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Mathematical Biosciences 164 (2000) 103±137
www.elsevier.com/locate/mbs

Analysis of a mathematical model of the e€ect of inhibitors on
the growth of tumors
Shangbin Cui a,b, Avner Friedman c,*
a

Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, People's Republic of China
Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA
Department of Mathematics, University of Minnesota, 206 Church Street, SE, Minneapolis, MN 55455-0488, USA
b

c

Received 21 August 1998; received in revised form 23 September 1999; accepted 18 November 1999

Abstract
In this paper, we study a model of tumor growth in the presence of inhibitors. The tumor is assumed to
be spherically symmetric and its boundary is an unknown function r ˆ R…t†. Within the tumor the concentration of nutrient and the concentration of inhibitor (drug) satisfy a system of reaction±di€usion
equations. The important parameters are K0 (which depends on the tumor's parameters when no inhibitors

are present), c which depends only on the speci®c properties of the inhibitor, and b which is the (normalized) external concentration of the inhibitor. In this paper, we give precise conditions under which there
exist one dormant tumor, two dormant tumors, or none. We then prove that in the ®rst case, the dormant
tumor is globally asymptotically stable, and in the second case, if the radii of the dormant tumors are
‡
ÿ
‡
ÿ
denoted by Rÿ
s ; Rs with Rs < Rs , then the smaller one is asymptotically stable, so that limt!1 R…t† ˆ Rs ,
‡
‡
provided the initial radius R0 is smaller than Rs ; if however R0 > Rs then the initial tumor in general grows
unboundedly in time. The above analysis suggests an e€ective strategy for treatment of tumors. Ó 2000
Elsevier Science Inc. All rights reserved.
Keywords: Tumors; Inhibitors; Parabolic equations; Free boundary problems

1. The model
In this paper, we study a model recently proposed by Byrne and Chaplain [1] for the growth of
tumor in the presence of inhibitors. The tumors consists of life cells (non-necrotic tumor) and
receives blood supply through a developed network of capillary vessels (vascularized tumor). The

*

Corresponding author. Tel.: +1-612 625 3377; fax: +1-612 624 2333.
E-mail address: friedman@math.umn.edu (A. Friedman).

0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 9 9 ) 0 0 0 6 3 - 2

104

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

blood supply provides the tumor with nutrients as well as inhibitors. Inhibitors may also di€use
into the tumor from neighboring tissues. The inhibitors may develop from the immune system of
healthy cells, as well as from drugs administered for therapy.
The paper develops mathematical techniques for rigorous analysis of transient and stationary
solutions to such models. In the particular model under study they allow us to con®rm, but also
signi®cantly extend, the results obtained in [1] through numerical studies.
As in other models developed over the last 30 years (see, e.g., [2±6] and the references cited

therein), Byrne and Chaplain represent the tumor's evolution in the form of a free-boundary
problem whereby its growth is determined by the levels of di€using nutrient and inhibitor concentrations. In contrast with previous ones, however, their model departs from in vitro growth
scenarios by taking into account the possible blood±tissue nutrient transfer that occurs in vivo
through angiogenesis as described and modelled in [7,8, Ch. 5]. (Angiogenesis is a process by
which tumors induce blood vessels to sprout capillary tips which migrate toward, and penetrate
into, the tumor, thus providing it with circulating blood supply.) Further, the model includes a
well-motivated cell-loss mechanism, apoptosis, which implies the existence of dormant (stationary)
non-necrotic tumor states.
Following [1], we shall assume the tumor to be spherically symmetric and to occupy a region
fr < R…t†g …r ˆj x j; x ˆ …x1 ; x2 ; x3 ††
at each time t; the boundary of the tumor is given by r ˆ R…t†, an unknown function of t. Then,
after non-dimensionalization, the (dimensionless) nutrient concentration r^…r; t† will satisfy a
reaction±di€usion equation of the form
!
o^
r 1 o
r
2 o^
‡ C1 …rB ÿ r^† ÿ k0 r^ ÿ c1 b^ if r < R…t†; t > 0;
r

c ˆ 2
…1:1†
ot r or
or
where b^ is the (dimensionless) inhibitor concentration. Here the constants rB and C1 denote the
(dimensionless) nutrient concentration in the vasculature and the rate of nutrient-in-blood±tissue
transfer per units length, respectively. Thus C1 …rB ÿ r† accounts for the transfer of nutrient by
means of the vasculature, whose presence stems from angiogenesis. The term k0 r^ is the nutrient
consumption rate, c1 b^ is the inhibitor consumption rate, and c ˆ Tdiffusion =Tgrowth is the ratio of the
nutrient di€usion time scale to the tumor growth (e.g., tumor doubling) time scale. Note that,
typically, Tdiffusion  1 min (see [8, pp. 194±195]) while Tgrowth  1 day, so that c  1.
The (dimensionless) inhibitor concentration b^ satis®es a similar reaction±di€usion equation [1]
!
^
ob^ D2 o
o
b
^ ÿ c b^ if r < R…t†; t > 0;
‡ C2 …bB ÿ b†
…1:2†

ˆ
r2
c2
2
ot
or
r2 or
where the constant bB denotes the (dimensionless) inhibitor concentration in the vasculature, C2 is
inhibitor-in-blood±tissue transfer per unit length, and c2 b^ is the inhibitor consumption rate. Here
D2 is the (dimensionless) di€usion coecient of the inhibitor concentration, and c2 =D2 is the
quotient of the inhibitor di€usion time scale to the tumor growth time scale; typically D2  1 so
that c2  1. Actually C2 ˆ 0 in [1]; however if the inhibitor is partially fed through the vasculature, then C2 > 0.
Note that in (1.1) and (1.2)

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

105

1 o 2 o
r

r2 or or
is meant to denote the radial part of the Laplace operator (in 3 dimensions), and therefore one
must include the conditions
o^
r
…0; t† ˆ 0;
or

ob^
…0; t† ˆ 0:
or

…1:3†

Assuming that the mass density of cells is constant, the principle of conservation of mass
coincides with the principle of conservation of volume. A reasonable simpli®ed approach to this
principle, developed in [2], gives the relation

 Z 2p Z p Z R…t†
d 4 3

^ 2 sin h dr dh du;
pR …t† ˆ
S…^
r; b†r
…1:4†
dt 3
0
0
0
^ denotes the cell proliferation rate within the tumor. For simplicity we restrict
where S…^
r; b†
ourselves to the inhibitor-free proliferation rate [1]
~~†;
S…r† ˆ l…^
rÿr

…1:5†

~~ are positive constants. This means that the cell birth-rate is l^

where l and r
r while the death-rate
~~. Finally, the external nutrient concentration is assumed to be a constant
(apoptosis) is given by lr
 so that
 and the external inhibitor concentration is assumed to be a constant b,
r
;
r^ ˆ r

 on r ˆ R…t†:
b^ ˆ b

…1:6†

The case where inhibitors are absent was recently studied by Friedman and Reitich [9] under the
assumption that
~~ > C1 rB :
>r
r

C1 ‡ k 0

…1:7†

Extending the results obtained in [1] by perturbative and numerical studies, they proved by rigorous mathematical analysis that there exists a unique stationary solution R…t†  Rs and that this
solution is asymptotically stable for the time-dependent problem, provided c is suciently small;
the asymptotic stability was only formally proved in [1] and only so in the limit case c ˆ 0.
The present work includes the presence of inhibitor, and our interest is to study the e€ect of the
inhibitor on the tumor's growth. We shall not make the assumption (1.7), but for simplicity we
shall require that
 > C2 bB
b
C2 ‡ c 2

…1:8†

or, equivalently, that

 < c b:
C2 …bB ÿ b†

2

…1:9†

This means that the level of transfer rate of the inhibitor (by means of the vasculature) at the
tumor's boundary is smaller than the inhibitor consumption rate at the boundary. We note that in
^ does not appear at all and so (1.8) is automatically satis®ed.
[1] the term C2 …bB ÿ b†

106

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

Remark 1.1. The methods presented in this paper can be extended to the case where (1.8) is not
satis®ed, but the results will be di€erent.
It will be convenient to simplify the system (1.1)±(1.5) by introducing the following notation:
C2 ‡ c2
; k ˆ C1 ‡ k 0 ;
D2
c

C2 bB
C r^
ÿ 1 B ;
r^B ˆ rB ÿ 1

; r ˆ r
C1 C2 ‡ c2
C1 ‡ k0


C1 r^B
C2 bB



~
:
; b ˆ c1 b ÿ
r ˆ r~ ÿ
C1 ‡ k0
C2 ‡ c 2

c0 ˆ

c2
;
D2

cˆ

…1:10†

The speci®c value of l (in (1.5)) will not a€ect the results of this paper. In order to slightly simplify
the calculations, we shall take l ˆ 3. Then, introducing the normalized nutrient and inhibitor
concentrations, r and b, by


C1 rB
C2 bB
^
;
…1:11†
r ˆ r^ ÿ
; b ˆ c1 b ÿ
C1 ‡ k0
C2 ‡ c 2
the system (1.1)±(1.6) reduces to


or 1 o
2 or
c ˆ 2
ÿ kr ÿ b
r
ot r or
or


ob 1 o
2 ob
ˆ
r
ÿ cb
c
ot r2 or
or
0

if r < R…t†; t > 0;

…1:12†
…1:13†

if r < R…t†; t > 0;

or
…0; t† ˆ 0;
or

r…R…t†; t† ˆ r if t > 0;

…1:14†

ob
…0; t† ˆ 0;
or

b…R…t†; t† ˆ b if t > 0;

…1:15†

dR…t†
3
ˆ 2
dt
R …t†

Z

R…t†
0

…r…r; t† ÿ r~†r2 dr

…1:16†

if t > 0:

Note that the assumption (1.8) means that
b > 0;

…1:17†

as mentioned above, we can actually allow b to be any real number though we shall not do so in
this paper.
Finally we have initial conditions
r…r; 0† ˆ u0 …r†;
ˆ0

b…r; 0† ˆ w0 …r† if 0 6 r 6 R0 ;

and R…0† ˆ R0 ;

ou0
ow
…0† ˆ 0 …0†
or
or

where u0 and w0 are continuously di€erentiable functions.

…1:18†

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

107

We may view k and c as the consumption±transfer coecients of the nutrient and inhibitor,
respectively, and b as the normalized external concentration of the inhibitor.
In this paper, we shall prove that there exists a unique solution to the system (1.12)±(1.18) for
all t > 0. However our main interest is to establish the existence of stationary solutions and to
study their asymptotic stability with respect to the non-stationary solution. Our results will depend on just four constants
r
1 r~
c
b
b
; r0 ˆ ÿ :
; K1 ˆ
; /ˆ
…1:19†
K0 ˆ
3 r
k
…c ÿ k†
r
k
We shall always assume that
r; 0†;
min…
r; r0 † 6 u0 …r† 6 max…

0 6 w0 …r† 6 b for 0 6 r 6 R0 :

…1:20†

For simplicity we always assume that
c 6ˆ k;

…1:21†

so that K1 is well-de®ned, and K1 6ˆ 0; the case c ˆ k is brie¯y considered in Remark 6.3.
We ®rst show that if r~ < min…
r; r0 †, then R…t† ! 1 as t ! 1, whereas if r~ > max…
r; 0†, then
R…t† ! 0 as t ! 1. The function


g coth g ÿ 1
p…g† ˆ
f …g† ˆ …1 ÿ K1 †p…g† ‡ K1 p…/g†
g2
will play a fundamental role in studying all the remaining cases.
We shall prove the following results:
…A1 † If 0 < K0 < 13 then there exists a unique stationary solution …rs ; bs ; Rs † which is asymptotically stable if r P 0 and unstable if r < 0, with respect to the time-dependent solutions of
(1.12)±(1.18).
…A2 † If ÿ…1=…/ ‡ 1†† 6 …/ ÿ 1†K1 6 /, and K0 62 …0; 13† then no stationary solutions exist.
…A3 † If …/ ÿ 1†K1 > / then K0 ˆ ming>0 f …g† < 0, and when K0 P 13; or K0 6 K0 , no stationary
ÿ
ÿ
solutions exist. However when K0 < K0 < 0 then there exist two stationary solutions, …rÿ
s ; bs ; Rs †
‡
‡
‡
ÿ
‡
ÿ
‡
and …rs ; bs ; Rs †, with Rs < Rs ; Rs is asymptotically stable whereas Rs is unstable.

…A4 † If …/ ÿ 1†K1 < ÿ…1=…/ ‡ 1†† then K
0 ˆ maxg>0 f …g† > 0, when K0 6 0, or K0 P K0 , no

1
stationary solution exist. However, when 3 < K0 < K0 then there exist two stationary solutions as
‡
in case …A3 †, and again Rÿ
s is asymptotically stable whereas Rs is unstable.
The above results are proved under the assumption that c and c0 are suciently small. We shall
also prove that when no stationary solutions exist then there are initial data for which
lim R…t† ˆ 1:

t!1

The mathematical results of this paper have implications for the treatment of tumor. Consider for
example the case where r~ < 0 < r, so that with no inhibitors the tumor will grow unboundedly.
Then, by administering inhibitor (drug) with suciently large external (normalized) concentration
b we are able to contain the tumor. In fact, there are two critical parameters, b and b , with
b < b , where b does not depend on the initial conditions and b depends on the initial
conditions, such that the following holds: If b > b , then a dormant situation can be achieved

108

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137



provided the initial size of the tumor is below a certain radius R‡
s …b †, whereas if b is further

increased so that it becomes larger than b then the tumor will de®nitely be contained and it will
evolve into a dormant one with radius Rÿ
s , more details will be given in Section 7.

2. Global existence and uniqueness
Theorem 2.1. The system (1.12)±(1.18) has a unique solution …r; b; R† for all t > 0 and
0 < b…r; t† < b for 0 6 r < R…t†; t > 0;

…2:1†

min…
r; r0 † < r…r; t† < max…
r; 0†

…2:2†

for 0 6 r < R…t† t > 0;

r; r0 † ÿ r~Šg 6 R…t† 6 R0 expft‰max…
r; 0† ÿ r~Šg for t P 0;
R0 expft‰min…
‰min…
r; r0 † ÿ r~Š 6

_
R…t†
6 ‰max…
r; 0† ÿ r~Š
R…t†

for t P 0:

…2:3†
…2:4†

Proof. Local existence and uniqueness can be proved as in the case of the Stefan problem [1]. If we
can prove the a priori bounds (2.3) and (2.4), then the solution can be continued for all t > 0. On
the other hand (2.4), and consequently also (2.3), follows from (2.2) and (1.16). Thus it remains to
prove (2.2) and (2.1). These assertions follow by the maximum principle. Indeed, (2.1) is rather
immediate; as for (2.2), by comparison we have that r1 …r; t† 6 r…r; t† 6 r2 …r; t†, where


or1
1 o
2 or1

r
…2:5†
ˆ 2
c
ÿ kr1 ÿ b;
r or
ot
or


or2
1 o
2 or2
r
ˆ 2
c
ÿ cr2
r or
ot
or
for 0 < r < R…t†; t > 0 and
r1 …R…t†; t† ˆ r2 …R…t†; t† ˆ r if t > 0;
r1 …r; 0†  min…
r; r0 †;

r2 …r; 0†  max…
r; 0† if 0 6 r < R…0†:

By comparison
r; 0†
r2 …r; t† 6 max…

and also (using the relation kr0 ‡ b ˆ 0)
r1 …r; t† P min…
r; r0 †
so that (2.2) holds.
From (2.3) we get



…2:6†

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

109

Corollary 2.2.
(i) If max…
r; 0† < r~ then limt!1 R…t† ˆ 0;
(ii) If min…
r; r0 † > r~ then limt!1 R…t† ˆ 1.
Therefore in studying the asymptotic behavior of R…t† we shall concentrate just on the case
where
min…
r; r0 † 6 r~ 6 max…
r; 0†:

…2:7†

Henceforth it will be assumed that (2.7) is satis®ed.

3. Steady-state solutions
The steady-state solution, if existing, is determined by the system


1 o
2 ors
r
ÿ krs ÿ bs ˆ 0; 0 < r < Rs ;
r2 or
or


1 o
2 obs
r
ÿ cbs ˆ 0; 0 < r < Rs ;
r2 or
or
ors
…0† ˆ 0;
or

rs …Rs † ˆ r;

obs

…0† ˆ 0; bs …Rs † ˆ b;
or
and the equation
Z Rs
3
r…r†r2 dr ˆ r~Rs :
R2s 0

…3:1†
…3:2†
…3:3†
…3:4†

…3:5†

For a given Rs > 0, the solution of (3.1)±(3.4) is
p
 s
sinh… cr†
bR
bs ˆ
;
p
sinh… cRs †
r

p
p
sinh… cr†
sinh… kr†
rRs
rRs
p
‡ K1
:
rs ˆ …1 ÿ K1 †
p
r
sinh… cRs †
r
sinh… kRs †

Substituting (3.6) and (3.7) into (3.5) we obtain
p
p
p
p
cRs coth… cRs † ÿ 1 1
kRs coth… kRs † ÿ 1
…1 ÿ K1 †
ˆ r~Rs ;
‡ K1 r
r
cRs
3
kRs
this equation was derived in [1].

…3:6†
…3:7†

…3:8†

110

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

It will be convenient to introduce the variable
p
gs ˆ kRs ;

…3:9†

and the functions

g coth g ÿ 1
;
g2
f …g† ˆ …1 ÿ K1 †p…g† ‡ K1 p…/g†:

…3:10†

p…g† ˆ

…3:11†

Dividing (3.8) by rRs , we obtain the relation
f …gs † ˆ K0 :

…3:12†

Thus we have proved:

Lemma 3.1. The system (3.1)±(3.5) has a solution if and only if Eq. (3.12) has a positive solution gs ,
and, in that case, the solution Rs ; rs ; bs is given by (3.9), (3.7) and (3.6).
The remaining of this section is devoted to determining how many solutions Eq. (3.12) admits.
We ®rst need two lemmas.
Lemma 3.2.
(i) p0 …g† < 0 for all g > 0;
(ii) limg!0 p…g† ˆ 13; limg!1 p…g† ˆ 0; limg!1 gp…g† ˆ 1;
(iii) 0 < p…g† < 13; 0 < gp…g† < 1 for all g > 0.
The ®rst part is proved in [9], and the other parts are rather immediate.
Lemma 3.3. The function
k…g† ˆ

gp00 …g†
p0 …g†

is strictly monotone decreasing and in fact, k 0 …g† < 0 for all g > 0.
Proof. By direct computation
2…sinh3 g ÿ g3 cosh g†
k…g† ˆ
ÿ2
‰g2 ‡ g cosh g sinh g ÿ 2 sinh2 gŠ sinh g
and
k 0 …g† ˆ ÿ
where

2g…g†
‰g2 ‡ g cosh g sinh g ÿ 2 sinh2 gŠ2 sinh2 g

;

g…g† ˆ sinh5 g cosh g ‡ g sinh4 g ‡ 6g3 sinh2 g cosh2 g ‡ 2g3 sinh2 g ‡ g4 sinh g cosh g ÿ 8g2
 sinh3 g cosh g ÿ 2g4 sinh g cosh3 g ÿ g5 :

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

111

Thus it remains to show that g…g† > 0 for all g > 0.
Writing




1
1
1
1 2 3 3 1 4 ÿ2g
5 1
1 3 ÿ4g
ÿ6g
2
ÿ
g…g†e ˆ
ÿ g‡ g ÿ g ‡ g e ‡
ÿ g‡g ‡ g e
64
16 16
2
8
8
64 4
2




3
7
5 1
1
g ÿ g3 ÿ g5 eÿ6g ‡ ÿ ÿ g ÿ g2 ‡ g3 eÿ8g
‡
8
4
64 4
2


1
1
1
3
1
1
‡ g ‡ g2 ‡ g3 ‡ g4 eÿ10g ÿ eÿ12g ;
‡
16 16
2
8
8
64
one can easily verify that, for g P 2,


1
1
1
1 2 3 3 1 4 ÿ2g
ÿ6g
ÿ
g…g†e P
ÿ g ‡ g ÿ g ‡ g e ÿ g5 eÿ6g  h…g†:
64
16 16
2
8
8

But, as easily seen, the function g5 eÿ6g is monotone decreasing if g P 5=6 and the function


1
1
1 2 3 3 1 4 ÿ2g
ÿ g‡ g ÿ g ‡ g e
16 16
2
8
8

is monotone decreasing for g P 2. Since also h…2:25† ˆ 1:57163  10ÿ4 > 0, it follows that g…g† > 0
if g P 2:25.
For g < 2:25 we write
7
7
X
g2k
sinh g X
g2k
g16
<
<
‡
cosh…2:25†;
g
…2k ‡ 1†!
…2k ‡ 1†! 17!
kˆ0
kˆ0
7
7
X
X
g2k
g2k
g16
< cosh g <
‡
cosh…2:25†:
…2k†!
…2k†! 17!
kˆ0
kˆ0

Applying the lower bounds to the ®ve positive terms in g…g†=g5 and the upper bounds to the
®rst two negative terms in g…g†, we obtain after some calculations,


g…g†
34 2 82 4 292 6
4058 8
17 692 10
62 492
198 944
12
14
> 11 ‡ g ‡ g ‡
g ‡
g ‡
g ‡
g ‡
g
g5
3
15
189
14 175
467 775
16 372 125
63 8512 875

34
82
292 6
4058 8
1156 10
2132 12
g ‡
g ‡
g ‡
g
ÿ 11 ‡ g2 ‡ g4 ‡
3
15
189
14 175
31 185
606 375

2944
14
16
g ‡ `…g†g ;
‡
11 609 325
where `…g† is a polynomial of degree 48 with positive coecients, so that
`…g† < `…2:25† ˆ 1:83644  10ÿ5

if 0 < g < 2:25:

For such g we then get, by computing the coecients of each power of g in the above expression,
g…g†
32
64
2848
>
‡
g2 ‡
g4 ÿ 1:83644  10ÿ5 g6
15
g
42 525 212 625
49 116 375
ˆ 10ÿ5 ‰75:2499 ‡ 30:0999g2 ‡ …5:79847 ÿ 1:83644g2 †g4 Š

112

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

and the expression in parenthesis is > ÿ3:4985 if 0 < g < 2:25. Hence
g…g† 5
10 P 75:2499 ‡ …30:0999 ÿ 3:4985g2 †g2 > 0
g15
if 0 < g < 2:25 since the expression in parenthesis is positive. 
Corollary 3.4.
If / > 1 …0 < / < 1† then
d p0 …/g†
0† for all g > 0:

…3:13†

Proof. Clearly
d p0 …/g† p0 …g†
/p00 …/g† ÿ p0 …/g†
p00 …g†
ˆ
ˆ
dg p0 …g†
…p0 …g††2



/gp00 …/g† gp00 …g†
ÿ 0
p0 …/g†
p …g†



p0 …/g†
:
gp0 …g†

Since the last factor is positive by Lemma 3.2, the assertion follows from Lemma 3.3.



In the next theorem we analyze the behavior of the function f …g† de®ned in (3.11), note that
1
f …0† ˆ ;
3

lim f …g† ˆ 0:

…3:14†

g!1

Theorem 3.5.
1
(i) If ÿ /‡1
< …/ ÿ 1†K1 6 / then

f 0 …g† < 0 for all g > 0:
(ii) If …/ ÿ 1†K1 > / then there is a unique g0 such that
f 0 …g† < 0

for 0 < g < g0 ;

and, by (3.14),

f 0 …g† > 0

for g > g0

…3:15†
…3:16†

K0 ˆ f …g0 † ˆ minf …g† < 0:
g>0

1
then there exists a unique g0 > 0, such that
(iii) If …/ ÿ 1†K1 < ÿ /‡1

f 0 …g† > 0

for 0 < g < g0 ;

and, by (3.14),

f 0 …g† < 0

for g > g0

…3:17†

1
K
0 ˆ f …g0 † ˆ maxf …g† > :
g>0
3
Proof. Note that


p0 …/g†
f …g† ˆ p …g† …1 ÿ K1 † ‡ /K1 0
:
p …g†
0

0

…3:18†

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

113

Also, by simple calculation,
p0 …/g†
ˆ /;
g!0 p0 …g†

p0 …/g†
1
ˆ 2:
g!1 p 0 …g†
/

lim

…3:19†

lim

Consider ®rst the case 0 6 …/ ÿ 1†K1 6 /. Then K1 P 0 (respectively, K1 < 0) implies / P 1
(respectively, / < 1), so that, by Corollary 3.4, p0 …/g†=p0 …g† is strictly monotone decreasing (respectively, increasing). Consequently, by (3.19),


p0 …/g†
1
1
respectively; < 2
> 2
p0 …g†
/
/
so that the expression in brackets in (3.18) is > ‰…1 ÿ K1 † ‡ /K1 =/2 Š. Since p0 …g† < 0, we get from
(3.18),


/K1
/ ÿ …/ ÿ 1†K1
0
0
60
f …g† < p …g† …1 ÿ K1 † ‡ 2 ˆ p0 …g†
/
/

and (3.15) follows. The remaining case of (i), namely, ÿ1=…/ ‡ 1† 6 …/ ÿ 1†K1 < 0, can be proved
similarly.
To prove (ii), note that …/ ÿ 1†K1 > / implies that
1
K1 ÿ 1
< / if K1 > 0 … () / > 1†;
<
2
/K1
/
/<

K1 ÿ 1
1
< 2
/K1
/

if K1 < 0 … () / < 1†:

…3:20†
…3:21†

From Lemma 3.2(i) and (3.18)±(3.20) we see that if K1 > 0 then
f 0 …g†  p0 …g†‰…1 ÿ K1 † ‡ K1 /2 Š < 0
f 0 …g†  p0 …g†‰…1 ÿ K1 † ‡

K1
Š>0
/

if g is near 0;

if g is near 1:

Similarly if K1 < 0 then (using (3.21))
f 0 …g† < 0

if

g is near 0;

f 0 …g† > 0

if

g is near 1:

Hence in both cases there exists a point g0 such that f 0 …g0 † ˆ 0. Since p0 …g† never vanishes whereas
the expression in brackets in (3.18) has everywhere negative derivative if / > 1 and everywhere
positive derivative if / < 1 (by Corollary 3.4), we deduce that f 0 …g† has a unique zero and (ii)
readily follows. Finally, (iii) is established by the same argument as (ii). 
From Lemma 3.1 and Theorem 3.5 we obtain the following results concerning the existence of
stationary solution of (3.1)±(3.5):
Theorem 3.6.
(i) If ÿ1=…/ ‡ 1† < …/ ÿ 1†K1 6 /, then for 0 < K0 < 13 there exists a unique stationary solution
…rs ; bs ; Rs †, while for K0 62 …0; 13† there are no stationary solutions.

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S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

(ii) If …/ ÿ 1†K1 > / then for 0 6 K0 < 13 there exists a unique stationary solution (rs ; bs ; Rs †; for
ÿ
ÿ
K0 < K0 < 0…K0 ˆ ming>0 f …g† < 0† there are two stationary solutions …rÿ
s ; bs ; Rs † and
‡

 1
‡
‡
ÿ
‡
…rs ; bs ; Rs † with Rs < Rs ; for K0 ˆ K0 the two solutions coincide, and for K0 62 ‰K0 ; 3† there
are no stationary solutions.
(iii) If …/ ÿ 1†K1 < ÿ1=…/ ‡ 1† then for 0 < K0 6 13 there exists a unique stationary solution

1
…rs ; bs ; Rs †; for 13 < K0 < K
0 …K0 ˆ maxg>0 f …g† > 3) there are two stationary solutions
ÿ
‡
ÿ
ÿ
‡
‡
ÿ
‡
…rs ; bs ; Rs † and …rs ; bs ; Rs † with Rs < Rs ; for K0 ˆ K
0 the two solutions coincide, and for K0
Š
there
are
no
stationary
solutions.
62 …0; K
0
We shall be interested in the asymptotic behavior of the solution of (1.12)±(1.18). In order to
gain some insight about what to expect, we brie¯y consider the limiting case where c ˆ c0 ˆ 0.
Then, for each t > 0,
p

sinh… cr†
bR…t†
;
…3:22†
b…r; t† ˆ
p
r
sinh… cR…t††

Fig. 1. 0 6 …/ ÿ 1†K1 6 /.

Fig. 2. …/ ÿ 1†K1 > /.

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

p
p
sinh… cr†
sinh… kr†
rR…t†
rR…t†
p
‡ K1
r…r; t† ˆ …1 ÿ K1 †
:
p
r
sinh… cR…t††
r
sinh… kR…t††
p
Substituting the last expression into (1.16) and setting g…t† ˆ kR…t†, we get
dg
ˆ rg‰f …g† ÿ K0 Š:
dt

115

…3:23†

…3:24†

By applying Theorems 3.5 and 3.6 we easily obtain the following conclusions:
(R1 ) If 0 6 …/ ÿ 1†K1 6 / (which implies r P 0), then for 0 < K0 < 13; limt!1 R…t† ˆ Rs for any
initial radius R0 , that is, …rs ; bs ; Rs † is globally asymptotically stable (Fig. 1); for K0 P 13 we have
limt!1 R…t† ˆ 0, and for K0 6 0 we have limt!1 R…t† ˆ 1.
(R2 ) If …/ ÿ 1†K1 > / (which implies r > 0) then for 0 6 K0 < 13 the solution …rs ; bs ; Rs † is again
ÿ
ÿ
asymptotically stable (Fig. 2(a)); for K0 < K0 < 0 the solution …rÿ
s ; bs ; Rs † is locally asymptoti‡
‡
cally stable with the attraction region of Rÿ
s being …0; Rs †, and for any R0 > Rs we have
1
limt!1 R…t† ˆ 1 (Fig. 2(b)). For K0 P 3 we have limt!1 R…t† ˆ 0, and for K0 < K0 we have
limt!1 R…t† ˆ 1.
(R3 ) If ÿ1=…/ ‡ 1† 6 …/ ÿ 1†K1 < 0 (which implies r < 0), then for 0 < K0 < 13 we have
limt!1 R…t† ˆ 0 if R0 < Rs ; limt!1 R…t† ˆ 1 if R0 > Rs , hence …rs ; bs ; Rs † is unstable (Fig. 3); for
K0 P 13 we have limt!1 R…t† ˆ 1 and for K0 6 0 we have limt!1 R…t† ˆ 0.

1
Fig. 3. ÿ /‡1
6 …/ ÿ 1†K1 < 0.

1
.
Fig. 4. …/ ÿ 1†K1 < ÿ /‡1

116

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

(R4 ) If …/ ÿ 1†K1 < ÿ1=…/ ‡ 1† (which implies r < 0) then, for 0 < K0 6 13 we have the same
ÿ
ÿ
ÿ
conclusion as in (R3 ) (Fig. 4(a)); for 13 < K0 < K
0 the solution …rs ; bs ; Rs † is locally asymptotiÿ
‡
cally stable with the domain of attraction of Rs being …0; Rs †, and for R0 > R‡
s we have
limt!1 R…t† ˆ 1 (Fig. 4(b)). For K0 > K
we
have
lim
R…t†
ˆ
1
and
for
K
6
0 we have
t!1
0
0
limt!1 R…t† ˆ 0.
The rest of the paper is devoted to the extension of the above results to the non-degenerate case
where c > 0; c0 > 0.

4. A non-extinction theorem
In cases (R1 ), (R2 ), and (R4 (b)) (see Figs. 1, 2 and 4(b)) we have limt!1 R…t† > 0, so that R…t†
does not become extinct as t ! 1. In this section, we shall extend this result to non-vanishing
c; c0 , proving that lim inf t!1 R…t† > 0. Note that in cases (R1 ) and (R2 ), r > 0; r~ < r, whereas in
case (R4 (b)) r~ < r < 0. Since we always assume that (2.7) holds (recall Corollary 2.2), in all these
three cases we have r0 < r~ < r.
We now state the non-extinction result:
Theorem 4.1. If r0 < r~ < r then for any e > 0 there exist positive constants d0 ˆ d0 …e† and
T0 ˆ T0 …e; R0 † such that, if 0 < c 6 e, then
R…t† P d0

for all t P T0 :

…4:1†

Remark 4.1. It is important to note that d0 is independent of R0 .
Proof. The proof is an extension and some simpli®cation of the corresponding proof for the inhibitor-free case [9]. We choose r < 0 such that
kr ‡ b 6 0

and r2  r ÿ r > 0

and introduce the function
v…r; t† ˆ v…r; t† ‡ r  r2

R…t†
sinh…Mr†
‡ r :
sinh…MR…t††
r

…4:2†

Note that
vt ˆ ÿ
v

_
R…t†
‰MR…t† coth…MR…t†† ÿ 1Š
R…t†

and, since kr ‡ b 6 0,

(

)
_
R…t†
‰MR…t† coth…MR…t†† ÿ 1Š ‡ k ÿ M 2 :
cvt ÿ Dv ‡ kv ‡ b 6 v ÿ c
R…t†

…4:3†

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

117

Recalling that
ÿ

_
R…t†
6 r~ ÿ min…
r; r0 † ˆ r~ ÿ r0
R…t†

and using the estimate
1
0 < n coth n ÿ 1 < n2 8 n > 0;
3
we get, for c 6 e,


1
cvt ÿ Dv ‡ kv ‡ b 6 v e…~
r ÿ r0 †M 2 R2 …t† ‡ k ÿ M 2 :
3

…4:4†

…4:5†

We shall now use the function v to show that if R1 is small enough then, for any T1 > 0, the
inequality
R…t† 6 R1

cannot hold for all t P T1 ;

here R1 is independent of the initial radius R0 .
Indeed, otherwise we deduce from (4.5), with M 2 ˆ k ‡ 1, that
cvt ÿ Dv ‡ kv ‡ b 6 0 if 0 6 r < R…t†; t > T1 :

…4:6†

Since further v ˆ r on r ˆ R…t†, the function
v…r; t† ÿ Aeÿk=c…tÿT1 †

…A ˆ r2 ‡ j r j†

is a subsolution for t P T1 and, by the maximum principle,
r…r; t† P v…r; t† ÿ Aeÿ…k=c†…tÿT1 † :

_
Using this in (1.16) we ®nd (as in [9]) that R…t†
> 0 if t ÿ T1 is suciently large, and this leads to a
contradiction (as in [9]).
Having proved (4.6) we conclude that for any T1 > 0 there exists a T2 …> T1 † such that
R…T2 † > R1 .
We choose positive constants d1 and M such that
…4:7†

d1 6 R1
and
1
e…~
r ÿ r0 †d21 < 1;
3

M2 >

1ÿ

k
:
ÿ r0 †d21

1
e…~
r
3

…4:8†

Note that
1
n coth n ˆ 1 ‡ n2 ‡ O…n3 †
3

if n ! 0

so that, for some small positive constant d2 ,
1
n coth n > 1 ‡ …1 ÿ j†n2
3

if 0 < n 6 d2 ;

…4:9†

118

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

where
jˆ

r ÿ r~
;
2
r2 ‡ j r j

…4:10†

0 < j < 1:

We shall now prove the assertion (4.1) with


d2
…~
rÿr0 †=k
;
d0 ˆ min
; d1 j
M

…4:11†

note that d0 < d1 6 R1 .
Indeed, suppose (4.1) is not true. Then for any T2 > 0 there exists t0 > T2 such that R…t0 † < d0 .
By the assertion we proved previously, we may choose T2 such that R…T2 † > R1 . It then follows
that there exists t1 2 …T2 ; t0 † such that R…t1 † ˆ d1 and R…t† 6 d1 for all t1 6 t < t0 . We claim that
r…r; t† P v…r; t† ÿ Aeÿ…k=c†…tÿt1 †

for 0 6 r 6 R…t†; t1 6 t 6 t0 ;

…4:12†

where v is the function de®ned in (4.2) and A ˆ r2 ‡ j r j. Indeed, using (4.5) and the second
inequality in (4.8) we see that the right-hand side of (4.12) is a subsolution to r, so that (4.12)
follows by comparison.
_ 2 † 6 0. Since
Now let t2 2 …t1 ; t0 † be such that R…t2 † ˆ d0 and R…t† < d0 for all t 2 …t2 ; t0 †; then R…t
_R…t†=R…t† P ÿ …~
r ÿ r0 †, we have
R…t† P R…t1 †eÿ…~rÿr0 †…tÿt1 †

if t P t1

and consequently
 1=…~rÿr0 †
d1
P t1 ‡ log jÿ1=k
t2 > t1 ‡ log
d0

…4:13†

by (4.11). Substituting (4.12) into (1.16) we get
Z R…t†
3
_
R…t†
ˆ
…r…r; t† ÿ r~†r2 dr
2
…R…t†† 0
Z R…t†
3
r2
r sinh…Mr† dr ÿ …~
r ÿ r †R…t† ÿ AR…t†eÿk…tÿt1 †
P
R…t† sinh…MR…t†† 0
3
r2
ˆ 2
‰MR…t† coth…MR…t†† ÿ 1Š ÿ …~
r ÿ r †R…t† ÿ AR…t†eÿk…tÿt1 † :
M R…t†
Since R…t† 6 d0 for t2 6 t 6 t0 , we have MR…t† 6 Md0 < d2 by (4.11). Using also (4.9), (4.11) and
(4.13), we conclude that
_
r ÿ r †R…t† ÿ jAR…t† if t2 6 t 6 t0 :
R…t†
> …1 ÿ j†…
r ÿ r †R…t† ÿ …~
But since the right-hand side vanishes at t ˆ t2 , by the choice j in (4.10), this is a contradiction to
R_ 2 …t0 † 6 0. 

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

119

5. Asymptotic stability
In this section we shall extend the stability results of …R1 †±…R4 † to the case where c and c0 are
non-zero, but small. Thus we shall show that in cases …R1 †; …R2 †; and …R4 † (see Figs. 1±4) the
ÿ
ÿ
solutions …rs ; bs ; Rs † and respectively, …rÿ
s ; bs ; Rs †, are globally asymptotically stable and, respectively, locally asymptotically stable with domain of attraction …0; R‡
s ÿ d† for any d > 0,
provided c and c0 are positive and suciently small.
We ®rst need to establish an upper bound on the solution, and this requires the following
lemma:
Lemma 5.1. Let …r…r; t†; b…r; t†; R…t†† be the solution of (1.12)±(1.18) for 0 6 t; T0 …0 < T0 6 1†
and set
p

sinh… cr†
bR…t†
;
…5:1†
w…r; t† ˆ
p
r
sinh… cR…t††
v…r; t† ˆ …1 ÿ K1 †

p
p
sinh… cr†
sinh… cr†
rR…t†
rR…t†
‡ K1
:
p
p
r
r
sinh… cR…t††
sinh… cR…t††

…5:2†

_
Assume that j R…t†
j 6 L for 0 6 t < T0 and
j u0 …r† ÿ v…r; 0† j 6 M;

j u0 …r† ÿ w…r; 0† j 6 M

for 0 6 r 6 R0 :

…5:3†

 and a (depending only on k; c) such
Then there exist positive constants C (depending only on k; c; r; b†
that
0

j b…r; t† ÿ w…r; t† j 6 C…Lc0 ‡ Meÿct=c †;
00

j r…r; t† ÿ v…r; t† j 6 C…Lc00 ‡ Meÿat=c †

for 0 6 r 6 R…t†; 0 6 t < T0 .

…5:4†
…c00 ˆ c ‡ c0 †

…5:5†

Proof. As easily veri®ed
_
cR…t†p…pcR…t††
w;
c0 wt ÿ Dw ‡ cw ˆ ÿc0 R…t†

where the function p…g† is de®ned in (3.10). By Lemma 3.2
p
p
0 < cR…t†p… cR…t†† 6 c;
 we get
and since 0 < w 6 b,

p
j c0 wt ÿ Dw ‡ cw j 6 Lb cc0 :

Consequently, by comparison,
p
Lb cc0
…b…r; t† ÿ w…r; t†† 6
;
c

which yields (again by comparison) the assertion (5.4).

…5:6†

120

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

Similarly,

p
p
p
sinh… kr†
rR
_
p
cvt ÿ Dv ‡ kv ‡ w ˆ ÿ…1 ÿ K1 †cR
kRp… kR†

ÿ K1 cR_
cRp… cR†
r
sinh… kR†
p
p
sinh… cr†
sinh… cr†
R
rR
‡ K1 …k ÿ c†
‡ w;
r

p
p
sinh… cR†
r
r
sinh… cR†
and since K1 …k ÿ c†
r ‡ b ˆ 0, we obtain
p
p
j vt ÿ Dv ‡ kv ‡ w j 6 LC j r j c …C ˆ …1‡ j K1 j† k‡ j K1 j c†:
On the other hand, by (5.4),

0
Lbc
0
j crt ÿ Dr ‡ kr ‡ w j 6 p ‡ Meÿct=c
c
so that

0

0

j c…r ÿ v†t ÿ D…r ÿ v† ‡ k…r ÿ v† j 6 CLc00 ‡ Meÿct=c 6 CLc00 ‡ Meÿ…kct†=…kc ‡2cc† :
This implies that
…cV ÿ DV ‡ kV † 6 CLc00 ;

where

0

V ˆ r ÿ v ÿ M1 eÿ…kct†=…kc ‡2cc†

and M1 ˆ M max…1; 2=k†. By comparison (as in the derivation of (5.6)) we then get
CLc00
V 6
;
k
and (5.5) follows. 
We next establish, in all cases (R1 )±(R4 ), a uniform bound on R…t† which is compatible with
Figs. 1±4.
‡
Theorem 5.2. Let …r…r; t†; b…r; t†; R…t†† be the solution of (1.12)±(1.18) and let Rs ; Rÿ
s ; Rs be as in
Theorem 3.6. Let K and d be positive constants such that one of the following three conditions holds:
(i) K P d ‡ maxfR0 ; Rs g if either 0 6 …/ ÿ 1†K1 6 /; 0 < K0 < 13 or …/ ÿ 1†K1 > /; 0 6 K0 < 13;

‡
(ii) maxfR0 ; Rÿ
s g ‡ d 6 K 6 Rs ÿ d if either …/ ÿ 1†K1 > /; K0 < K0 < 0 or …/ ÿ 1†K1 < ÿ1=

1
…/ ‡ 1†; 3 < K0 < K0 ;
(iii) R0 ‡ d < K < Rs ÿ d if either ÿ1=…/ ‡ 1† 6 …/ ÿ 1†K1 < 0; 0 < K0 < 13 or …/ ÿ 1†K1 <
ÿ1=…/ ‡ 1†; 0 < K0 6 13.
 r~; d; K such
Then there exists a positive constant e0 depending only on M (in (5.3)) and on k; c; r; b;
0
that if c ‡ c 6 e0 then

R…t† 6 K

for all t P 0:

Proof. The assumptions on /; K1 imply that
A ˆ maxf
r; 0g ÿ r~ > 0;

…5:7†

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

121

and that
p
either r > 0 and f … kK† 6 K0 ÿ d~

p
~
or r < 0 and f … kK† P K0 ‡ d;

…5:8†

where d~ depends on the same constants on which e0 is asserted to depend on.
Suppose that (5.7) is not true. Then, since K > R0 , there exists a t0 > 0 such that (5.7) holds for
all t < t0 and R…t0 † ˆ K. Consequently,
_ 0† P 0
R…t

…5:9†

and, by Theorem 2.1,
t0 P

1
K
1
K
log > log
ˆ t > 0:
A
R0 A
K ÿd

…5:10†

By Theorem 2.1 we also have
_
j R…t†
j 6L

if

…5:11†

0 < t 6 t0 ;

where
L ˆ maxfj r ÿ r~ j K; j r~ ÿ r0 j Kg:

…5:12†

It follows that t and L depend on the same constants upon which e0 is asserted to depend on.
We now apply Lemma 5.1 to obtain the inequalities
00

00

v…r; t† ÿ C…c00 ‡ eÿat=c † 6 r…r; t† 6 v…r; t† ‡ C…c00 ‡ eÿat=c †

…5:13†

v is the function de®ned in (5.2). Substituting the upper bound on
for 0 6 r 6 R…t†; 0 < t 6 t0 , wherep
r into (1.16) and setting g…t† ˆ kR…t†, we get
dg…t†
00
6 rg…t†…f …g…t†† ÿ K0 † ‡ Cg…t†…c00 ‡ eÿat=c †:
dt

…5:14†

_ 0 † < 0 if c00 6 e0 and e0 is small enough, which is
Taking t ˆ t0 and using (5.8) and (5.10) we get g…t
a contradiction to (5.9). 
We now state the main result of this section which asserts, for c; c0 small enough, the same
stability results that hold in the case c ˆ c0 ˆ 0 (see Figs. 1±4).
‡
Theorem 5.3. Let …r…r; t†; b…r; t†; R…t†† and Rs ; Rÿ
s ; Rs be as in Theorem 5.2. Suppose the initial
radius R0 satisfies, for some small d > 0, one of the following three conditions:
(i) 0 < R0 6 1=d if either 0 6 …/ ÿ 1†K1 6 /; 0 < K0 < 13 or …/ ÿ 1†K1 > /; 0 6 K0 < 13;
if either …/ ÿ 1†K1 > /; K0 < K0 < 0 or …/ ÿ 1† < ÿ1=…/ ‡ 1†;
(ii) 0 < R0 6 R‡
s ÿd
1
< K0 < K
0 ;
3
(iii) 0 < R0 6 Rs ÿ d if either ÿ1=…/ ‡ 1† 6 …/ ÿ 1†K1 < 0; 0 < K0 < 13 or …/ ÿ 1†K1 < ÿ1=
…/ ‡ 1†; 0 < K0 6 13.
Then there exists a positive constant e0 depending only on M (in (5.3)), k; c; r; r~; b and d such that if
c ‡ c0 6 e0 then

122

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

8
< Rs
lim R…t† ˆ Rÿ
t!1
: s
0

in case …i†;
in case …ii†;
in case …iii†;

…5:15†

moreover, the convergent is exponentially fast.
We shall ®rst prove Theorem 5.3 in case (iii).
Proof of (5.15) in case (iii). Take a constant K such that it satis®es the conditions in Theorem 5.2
(with d replaced by d=2). By Theorem 5.2, R…t† 6 K for all t > 0 provided c ‡ c0 6 e0 ; e0 suciently
small, and then (5.11)±(5.14) follow as before. Recall that in case (iii)
p
r 6 r~ < 0 and f … kR0 † > K0 :
Fix t0 suciently small (independently of e0 ) so that, by Theorem 2.1,
R…t† 6 R0 eÿ~rt < K

for

0 6 t 6 t0 :

Then
f …g…t†† ÿ K0 P b0 > 0

…b0 constant†

…5:16†

for all 0 6 t 6 t0 . On the other hand from (5.14) we get
_ 6 rg…t†‰f …g…t†† ÿ K0 Š ‡ Ce0 g…t†
g…t†

if t P t0 :

_ p<
_ < 0 in
Choosing e0 0 there exist positive
 d and a0 , such that the folconstants C,b and (sufficiently small) e0 depending only on k; c; r; r~; b;
00
0
lowing is true if c ˆ c ‡ c 6 e0 : For any 0 < a 6 a0 , if the inequalities
j R…t† ÿ Rs j 6 a;

_
j R…t†
j 6 a;

j r…r; t† ÿ rs …r† j 6 a;

j b…r; t† ÿ bs …r† j 6 a

…5:18†

hold for all 0 6 r 6 R…t†; t P 0 then also the inequalities
j R…t† ÿ Rs j 6 Ca…c00 ‡ eÿbt †;

_
j R…t†
j 6 Ca…c00 ‡ eÿbt †;

j r…r; t† ÿ rs …r† j 6 Ca…c00 ‡ eÿbt †;

j b…r; t† ÿ bs …r† j 6 Ca…c00 ‡ eÿbt †

holds for all 0 6 r 6 R…t†; t P 0. Here rs …r†; bs …r† are defined by (3.6), (3.7) for all r > 0.

…5:19†

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

123

Proof. From Lemma 5.1 we get
00

j r…r; t† ÿ v…r; t† j 6 Ca…c00 ‡ eÿat=c †

8t P 0;

…5:20†

00

j b…r; t† ÿ w…r; t† j 6 Ca…c00 ‡ eÿct=c †

8t P 0:
p
Substituting (5.20) into (1.16) and setting g…t† ˆ kR…t†, we get
00

_ ÿ rg…t†‰f …g…t†† ÿ K0 Š j 6 Cag…t†…c00 ‡ eÿat=c †
j g…t†

8t P 0:

Consider ®rst the case 0 6 …/ ÿ 1†K1 6 /. Then r > 0 and
f 0 …g† < 0

for all g > 0:

…5:21†
…5:22†
…5:23†

By Theorems 4.1 and 5.2 we know that there are positive constants e0 ; T0 ; d0 and K independent
of c; c0 and a (but dependent on a0 ) such that if c00 6 e0 then
d0 6 g…t† 6 K

…5:24†

for all t P T0 :

Consequently, by the mean value theorem,

6 ÿ C0 …g…t† ÿ gs †
if g…t† P gs ;
…5:25†
rg…t†‰f …g…t†† ÿ K0 Š
if g…t† < gs ;
P ÿ C0 …g…t† ÿ gs †
p
where gs ˆ kRs and C0 is a positive constant depending only on r; d0 ; K and the coecients of
f …g†. We shall use this inequality to prove that there exist positive constants m; B independent of
c; c0 , and a such that
j g…t† ÿ gs j< Ba…c00 ‡ eÿmt †

…5:26†

for all t P 0:

It is clear that (5.26) holds for all 0 6 t 6 T0 if, for ®xed m > 0; B is chosen suciently large.
Therefore, if (5.26) is not true, then there exists a t0 > T0 such that
gs ÿ Ba…c00 ‡ eÿmt † < g…t† < gs ‡ Ba…c00 ‡ eÿmt †

for 0 6 t < t0

but not for t ˆ t0 ; for de®niteness suppose that
g…t0 † ˆ gs ‡ Ba…c00 ‡ eÿmt0 †;

…5:27†

_ 0 † ‡ mBaeÿmt0 P 0:
g…t

…5:28†

then also

On the other hand, by (5.22) and (5.25),
_ 0 † 6 ÿ C0 …g…t0 † ÿ gs † ‡ Cag…t0 †…c00 ‡ eÿat0 =e0 †:
g…t

Substituting (5.27) and (5.28) into this inequality and using the fact g…t0 † 6 K we get
ÿmBaeÿmt0 6 ÿ C0 Ba…c00 ‡ eÿmt0 † ‡ CKa…c00 ‡ eÿat0 =e0 †;

which is a contradiction if we choose m suciently small, say m 6 min…C0 =2; a=e0 †, and B correspondingly large. Having proved (5.26), the other estimates in (5.19) easily follow.
It remains to consider the case …/ ÿ 1†K1 > /. Again we have r > 0. But now (5.23) is not
valid. However, if we denote by g0 the stationary point of f …g†, then we still have
f 0 …g† < 0

for all g 2 …0; g0 †;

124

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

p
so that if R0 < R  g0 = k then the previous argument still works for the present situation. If on
the other hand R0 P R , then f …g…0†† < 0 and thus g…t† remains in the region where f …g† is
negative when t varies in a small interval ‰0; t0 Š. Using (5.22) for t P t0 we conclude that if
c00 6 e0 ; e0 is suciently small, then as long as g…t† P g0 ÿ 2d0 (for some small d0 > 0) we have
_ 6 ÿ b0 g…t†
g…t†

(for some positive constant b0 ). Thus there exists a time t P t0 such that g…t† lies in …0; g0 ÿ d0 Š at
t ˆ t . We can then proceed as in the former case to establish (5.26). 
Having proved Lemma 5.4, we can now apply it successively in [9] (over intervals ‰tn ; 1) with
increasing n) and establish (5.15) in case (i).
We shall next extend Lemma 5.4 to case (ii).
Lemma 5.5. Consider case (ii) of Theorem 5.3. Then for an arbitrary a0 there exist positive con d and a0 such that the following is true
stants C,b and (sufficiently small) e0 depending on k; c; r; r~; b;
00
0
if c ˆ c ‡ c 6 e0 : For any 0 < a < a0 , if the inequalities
_
j R…t†
j 6 a;
j R…t† ÿ Rÿ
s j 6 a;
ÿ
j r…r; t† ÿ rs …r† j 6 a; j b…r; t† ÿ bÿ
s …r† j 6 a

…5:29†

hold for all 0 6 r 6 R…t†; t P 0 then also the inequalities
00
ÿbt
_
j R…t† ÿ Rÿ
†; j R…t†
j 6 Ca…c00 ‡ eÿbt †;
s j 6 Ca…c ‡ e
00
ÿbt
00
ÿbt
j r…r; t† ÿ rÿ
†; j b…r; t† ÿ bÿ
†
s …r† j 6 Ca…c ‡ e
s …r† j 6 Ca…c ‡ e

…5:30†

hold for all 0 6 r 6 R…t†; t P 0.
Proof. The proof is similar to the proof of Lemma 5.4 for the case …/ ÿ 1†K1 > /.



Again, by using Lemma 5.5 and following a similar argument as in [9] we can derive the assertion (5.15) in case (ii).
6. Instability: unboundedness of R(t)
We shall use the notation
wl …r; t† ˆ

p
sinh… lr†
R…t†
p
r
sinh… lR…t††

for any l > 0;

where …r…r; t†; b…r; t†; R…t†† is the solution of (1.12)±(1.18). We set
 c …r; t†;
w…r; t† ˆ bw

…6:1†

v…r; t† ˆ rf…1 ÿ K1 †wk …r; t† ‡ K1 wc …r; t†g

…6:2†
…6:3†

w0 …r† ˆ w…r; 0†;

…6:4†

and
v0 …r† ˆ v…r; 0†:

 R0 g.
 R0 by A, i.e., A ˆ fk; c; r; r~; b;
We shall denote the set of parameters k; c; r; r~; b;

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

125

In this section, we consider essentially all the cases that were not covered by the stability results
of Section 5. We can divide them into four disjoint cases:
(i) 0 6 …/ ÿ 1†K1 6 /; K0 < 0; 0 < R0 < 1;

(ii) …/ ÿ 1†K1 > / and either K0 < K0 < 0; R0 > R‡
s or K0 < K0 ; 0 < R0 < 1;
1
(iii) ÿ1=…/ ‡ 1† 6 …/ ÿ 1†K1 < 0 and either K0 P 3; 0 < R0 < 1, or 0 < K < 13; R0 > Rs , and
‡
or
(iv) …/ ÿ 1†K1 < ÿ1=…/ ‡ 1† and 0 < K0 6 13; R0 > Rs , or 13 < K0 < K
0 ; R0 > Rs ,

K0 > K0 ; 0 < R0 < 1.
We want to show that if one of these conditions is satis®ed then there are initial data u0 …r†; w0 …r†
for which R…t† ! 1 as t ! 1.
We take u0 ; w0 such that
!
p
b
1
R0 sinh… kr†
p
u0 …r† ÿ
;
…6:5†
w …r† P r ÿ
c ÿ k r sinh… kR0 †
cÿk 0

and

8
 0 sinh…pcr†
bR
>
>
>
< P r sinh…pcR0 †
w0 …r†
 0 sinh…pcr†
> bR
>
>
:6
p
r sinh… cR0 †

if c > k;
if c > k

Dw0 …r† ÿ cw0 …r† 6 0

…6:6†

…6:7†

for 0 6 r 6 R0 .
Theorem 6.1. Assume that one of the conditions …i†; . . . ; …iv† is satisfied and that (6.5)±(6.7) hold. If
0 < c0 6 c 6 e0 when e0 is sufficiently small depending on the parameter set A, then
_
R…t†
>0

…6:8†

for all t > 0;

R…t† ! 1 if t ! 1:

…6:9†

Remark 6.1. The condition c0 6 c is a technical limitation of the proof. It means that the inhibitor
di€uses faster than the nutrient.
Remark 6.2. From the proof of the theorem it follows that if one of the conditions (i)±(iv) is
satis®ed with R0 ÿ d instead of R0 , for some d > 0, then e0 depends on d but not on the speci®c R0 .
We shall need the following lemma:
Lemma 6.2. Under assumptions of Theorem 6.1, for any given M1 > R0 there exist e0 > 0 and T0 > 0
such that
_
R…t†
>0

if

provided c ‡ c0 6 e0 .

0 6 t 6 T0

and

R…T0 † P M1 ;

126

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

Proof. From (6.5) and (6.6) it follows that
p
p
rR0 sinh… cr†
rR0 sinh… kr†
p
‡ K1
u0 …r† P …1 ÿ K1 †
:
p
r sinh… cR0 †
r sinh… kr†
Substituting this into (1.16) we get
p
_
R…0†
P rR0 ‰f … kR0 † ÿ K0 Š > 0:

_
Hence there exists a t0 > 0 such that R…t†
> 0 for 0 6 t 6 t0 . Now, for a given M1 > R0 , let
T0 ˆ …1=l0 † log…M1 =R0 † ‡ t0 , where
p
1
l0 ˆ r‰f … kR0 † ÿ K0 Š > 0:
2

From Theorem 2.1 we see that for all 0 6 t 6 T0 ,
R…t† 6 R0 eAT0
which implies

…A ˆ max…
r; 0† ÿ r~ > 0†;

_
j R…t†
j 6 A1 R0 eAT0

for all 0 6 t 6 T0 ;

where A1 is a constant depending only on r; r~ and r0 . Therefore, applying Lemma 5.1 (taking
 we get
M ˆ b)
r…r; t† P v…r; t† ÿ Cc00
00

for all t0 6 t 6 T0 ;

0

where c ˆ c ‡ c and C is a constant depending only on A; t0 and T0 . Substituting this estimate
into (1.16) we ®nd, as before, that
p
_ P R…t†f
R…t†
r‰f … kR…t†† ÿ K0 Š ÿ Cc00 g for all t0 6 t 6 T0 :
_
remains positive for
From this inequality, it follows that, provided c00 6 e0  l0 =C, as long as R…t†
t P t0 and 6 T0 we have R…t† > R0 , so that
p
r…f … kR…t†† ÿ K0 † P 2l0
_ P l0 R…t† > 0. Finally, integrating the last inequality in the interval t0 6 t 6 T0 and
and thus R…t†
using the de®nition of T0 we obtain the assertion R…T0 † P M1 . 
Proof of Theorem 6.1. We shall specify M1 later on, and then e0 and T0 will be chosen, accordingly,
as in Lemma 6.2. Let t be any number larger than T0 such that
_
R…t†
>0

if 0 6 t < t :
…6:10†
_  † > 0, then a continuity argument shows that (6.8) is satis®ed. Then also (6.9)
If we prove that R…t
holds since, otherwise, R ˆ limt!1 R…t† < 1 and the corresponding limits of r…r; t†; b…r; t† as
t ! 1 (which exist by standard parabolic theory [10]) form a stationary solution, which is a
contradiction.
Notice that (6.10) implies that
R0 6 R…t† 6 R…t †  M …0 6 t 6 t †

and M P M1 :

…6:11†

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

127

_  † > 0 we consider ®rst the case r > 0 (Figs. 1 and 2) and divide it into three
To prove that R…t
cases. The ®rst one is
b
;
…a†:
c > k; r P
cÿk
_ P 0 for 0 6 t 6 t ,
i.e., / > 1; 0 6 K1 6 1 (for this case Lemma 6.2 is not needed). Since R…t†
c0 wt ÿ Dw ‡ cw 6 0

if 0 6 r 6 R…t†; 0 6 t 6 t :

We claim that

bt …r; t† 6 0

if 0 6 r 6 R…t†; 0 6 t 6 t :

Indeed, the function u ˆ bt satis®es
c0 ut ˆ Du ÿ cu

…6:12†
…6:13†

and by di€erentiating (1.14),
ou
_ 6 0;
…0; t† ˆ 0; u…R…t†; t† ˆ ÿbr …R…t†; t†R…t†
or
_ P 0 and br …R…t†; t† > 0 by the maximum principle. Finally (6.7) ensures that u…r; 0† 6 0
since R…t†
and, then, (6.13) follows by the maximum principle applied to u.
From (1.13) and (6.13) we deduce that
cbt ÿ Db ‡ cb ˆ …c ÿ c0 †bt 6 0
…6:14†
so that, by comparison,
b…r; t† P w…r; t†:
Consider next the function
1
b…r; t†:
z…r; t† ˆ r…r; t† ÿ
cÿk
By (1.12) and (6.14) we have

czt ÿ Dz ‡ kz P 0
 ÿ k† P 0,
and since r ÿ b=…c
!
b
z…r; t† P r ÿ
wk …r; t†
cÿk
by comparison. Combining this with (6.15) we conclude that
1
r…r; t† ˆ z…r; t† ‡
b…r; t† P v…r; t†:
cÿk
Substituting this estimate into (1.16) we obtain the inequality
p
_ P rR…t†‰f … kR…t†† ÿ K0 Š
R…t†
for 0 6 t 6 t ;
and since
p
f … kR…t†† ÿ K0 > 0;
_  † > 0.
it follows that R…t

…6:15†
…6:16†

128

S. Cui, A. Friedman / Mathematical Biosciences 164 (2000) 103±137

In the sequel we shall use the fact that
p
p
R sinh… lr†
R sinh… kr†
p
if l < k; 0 6 r 6 R:
p P
r sinh… lR†
r sinh… kR†

…6:17†

The proof is by comparison. If we denote the right-hand side by u and the left-hand side by v, then
Dv ÿ lv ˆ 0;

Du ÿ lu ˆ …k ÿ l†u P 0 if

0 6 r 6 R;

since further u ˆ v on r ˆ R, the assertion (6.17) follows.
Consider next the case:
b
;
…b† : c > k;
r <
cÿk
i.e., / > 1; K1 > 1.
By Theorem 2.1
_ 6 AR…t†
R…t†

for all t > 0;

…6:18†

where A ˆ r ÿ r~ > 0. Since limg!1 p…g† ˆ 0 and K0 < 0, we can choose M1 suciently large and
e0 small enough so that


kM1
p > K0
…1 ÿ K1 †p
e0 AM1 ‡ k

(e0 has also to be small enough as required by Lemma 6.2). Recalling that K1 is positive, p is
monotone decreasing and that M P M1 , we then also have


p
kM
p


…6:19†
‡ K1 p… kM† > K0 :
…1 ÿ K1 †p
e0 AM ‡ k
Let

k1 ˆ



k
e0 AM ‡

p
k

2

:

pp
p
Clearly 0 < k1 < k and, since k ˆ e0 k1 AM ‡ k1 k,
p
k ÿ k1 ÿ e0 k1 AM P 0:

Introduce the function
!
b
wk1 …r; t†:
Z ˆ r ÿ
cÿk

…6:20†

…6:21†

…6:22†

It satis®es
!
p
p


p
b
k
R…t†
coth…
k
R…t††
ÿ
1
1
1
_
p
ÿ k1 ‡ k r ÿ