OF NONLINEAR SIZE-STRUCTURED POPULATIONS
A.S. Ackleh
, H.T. Banks y
and K. Deng
Abstract: We study a quasilinear nonlocal hyperbolic initial-boundary value
problemthat models the evolution of N size-structured subpopulations
compet-ing for common resources. We develop an implicit nite dierence scheme to
approximate the solution of this model. The convergence of this approximation
to aunique bounded variationweak solution is obtained. The numericalresults
for a special case of this model suggest that when subpopulations are closed
underreproduction, one subpopulationsurvivesand the othersgoto extinction.
Moreover, inthecaseofopenreproduction,survivalofmorethanonepopulation
is possible.
AMS subject classication. 92D25,35A40, 65M06
1. Introduction
In this paper, we consider the following initial boundary value problem that describes
thedynamicsofcoupledsize-structuredsubpopulationswithnonlineargrowth,reproduction
DepartmentofMathematics,UniversityofLouisianaatLafayette,Lafayette,Louisiana70504.
y
CenterforResearchinScienticComputation,NorthCarolinaStateUniversity,Raleigh,NorthCarolina
8
>
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>
>
>
<
>
>
>
>
>
>
>
: u
I
t +(g
I
(x;P(t))u I
)
x +m
I
(x;P(t))u I
=0; (x;t)2(0;x
max
](0;T];
g I
(0;P(t))u I
(0;t)=C I
(t)+ N
X
J=1 Z
x
max
0
I;J
J
(x;P(t))u J
(x;t)dx; t2(0;T];
u I
(x;0)=u I;0
(x); x2[0;x
max ]:
(1)
Here u I
(x;t); I = 1;:::;N; is the density of individuals in the I-th subpopulation having
size x at time t, and
P(t)= N
X
J=1 Z
xmax
0 w
J
(x)u J
(x;t)dx
is a weighted total population at time t. The function m I
denotes the mortality rate of an
individualinthe I-thsubpopulation,and I
isthe reproductionrate of anindividual inthe
I-thsubpopulation. Theconstantparameters0 I;J
1representstheprobabilitythatan
individualofthe J-thsubpopulationwillreproduceanindividualofthe I-thsubpopulation.
The function g I
denotes the growth rate of an individual in the I-th subpopulation, and
C I
(t) represents the inow rate of the I-th subpopulation of zero-size individuals from an
external source.
The model(1) isa generalizationof several size-structured populationmodels (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 factthat, inaddition toobservable characteristics
such as size or age of individuals, non-observable genetic characteristics may often play a
criticalrole inthedevelopmentof the individuals,researchers in[8]presented the rstsuch
generalization of the classical Sinko-Streifer model. There, the population under
consider-ation was treated as being composed of several subpopulations with dierent growth rates,
i.e., thereare inherentdierences ingrowth between the individualsofthe population. This
results in a system of equations similar to (1) with the parameters g I
; I
and m I
being
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 classicalmodels are in conictwith most of the eld data collected by
exper-imentalbiologists(see [8] for more details). In [9]anapproximationmethod forthe inverse
problemofidentifyingthegrowthratedistributionwasstudiedandconvergence resultswere
presented. This method was subsequently applied [18] to a semilinear model where only
the mortality rate m I
depends on the total population due to competition. Moreover, the
convergence results forthe inverseproblemwere extended tothis setting. In[10] the inverse
problem technique was used to t eld data (mosquitosh data which attains dispersion of
the density) tothe generalizedlinearmodel. The resultingdatatin[10] indicatesthat the
need for suchmodicationis crucial if these models were to be used asprediction 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.
Forthe linear and semilinearformsof such amodel(where g =g(x) and =(x)), several
approaches have been developed in the literature for establishing existence-uniqueness of
solutions. Forexample, in [11, 12, 19]the semigroup of linear operators theoretic approach
wasusedtoobtainsuchresults. Monotoneapproximationsare developed in[1,2],andupon
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 =(x;P)), the
well-posedness has been discussed in [3, 13], wherein completely dierent techniques were
used for establishing the existence of a unique solution to this model. In [13] the method
of characteristics together with a xed point argument, is employed to prove this result.
A dierence 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
are not available inthe literature.
In this paper, we develop an implicit nite dierence approximation for problem (1).
Techniques in the spirit of those in [14, 23] are used toobtain existence-uniqueness of weak
solutionsaswellasconvergenceofthedierenceapproximations. Byaweaksolutionto
prob-lem(1)wemeanaboundedandmeasurablefunctionu(x;t)=(u 1
(x;t);u 2
(x;t);:::;u N
(x;t))
satisfying
Z
xmax
0 u
I
(x;t)'(x;t)dx Z
xmax
0 u
I;0
(x)'(x;0)dx
= Z
t
0 Z
x
max
0 (u
I
'
s +g
I
u I
'
x m
I
u I
')dxds
+ Z
t
0
'(0;s) C I
(s)+ N
X
J=1 Z
xmax
0
I;J
J
(x;P(s))u J
(x;s)dx !
ds
(2)
for t2[0;T]; I =1;:::;N; and every test function '2C 1
((0;x
max
)(0;T)).
The followingregularityconditionswillbeimposedonour modelparametersthroughout
the paper: for any I =1;:::;N
(H1) u I;0
(x)2BV(0;x
max )\L
1 (0;x
max
)and u I;0
(x)0.
(H2) m I
(x;P) is a nonnegative continuously dierentiable function with respect to x and
P.
(H3) I
(x;P) isanonnegativecontinuously dierentiablefunctionwithrespect toxandP.
(H4) g I
(x;P)isatwicecontinuouslydierentiablefunctionwithrespecttoxandP,g I
(x;P)>
0 forx2[0;x
max
); and g I
(x
max
;P)=0.
(H5) C I
isa nonnegative continuously dierentiable function.
(H6) sup
(x;P)2[0;x
max )[0;1)
I
(x;P)!
sup
(x;P)2[0;l )[0;1)
g
I
(x+Æ;P) g I
(x;P)
Æ
+m I
(x;P)
!
2 :
(H8) w I
is anonnegative continuously dierentiable function.
The paper is organized as follows. In Section 2, we develop a numerical scheme for
computing the solutionof (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 showthe continuity of the weak solutionunder additionalconditions onthe initialdata.
Concluding remarks are given inSection 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 nite dierence approximation to conservation laws. However, it is worth
pointing out that there are some major dierences between problem (1) and a classical
system of conservation laws. In particular, the ux in (1) is a nonlocal nonlinear function
of the solution u I
(i.e., g I
= g I
(x; P
N
J=1 Z
xmax
0 w
I
(x)u I
dx)); whereas it is a local nonlinear
functioninclassicalconservationlaws. Furthermore,problem(1)isconsideredonabounded
domain[0;x
max
] with a boundary term that is anonlocalnonlinear function of the solution
u;versus anunbounded domainR fora classicalconservation lawsystem. In the sequel, we
shallshowthatsuchdierencesresultintwoproblemsthatareverydierentmathematically.
In particular,it iswellknown that fora conservationlawsystem itisgenerally not possible
toobtainaboundonthetotalvariationfortheapproximatingsolutions,andhencetoobtain
convergence oneresorts tothe compensated compactnessmethod(see, e.g.,[23] fordetails).
However, a bound for the total variation of the approximating solutions of problem (1) is
The following notation will be used throughout this paper: x = max
n
and t =
m
denotethespatialandtimemeshsize,respectively. Themeshpointsaregivenby: x
j
=jx,
j = 0;1;2; ;n and t
k
= kt, k = 0;1;2;;m. We denote by u I;k
j
and P k
the nite
dierenceapproximations ofu I
(x
j ;t
k
) and P(t
k
); respectively, and we let
g I;k j =g I (x j ;P k ); I;k j = I (x j ;P k ); m I;k j =m I (x j ;P k ); w I j =w I (x j
) and C I;k =C I (t k ):
Wedene the dierence operator
D h u I;k j = u I;k j u I;k j 1 x
; 1j n
and the l 1
and l 1
norm of u I;k by ku I;k k 1 = P n j=1 ju I;k j jx ku I;k k 1 =max j=0;1;2;;n ju I;k j j:
We then discretize the partial dierential equation in (1) using the following implicit nite
dierenceapproximation 8 > > > > > > < > > > > > > : u I;k+1 j u I;k j t + g I;k j u I;k+1 j g I;k j 1 u I;k+1 j 1 x +m I;k j u I;k+1 j
=0; 1j n
g I;k 0 u I;k+1 0 =C I;k + P N J=1 P n i=1 I;J J;k i u J;k i x P k+1 = P N I=1 P n i=1 w I i u I;k+1 i x (3)
with the initialcondition
u I;0 j = 1 x Z jx
( j 1)x u
I;0
(x)dx; j =1; ;n; I =1;:::;N:
If wedene
d I;k
j
=1+ t
x g
I;k
j
+tm I;k
j
; 1j n; I =1;:::;N;
then (3) can be equivalently written as the following system of linear equations for ~u k+1 = [u 1;k+1 0 ;u 1;k+1 1
;:::;u 1;k+1 n ;u 2;k+1 0 ;u 2;k+1 1
;:::;u 2;k+1
n
;:::;u N;k+1
0 ;u
N;k+1
1
~ f k =[C 1;k + N X J=1 n X i=1 1;J J;k i u J;k i x;u 1;k 1
;:::;u 1;k n ;C 2;k + N X J=1 n X i=1 2;J J;k i u J;k i x; u 2;k 1
;:::;u 2;k
n
;:::;C N;k + N X J=1 n X i=1 N;J J;k i u J;k i x;u N;k 1
;:::;u N;k n ] T and A k
isthe following block diagonal matrix
A k = 0 B B B B @ A 1;k
0 0 0
0 A 2;k
0 0
0 0 A
3;k
0
...
0 0 0 0 A
N;k 1 C C C C A
with the lowertriangular matrix
A I;k = 0 B B B B @ g I;k 0
0 0 0
t x g I;k 0 d I;k 1
0 0
0 t x g I;k 1 d I;k 2 0 ...
0 0 0
t x g I;k N 1 d I;k n 1 C C C C A :
Note that using the assumptions on our parameters one can easily show that equation
(4) has a unique solution satisfying ~u k+1
0; k = 0;:::;m. Next we will show that the
dierenceapproximation isbounded inl 1
norm.
Lemma 1 The following estimate holds:
N X I=1 ku I;k k 1
(1+N !
1 t) k N X I=1 ku I;0 k 1 + k X i=1
(1+N !
1 t) k i N X I=1 jC I;i 1 jt; and thus P k P max = max I=1;:::;N jjw I jj 1
(1+N !
1 t) m N X I=1 ku I;0 k 1 + N X I=1 m X i=1
(1+N !
N X I=1 ku I;k+1 k 1 N X I=1 " ku I;k k 1
+t C
I;k + N X J=1 n X i=1 I;J J;k i u J;k i x ! # N X I=1 " ku I;k k 1
+t C
I;k + N X J=1 k J k 1 ku J;k k 1 !# = N X I=1 ku I;k k 1 + N X I=1 tC I;k +tN N X J=1 k J k 1 ku J;k k 1 N X I=1 ku I;k k 1 +t N X I=1 C I;k
+tN max
I=1;:::;N jj I jj 1 N X I=1 ku I;k k 1 : Since max I I
(x;P)!
1
,it follows that
N X I=1 ku I;k+1 k 1
(1+N!
1 t) N X I=1 ku I;k k 1 +t N X I=1 jC I;k j;
whichimplies the estimate.
We thenestablish anl 1
boundon the dierence approximation.
Lemma 2 Assume that t ischosen to satisfy !
2
t<1. Thenwe have the estimate
ku I;k k 1 max ( 1 1 ! 2 t k ku I;0 k 1 ; jjC I jj 1 +! 1 P N I=1 ku I;k 1 k 1 1 ) ; where 1 g I
(0;P);I =1;:::;N:
Proof. We rst note that if max
i u
I;k+1
i
occurs at the left boundary, then from the second
equation of (3)
g I;k 0 ju I;k+1 0
jjC I;k
j+!
1 N X I=1 ku I;k k 1 :
Otherwise, suppose that for some 1 j n; u I;k+1 j =max i u I;k+1 i
: Then from the dierence
equation (3)we have that
Rearranging terms and using the inequalityu
j 1 u
j
; we nd
(1+tm I;k
j )u
I;k+1
j
+t g
I;k
j g
I;k
j 1
x u
I;k+1
j
u I;k
j :
Hence, by (H7) we obtain
(1 !
2 t)u
I;k+1
j
u I;k
j
max
i u
I;k
i ;
whichimplies the estimate.
Multiplying equation (3) by w I
j
, summing over j = 1;:::;n; I = 1;:::;N; and using
Lemmas 1-2one can easily showthat there exists a c~>0 such that
P
k+1
P k
t
~c: (5)
The bound (5)willbeused inthe proof of the next lemmawhere weshow that our
approx-imations u I;k
j
have bounded total variation. This result plays a crucial role in establishing
the subsequentialconvergence of the dierence approximation (3)toa weaksolution of (1).
Weremark again that such a bound isnot possible, ingeneral, fora system of conservation
laws (see [23]).
Lemma 3 Assume t satises maxf!
1 ;!
2
g t < 1. Then there exists a constant c =
c(max
I jju
I;0
jj
BV ;max
I jjC
I
jj
C 1
(0;T)
) such that for all k = 1;;m, kD
h u
I;k
k
1
c; I =
1;:::;N.
Proof. Set I;k
j
=D
h
u I;k
j
and apply the operatorD
h
toequation (3)to get
I;k+1
j
I;k
j
t
+D
h
g I;k
j u
I;k+1
j
g I;k
j 1 u
I;k+1
j 1
x
+D
h (m
I;k
j u
I;k+1
j
)=0; 2j n
and for j =1wehave that
I;k+1
1
I;k
1
t
= 1
t u
I;k+1
1
u I;k+1
0
x
u I;k
1 u
I;k
0
x !
= 1
x u
I;k+1
0
u I;k
0
t
+D
h (g
I;k
1 u
I;k+1
1
)+m I;k
1 u
I;k+1
1 !
Multiplyingeach equationby xsgn (
j
), usingthe fact that I;k
sgn(
j
) j
I;k
j,
and summingover the indices, j =1;2; ;n, we nd
k I;k+1 k 1 k I;k k 1 t + n X j=1 " D h g I;k j u I;k+1 j g I;k j 1 u I;k+1 j 1 x ! +D h (m I;k j u I;k+1 j ) # sgn ( I;k+1 j
)x0;
where we set m I;k
0
=0and
D h g I;k 0 u I;k+1 0 = u I;k+1 0 u I;k 0 t :
Now, simple calculations yield
n X j=1 D h g I;k j u I;k+1 j g I;k j 1 u I;k+1 j 1 x ! sgn( I;k+1 j )x n X j=2 D h g I;k j g I;k j 1 x u I;k+1 j 1 ! sgn( I;k+1 j
)x+ u I;k+1 0 u I;k 0 t sgn( I;k+1 1 ) +D h g I;k 1 u I;k+1 0 sgn ( I;k+1 1 ): Thus, k I;k+1 k 1 k I;k k 1 t max j (D h g I;k j +m I;k j )k I;k+1 k 1 +max j jD h (D h g I;k j +m I;k j )j ku I;k+1 k 1 + u I;k+1 0 u I;k 0 t + D h g I;k 1 ju I;k+1 0 j: (6)
From Lemmas 1-2, it suÆces to obtain a bound for the term u I;k+1 0 u I;k 0 t
. To this end,
consider the boundary condition
g I;k 0 u I;k+1 0 u I;k 0 t ! + g I;k 0 g I;k 1 0 t ! u I;k 0 C I;k C I;k 1 t = N X J=1 n X j=1 I;J " J;k j u J;k j u J;k 1 j t ! + J;k j J;k 1 j t ! u J;k 1 j # x N X J=1 n X j=1 J;k j D h (g J;k 1 j u J;k j
)+m J;k 1 j u J;k j + J P (x; P) P k P k 1 t u J;k 1 j x N X J=1 n X j=1 (D h ( J;k j )g J;k 1 j J;k j m J;k 1 j )u J;k j + J P (x; P) P k P k 1 t u J;k 1 j x + N X J=1 J;k 1 g J;k 1 0 u J;k 0 ; where
P is between P k 1
and P k
: Hence, using the bound given in(5) we nd
g I;k 0 u I;k+1 0 u I;k 0 t ! + g I;k 0 g I;k 1 0 t ! u k 0 C I;k C I;k 1 t N X J=1 sup j jD h ( J;k j )g J;k 1 j
j+sup
j j J;k j m J;k 1 j j jju J;k jj 1 + N X J=1 ~ c sup (x; P)2[0;xmax][0;Pmax] J P (x; P) ku J;k 1 k 1 + N X J=1 j J;k 1 j C J;k 1 + N X L=1 sup j L;k 1 j ku L;k 1 k 1 ! :
Now, Lemmas 1-2 imply that there exists a constant > 0 such that u I;k+1 0 u I;k 0 t .
Applying this bound to(6), we conclude that there exists aconstant !
3
>0such that
k I;k+1 k 1 k I;k k 1 t ! 2 k k+1 k 1 +! 3 ;
and the result is established.
The next result shows that the dierence approximationssatisfy a Lipschitz-type
1 2
for any m >p
n X j=1 j u I;m j u I;p j t
jxA(m p); I =1;:::;N:
Proof. Summingthe rst equationin (3) over jand multiplying by xwe obtain
n X j=1 j u I;k+1 j u I;k j t jx= n X j=1 g I;k j 1 u I;k+1 j u I;k+1 j 1 x u I;k+1 j g I;k j g I;k j 1 x m I;k j u I;k+1 j x sup j g I;k j n X j=1 j u I;k+1 j u I;k+1 j 1 x
jx+sup
j j g I;k j g I;k j 1 x +m I;k j jku I;k+1 k 1 sup (x;P)2[0;xmax][0;Pmax] g I
(x;P) k I;k+1 k 1 +! 2 ku I;k+1 k 1 A: Hence n X j=1 j u I;m j u I;p j t jx m X k=p n X j=1 j u I;k+1 j u I;k j t
jxA(m p):
Following[23] we dene a familyof functions U I x;t by U I x;t
(x;t)=u I;k
j
for x2[x
j 1 ;x
j
); t 2[t
k 1 ;t
k
); j =1;:::;n; k =1;:::;m; I =1;:::;N:
Then,the set offunctions
U I
x;t
is compactinthe topologyof L 1
((0;x
max
)(0;T));and
we havethe following lemma.
Lemma 5 For I =1;:::;N there exists a sequence U I x i ;t i U I x;t which converges
to a BV([0;x
max
][0;T])function u I
(x;t) in the sense that for all t>0
Z xmax 0 jU I x i ;t i
(x;t) u I
(x;t)jdx!0;
and Z T 0 Z x max 0 jU I xi;ti
(x;t) u I
jju I
jj
BV([0;xmax][0;T])
c(jju I;0
jj
BV ,jjC
I
jj
C 1
(0;T) ).
Proof. TheresultfollowsfromLemmas1-4andthe proofofLemma16.7(page276)in[23].
The nexttheorem willshowthatthe limitfunction,u=(u 1
;u 2
;:::;u N
),constructedvia
our dierence scheme is actuallya weak solutionof problem (1).
Theorem6Anylimit u(x;t)=(u 1
(x;t);u 2
(x;t);:::;u N
(x;t))denedinLemma5isaweak
solution of (1) and satises
P(t) max
I=1;:::;N jjw
I
jj
1 N
X
I=1 ku
I
(t)k
1
max
I=1;:::;N jjw
I
jj
1 e
!
1 NT
N
X
I=1 ku
I;0
k
1 +
N
X
I=1 Z
T
0 e
!
1 (T s)
C I
(s)ds !
=
P
and
ku I
k
L 1
((0;xmax)(0;T))
max (
e !2T
ku I;0
k
1 ;
kC I
k
1 +!
1 P
N
I=1 ku
I
(t)k
1
1
)
;
where
1 g
I
(0;P) for P 2[0;
P];I=1,:::;N.
Proof. This result can be easily established by using similar techniques as in the proof of
Lemma 16.10 (page279) in [23].
The following theorem guarantees the continuous dependence of the solution fu I;k
j g to
(3)with respect to the initialcondition u I;0
.
Theorem 7 Letfu I;k
j
g and f^u I;k
j
g bethe solutionsof (3)correspondingtothe initial
condi-tions u I;0
j
andu^ I;0
j
, respectively. Then there exists a
=(max
k;I jj^u
I;k
k
1 ;max
k;I ku^
I;k
k
1 ;max
k;I kD
h ^ u
I;k
k
N X I=1 ku I;k+1 ^ u I;k+1 k 1
[1+(!
1
+max
I jjw I jj)t] N X I=1 ku I;k ^ u I;k k 1
for allk 0.
Proof. Letv I;k j =u I;k j ^ u I;k j
for0k mand 0j n. Then v I;k
j
satises the following
8 > > > > > > > > > > > < > > > > > > > > > > > : v I;k+1 j v I;k j t +D h g I;k j u I;k+1 j ^ g I;k j ^ u I;k+1 j +m I;k j v I;k+1 j +(m I;k j ^ m I;k j )^u I;k+1 j
=0; 1j n
g I;k 0 u I;k+1 0 ^ g I;k 0 ^ u I;k+1 0 = P N J=1 P n i=1 I;J J;k i v J;k i x + P N J=1 P n i=1 I;J J;k i ^ J;k i ^ u J;k i x; (7)
where g^ I;k j =g I (x j ; ^ P k
), and similar notation is used for the rest of the parameters.
Multi-plyingeachequationof (7) by xsgn (v I;k+1
j
),summingover j =1;:::;n, I =1;:::;N; and
using the followingfact:
P n j=1 D h g I;k j u I;k+1 j ^ g I;k j ^ u I;k+1 j sgn (v I;k+1 j
)x g I;k 0 jv I;k+1 0 j + P n j=1 D h h (g I;k j ^ g I;k j )^u I;k+1 j i sgn (v I;k+1 j )x;
we get that
N X I=1 kv I;k+1 k 1 kv I;k k 1 t N X I=1 n X j=1 D h h (g I;k j ^ g I;k j )^u I;k+1 j i sgn(v I;k+1 j )x + N X I=1 g I;k 0 jv I;k+1 0 j N X I=1 n X j=1 (m I;k j ^ m I;k j )^u I;k+1 j sgn(v I;k+1 j )x N X I=1 n X j=1 m I;k j jv I;k+1 j jx:
Now, using assumption (H4)we can easily obtainthe following
(H6)that
P
N
I=1 g
I;k
0 jv
I;k+1
0 j
P
N
I=1 P
n
j=1 (m
I;k
j ^ m
I;k
j )^u
I;k+1
j
sgn(v I;k+1
j
)x
P
N
I=1 P
n
j=1 m
I;k
j jv
I;k+1
j
jx!
1 P
N
I=1 kv
I;k
k
1
+(c
4 max
I jj^u
I;k+1
jj
1 +c
5 max
I k^u
I;k+1
k
1 ) jP
k
^
P k
j:
Hence, choosing >0so that
(c
1 +c
4 )max
I k^u
k+1
k
1 +(c
2 +c
5 )max
I k^u
k+1
k
1 +c
3 max
I kD
h ^ u
k+1
k
1 ;
we obtain
N
X
I=1 kv
I;k+1
k
1
[1+(!
1
+max
I jjw
I
jj
1 )t]
N
X
I=1 kv
I;k
k
1 ;
whichimplies the theorem.
Next, we prove that the BV solution dened in Lemma 5and Theorem 6 is unique. To
this end,assume thatP(t)2C 1
(0;T)and B I
(t)2C(0;T)aregiven functionsand consider
the followinginitial-boundaryvalue problem:
8
>
>
>
>
<
>
>
>
>
: u
I
t +(g
I
(x;P(t))u I
)
x +m
I
(x;P(t))u I
=0; (x;t)2(0;x
max
](0;T];
g I
(0;P(t))u I
(0;t)=B I
(t); t2(0;T];
u I
(x;0)=u I;0
(x); x2[0;x
max ]:
(8)
Then one can easily show that (8) has a unique weak solution (note that this system is
uncoupled and has a local boundary condition). In fact, a weak solution can be dened as
a limit of the nite dierence approximation (3) with the given numbers P k
= P(t
k ) and
uniqueness can be established by using a similartechnique asin [23, Page 282]. Hence, the
nitedierence solutionto(3)withgiven numbers P k
=P(t
k
)and B I;k
=B I
(t
k
to the unique solution of (1) with the given P 2 C (0;T) and B 2 C(0;T). In addition,
from the proof of Theorem 7 we can easily show that if fu k
g and f^u k
g are the solutionsto
(3)corresponding to given functions (P k
;B I;k
) and ( ^ P k ; ^ B I;k
), respectively, then wehave
N X I=1 kv I;k+1 k 1 N X I=1 kv I;k k 1
+tjP k
^
P k
j+t N X I=1 jB I;k ^ B I;k j; where v I;k =u I;k ^ u I;k
. Equivalently,
N X I=1 kv I;k k 1 N X I=1 kv I;0 k 1 + k 1 X i=0 " jP i ^ P i j + N X I=1 jB I;i ^ B I;i j # t: (9)
Now, since from Theorem 6 for I = 1;:::;N; fU I
x;t
g converges to u I
(x;t) and f ^
U I
x;t g
converges tou^ I
(x;t) strongly inC([0;T];L 1
(0;x
max
)),taking the limitin(9)weobtain
N X I=1 kv I (t)k 1 N X I=1 kv I (0)k 1 + Z t 0 " jP(s) ^
P(s)j + N X I=1 jB I (s) ^ B I (s)j # ds; (10)
where u(x;t); u(x;^ t) are the unique solutions to (8) given (P(t);B I
(t)) and ( ^ P(t); ^ B I (t)),
respectively, and v I
(t)=u I
(;t) u^ I
(;t). Then, applyingthe estimategiven in(10) for the
corresponding solutionsto (8)where
P(t)= N X I=1 Z xmax 0 w I (x)u I
(x;t)dx;
B I
(t)=C I (t)+ N X J=1 Z xmax 0 I;J J
(x;P(t))u J
(x;t)dx;
^
P(t)= N X I=1 Z xmax 0 w I
(x)u^ I
(x;t)dx;
^
B I
(t)=C I (t)+ N X J=1 Z xmax 0 I;J J (x; ^
P(t))^u J
(x;t)dx
are dened inTheorem 6, we obtainthe following result.
Theorem 8 Suppose that u and u^ are two bounded variation weak solutions of (1)
corre-sponding to initial conditions u 0
and u^ 0
, respectively. Then
N
X
I=1 ku
I
(t) u^ I
(t)k
1 e
[(+NmaxIjjPujj1)^ maxIjjw I jj1+N!1]t N X I=1 ku I
(0) u^ I
(0)k
Hence, from Theorem 8 it follows that the nite dierence solution converges to the
unique bounded variation solutionof (1).
3. Numerical Results
Inthissection,weprovidesomenumericalresultsthatcorroboratetheconvergencetheory
presented in Section 2 and demonstrate the eect of the probability function I;J
on the
dynamics of this system. Forthe rest of this section we assume that x
max
= 1;N =2; and
the weight functions w I
= 1. This implies that P = R
1
0 [u
1
(x;t)+u 2
(x;t)]dx. We choose
the parameters g I
, I
, m I
, and C I
as follows:
g I
(x;P)=g I 0 k I f I
(P)(1 x); I
(x;P)= I 0 (1 k I )f I
(P)x; m I
(x;P)=m I
(P); C I
(t)=0;
where k I
2(0;1);I =1;2. Hence, our modelreduces to
8 > > < > > : u I t +g I 0 k I f I
(P)((1 x)u I ) x +m I (P)u I
=0; x2(0;1];t>0;
g I 0 k I u I
(0;t)= P 2 J=1 I;J J 0 (1 k J ) Z 1 0 xu J
(x;t)dx ; t>0;
u I
(x;0)=u I;0
(x); x2[0;1]; I =1;2:
(11)
Integrating (11) andmultiplying(11)byxand integrating onceagain,wereadily obtainthe
followingsystem of dierentialequations:
( P I 0 = X 2 J=1 I;J J 0 (1 k J )f J (P)Q J m I (P)P I
I =1;2;
Q I 0 =g I 0 k I f I (P)(P I Q I ) m I (P)Q I
; I =1;2;
(12) whereP I = R 1 0 u I
(x;t)dxand Q I = R 1 0 xu I
(x;t)dx;I =1;2:Inournumericalsimulationswe
have used f 1
(P)=e 0:5P
, f 2
(P)=e 0:1P
; m 1
(P) =0:3P, m 2
(P) =2P=(1+P), k 1 = 0:5; k 2 =0:7; I 0
=1, g I
0
=1,t2 [0;10]and
u 1;0
(x)=
5 x2[0;0:3]
0 x2(0:3;1] ; u
2;0
(x)=
8 x2[0;0:2]
0 x2(0:2;1] :
To test ourcode we compute (P I
;Q I
), I =1;2;thesolutionof (12)using a4 5 th
order
P I
x
(t); and the total biomass, Q I
x
(t); that result from the nite dierence system
describedinSection2(withx=0:02;andt=0:01)tothenumericalsolution P I
;Q I
of
the dierentialequation system (12) that resultsfrom the Runge Kuttaroutine. In Figures
1-2 we present the dierences P I
x
(t) P I
(t) and Q I
x
(t) Q I
(t), respectively. The
gures indicate that the nite dierence approximation provides a good approximation to
the solution of (11). Next we present two dierent dynamics for the total population and
total mass dependingon the choice of I;J
.
3.1. Closed reproduction case
Inthiscase,weassumethatreproductionisclosedundersubpopulations,thatis,
individ-uals in the I-thsubpopulation only produce individuals in the I-th subpopulation. Hence,
I;I
= 1; I = 1;2; and I;J
= 0, I 6= J. Solving (11) we present the total populations
P I
x
(t); I = 1;2; and total mass Q I
x
(t); I =1;2, in Figures 3-4, respectively. Note
that in this case the second subpopulation goes to extinction. Similar phenomena have
been known to occur in dierent structured and non-structured population models, where
the surviving subpopulation is often referred to as the ttest among the others (e.g., see
[4,17]).
3.2. Open reproduction case
In this case, we assume that reproduction is open under subpopulations, that is, a
sub-populationoftypeI alsoreproducesindividualsoftypeJ. Forournumericalexampleweset
I;J
= 1
2
; I;J =1;2. However, we have performed many other numericalexperiments with
dierentpositive I;J
;and the dynamics are essentially the same. In Figures5-6we present
the total subpopulations P I
x
(t); I =1;2; and total mass Q I
x
(t), respectively. Note
0
5
10
15
20
25
30
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
x 10
−3
t
difference in total number
_____ Population # 1
−−−−− Population # 2
Figure 1: The dierence between the total population resulting from the nite dierence
scheme and the total population resulting from solving the dierential equations system
using Runge-Kuttaroutine.
0
5
10
15
20
25
30
−2
−1
0
1
2
3
4
x 10
−3
t
difference in total mass
_____ Population # 1
−−−−− Population # 2
Figure 2: The dierence between the total mass resulting from the nite dierence scheme
andthetotalmassresultingfromsolvingthedierentialequationssystemusingRunge-Kutta
0
5
10
15
20
25
30
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
t
population total number
_____ Population #1
−−−−− Population # 2
Figure 3: The computed total population P I
x
(t); I =1;2;onthe interval [0;30]:
0
5
10
15
20
25
30
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
population total mass
t
_____ Population #1
−−−−− Population # 2
Figure4: The computed total mass Q I
x
0
5
10
15
20
25
30
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
t
population total number
_____ Population # 1
−−−−− Population # 2
Figure 5: The computed total population(P I
) x
(t); I =1;2;onthe interval[0;30]:
0
5
10
15
20
25
30
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
t
population total mass
_____ Population # 1
−−−−− Population # 2
Figure6: The computed total mass Q I
x
Thegoalofthissectionistoshowthattheboundedvariationweaksolutionofproblem(1)
is continuous provided that the initialdistributions u I;0
; I =1;:::;N; satisfy the following
compatibilityconditions:
(H9) Let P 0
= N
P
I=0 R
x
max
0 w
I
(x)u I;0
(x)dx, g I;0
(x) = g I
(x;P 0
), I;0
(x) = I
(x;P 0
) and
m I;0
(x) = m I
(x;P 0
), I = 1;:::;N. Assume that the function u I;0
is a nonnegative
continuous function and satises
g I;0
(0)u I;0
(0)=C I
(0)+ N
X
J=1 Z
xmax
0
I;J
J;0
(x)u J;0
(x)dx; I =1;:::N:
It isworthpointingout that the following resultdemonstratesthe remarkabledierence
between nonlocal quasilinear hyperbolicinitial-boundary value problems similar to (1) and
local quasilinearhyperbolicproblems since it is wellknown that solutionsof localproblems
can attain discontinuities in nite time.
Theorem 9 Under the additional assumption (H9) the bounded variation weak solution of
problem (1) is continuous.
Proof. Let ^
I
andm^ I
benonnegativecontinuously dierentiable functions. Furthermore,
assume that ^g I
is twice continuously dierentiable in x and continuously dierentiable in
t; g^ I
(x;t) > 0; x 2 [0;x
max
) and g^ I
(x
max
;t) = 0, t 2 [0;T]. Consider the following
initial-boundary value problem:
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
: v
I
t +(^g
I
(x;t)v I
)
x +m^
I
(x;t)v I
=0; (x;t)2(0;x
max
)(0;T);
^ g I
(0;t)v I
(0;t)=C I
(t)+ N
X
J=1 Z
x
max
0
I;J
^
J
(x;t)v J
(x;t)dx; t2(0;T);
v I
(x;0)=u I;0
(x); x2[0;x
max ]:
weak solution. Furthermore, using standard arguments (e.g., see [1, 8, 12, 19]) one can see
thatacharacteristiccurvepassingthrough(bx; b
t)isgivenby(X I
(t;bx; b
t);t),whereX I
satises
d
dt X
I
(t;x;b b
t)=^g I
(X(t;x;b b
t);t)
and X I
( b
t;bx; b
t) = bx. By the assumptions on ^g I
the function X I
is a strictly increasing
function,andthereforeauniqueinversefunction I
(x;x;b b
t)exists. HenceifwedeneG I
(x)=
I
(x;0;0),then(x;G I
(x))represents thecharacteristiccurvepassingthrough (0;0)and this
curve divides the (x;t)-plane intotwoparts. Andthe weak solutionv I
of problem(13) has
the followingimplicit representation
v I
(x;t)= 8
>
>
>
>
>
>
<
>
>
>
>
>
>
: u
I;0
(X I
(0;x;t))
exp n
R
t
0 [^g
I
x (X
I
(s;x;t);s)+m I
(X I
(s;x;t);s)]ds o
tG I
(x);
R I
( I
(0;x;t))
exp n
R
t
I
(0;x;t) [^g
I
x (X
I
(s;x;t);s)+m^ I
(X I
(s;x;t);s)]ds o
t>G I
(x);
(14)
where R I
(t)=1=^g I
(0;t) h
C I
(t)+ P
N
J=1 R
xmax
0
I;J
^
J
(x;t)v J
(x;t)dx i
.
Now, let P(t) = P
N
J=1 R
xmax
0 w
I
(x)u I
(x;t)dx where u I
is the unique bounded variation
weak solutionof problem(1). Using(H2)-(H4) we see that g^ I
(x;t)g I
(x;P(t)), ^
I
(x;t)
I
(x;P(t))andm^ I
(x;t)m I
(x;P(t))satisfytheaboverequirements. Then,using(H8), the
solutionrepresentation (14)andstandard argumentsasin[1]itfollowsthat v I
iscontinuous
for this choice of parameters. Furthermore, by uniqueness of solutions, v I
coincides with
the solution component u I
of the nonlinear problem(1), I =1;:::;N. This establishes the
In this paper we presented a model that describes the evolution of N subpopulations
competing for common resources. Our numerical results indicate that the parameters I;J
play a crucial role in the dynamics of these subpopulations. Several interesting questions
about the model (1) arise naturally: What is a good measure (in terms of the rates g I
,
m I
and I
) that will lead to the survival of the ttest in a closed reproduction case? In
an open reproductioncase, which populations willsurvive and which will goto extinction?
We mention that for special cases of structured age and age-size population models, it was
proved in[17]thatunderclosedreproductionagoodmeasureofspeciestnessistheproduct
of the birth rate function and the survivorship function. To our knowledge, however, no
resultsconcerningtheopenreproductioncase forstructured populationsareavailable. Fora
classicalLotka-VolterracompetitionmodelwhichisrepresentedbyasystemofN dierential
equations, conditions on the growth and mortality rates of each populationthat willresult
in its survival or extinction have been discussed recently by several researchers (e.g., see
[5,6,7, 21, 22]). Ourfuture eorts willfocus ongeneralizing suchresults tothe distributed
rate structured modelpresented in(1).
ACKNOWLEDGMENTS
The research of the rst author was supported in part by the Louisiana Education Quality
Support Fund under grant LEQSF(1996-99)-RD-A-36, whilethe research of the second
au-thorwassupportedinpartbytheAirForceOÆceofScienticResearchundergrantAFOSR
[1] A.S. Ackleh and K. Deng, A Monotone Approximation for the Nonautonomous
Size-Structured Population Model, Quart.Appl. Math., 57 (1999), 261-267.
[2] A.S.AcklehandK.Deng,A Monotone Approximationfora NonlinearNonautonomous
Size-Structured Population Model, Appl. Math. Comput., 108 (2000), 103-113.
[3] A.S. Ackleh and K. Ito, An Implicit Finite Dierence Scheme for the Nonlinear Size
Structure Model, Numer. Funct. Anal.Optim., 18 (1997),865-884.
[4] A.S. Ackleh, D. Marshall, B.G. Fitzpatrick and H.E. Heatherly, Survival of the Fittest
in aGeneralizedLogisticModel,Math.ModelsMethodsAppl.Sci.,9(1999),1379-1391.
[5] S. Ahmed, Extinction of Species in Nonautonomous Lotka-Volterra Systems, Proc.
Amer. Math. Soc., 127 (1999), 2905-2910.
[6] S. Ahmed and A.C. Lazer, One Species Extinction in an Autonomous Competition
Model,Proc.FirstWorldCongressNonlinearAnalysts,WalterDeGruyter,Berlin,1995.
[7] S. Ahmed and F. Montes de Oca, Extinction in Nonautonomous T-periodic
Lotka-Volterra System, Appl. Math. Comput., 90 (1998), 155-166.
[8] H.T. Banks, L.W. Botsford,F. Kappeland C. Wang, Modeling and Estimation in Size
Structured Population Models, Math. Ecology,T.G. Hallam,L.J. Gross and S.A.Levin
(Eds.), Singapore: World Scientic, 1988, pp. 521-541.
[9] H.T.BanksandB.G.Fitzpatrick,Estimationof GrowthRateDistributioninSize
Struc-ture Population Models,Quart. Appl. Math., 49 (1991), 215-235.
[10] H.T.Banks,B.G.Fitzpatrick,L.K.Potter,andY.Zhang,EstimationofProbability
Stochas-(Eds.), Birkhauser, 1998, pp. 353-371.
[11] H.T. BanksandF. Kappel,Transformation Semigroupsand L 1
-Approximationfor Size
Structure Population Models, Semigroup Forum,38 (1989), 141-155.
[12] H.T. Banks, F. Kappel and C. Wang, Weak Solutions and Dierentiability for Size
Structured Population Models, Birkhauser Internat. Ser. Numer. Math., 100 (1991),
35-50.
[13] A. Calsinaand J.Saldana,A Model of PhysiologicallyStructured PopulationDynamics
with a Nonlinear Growth Rate, J. Math. Biol.,33 (1995), 335-364.
[14] M.G.CrandallandA. Majda,Monotone Dierence Approximationsfor Scalar
Conser-vation Laws, J. Math. Comp., 34 (1980),1-21.
[15] B.G. Fitzpatrick, Vector Valued Measure Approach for a Size Structured Population
Model, J. Math. Anal.Appl., 172 (1993),73-91.
[16] B.G.Fitzpatrick,Rate DistributionModeling forStructured HeterogeneousPopulations,
Birkhauser Internat.Ser. Numer. Math., 118 (1994),131{141.
[17] S.H. Henson and T.G. Hallam, Survival of the Fittest: Asymptotic Competitive
Ex-clusion in Structured Population and Community Models, Nonlinear World, 1 (1994),
385-402.
[18] W. Huyer, A Size Structured Population Model with Dispersion, J. Math. Anal. Appl.,
181 (1994), 716{754.
[19] K. Ito, F. Kappel and G. Peichl, A Fully Discretized Approximation Scheme for
[21] F. Montes de Oca and M.L. Zeeman, Balancing Survival and Extinction in
Nonau-tonomous Competitive Lotka-Volterra Systems, J. Math. Anal.Appl., 192 (1995),
360-370.
[22] F. Montes deOca andM.L. Zeeman,Extinction in NonautonomousCompetitive
Lotka-Volterra Systems, Proc. Amer. Math. Soc., 124 (1996), 3677-3687.
[23] J.Smoller,ShockWavesandReaction-DiusionEquations,Springer-Verlag, New York,
