Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol165.Issue1.May2000:

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

b

Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA

c

Department of Mathematics, University of Minnesota, 206 Church Street, SE, Minneapolis, MN 55455-0488, USA

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 rˆR…t†. Within the tumor the con-centration of nutrient and the concon-centration of inhibitor (drug) satisfy a system of reaction±di€usion equations. The important parameters areK0(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 bwhich is the (nor-malized) 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 denoted byRÿ

s;R‡s withRÿs <Rs‡, then the smaller one is asymptotically stable, so that limt!1R…t† ˆRÿs,

provided the initial radiusR0is smaller thanR‡

s; if howeverR0>R‡s 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

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*_{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.}

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

con-centrations. 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 ofdormant(stationary) non-necrotic tumor states.

Following [1], we shall assume the tumor to be spherically symmetric and to occupy a region

fr<Rt_†gr_ˆjx_j; x_ˆx1;x2;x3_††

at each time t; the boundary of the tumor is given by r_ˆRt_†, an unknown function oft. Then, after non-dimensionalization, the (dimensionless) nutrient concentration r^r;t_† will satisfy a reaction±di€usion equation of the form

cor^

o_t ˆ

1

r2

o

o_r r

2or^

o_r !

‡C1…rBÿr^† ÿk0r^ÿc1b^ if r<R…t†; t>0; …1:1†

whereb^is the (dimensionless) inhibitor concentration. Here the constantsrB 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…rBÿr† accounts for the transfer of nutrient by

means of the vasculature, whose presence stems from angiogenesis. The termk0r^is the nutrient

consumption rate,c₁b^is the inhibitor consumption rate, andc_ˆTdiffusion=Tgrowth is the ratio of the nutrient di€usion time scale to the tumor growth (e.g., tumor doubling) time scale. Note that, typically, Tdiffusion1 min (see [8, pp. 194±195]) while Tgrowth 1 day, so that c1.

The (dimensionless) inhibitor concentrationb^satis®es a similar reaction±di€usion equation [1]

c2o

^ b o_t ˆ

D2 r2

o

o_r r

2ob^

o_r !

‡C2…bBÿb^† ÿc2b^ if r<R…t†; t>0; …1:2†

where the constantb_Bdenotes the (dimensionless) inhibitor concentration in the vasculature,C2is

inhibitor-in-blood±tissue transfer per unit length, andc₂b^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; typicallyD2 1 so that c2 1. Actually C2 ˆ0 in [1]; however if the inhibitor is partially fed through the

vascu-lature, then C2 >0.

1

r2

o

o_rr

2 o

o_r

is meant to denote the radial part of the Laplace operator (in 3 dimensions), and therefore one must include the conditions

o_r^

o_r…0;t† ˆ0; o_b^

o_r…0;t† ˆ0: …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

d dt

4 3pR

3

…t_†

ˆ

Z 2p

0

Z p

0

Z R…t†

0

Sr^;b^_†r2sinhdrdhdu; 1:4_†

where Sr^;b^_† denotes the cell proliferation rate within the tumor. For simplicity we restrict ourselves to the inhibitor-free proliferation rate [1]

Sr_{† ˆ}lr^_ÿr~~_†; 1:5_†

whereland~~rare positive constants. This means that the cell birth-rate islr^while the death-rate (apoptosis) is given bylr~~. Finally, the external nutrient concentration is assumed to be a constant

rand the external inhibitor concentration is assumed to be a constant b, so that

^

r_ˆr; b^_ˆb on r_ˆRt_†: 1:6_†

The case where inhibitors are absent was recently studied by Friedman and Reitich [9] under the assumption that

r>r~~> C1rB

C1‡k0

: 1:7_†

Extending the results obtained in [1] by perturbative and numerical studies, they proved by rig-orous mathematical analysis that there exists a unique stationary solutionRt_† Rs and that this solution is asymptotically stable for the time-dependent problem, providedcis suciently small; the asymptotic stability was only formally proved in [1] and only so in the limit casec_ˆ0.

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

b> C2bB

C2‡c2 …

1:8_†

or, equivalently, that

C2…bBÿb†<c2b: …1:9†

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:

c0_ˆ c2

The speci®c value ofl(in (1.5)) will not a€ect the results of this paper. In order to slightly simplify the calculations, we shall take l_ˆ3. Then, introducing the normalized nutrient and inhibitor concentrations,r and b, by

r_ˆr^_ÿ C1rB

the system (1.1)±(1.6) reduces to

cor

Note that the assumption (1.8) means that

b>0; 1:17_†

as mentioned above, we can actually allow bto be any real number though we shall not do so in this paper.

Finally we have initial conditions

rr;0_{† ˆ}u₀r_†; br;0_{† ˆ}w₀r_† if 06r6R0; ou0

We may view k and c as the consumption±transfer coecients of the nutrient and inhibitor, respectively, andbas 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 de-pend on just four constants

K0 ˆ

1 3

~ r

r; K1 ˆ b

…c_ÿk_†r; /ˆ

 c k r

; r0 ˆ ÿ

b

k: …1:19†

We shall always assume that

minr;r0†6u0…r†6max…r;0†;

06w₀r_†6b for 06r6R0: …1:20†

For simplicity we always assume that

c_6ˆk; 1:21_†

so thatK1 is well-de®ned, andK1 6ˆ0; the case cˆk is brie¯y considered in Remark 6.3.

We ®rst show that ifr~<minr;r0†, then R…t† ! 1ast! 1, whereas ifr~>max…r;0†, then Rt_{† !}0 as t_{! 1}. The function

fg_{† ˆ}1_ÿK1†p…g† ‡K1p…/g† p…g†

ˆg coth_g2gÿ1

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 asymp-totically stable if rP0 and unstable if r<0, with respect to the time-dependent solutions of (1.12)±(1.18).

…A2† Ifÿ…1=…/‡1††6…/ÿ1†K16/, and K062 …0;1₃†then no stationary solutions exist.

…A3†If…/ÿ1†K1>/thenK0ˆming>0f…g†<0, and whenK0P1₃; or K06K0, no stationary

solutions exist. However whenK₀ <K0 <0 then there exist two stationary solutions,…rÿs;b

ÿ

s;Rÿs† and r‡_s;b‡_s;R‡_s†, withRÿ_s <R‡_s;Rÿ_s is asymptotically stable whereasR‡_s is unstable.

…A4† If …/ÿ1†K1 <ÿ…1=…/‡1†† then K₀ ˆmaxg>0f…g†>0, when K060, or K0PK₀ , no

stationary solution exist. However, when1

3<K0 <K

0 then there exist two stationary solutions as

in caseA3†, and againRÿ_s is asymptotically stable whereas R‡_s is unstable.

The above results are proved under the assumption thatcandc0are suciently small. We shall also prove that when no stationary solutions exist then there are initial data for which

lim

t!1R…t† ˆ 1:

The mathematical results of this paper have implications for the treatment of tumor. Consider for example the case wherer~<0<r, so that with no inhibitors the tumor will grow unboundedly. Then, by administering inhibitor (drug) with suciently large external (normalized) concentration

provided the initial size of the tumor is below a certain radius R‡

s…b

†, whereas if bis further increased so that it becomes larger thanbthen the tumor will de®nitely be contained and it will evolve into a dormant one with radiusRÿ

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 solutionr;b;R_† for allt>0 and

0<br;t_†<b for06r<Rt_†; t>0; 2:1_†

minr;r0†<r…r;t†<max…r;0† for06r<R…t†t>0; …2:2†

R0exp_ft_‰minr;r0† ÿr~Šg6R…t†6R0expft‰max…r;0† ÿr~Šg for tP0; …2:3†

‰minr;r0† ÿr~Š6

_

Rt_†

Rt_† 6‰max…r;0† ÿr~Š for tP0: …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 allt>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†6r…r;t†6r2…r;t†, where

cor1

o_t ˆ

1

r2

o

o_r r

2or1

o_r

ÿkr1ÿb; …2:5†

cor2

o_t ˆ

1

r2

o

o_r r

2or2

o_r

ÿcr2 …2:6†

for 0<r <Rt_†; t>0 and

r1…R…t†;t† ˆr2…R…t†;t† ˆr if t>0;

r1…r;0† min…r;r0†; r2…r;0† max…r;0† if 06r<R…0†:

By comparison

r2…r;t†6 max…r;0†

and also (using the relationkr0‡bˆ0)

r1…r;t†P min…r;r0†

so that (2.2) holds.

Corollary 2.2.

(i) Ifmaxr;0_†<r~ thenlimt!1R…t† ˆ0;

(ii)Ifminr;r0†>r~ thenlimt!1R…t† ˆ 1.

Therefore in studying the asymptotic behavior of Rt_† we shall concentrate just on the case where

minr;r0†6r~6max…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

Substituting (3.6) and (3.7) into (3.5) we obtain

…1_ÿK1†r

It will be convenient to introduce the variable

g_{s ˆ}pkRs; …3:9†

and the functions

pg_{† ˆ}gcothgÿ1

g2 ; …3:10†

fg_{† ˆ}1_ÿK1†p…g† ‡K1p…/g†: …3:11†

Dividing (3.8) by rRs, we obtain the relation

fg_{s† 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 solutiong_s, 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†<0for allg>0;}

(ii) limg!0p…g† ˆ1₃; limg!1p…g† ˆ0; limg!1gp…g† ˆ1;

(iii) 0<pg_†<1

3; 0<gp…g†<1for all g>0.

The ®rst part is proved in [9], and the other parts are rather immediate.

Lemma 3.3. The function

kg_{† ˆ}gp

00_g†

p0_g†

is strictly monotone decreasing and in fact, k0_{g†<0 for allg>0.}

Proof.By direct computation

kg_{† ˆ} 2…sinh

3

g_ÿg3_coshg†

‰g2_{‡gcoshgsinhgÿ2 sinh}2_g

Šsinhgÿ2 and

k0g_{† ˆ ÿ} 2g…g†

‰g2_{‡gcoshgsinhgÿ2 sinh}2_gŠ2_sinh2_g;

where

Thus it remains to show thatgg_†>0 for all g>0.

one can easily verify that, for gP2,

gg_†eÿ6gP 1

Applying the lower bounds to the ®ve positive terms in gg_†=g5 _{and the upper bounds to the}

®rst two negative terms ingg_†, we obtain after some calculations,

gg_†

where`g_† is a polynomial of degree 48 with positive coecients, so that

`<sub>…</sub>g<sub>†</sub>< `2:25_{† ˆ}1:8364410ÿ5 _{if 0<g<2:25:}

For suchgwe then get, by computing the coecients of each power ofgin the above expression,

and the expression in parenthesis is>_ÿ3:4985 if 0<g<2:25. Hence

gg_† g15 10

5_P75:2499

‡ …30:0999_ÿ3:4985g2_†g2 >0

if 0<g<2:25 since the expression in parenthesis is positive.

Corollary 3.4.

If/>1 0</<1_† then

d dg

p0_/g†

p0_g† <0 …>0† for all g>0: …3:13†

Proof.Clearly

d dg

p0_/g† p0_g† ˆ

p0_{g† /p}00_{/g† ÿp}0_{/g† p}00_g†

…p0_g††2 ˆ

/gp00_/g† p0_/g†

ÿgp

00_g†

p0_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 fg_†de®ned in (3.11), note that

f0_{† ˆ}1

3; glim!1f…g† ˆ0: …3:14†

Theorem 3.5. (i) If_ÿ 1

/‡1<…/ÿ1†K16/then

f0g_†<0 for all g>0: 3:15_†

(ii)If /_ÿ1_†K1 >/ then there is a uniqueg0 such that

f0g_†<0 for 0<g<g₀; f0g_†>0 forg>g₀3:16_†

and, by (3.14),

K_0ˆfg_{0† ˆ}min

g>0f…g†<0:

(iii) If/_ÿ1_†K1<ÿ_/‡1₁ then there exists a uniqueg0>0, such that

f0g_†>0 for 0<g<g₀; f0g_†<0 forg>g₀3:17_†

and, by (3.14),

K_{0 ˆ}fg_{0† ˆ}max

g>0 f…g†>

1 3:

Proof.Note that

f0g_{† ˆ}p0g_†1

ÿK1† ‡/K1 p0_/g†

p0_g†

Also, by simple calculation,

lim

g!0 p0/g_†

p0_g† ˆ/; glim!1 p0/g_†

p0_g† ˆ

1

/2: …3:19†

Consider ®rst the case 06/_ÿ1_†K16/. Then K1P0 (respectively, K1 <0) implies /P1

(respectively, /<1), so that, by Corollary 3.4, p0_/g†=p0_{g† is strictly monotone decreasing}

(re-spectively, increasing). Consequently, by (3.19),

p0/g_†

p0_g† >

1

/2 respectively;

< 1

/2

so that the expression in brackets in (3.18) is>_‰1_ÿK1† ‡/K1=/2Š. Sincep0…g†<0, we get from

(3.18),

f0g_†<p0g_†1

ÿK1† ‡

/K1

/2

ˆp0g_†/ÿ …/ÿ1†K1

/ 60

and (3.15) follows. The remaining case of (i), namely,_ÿ1=/_‡1_†6/_ÿ1_†K1 <0, can be proved

similarly.

To prove (ii), note that /_ÿ1_†K1 >/ implies that

1

/2 < K1ÿ1

/K1

</ if K1 >0 … () />1†; …3:20†

/<K1ÿ1

/K1

< 1

/2 if K1 <0 … () /<1†: …3:21†

From Lemma 3.2(i) and (3.18)±(3.20) we see that ifK1>0 then

f0g_† p0g_†‰1_ÿK1† ‡K1/2Š<0 if g is near 0;

f0g_† p0g_†‰1_ÿK1† ‡

K1

/Š>0 if g is near 1:

Similarly if K1<0 then (using (3.21))

f0g_†<0 if g is near 0;

f0g_†>0 if g is near ₁:

Hence in both cases there exists a pointg₀such thatf0g_{0† ˆ}0. Sincep0g_†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 f0g_† 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_†</_ÿ1_†K16/, then for0<K0 <1₃there exists a unique stationary solution

(ii)If/_ÿ1_†K1>/ then for06K0 <1₃ there exists a unique stationary solution (rs;bs;Rs†; for K₀ <K0<0…K0ˆming>0f…g†<0† there are two stationary solutions …rÿs;bÿs ;Rÿs† and …r‡

s;b

‡

s;R‡s† with Rÿs <R‡s; for K0ˆK0 the two solutions coincide, and for K062 ‰K0; 1 3† there

are no stationary solutions.

(iii) If /_ÿ1_†K1 <ÿ1=…/‡1† then for 0<K061₃ there exists a unique stationary solution

…rs;bs;Rs†; for 1₃<K0<K0 …K0 ˆmaxg>0f…g†>1₃) there are two stationary solutions

…rÿ

s;b

ÿ

s;Rÿs† and…r‡s;b

‡

s;R‡s†with Rÿs <R‡s; forK0 ˆK0 the two solutions coincide, and forK0

62 …0;K_{0 Š} there are no stationary solutions.

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_ˆc0 _ˆ0. Then, for each t>0,

br;t_{† ˆ}

bRt_†

sinhp_cR

…t_††

sinhp_cr

†

r ; …3:22†

Fig. 1. 06/ÿ1†K16/.

rr;t_{† ˆ}1_ÿK1†

rRt_†

sinh  k p

Rt_††

sinh  k p

r_†

r ‡K1

rRt_†

sinhp_cR

…t_††

sinhp_cr

†

r : …3:23†

Substituting the last expression into (1.16) and setting gt_{† ˆ}pkRt_†, we get

dg

dt ˆrg ‰f…g† ÿK0Š: …3:24†

By applying Theorems 3.5 and 3.6 we easily obtain the following conclusions:

(R1) If 06…/ÿ1†K16/(which implies rP0), then for 0<K0<1₃; limt!1R…t† ˆRs for any initial radiusR0, that is, rs;bs;Rs†is globally asymptotically stable (Fig. 1); for K0P1₃ we have

limt!1R…t† ˆ0, and forK060 we have limt!1R…t† ˆ 1.

(R2) If…/ÿ1†K1 >/(which impliesr>0) then for 06K0 <1₃the solution…rs;bs;Rs†is again asymptotically stable (Fig. 2(a)); for K₀ <K0 <0 the solution …rÿs;b

ÿ

s;Rÿs† is locally asymptoti-cally stable with the attraction region of Rÿ_s being 0;R‡_s†, and for any R0 >R‡_s we have limt!1R…t† ˆ 1 (Fig. 2(b)). For K0P1₃ we have limt!1R…t† ˆ0, and for K0 <K0 we have

limt!1R…t† ˆ 1.

(R3) If ÿ1=…/‡1†6…/ÿ1†K1 <0 (which implies r<0), then for 0<K0 <1₃ we have

limt!1R…t† ˆ0 ifR0 <Rs; limt!1R…t† ˆ 1if R0>Rs, hence…rs;bs;Rs† is unstable (Fig. 3); for K0P1₃we have limt!1R…t† ˆ 1and forK060 we have limt!1R…t† ˆ0.

Fig. 3.ÿ 1

/‡16…/ÿ1†K1<0.

(R4) If …/ÿ1†K1<ÿ1=…/‡1† (which impliesr<0) then, for 0<K061₃ we have the same

conclusion as in (R3) (Fig. 4(a)); for 1₃<K0 <K0 the solution …rÿs;b

ÿ

s;Rÿs† is locally asymptoti-cally stable with the domain of attraction of Rÿ_s being 0;R‡_s†, and for R0 >R‡_s we have limt!1R…t† ˆ 1 (Fig. 4(b)). For K0 >K₀ we have limt!1R…t† ˆ 1 and for K060 we have

limt!1R…t† ˆ0.

The rest of the paper is devoted to the extension of the above results to the non-degenerate case wherec>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!1R…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 inft!1R…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 haver0 <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 ˆd0…e† and T0_ˆT0e;R0_† such that, if0<c6e, then

Rt_†Pd0 for all tPT0: …4:1†

Remark 4.1.It is important to note that d0 is independent ofR0.

Proof.The proof is an extension and some simpli®cation of the corresponding proof for the in-hibitor-free case [9]. We choose r <0 such that

kr_‡b60 and r2 rÿr >0

and introduce the function

vr;t_{† ˆ}vr;t_{† ‡}r r2

Rt_†

sinhMRt_††

sinhMr_†

r ‡r: …4:2†

Note that

vt ˆ ÿv _

Rt_†

Rt_†‰MR…t†coth…MR…t†† ÿ1Š

and, since kr_‡b60,

cvtÿDv‡kv‡b6v (

ÿcR_…t†

Rt_†‰MR…t†coth…MR…t†† ÿ1Š ‡kÿM 2

)

Recalling that

ÿR_R_…t†

…t_† 6r~ÿmin…r;r0† ˆr~ÿr0

and using the estimate

0<ncothn_ÿ1<1

3n

2

8n>0; 4:4_†

we get, forc6e,

cvtÿDv‡kv‡b6v 1 3e…r~

ÿr0†M2R2…t† ‡kÿM2

: 4:5_†

We shall now use the function v to show that if R1 is small enough then, for any T1 >0, the

inequality

Rt_†6R1 cannot hold for all tPT1; 4:6_†

hereR1 is independent of the initial radiusR0.

Indeed, otherwise we deduce from (4.5), with M2 _{ˆk‡1, that}

cvtÿDv‡kv‡b60 if 06r<R…t†; t>T1:

Since further v_ˆron r_ˆRt_†, the function

vr;t_{† ÿ}Aeÿk=c…tÿT1† _Aˆr

2‡ jr j†

is a subsolution fortPT1 and, by the maximum principle,

rr;t_†Pvr;t_{† ÿ}Aeÿ…k=c†…tÿT1†_:

Using this in (1.16) we ®nd (as in [9]) thatR_t_†>0 ift_ÿT1 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

RT2†>R1.

We choose positive constants d1 andM such that

d16R1 …4:7†

and

1

3e…r~ÿr0†d

2

1<1; M

2_> k

1_ÿ1

3e…r~ÿr0†d

2 1

: 4:8_†

Note that

ncothn_ˆ1_‡1 3n

2

‡On3_† if n_!0

so that, for some small positive constantd2,

ncothn>1_‡1

3…1ÿj†n

2 _{if 0}

where

j_ˆ rÿr~ 2r2‡ jr j

; 0<j<1: 4:10_†

We shall now prove the assertion (4.1) with

d0 ˆmin

d2 M; d1j

…r~ÿr0†=k

; 4:11_†

note that d0 <d16R1.

Indeed, suppose (4.1) is not true. Then for anyT2 >0 there existst0 >T2 such thatRt0_†<d0.

By the assertion we proved previously, we may choose T2 such that R…T2†>R1. It then follows

that there exists t1₂T2;t0_† such that Rt1_{† ˆ}d1 and R…t†6d1 for all t16t<t0. We claim that

rr;t_†Pv…r;t† ÿAeÿ…k=c†…tÿt1† _{for 06r6R…t†; t16t6t0; …4:12†}

where v is the function de®ned in (4.2) and A_ˆr2‡ jr 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.

Now lett2 ₂t1;t0_†be such thatRt2_{† ˆ}d0 andR…t†<d0 for allt2 …t2;t0†; thenR_…t2†60. Since

_

Rt_†=Rt_†P ÿ …r~ÿr₀†, we have

Rt_†PRt1_†eÿ…r~ÿr0†…tÿt1† _{if tPt1}

and consequently

t2 >t1_‡log d1 d0

1=…r~ÿr0†

Pt1_‡logjÿ1=k

…4:13_†

by (4.11). Substituting (4.12) into (1.16) we get

_

Rt_{† ˆ} 3

…Rt_††2

Z R…t†

0 …

rr;t_{† ÿ}r~_†r2dr

P 3r2

Rt_†sinhMRt_††

Z R…t†

0

rsinhMr_†dr_ÿr~_ÿr_†Rt_{† ÿ}ARt_†eÿk…tÿt1†

ˆ_M32r_R2_t†‰MR…t†coth…MR…t†† ÿ1Š ÿ …r~ÿr†R…t† ÿAR…t†e ÿk…tÿt1†_:

Since Rt_†6d0 fort26t6t0, we have MR…t†6Md0 <d2 by (4.11). Using also (4.9), (4.11) and

(4.13), we conclude that

_

Rt_†>1_ÿj_†r_ÿr_†Rt_{† ÿ}r~_ÿr_†Rt_{† ÿ}jARt_† if t26t6t0:

But since the right-hand side vanishes att_ˆt2, by the choicejin (4.10), this is a contradiction to

_

5. Asymptotic stability

In this section we shall extend the stability results of R1†±…R4†to the case wherec andc0 are

non-zero, but small. Thus we shall show that in cases R1†;…R2†; and …R4† (see Figs. 1±4) the

solutions rs;bs;Rs† and respectively, …rÿs;bÿs;Rÿs†, are globally asymptotically stable and, re-spectively, locally asymptotically stable with domain of attraction 0;R‡

s ÿd† for any d>0, provided cand 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 rr;t_†; br;t_†; Rt_†† be the solution of(1.12)±(1.18) for 06t; T0 0<T06_1†

and set

wr;t_{† ˆ}

bRt_†

sinhp_cR…t††

sinhp_cr†

r ; …5:1†

vr;t_{† ˆ}1_ÿK1†

rRt_†

sinhp_cR

…t_††

sinhp_cr

†

r ‡K1

rRt_†

sinhp_cR

…t_††

sinhp_cr

†

r : …5:2†

Assume that _jR_t_{† j} 6L for06t<T0 and

j u₀r_{† ÿ}vr;0_{† j} 6M; _ju₀r_{† ÿ}wr;0_{† j} 6M for 06r6R0: 5:3_†

Then there exist positive constants C (depending only onk;c;r;b_†and a (depending only onk;c) such that

jbr;t_{† ÿ}wr;t_{† j} 6CLc0_‡Meÿct=c0

†; 5:4_†

j rr;t_{† ÿ}vr;t_{† j} 6CLc00_‡Meÿat=c00_†c00_ˆc_‡c0_†5:5_† for 06r6Rt_†;06t<T0.

Proof.As easily veri®ed

c0wtÿDw‡cwˆ ÿc0R_…t† cR…t†p…pcR…t†† w;

where the functionpg_† is de®ned in (3.10). By Lemma 3.2

0<cRt_†pp_cR

…t_††6p_c;

and since 0<w6b, we get

j c0wtÿDw‡cwj 6Lbpcc0: Consequently, by comparison,

…br;t_{† ÿ}wr;t_††6 L

bp_cc0

c ; …5:6†

Similarly,

On the other hand, by (5.4),

jcrtÿDr‡kr‡wj 6 Theorem3.6.Let K anddbe positive constants such that one of the following three conditions holds: (i) KPd_‡max_fR0;Rsgif either 06…/ÿ1†K16/; 0<K0 <1₃ or…/ÿ1†K1>/; 06K0 <1₃;

Proof.The assumptions on /;K1 imply that

and that

either r>0 and fpkK_†6K0ÿd~ or r<0 and f…

 k p

K_†PK0‡d~; …5:8†

whered~depends on the same constants on which e0 is asserted to depend on.

Suppose that (5.7) is not true. Then, sinceK>R0, there exists at0 >0 such that (5.7) holds for allt<t0 and R…t0† ˆK. Consequently,

_

Rt0_†P0 5:9_†

and, by Theorem 2.1,

t0P 1

Alog

K R0 >

1

Alog

K

K_ÿdˆt >0: …5:10†

By Theorem 2.1 we also have

jR_t_{† j} 6L if 0<t6t0; 5:11_†

where

L_ˆmax_fjr_ÿr~_jK; _jr~_ÿr0 jKg: …5:12†

It follows thatt and Ldepend on the same constants upon whiche0 is asserted to depend on.

We now apply Lemma 5.1 to obtain the inequalities

vr;t_{† ÿ}Cc00_‡eÿat=c00_†6rr;t_†6vr;t_{† ‡}Cc00_‡eÿat=c00_†5:13_†

for 06r6Rt_†;0<t6t0, wherevis the function de®ned in (5.2). Substituting the upper bound on rinto (1.16) and setting gt_{† ˆ}pkRt_†, we get

dgt_†

dt 6rg …t†…f…g…t†† ÿK0† ‡Cg…t†…c

00_‡eÿat=c00

†: 5:14_†

Takingt_ˆt0 and using (5.8) and (5.10) we getg_t0_†<0 ifc006e0 ande0is small enough, which is

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 casec_ˆc0_ˆ0 (see Figs. 1±4).

Theorem 5.3. Let rr;t_†; br;t_†;Rt_†† and Rs;Rÿs;R‡s be as in Theorem 5.2. Suppose the initial radiusR0 satisfies, for some smalld>0, one of the following three conditions:

(i) 0<R061=d if either 06/_ÿ1_†K16/; 0<K0 <1₃ or …/ÿ1†K1 >/; 06K0 <1₃;

(ii) 0<R06R‡_{s ÿ}d if either /_ÿ1_†K1 >/; K0 <K0<0 or …/ÿ1†<ÿ1=…/‡1†; 1

3<K0<K0 ;

(iii) 0<R06Rsÿd if either ÿ1=…/‡1†6…/ÿ1†K1 <0; 0<K0 <1₃ or …/ÿ1†K1 <ÿ1=

…/_‡1_†; 0<K061₃.

lim

t!1R…t† ˆ

Rs in case…i†;

Rÿ

s in case…ii†;

0 in caseiii_†;

8 <

:

…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 constantKsuch that it satis®es the conditions in Theorem 5.2 (withdreplaced byd=2). By Theorem 5.2,Rt_†6Kfor allt>0 providedc_‡c0_6e

0; e0suciently

small, and then (5.11)±(5.14) follow as before. Recall that in case (iii)

r6r~<0 and fpkR0_†>K0:

Fixt0 suciently small (independently of e0) so that, by Theorem 2.1,

Rt_†6R0eÿr~t _{<K for 06t6t0:}

Then

fgt_{†† ÿ}K0Pb0>0 …b0 constant† …5:16†

for all 06t6t0. On the other hand from (5.14) we get

_

gt_†6rg t_†‰fgt_{†† ÿ}K0Š ‡Ce0g…t† if tPt0:

Choosing e0 <jrjb0=…2C† we deduce thatg_…t†<0 at tˆt0 and then, by continuity, g_…t†<0 in

some interval _‰t0;t0‡sŠ, so that g…t†6g…t0†<

 k p

K and

_

gt_†<rg t_†‰fgt_{†† ÿ}K0Š ÿ

1

2rb 0g…t† …5:17† in this interval. We can now repeat the previous step and deduce, step-by-step, thatg_t_†<0 and (5.17) holds for all tPt0, so that gt_{† !}0 as t_{! 1} (and, in fact, exponentially fast since the right-hand side of (5.17) remains 6 _ÿ1=2_†rb ₀gt_††.

Remark 5.1.The above proof shows thatRt_† decreases monotonically (to zero) for alltP0.

We next consider case (i) of Theorem 5.3.

Lemma 5.4. Consider case (i) of Theorem 5.3. Then for an arbitrary a0 >0 there exist positive

constants C,b and (sufficiently small) e0 depending only on k;c;r;r~;b;d anda0, such that the

fol-lowing is true ifc00_ˆc_‡c06e0: For any0<a6a0, if the inequalities

jRt_{† ÿ}Rs j 6a; jR_…t† j 6a;

jrr;t_{† ÿ}rs…r† j 6a; jb…r;t† ÿbs…r† j 6a

…5:18_†

hold for all 06r6Rt_†; tP0then also the inequalities

jRt_{† ÿ}Rs j 6Ca…c00‡eÿbt†; jR_…t† j 6Ca…c00‡eÿbt†; jrr;t_{† ÿ}rsr_{† j} 6Cac00_‡eÿbt_{†; jb…r;t† ÿb}

s…r† j 6Ca…c00‡eÿbt†

…5:19_†

Proof.From Lemma 5.1 we get

j rr;t_{† ÿ}vr;t_{† j} 6Cac00_‡eÿat=c00_{† 8}tP0; 5:20_†

j br;t_{† ÿ}wr;t_{† j} 6Cac00_‡eÿct=c00

† 8tP0: …5:21†

Substituting (5.20) into (1.16) and settinggt_{† ˆ}pkRt_†, we get

j g_t_{† ÿ}rg t_†‰fgt_{†† ÿ}K0Š j 6Cag…t†…c00‡eÿat=c 00

† 8tP0: …5:22†

Consider ®rst the case 06/_ÿ1_†K16/. Then r>0 and

f0g_†<0 for all g>0: 5:23_†

By Theorems 4.1 and 5.2 we know that there are positive constantse0; T0; d0 andKindependent

ofc; c0 _{anda (but dependent on a}

0) such that ifc006e0 then

d06g…t†6K for all tPT0: …5:24†

Consequently, by the mean value theorem,

rgt_†‰fgt_{†† ÿ}K0Š

6 _ÿC0gt_{† ÿ}g_s† if gt_†Pg_s; P ÿC0…g…t† ÿg_s† if g…t†<g_s;

…5:25_†

whereg_sˆpkRs andC0 is a positive constant depending only onr;d0;K and the coecients of fg_†. We shall use this inequality to prove that there exist positive constantsm; Bindependent of

c; c0_{, and a such that}

jgt_{† ÿ}g_{s j}<Bac00_‡eÿmt

† for alltP0: …5:26† It is clear that (5.26) holds for all 06t6T0 if, for ®xed m>0;B is chosen suciently large. Therefore, if (5.26) is not true, then there exists at0 >T0 such that

g_sÿBac00_‡eÿmt

†<gt_†<g_s‡Bac00_‡eÿmt

† for 06t<t0

but not fort_ˆt0; for de®niteness suppose that

gt0_{† ˆ}g_s‡Bac00_‡eÿmt0_{†; …5:27†}

then also

_

gt0_{† ‡}mBaeÿmt0_{P0: …5:28†}

On the other hand, by (5.22) and (5.25),

_

gt0_†6 _ÿC0gt0_{† ÿ}g_{s† ‡}Cagt0_†c00_‡eÿat0=e0_†:

Substituting (5.27) and (5.28) into this inequality and using the factgt0_†6K we get

ÿmBaeÿmt0_{6 ÿC0Ba…c}00_‡eÿmt0_{† ‡CKa…c}00_‡eÿat0=e0_†;

which is a contradiction if we choose m suciently small, say m6minC0=2;a=e0†, and B

corre-spondingly large. Having proved (5.26), the other estimates in (5.19) easily follow.

It remains to consider the case /_ÿ1_†K1 >/. Again we have r>0. But now (5.23) is not

valid. However, if we denote byg₀ the stationary point offg_†, then we still have

so that ifR0 <R g0=

 k p

then the previous argument still works for the present situation. If on the other hand R0PR, then fg0_††<0 and thus gt_† remains in the region where fg_† is negative when t varies in a small interval _‰0;t0_Š. Using (5.22) for tPt0 we conclude that if

c006e0; e0 is suciently small, then as long asg…t†Pg0ÿ2d0 (for some smalld0>0) we have

_

gt_†6 _ÿb0g…t†

(for some positive constantb₀). Thus there exists a timetPt0 such thatgt_†lies in0;g_0ÿd0Šat t_ˆt. 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-stants C,b and (sufficiently small)e0 depending onk;c;r;r~;b;danda0such that the following is true

if c00_ˆc‡c0_6e

0: For any0<a<a0, if the inequalities

jRt_{† ÿ}Rÿ_{s j} 6a; _jR_t_{† j} 6a;

jrr;t_{† ÿ}rÿ

s…r† j 6a; jb…r;t† ÿbÿs…r† j 6a

…5:29_†

hold for all 06r6Rt_†; tP0then also the inequalities

jRt_{† ÿ}Rÿ

s j 6Ca…c00‡eÿ

bt_{†; jR_…t† j 6Ca…c}00_‡eÿbt_†;

jrr;t_{† ÿ}rÿ_sr_{† j} 6Cac00_‡eÿbt_{†; jb…r;t† ÿb}ÿ

s…r† j 6Ca…c00‡eÿ

bt_† …5:30†

hold for all 06r6Rt_†; tP0.

Proof.The proof is similar to the proof of Lemma 5.4 for the case /_ÿ1_†K1 >/.

Again, by using Lemma 5.5 and following a similar argument as in [9] we can derive the as-sertion (5.15) in case (ii).

6. Instability: unboundedness ofR₍t₎

We shall use the notation

wl…r;t† ˆ

Rt_†

sinhp_lR…t††

sinhp_lr

†

r for any l>0; …6:1†

whererr;t_†;br;t_†;Rt_††is the solution of (1.12)±(1.18). We set

wr;t_{† ˆ}bwc…r;t†; …6:2†

vr;t_{† ˆ}r_f1_ÿK1†wk…r;t† ‡K1wc…r;t†g …6:3†

and

w0r_{† ˆ}wr;0_†; v0r_{† ˆ}vr;0_†: 6:4_†

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) 06/_ÿ1_†K16/; K0 <0; 0<R0 <1;

(ii)/_ÿ1_†K1>/and either K₀ <K0<0; R0>R‡_s or K0<K₀; 0<R0<1;

(iii)_ÿ1=/_‡1_†6/_ÿ1_†K1 <0 and either K0P1₃; 0<R0<1, or 0<K<1₃; R0>Rs, and (iv) /_ÿ1_†K1 <ÿ1=…/‡1† and 0<K06₃1; R0>Rs, or 13<K0 <K0 ; R0 >R‡s, or K0>K0 ; 0<R0<1.

We want to show that if one of these conditions is satis®ed then there are initial datau0…r†;w0…r†

for whichRt_{† ! 1}ast_{! 1}. We take u₀;w₀ such that

u₀r_{† ÿ} 1

c_ÿkw0…r†P rÿ b c_ÿk

!

R0sinh  k p

r_†

rsinhpkR0_†; …6:5†

w₀r_†

P

bR0sinhp_cr

†

rsinhp_cR0† if c>k;

6

bR0sinhp_cr†

rsinhp_cR0† if c>k 8

> > > <

> > > :

…6:6_†

and

Dw₀r_{† ÿ}cw₀r_†60 6:7_†

for 06r6R0.

Theorem 6.1. Assume that one of the conditionsi_†;. . .;iv_†is satisfied and that(6.5)±(6.7)hold. If

0<c06c6e0 whene0 is sufficiently small depending on the parameter setA, then

_

Rt_†>0 for all t>0; 6:8_†

Rt_{† ! 1} if t_{! 1}: 6:9_†

Remark 6.1.The conditionc06cis 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 withR0_ÿdinstead ofR0, for somed>0, thene0 depends ondbut not on the speci®cR0.

We shall need the following lemma:

Lemma 6.2. Under assumptions of Theorem6.1,for any givenM1 >R0there existe0>0andT0 >0

such that

_

Rt_†>0 if 06t6T0 and RT0_†PM1;

Proof.From (6.5) and (6.6) it follows that

u₀r_†P…1ÿK₁†rR0sinh…  k p

r_†

rsinhpkr_† ‡K1

rR0sinhp_cr

†

rsinhp_cR0

† : Substituting this into (1.16) we get

_

R0_†PrR0_‰fpkR0_{† ÿ}K0Š>0:

Hence there exists a t0 >0 such that R_t_†>0 for 06t6t0. Now, for a given M1>R0, let

T0ˆ …1=l0†log…M1=R0† ‡t0, where

l0 ˆ

1 2r‰f…

 k p

R0_{† ÿ}K0Š>0:

From Theorem 2.1 we see that for all 06t6T0,

Rt_†6R0eAT0 _{Aˆmax…r;0† ÿr~>0†;}

which implies

jR_t_{† j} 6A1R0eAT0 _{for all 06t6T0;}

where A1 is a constant depending only onr; r~ and r0. Therefore, applying Lemma 5.1 (taking M _ˆb) we get

rr;t_†Pvr;t_{† ÿ}Cc00 for all t06t6T0;

wherec00_ˆc‡c0 _{andCis a constant depending only on}A_{; t0 and T0. Substituting this estimate}

into (1.16) we ®nd, as before, that

_

Rt_†PRt_†fr_‰fpkRt_{†† ÿ}K0Š ÿCc00g for all t06t6T0:

From this inequality, it follows that, providedc00_6e

0 l0=C, as long asR_…t†remains positive for tPt0 and 6T0 we have Rt_†>R0, so that

rfpkRt_{†† ÿ}K0†P2l0

and thus R_t_†Pl₀Rt_†>0. Finally, integrating the last inequality in the interval t06t6T0 and using the de®nition ofT0 we obtain the assertion RT0_†PM1.

Proof of Theorem 6.1.We shall specifyM1later on, and thene0andT0 will be chosen, accordingly,

as in Lemma 6.2. Let t be any number larger than T0 such that

_

Rt_†>0 if 06t<t: 6:10_†

If we prove thatR_t_†>0, then a continuity argument shows that (6.8) is satis®ed. Then also (6.9) holds since, otherwise, R_ˆlim

t!1R…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

To prove thatR_t_†>0 we consider ®rst the caser>0 (Figs. 1 and 2) and divide it into three cases. The ®rst one is

…a_†: c>k; rP b c_ÿk;

i.e.,/>1; 06K161 (for this case Lemma 6.2 is not needed). Since R_…t†P0 for 06t6t,

c0wtÿDw‡cw60 if 06r6R…t†; 06t6t: …6:12†

We claim that

b_tr;t_†60 if 06r6Rt_†; 06t6t: 6:13_†

Indeed, the functionu_ˆb_t satis®es

c0utˆDuÿcu

and by di€erentiating (1.14),

o_u

o_r…0;t† ˆ0; u…R…t†;t† ˆ ÿbr…R…t†;t†R_…t†60;

sinceR_t_†P0 andb_r…Rt_†;t_†>0 by the maximum principle. Finally (6.7) ensures thatur;0_†60 and, then, (6.13) follows by the maximum principle applied tou.

From (1.13) and (6.13) we deduce that

cb_tÿDb_‡cb_ˆc_ÿc0_†b_t60 6:14_†

so that, by comparison,

br;t_†Pwr;t_†: 6:15_†

Consider next the function

zr;t_{† ˆ}rr;t_{† ÿ} 1

c_ÿkb…r;t†: …6:16†

By (1.12) and (6.14) we have

cztÿDz‡kzP0

and sincer_ÿb=c_ÿk_†P0,

zr;t_†P rÿ b c_ÿk

!

wk…r;t†

by comparison. Combining this with (6.15) we conclude that

rr;t_{† ˆ}zr;t_{† ‡} 1

c_ÿkb…r;t†Pv…r;t†:

Substituting this estimate into (1.16) we obtain the inequality

_

Rt_†PrRt_†‰fpkRt_{†† ÿ}K0Š for 06t6t;

and since

fpkRt_{†† ÿ}K0 >0;

In the sequel we shall use the fact that

The proof is by comparison. If we denote the right-hand side byuand the left-hand side byv, then

Dv_ÿlv_ˆ0; Du_ÿlu_ˆk_ÿl_†uP0 if 06r6R;

e0 small enough so that

…1_ÿK1†p

kM1

e0AM1_‡pk

>K0

(e0 has also to be small enough as required by Lemma 6.2). Recalling that K1 is positive, p is

monotone decreasing and thatMPM1, we then also have

by condition (b), (6.18) and (6.21) and the inequalityncothn_ÿ16n. Also

Substituting this estimate into (1.16) and using (6.19) we obtain, att_ˆt,

_

e0 is also chosen small enough as required in Lemma 6.2.

Let

and proceeding as in the case ofZ above, we deduce, using (6.24), that

Since also

In case (a0_{) we chooseM1 suciently large ande}

0 small so that, forMPM1,

Next we proceed to estimatezfrom below by Zand use (6.15), in order to derive a lower bound onr, which leads to R_t_†>0 (by (1.16) and (6.29)).

We ®nally note that the case r_ˆ0 can be handled similarly to the case r>0, orr<0:

Remark 6.3.Throughout this paper we assume thatc_6ˆk. For completeness we mention that the casec_ˆk or /_ˆ1, can be handled by letting /_!1. By L'Hospital's rule we ®nd that

fg_{† ˆ}pg_{† ‡} b kr

gp0g_† 2

and the stability and instability of stationary solutions follows by the same analysis as above applied to this limit function f.

7. The e€ect of inhibitor's parameters on tumor's growth

We may view the parameterK0as the tumor's characteristic (without inhibitors), the parameter

c as the inhibitor's (drug) characteristic, and the parameterbas the external inhibitor (normal-ized) concentration. In this section, building upon the mathematical results of Sections 3±6, we shall determine how the tumor's growth depends upon the parameters band c (or /_†; this will suggest a strategy for drug treatment.

Note that we are concerned here with large time progress of the therapy and therefore the parametersc and c0 will not play a role in the discussion of the drug treatment, we assume that they are small enough as in Sections 5 and 6.

By (1.19) we can write

K1 ˆ

1 kr

b

/2_ÿ1; /ˆ

 c k r

: 7:1_†

We shall ®rst consider the case

r>0; 7:2_†

we are then in the situations described in Figs. 1 and 2. Assumption (7.2) implies that

K1 >0 if and only if />1;

K1 <0 if and only if /<1; …

7:3_†

a property that we shall often be using in the sequel. According to Section 3, stationary solutions exist in just two cases:

CaseA. 0<K0<1₃ (Figs. 1 and 2(a)).

CaseB. K₀<K0 <0 (Fig. 2(b))

(we omit for simplicity that casesK0 ˆ0;K0 ˆK0).

gÿ _ˆpkRÿ_s; g‡_ˆpkR‡_sRÿ_s <R‡_s†;

the one with smaller radius is asymptotically stable if R0 <Rÿ_s and the one with larger radius is unstable and, in fact, for the time-dependent solution, if R0 >R‡s then R…t† may grow to 1 as

t_{! 1}.

Taking (7.1) into consideration, we can rewrite the functionfg_†de®ned by (3.11) and (3.10) as follows:

fg_{† ˆ}pg_{† ‡} b kr

p/g_{† ÿ}pg_†

/2_ÿ1 ; /ˆ

 c k r

: 7:4_†

We shall writefg_†asfg;b;/_†to emphasize the dependence on the relevant parameters regarded as independent variables. We shall indicate the dependency ofg and g _{on b;/;c by writing}

g_ˆgb;/_{† ˆ}g_‰b;c_Š;

g_ˆgb;/_{† ˆ}g_‰b;c_Š: …7:5†

At the point g₀, wheref takes its minimum (Theorem 3.5(ii)) we have

o_f

o_g gˆg₀

ˆp0g_{0† ‡} b kr

/p0/g_{0† ÿ}p0g_0†

/2_ÿ1 ˆ0: …7:6†

We shall write g_{0 ˆ}g₀b;/_†and denote the minimum of f by K_0ˆK₀b;/_{† ˆ}K_0‰b;c_Š, so that

K_0ˆK₀b;/_{† ˆ}fg₀b;/_†;b;/_†: 7:7_†

The following lemma will play an important role in our discussion:

Lemma 7.1. For each fixedg>0; the function

hg…/† ˆ …p…/g† ÿp…g††=…/

2

ÿ1_† for /_6ˆ1/>0_†;

…1=2_†gp0_{g† for /ˆ1}

…7:8_†

is continuous and strictly monotone increasing for />0.

Proof.The continuity ofhg…/†follows by L'Hospital's rule (cf. Remark 6.3). To prove thathg…/†

is strictly monotone increasing we compute its derivative in /,

h0_g/_{† ˆ}…/

2

ÿ1_†gp0_{/g† ÿ2/‰p…/g† ÿp…g†Š}

…/2_ÿ1_†2 for /6ˆ1: …7:9†

Thus we only need to prove that for all /_6ˆ1 />0_†,

gg…/† …/2ÿ1†gp0…/g† ÿ2/‰p…/g† ÿp…g†Š>0: …7:10†

We compute

g0_g/_{† ˆ}/2_ÿ1_†g2p00/g_{† ÿ}2_‰p/g_{† ÿ}pg_†Š; 7:11_†

lim

/!1

gg…/†

…/_ÿ1_†2 ˆg‰gp

From Lemma 3.3 and the fact that limg!0k…g† ˆ1 we see thatk…g†<1 if g>0, i.e.,

gp00g_†

p0_g† <1 if g>0: …7:13†

Hence (7.12) yields

gg…/†>1₂…/ÿ1†

2

g_‰gp00g_{† ÿ}p0g_†Š>0;

if/₂1_ÿd;1_‡d_{† n f}1_g, for some smalld>0.

Assume now that (7.10) is not true for all/P1‡d. Then there exists a /0>1‡dsuch that gg…/†>0 if 1</</0, andgg…/0† ˆ0; this implies thatg0g…/0†60. On the other hand we have,

by (7.11),

g_g0/_{0† ˆ}/₀2_ÿ1_†g2_p00_/

0g† ÿ2‰p…/0g† ÿp…g†Š

ˆ …/2_0ÿ1_†g2p00/0g† ÿ

/2_0ÿ1 /₀ gp

0_/

0g† …since gg…/0† ˆ0†

ˆ …/

2

0ÿ1†g

/₀ ‰/0gp

00_/

0g† ÿp0…/0g†Š>0 by …7:13†;

a contradiction.

Similarly one can prove that (7.10) is true for all 0</<1.

Theorem 7.2. The following properties hold:

o

o_bK

0‰b;cŠ<0; …7:14†

o

o_cK

0‰b;cŠ>0; …7:15†

lim

b!1

K_0‰b;c_{Š ˆ ÿ1}: 7:16_†

Proof.Since

o_f

o_g gˆg₀

ˆ0

we have, by (7.7) and (7.4),

o_K

0

ob ˆ o_f

ob…g0;

b;/_{† ˆ} 1 kr

p/g_{0† ÿ}pg_0† /2_ÿ1 <0;

o_K

and (7.15) follows. Finally, to prove (7.16) we take any 0<g<₁ and write

K_0ˆfg₀;b;/_†6fg;b;/_{† ˆ}pg_{† ‡} b kr:

p/g_{† ÿ}pg_† /2_ÿ1 :

Since pg_† is strictly monotone decreasing andr>0, we see that

1 kr

p/g_{† ÿ}pg_†

/2_ÿ1 <0 …7:17†

and (7.16) readily follows.

Theorem 7.3. Assume that (7.2)holds and that either 0<K0 <1₃ or K0<K0 <0: Then

Di€erentiating this equation with respect to band/, respectively, we get

Sinceo_f=o_g

gˆg…_b;/†<0

(by Theorem 3.5), the inequalities in (7.18) follow immediately from (7.8)

and (7.17) and from Lemma 7.1. The inequalities (7.19) and (7.20) can be proved in a similar way. Next let us prove (7.21). From (7.18) we see that

g lim

b!1

g_‰b;c_Š

exists and is non-negative and ®nite. If this number is positive then, by dividing (7.24) byband letting b_{! 1}, we obtain

1 kr

p/g_{† ÿp…g†}

/2_ÿ1 ˆ0

which is a contradiction sincepg_† is strictly monotone decreasing. Hence we must haveg_ˆ0,

and (7.21) is proved. The relations (7.22) and (7.23) can be proved in the same way.

Let us now examine the results obtained above from the point of view of medical treatment of tumours. If 0<K0 <1₃then there exists a unique dormant tumor, and it is globally asymptotically

stable; Theorem 7.3 tells us that its radius decreases as bis increased. If K0 <0 then the tumor

becomes unbounded (and Rt_{† ! 1} as t_{! 1}) when there is no inhibitor. However, in the presence of inhibitor, Theorem 7.2 tells us that if the external concentrationbexceeds a certain critical number b then two dormant states appear with radii Rÿ_s and R‡_s; Rÿ_s <R‡_s; Rÿ_s is as-ymptotically stable, and Rt_{† !}Rÿ_s as t_{! 1}, providedR0 <R‡_s. On the other hand if R0 >R‡_s

then (at least for some initial data) Rt_† grows to ₁ ast_{! 1}. Given initial radius R0, the last

conclusion of Theorem 7.3 asserts that we can increasebso thatR‡_{s ˆ}R‡_sb_†(which is dependent onb) will satisfyR‡

s…b†>R0and then indeedR…t† !Rÿs…b†ast! 1. The constantb

_{for which}

R‡_sb_†>R0whenb>bwill of course depend onR0, butbdoes not depend onR0. Finally, when

bkeeps increasing, Rÿ

s…b† keeps decreasing, and Rÿs…b† !0 as b! 1.

In conclusion, by increasing the amount of drug concentration b we can always decrease the tumour and in fact, render its limiting size arbitrarily small; the smallest concentration that ensures containment of the tumour isb_ˆb, and it is a function of its initial radius R0.

Theorems 7.2 and 7.3 also show that decreasingchas a similar e€ect as increasingb: the smaller thec, the more e€ective the drug is; if cis decreased then the limiting Rt_† is also decreased.

Remark 7.1.So far we have assumed that (7.2) holds. Consider next the case

r<0 Figs: 3 and 4_†: 7:25_†

If 0<K0 <1₃ then there is just one dormant state with normalized radiusg, and if1₃<K0 <K0

then there are two dormant states with normalized radiigÿ _andg‡_{, whereg}ÿ_<g‡_{. Proceeding as}

in the case of (7.2), one can prove that

o

obK

0 ‰b;cŠ>0;

o

o_cK

0 ‰b;cŠ<0;

lim

b!1K

o

e€ect: if bis increased beyond a critical number bso that

g

0 †. Here again a drug with smallercis more e€ective: ifcis decreased then the limiting Rt_†is also decreased.

We ®nally note that the caser_ˆ0 can be handled as a limit case of eitherr>0 orr<0.

8. Conclusion

We have show that, for ®xed K0, the number of dormant tumors depend on the parameters

K1;/de®ned in (1.19), or on the intrinsic inhibitor-parametercand its external concentrationb.

There may be one, two or no dormant states. In the cases where dormant states exist, we de-termined by rigorous mathematical analysis which of them is asymptotically stable and which is unstable; when two dormant states exist, the smaller one is stable and the larger one is unstable. We established monotonic dependence of the tumor's radius on the parameters b and c. Our analysis suggests how the external concentration bshould be chosen.