. Deng
Abstract: We study a quasilinear nonlocal hyperbolic initial-boundary value
problem that models the evolution of N size-structured subpopulations compet-
ing for common resources. We develop an implicit (cid:12)nite di(cid:11)erence scheme to
approximate the solution of this model. The convergence of this approximation
to a unique bounded variation weak solution is obtained. The numerical results
for a special case of this model suggest that when subpopulations are closed
under reproduction, one subpopulation survives and the others go to extinction.
Moreover, in the case of open reproduction, survival of more than one population
is possible.
AMS subject classi(cid:12)cation. 92D25, 35A40, 65M06
1. Introduction
In this paper, we consider the following initial boundary value problem that describes
the dynamics of coupled size-structured subpopulations with nonlineargrowth, reproduction
(cid:3)
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504.
y
CenterforResearchinScienti(cid:12)cComputation,NorthCarolinaStateUniversity,Raleigh,NorthCarolina
27695-8205.
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2000 2. REPORT TYPE 00-00-2000 to 00-00-2000
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A Finite Difference Approximation for a Coupled System of Nonlinear
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Size-Structured Populations
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unclassified unclassified unclassified
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
and mortality rates:
I I I I I
ut +(g (x;P(t))u )x +m (x;P(t))u = 0; (x;t) 2 (0;xmax](cid:2)(0;T];
8 N xmax
>> I I I I;J J J
> g (0;P(t))u (0;t) = C (t)+ (cid:13) (cid:12) (x;P(t))u (x;t)dx; t 2 (0;T]; (1)
>
>>>< XJ=1Z0
I I;0
>> u (x;0) = u (x); x 2 [0;xmax]:
>
>
>
> I
Her>:e u (x;t); I = 1;:::;N; is the density of individuals in the I-th subpopulation having
size x at time t, and
N xmax
J J
P(t) = w (x)u (x;t)dx
J=1Z0
X
I
is a weighted total population at time t. The function m denotes the mortality rate of an
I
individual in the I-th subpopulation, and (cid:12) is the reproduction rate of an individual in the
I;J
I-thsubpopulation. Theconstantparameters0 (cid:20) (cid:13) (cid:20) 1represents theprobabilitythatan
individual of the J-th subpopulation will reproduce an individual of the I-th subpopulation.
I
The function g denotes the growth rate of an individual in the I-th subpopulation, and
I
C (t) represents the in(cid:13)ow rate of the I-th subpopulation of zero-size individuals from an
external source.
The model (1) is a generalization of several size-structured population models (often re-
ferred to as distributed rate models) which have been widely investigated in recent years
(see [8, 9, 15, 16, 18]). Motivated by the fact that, in addition to observable characteristics
such as size or age of individuals, non-observable genetic characteristics may often play a
critical role in the development of the individuals, researchers in [8] presented the (cid:12)rst such
generalization of the classical Sinko-Streifer model. There, the population under consider-
ation was treated as being composed of several subpopulations with di(cid:11)erent growth rates,
i.e., there are inherent di(cid:11)erences in growth between the individuals of the population. This
I I I
results in a system of equations similar to (1) with the parameters g ;(cid:12) and m being in-
dependent of the total population (i.e., e(cid:11)ect of competition is not accounted for). In [8] it
2
was shown through numerical simulations that there is a crucial di(cid:11)erence in the dynamics
of the classical Sinko-Streifer models and those of the generalized models. In particular,
the classical models cannot have dispersion of the density of the population in age or size.
Therefore the classical models are in con(cid:13)ict with most of the (cid:12)eld data collected by exper-
imental biologists (see [8] for more details). In [9] an approximation method for the inverse
problem of identifyingthe growth rate distributionwas studied and convergence results were
presented. This method was subsequently applied [18] to a semilinear model where only
I
the mortality rate m depends on the total population due to competition. Moreover, the
convergence results for the inverse problem were extended to this setting. In [10] the inverse
problem technique was used to (cid:12)t (cid:12)eld data (mosquito(cid:12)sh data which attains dispersion of
the density) to the generalized linear model. The resulting data (cid:12)t in [10] indicates that the
need for such modi(cid:12)cation is crucial if these models were to be used as prediction tools.
When N = 1; problem (1) reduces to a classical nonlinear Sinko-Streifer model that
describes the evolution of one population with possible competition between individuals.
For the linear and semilinear forms of such a model (where g = g(x) and (cid:12) = (cid:12)(x)), several
approaches have been developed in the literature for establishing existence-uniqueness of
solutions. For example, in [11, 12, 19] the semigroup of linear operators theoretic approach
was used to obtain such results. Monotone approximations are developed in [1, 2], and upon
passing to the limit a solution to the problem is obtained, whereas uniqueness is obtained
via comparison results. For the quasilinear case (where g = g(x;P) and (cid:12) = (cid:12)(x;P)), the
well-posedness has been discussed in [3, 13], wherein completely di(cid:11)erent techniques were
used for establishing the existence of a unique solution to this model. In [13] the method
of characteristics together with a (cid:12)xed point argument, is employed to prove this result.
A di(cid:11)erence approximation is developed in [3], and upon passing to the limit a solution
to the model is obtained. Then the Holmogren Uniqueness Theorem is used to establish
uniqueness of this solution. To our knowledge, results concerning existence, uniqueness, and
3
convergence of approximations for the general quasilinear case given in (1) with arbitrary N
are not available in the literature.
In this paper, we develop an implicit (cid:12)nite di(cid:11)erence approximation for problem (1).
Techniques in the spirit of those in [14, 23] are used to obtain existence-uniqueness of weak
solutionsaswellasconvergence ofthedi(cid:11)erenceapproximations. Byaweaksolutiontoprob-
1 2 N
lem(1)wemeanaboundedandmeasurablefunctionu(x;t) = (u (x;t);u (x;t);:::;u (x;t))
satisfying
xmax xmax
I I;0
u (x;t)’(x;t)dx(cid:0) u (x)’(x;0)dx
0 0
Z Z
t xmax
I I I I I
= (u ’s +g u ’x (cid:0)m u ’)dxds
(2)
0 0
Z Z
t N xmax
I I;J J J
+ ’(0;s) C (s)+ (cid:13) (cid:12) (x;P(s))u (x;s)dx ds
Z0 J=1Z0 !
X
1
for t 2 [0;T]; I = 1;:::;N; and every test function ’ 2 C ((0;xmax)(cid:2)(0;T)).
The following regularity conditions willbe imposed on our model parameters throughout
the paper: for any I = 1;:::;N
I;0 I;0
(H1) u (x) 2 BV(0;xmax)\L1(0;xmax) and u (x) (cid:21) 0.
I
(H2) m (x;P) is a nonnegative continuously di(cid:11)erentiable function with respect to x and
P.
I
(H3) (cid:12) (x;P) is a nonnegative continuously di(cid:11)erentiable function with respect to x and P.
I I
(H4) g (x;P)isatwicecontinuouslydi(cid:11)erentiablefunctionwithrespecttoxandP,g (x;P) >
I
0 for x 2 [0;xmax); and g (xmax;P) = 0.
I
(H5) C is a nonnegative continuously di(cid:11)erentiable function.
I
(H6) sup (cid:12) (x;P) (cid:20) !1.
(x;P)2[0;xmax)(cid:2)[0;1)
4
(H7) For any suÆciently small Æ > 0
I I
g (x+Æ;P)(cid:0)g (x;P) I
sup +m (x;P) (cid:20) !2:
(x;P)2[0;l)(cid:2)[0;1) Æ
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
I (cid:12) (cid:12)
(H8) w is a nonnegative continuously di(cid:11)erentiable function.
The paper is organized as follows. In Section 2, we develop a numerical scheme for
computing the solution of (1) and prove the convergence of this scheme to a bounded total
variation function satisfying (2). In Section 3, we present numerical results. In Section 4,
we show the continuity of the weak solution under additional conditions on the initial data.
Concluding remarks are given in Section 5.
2. Convergence of Approximations
The techniques used in this section are in the spirit of those used in [14, 23] to obtain
convergence of (cid:12)nite di(cid:11)erence approximation to conservation laws. However, it is worth
pointing out that there are some major di(cid:11)erences between problem (1) and a classical
system of conservation laws. In particular, the (cid:13)ux in (1) is a nonlocal nonlinear function
xmax
I I I N I I
of the solution u (i.e., g = g (x; J=1 w (x)u dx)); whereas it is a local nonlinear
0
Z
function in classicalconservation lawPs. Furthermore, problem(1) isconsidered on a bounded
domain [0;xmax] with a boundary term that is a nonlocal nonlinear function of the solution
u;versus an unbounded domain R for a classical conservation law system. In the sequel, we
shallshowthatsuchdi(cid:11)erencesresultintwoproblemsthatarevery di(cid:11)erentmathematically.
In particular, it is well known that for a conservation law system it is generally not possible
toobtaina bound onthe totalvariationforthe approximatingsolutions, and hence toobtain
convergence one resorts to the compensated compactness method (see, e.g., [23] for details).
However, a bound for the total variation of the approximating solutions of problem (1) is
established (see Lemma 3 in this section).
5
xmax T
The following notation will be used throughout this paper: (cid:1)x = and (cid:1)t =
n m
denotethe spatialandtimemeshsize, respectively. The meshpointsaregivenby: xj = j(cid:1)x,
I;k k
j = 0;1;2;(cid:1)(cid:1)(cid:1) ;n and tk = k(cid:1)t, k = 0;1;2;(cid:1)(cid:1)(cid:1) ;m. We denote by uj and P the (cid:12)nite
I
di(cid:11)erence approximations of u (xj;tk) and P(tk); respectively, and we let
I;k I k I;k I k I;k I k I I I;k I
gj = g (xj;P ); (cid:12)j = (cid:12) (xj;P ); mj = m (xj;P ); wj = w (xj) and C = C (tk):
We de(cid:12)ne the di(cid:11)erence operator
I;k I;k
(cid:0) I;k uj (cid:0)uj(cid:0)1
Dh uj = ; 1 (cid:20) j (cid:20) n
(cid:1)x
(cid:16) (cid:17)
1 1 I;k
and the l and l norm of u by
I;k n I;k
ku k1 = j=1juj j(cid:1)x
I;k P I;k
ku k1 = maxj=0;1;2;(cid:1)(cid:1)(cid:1);njuj j:
We then discretize the partial di(cid:11)erential equation in (1) using the following implicit (cid:12)nite
di(cid:11)erence approximation
I;k+1 I;k I;k I;k+1 I;k I;k+1
uj (cid:0)uj gj uj (cid:0)gj(cid:0)1uj(cid:0)1 I;k I;k+1
+ +mj uj = 0; 1 (cid:20) j (cid:20) n
(cid:1)t (cid:1)x
8
>>>>> g0I;kuI0;k+1 = CI;k + NJ=1 ni=1(cid:13)I;J(cid:12)iJ;kuJi;k(cid:1)x (3)
>
<
k+1 N n PI I;kP+1
> P = I=1 i=1wiui (cid:1)x
>
>
>
>
with the i>:nitial condiPtion P
j(cid:1)x
I;0 1 I;0
uj = u (x)dx; j = 1;(cid:1)(cid:1)(cid:1) ;n; I = 1;:::;N:
(cid:1)x (j(cid:0)1)(cid:1)x
Z
If we de(cid:12)ne
I;k (cid:1)t I;k I;k
dj = 1+ gj +(cid:1)tmj ; 1 (cid:20) j (cid:20) n; I = 1;:::;N;
(cid:1)x
k+1
then (3) can be equivalently written as the following system of linear equations for ~u =
1;k+1 1;k+1 1;k+1 2;k+1 2;k+1 2;k+1 N;k+1 N;k+1 N;k+1 T N(cid:2)(n+1)
[u0 ;u1 ;:::;un ;u0 ;u1 ;:::;un ;:::;u0 ;u1 ;:::;un ] 2 R
k k+1 ~k
A ~u = f ; (4)
6
where
N n N n
~k 1;k 1;J J;k J;k 1;k 1;k 2;k 2;J J;k J;k
f = [C + (cid:13) (cid:12)i ui (cid:1)x;u1 ;:::;un ;C + (cid:13) (cid:12)i ui (cid:1)x;
J=1 i=1 J=1 i=1
XX XX
N n
2;k 2;k N;k N;J J;k J;k N;k N;k T
u1 ;:::;un ;:::;C + (cid:13) (cid:12)i ui (cid:1)x;u1 ;:::;un ]
J=1 i=1
XX
k
and A is the following block diagonal matrix
1;k
A 0 0 (cid:1)(cid:1)(cid:1) 0
2;k
0 A 0 (cid:1)(cid:1)(cid:1) 0
k 0 3;k 1
A = 0 0 A (cid:1)(cid:1)(cid:1) 0
B ....................... C
B N;k C
B 0 0 0 0 A C
B C
@ A
with the lower triangular matrix
I;k
g0 0 0 (cid:1)(cid:1)(cid:1) 0
(cid:1)t I;k I;k
(cid:0)(cid:1)xg0 d1 0 (cid:1)(cid:1)(cid:1) 0
I;k 0 (cid:1)t I;k I;k 1
A = 0 (cid:0)(cid:1)xg1 d2 (cid:1)(cid:1)(cid:1) 0 :
B .................................... C
B (cid:1)t I;k I;k C
BB 0 0 0 (cid:0)(cid:1)xgN(cid:0)1 dn CC
@ A
Note that using the assumptions on our parameters one can easily show that equation
k+1
(4) has a unique solution satisfying ~u (cid:21) 0; k = 0;:::;m. Next we will show that the
1
di(cid:11)erence approximation is bounded in l norm.
Lemma 1 The following estimate holds:
N N k N
I;k k I;0 k(cid:0)i I;i(cid:0)1
ku k1 (cid:20) (1+N !1(cid:1)t) ku k1 + (1+N !1(cid:1)t) jC j(cid:1)t;
I=1 I=1 i=1 I=1
X X X X
and thus
N
k I m I;0
P (cid:20) Pmax = max jjw jj1 (1+N !1(cid:1)t) ku k1
I=1;:::;N
I=1
X
N m
m(cid:0)i I;i(cid:0)1
+ (1+N !1(cid:1)t) jC j(cid:1)t :
I=1 i=1 !
XX
7
Proof. Multiply equation (3) by (cid:1)x; sum over j = 1;(cid:1)(cid:1)(cid:1) ;n and I = 1;:::;N to obtain
N N N n
I;k+1 I;k I;k I;J J;k J;k
ku k1 (cid:20) ku k1 +(cid:1)t C + (cid:13) (cid:12)i ui (cid:1)x
I=1 I=1 " J=1 i=1 !#
X X XX
N N
I;k I;k J J;k
(cid:20) ku k1 +(cid:1)t C + k(cid:12) k1ku k1
I=1 " J=1 !#
X X
N N N
I;k I;k J J;k
= ku k1 + (cid:1)tC +(cid:1)tN k(cid:12) k1ku k1
I=1 I=1 J=1
X X X
N N N
I;k I;k I I;k
(cid:20) ku k1 +(cid:1)t C +(cid:1)tN max jj(cid:12) jj1 ku k1:
I=1;:::;N
I=1 I=1 I=1
X X X
I
Since max(cid:12) (x;P) (cid:20) !1, it follows that
I
N N N
I;k+1 I;k I;k
ku k1 (cid:20) (1+N!1(cid:1)t) ku k1 +(cid:1)t jC j;
I=1 I=1 I=1
X X X
which implies the estimate.(cid:3)
1
We then establish an l bound on the di(cid:11)erence approximation.
Lemma 2 Assume that (cid:1)t is chosen to satisfy !2(cid:1)t < 1. Then we have the estimate
k I N I;k(cid:0)1
I;k 1 I;0 jjC jj1+!1 I=1ku k1
ku k1 (cid:20) max ku k1; ;
( 1(cid:0)!2(cid:1)t (cid:11)1 )
(cid:18) (cid:19) P
I
where (cid:11)1 (cid:20) g (0;P); I = 1;:::;N:
I;k+1
Proof. We (cid:12)rst note that if maxui occurs at the left boundary, then from the second
i
equation of (3)
N
I;k I;k+1 I;k I;k
g0 ju0 j (cid:20) jC j+!1 ku k1:
I=1
X
I;k+1 I;k+1
Otherwise, suppose that for some 1 (cid:20) j (cid:20) n; uj = maxui : Then from the di(cid:11)erence
i
equation (3) we have that
I;k (cid:1)t I;k I;k+1 (cid:1)t I;k I;k+1 I;k
(1+(cid:1)tmj + gj )uj (cid:0) gj(cid:0)1uj(cid:0)1 = uj :
(cid:1)x (cid:1)x
8
I;k+1 I;k+1
Rearranging terms and using the inequality uj(cid:0)1 (cid:20) uj ; we (cid:12)nd
I;k I;k
I;k I;k+1 gj (cid:0)gj(cid:0)1 I;k+1 I;k
(1+(cid:1)tmj )uj +(cid:1)t uj (cid:20) uj :
(cid:1)x
Hence, by (H7) we obtain
I;k+1 I;k I;k
(1(cid:0)!2(cid:1)t)uj (cid:20) uj (cid:20) max ui ;
i
which implies the estimate. (cid:3)
I
Multiplying equation (3) by wj, summing over j = 1;:::;n; I = 1;:::;N; and using
Lemmas 1-2 one can easily show that there exists a c~> 0 such that
k+1 k
P (cid:0)P
(cid:20) c~: (5)
(cid:1)t
(cid:12) (cid:12)
(cid:12) (cid:12)
The bound (5) will be used in the pro(cid:12)of of the ne(cid:12)xt lemma where we show that our approx-
(cid:12) (cid:12)
I;k
imations uj have bounded total variation. This result plays a crucial role in establishing
the subsequential convergence of the di(cid:11)erence approximation (3) to a weak solution of (1).
We remark again that such a bound is not possible, in general, for a system of conservation
laws (see [23]).
Lemma 3 Assume (cid:1)t satis(cid:12)es maxf!1;!2g (cid:1)t < 1. Then there exists a constant c =
I;0 I 1 (cid:0) I;k
c(maxjju jjBV;maxjjC jjC (0;T)) such that for all k = 1;(cid:1)(cid:1)(cid:1) ;m, kDh u k1 (cid:20) c; I =
I I
1;:::;N. (cid:0) (cid:1)
I;k (cid:0) I;k (cid:0)
Proof. Set (cid:17)j = Dh uj and apply the operator Dh to equation (3) to get
I;k+1 I;k (cid:16) (cid:17)I;k I;k+1 I;k I;k+1
(cid:17)j (cid:0)(cid:17)j (cid:0) gj uj (cid:0)gj(cid:0)1uj(cid:0)1 (cid:0) I;k I;k+1
+Dh +Dh(mj uj ) = 0; 2 (cid:20) j (cid:20) n
(cid:1)t (cid:1)x
(cid:18) (cid:19)
and for j = 1 we have that
I;k+1 I;k I;k+1 I;k+1 I;k I;k
(cid:17)1 (cid:0)(cid:17)1 1 u1 (cid:0)u0 u1 (cid:0)u0
= (cid:0)
(cid:1)t (cid:1)t (cid:1)x (cid:1)x
!
I;k+1 I;k
1 u0 (cid:0)u0 (cid:0) I;k I;k+1 I;k I;k+1
= (cid:0) +Dh(g1 u1 )+m1 u1 :
(cid:1)x (cid:1)t
!
9