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Mathematical Biosciences 163 (2000) 159±199
www.elsevier.com/locate/mbs

A mathematical model of cancer treatment by
immunotherapy q
Frank Nani a, H.I. Freedman a,b,*,1
a

Department of Mathematical Sciences, Applied Mathematics Institute, 632 Central Academic Building, University of
Alberta, Edmonton, Alberta, Canada T6G 2G1
b
School of Mathematical Sciences, Swinburne University of Technology, Hawthorn, Victoria, Australia
Received 12 September 1998; received in revised form 19 October 1999; accepted 20 October 1999

Abstract
In this paper, a detailed mathematical study of cancer immunotherapy will be presented. General
principles of cancer immunotherapy and the model equations and hypotheses will be discussed. Mathematical analyses of the model equations with regard to dissipativity, boundedness of solutions, invariance
of non-negativity, nature of equilibria, persistence, extinction and global stability will be analyzed. It will
also be shown that bifurcations can occur, and criteria for total cure will also be derived. Ó 2000 Elsevier
Science Inc. All rights reserved.
Keywords: Cancer treatment; Competition; Dynamic modelling; Hopf bifurcation; Immunotherapy; Periodicity;

Persistence and extinction; Stability

1. Introduction: The immune system and cancer
When cancer cells proliferate to a detectable threshold number in a given physiological space of
the human anatomy, the body's own natural immune system is triggered into a search-and-destroy mode. The spontaneous immune response is possible if the cancer cells possess distinctive
surface markers called tumor-speci®c antigens. Tumor cells which possess such antigens are called

q

This paper is derived from a thesis submitted to the Faculty of Graduate Studies and Research of the University of
Alberta in partial ful®llment of the PhD requirements.
*
Corresponding author. Tel.: +1-403 492 3396; fax: +1-403 492 6826.
E-mail address: hfreedma@math.ualberta.ca, mathsci@math.ualberta.ca (H.I. Freedman).
1
Research partially supported by the Natural Sciences and Engineering Research Council of Canada, Grant NSERC
OGP4823.
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 5 8 - 9

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F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

immunogenic cancers [1±5]. The immune response against cancer cells can be categorized into two
types: the cellular and the humoral immune response.
The cellular natural immune response is provided by (i) lymphocytes (ii) lymphokines/cytokines
and (iii) antigen-presenting cells.
The e€ector lymphocytes which are involved in anti-cancer mechanisms are T-cells, natural
killer (NK) cells, lymphokine activated killer (LAK) cells and K cells. The lymphokines or cytokines are biological response modi®ers or growth-stimulating substances biosynthesized by
certain immune cells. These include the interleukins and the interferons. In particular, interleukin2 (IL-2) is biosynthesized by an antigen-sensitized subset of the T-cells called Helper T-cells. IL-2
is responsible for stimulating antigen-sensitized NK cells, cytotoxic T-cells and LAK cells to
develop into mature anti-cancer e€ector lymphocytes and also provides the growth stimulus for
these lymphocytes to proliferate into a high enough cell number capable of mounting an e€ective
attack against the cancer cells. The antigen-presenting cells include macrophages and dendritic
cells. These cells are responsible for presenting cancer antigens to the T-cells such as to trigger the
immune response. The detailed description of the morphology and roles of the lymphocytes,
lymphokines and antigen-presenting cells can be obtained from the following Refs. [3±5].
The humoral immune response to cancer is provided by (i) B-lymphocytes and (ii) Immunoglobulins/Antibodies. There exists a mechanism in which both cellular and humoral responses
cooperate in providing an anti-cancer activity. This mechanism is called antibody-dependent

cancer cell destruction: cf. [1,3,5], and it involves K and NK cells as well as IL-2 and immunoglobulin-G.
The cellular response is most proli®c against most cancers and usually the ®rst line of action.
The basic steps involved in a cellular immune response are listed as follows:
s1 : Cancer cells develop in a given physiological space of the human anatomy. The cancer cells
could be immunogenic or non-immunogenic.
s2 : The cancer cells subvent the immuno-surveillance activity provided by NK cells (which can
kill cancer cells whether immunogenic or not).
s3 : The cancer cells proliferate above the subclinical threshold of 103 cells and reach 109 cells
which is the X-ray detectable threshold. Some cancer cells might have metastasized to other
physiological regions of the human anatomy.
s4 : The antigen-presenting cells, particularly macrophages, encounter the cancer cells. They internalize the cancer cells, dissolve them into fragments called epitopes. These epitopes bear the
cancer-associated antigens. The macrophages then exhibit the cancer antigens on their surfaces
and circulate into the vicinity of T-cells (particularly helper T-cells) and mechanistically present
these cancer antigens to them (cf. [4,5]).
s5 : The antigen-sensitized helper T-cells then release the immuno-stimulatory growth substance
called IL-2. This lymphokine, IL-2, then stimulates the cancer killing subset of the T-cells called
the cytotoxic T-cells, to mature and proliferate. In particular, the IL-2 also enhances the proliferation of NK and LAK cells.
s6 : The activated lymphocytes (LAK, T, NK cells) then engage in a search-and-destroy anticancer activity.
The process of natural immune attack against immunogenic cancers is not always sustainable
nor eventually successful and can always be terminated or downgraded due to one or all of the

following reasons: (cf. [4±7]).

F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

161

r1 : The initial numbers of the cancer-killing lymphocytes at the time of tumor diagnosis or initiation of therapy are insigni®cant and easily overwhelmed by the rapidly proliferating tumor
cells.
r2 : The cancer cells eventually evade the immune recognition mechanism by shedding, altering
or re-distributing its surface tumor-associated antigens, cf. [4,5]. The `stealth' tumor then becomes non-immunogenic.
r3 : Some of the shedded tumor antigens and receptors bind to form circulating immune complexes which interact destructively with the surface receptors on the cancer cells and thereby
e€ectively block the cancer-killing lymphocytes from getting access to the cancer cells [7].
r4 : Cancer cells release inhibitory substances which e€ectively reduce the therapeutic ecacy of
the cancer-killing lymphocytes (cf. [4]).
In view of the processes r1 ±r4 it is observed by clinical investigators and medical oncologists
that the natural immune system cannot provide a sustained and therapeutically successful anticancer attack (see [1,4,5]).
Further research by clinical oncologists including those at the National Cancer Institute led to
the development of several techniques and methodologies to enhance the natural immune response against cancer. Some of these novel approaches include (see [1,2,8±11]):
(i) non-speci®c cancer immunotherapy,
(ii) speci®c passive cancer immunotherapy (adoptive cancer immunotherapy),

(iii) speci®c active cancer immunotherapy,
(iv) gene therapy of cancer,
(v) monoclonal antibody mediated anti-tumor immunization of host via induction of idiotypeanti-idiotypic immune network.
In this paper, we present a model of cancer treatment by immunotherapy, treating normal cells
and cancer cells as competitors for common resources. The anti-cancer cells are thought of as
predators on the cancer cells.
This paper is organized as follows. In Section 2 we state the principles upon which our model is
based. In Section 3 our model will be derived, and some elementary properties discussed in
Section 4. The equilibria and their stabilities are given in Section 5 followed by a global stability
analysis of subsystems in Section 6. Section 7 deals with persistence and interior equilibria while in
Section 8 Hopf bifurcation is discussed. The question of cancer extinction is addressed in Section
9. A short discussion is given Section 10.

2. Clinical principles of adoptive cellular immunotherapy
Adoptive cellular immunotherapy is a relatively new immunotherapeutic modality for treating
advanced and metastatically disseminated human solid tumors (see [11±17]). It involves the use of
tumor-killing lymphocytes and lymphokines such as natural or cloned IL-2 (n-IL-2, or r-IL-2),
NK cells, tumor in®ltrating lymphocytes (TIL), interferon-c activated killer monocytes (AKM)
and LAK.
The mathematical models will be based on LAK ACI using IL-2. The use of LAK and IL-2

therapy has been given prominence due to the work of Rosenberg and colleagues at the
National Cancer Institute (see [11,15±17]). In LAK ACI, the LAK precursor mononuclear

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F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

lympthocytes are clinically extracted from the cancer patients' body by a process called
cytapheresis. The LAK cells are then incubated (outside the patient's body for at least 48 h)
using high dose IL-2. Two phenotypes of LAK cells are called NK-LAK or A-LAK and
T-LAK depending on their precursors are produced. The NK-LAK cells have been used for
adoptive immunotherapy of metastatic cancers (cf. [12±14]). T-LAK cells have also been used
in adoptive immunotherapy of ovarian cancer and malignant brain tumors [14,18]. In LAK
ACI, the LAK cells are incubated with high dose IL-2 until the number of LAK cells is of the
order of 107 ±108 cells. They are then re-transfused by intravenous injection infusion into the
patient in addition to continuous infusion of IL-2 in the order of 105 units=m2 =day or 106
units/kg/day of r-IL-2 (cf. [11,14]).
The choice of ACI is based on its current status as the most clinically successful and promising
during clinical trials and applications to advanced cancers [12,13,16,18±20].
3. The mathematical model of ACI for solid tumors

In this section, the mathematical model for ACI will be presented.
Notation
x1 : The concentration of normal/non-cancer cells in the physiologic space or organ of the human anatomy where cancer cells are localized.
x2 : The concentration of cancer cells in a given physiologic space or organ of the human
anatomy.
w: The concentration of cancer-killing lymphocyte binding sites such as LAK cells in the neighborhood of the cancer cells and normal cells.
z: The concentration of lymphokine (e.g., IL-2) in the neighborhood of the cancer cells and normal cells.
Q1 : The rate of external (adoptive) intravenous re-infusion of lymphocyte (LAK cells) into the
cancer patient.
Q2 : The rate of external (adoptive) intravenous re-infusion of lymphokines (IL-2) into the cancer patient.
S1 : The rate of internal production of lymphocytes (LAK cells).
S2 : The rate of internal production of lymphokines (IL-2).
The model equations are as follows:
x_ 1 ˆ B1 …x1 † ÿ D1 …x1 † ÿ x1 x2 q1 …x1 ; x2 †;
x_ 2 ˆ B2 …x2 † ÿ D2 …x2 † ÿ x1 x2 q2 …x1 ; x2 † ÿ h…x2 ; w†;
w_ ˆ S1 ‡ Q1 ÿ a1 e1 …w† ‡ f …x; z† ÿ bh…x2 ; w†;
z_ ˆ S2 ‡ Q2 ÿ a2 e2 …z† ÿ gf …w; z†;
xi …t0 † ˆ xi0 P 0;

i ˆ 1; 2;

w…t0 † ˆ w0 P 0;
z…t0 † ˆ z0 P 0;

…3:1†

F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

163

where Bi …xi † and Di …xi †, i ˆ 1; 2; are, respectively, the birth and death rates of xi ;, qi …x1 ; x2 † the
speci®c natural competition functions between cancer and normal cells, i.e., the functions representing suppression of growth, f …w; z† the rate of lymphocyte (LAK) proliferation due to induction by lymphokine (IL-2), h…x2 ; w† the rate of cancer cell destruction by (cancer killing)
lymphocytes, e1 …w†; e2 …z† the rates of degradation or elimination of lymphocytes (LAK) or lymphokine (IL-2), respectively, g; b are constants depicting binding stoichiometry, and ai are elimination coecients.
We assume that all functions are suciently smooth so that solutions to initial value problems
exist uniquely for all positive time.
We shall assume that Qi  Si are such that the process relies solely on the rate of adoptive
transfer of LAK cells and IL-2. Then Si is negligible and will subsequently be omitted. Furthermore, the toxicity to normal cells is assumed to be minimal and hence not represented in the
models. This can be achieved in practice by use of low dose IL-2 (see [14,19]).
Thus the ®nal form of the model equations are
x_ 1 ˆ B1 …x1 † ÿ D1 …x1 † ÿ x1 x2 q1 …x1 ; x2 †;

x_ 2 ˆ B2 …x2 † ÿ D2 …x2 † ÿ x1 x2 q2 …x1 ; x2 † ÿ h…x2 ; w†;
w_ ˆ Q1 ÿ a1 e1 …w† ‡ f …w; z† ÿ bh…x2 ; w†;
z_ ˆ Q2 ÿ a2 e2 …z† ÿ gf …w; z†;
xi …t0 † ˆ xi0 P 0
w…t0 † ˆ w0 P 0;

for i ˆ 1; 2;
z…t0 † ˆ z0 P 0:

…3:2†

The following additional hypotheses are assumed to hold:
4 :
P1 : The initial conditions are such that …x10 ; x20 ; w0 ; z0 † 2 R
‡
1
P2 : (a) f …w; z† 2 C …R‡  R‡ ; R‡ †,
(b) fw …w; z† > 0; w > 0; z > 0,
(c) fz …w; z† > 0; w > 0; z > 0,
(d) f …0; z† ˆ 0; f …w; 0† ˆ 0:

P3 : (a) h…x2 ; w† 2 C 1 …R‡  R‡ ; R‡ †,
(b) hx2 …x2 ; w† > 0; x2 > 0; w > 0,
(c) hw …x2 ; w† > 0; x2 > 0; w > 0,
(d) hw …0; w† ˆ 0; w > 0,
6 0; w > 0,
(e) hx2 …0; w† ˆ
(f) h…0; w† ˆ 0; h…x2 ; 0† ˆ 0:
x2
w
and (ii) c2 ‡cc13 xx22 w‡c4 w.)

bb21‡w
(Some plausible expressions for h…x2 ; w† include: (i) aa21‡x
2
P4 : (a) ei 2 C 1 …R‡ ; R†,
(b) ei …0† ˆ 0,
(c) e0 …w† > 0; w > 0,
(d) e0 …z† > 0; z > 0:
P5 : Bi …0† ˆ Di …0† ˆ 0; B0i …xi † > 0; D0i …xi † > 0; B0i …0† > D0i …0†; there exists Ki > 0 such that
Bi …Ki † ˆ Di …Ki † and B0i …Ki † < D0i …Ki †; i ˆ 1; 2:
i

…x1 ; x2 † P 0, i; j ˆ 1; 2:
P6 : qi …0; 0† > 0, oq
oxi

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F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

4. Boundedness, invariance of non-negativity, and dissipativity
In this section, we shall show that the model equations are bounded, positively (non-negatively)
invariant with respect to a region in R4‡ ; and dissipative.
Theorem 4.1. Let B be the region defined by

8
9
 0 6 x1 6 K1 ; 0 6 x2 6 K2 ;
>
>

<
=
4  0 6 w 6 ÿ Q1 ; where d < 0;
1
:
B ˆ …x1 ; x2 ; w; z† 2 R‡ 
d1

>
>
:
;
Q
 0 6 z 6 2 ; where d2 > 0
d2

…4:1†

Then
(i) B is positively invariant.
 4 are eventually uniformly bounded and are
(ii) All solutions of system (3.1) with initial values in R
‡
attracted into the region B:
(iii) System (3.1) is dissipative.

Proof. Let x10 > 0: Consider
x_ 1 ˆ B1 …x1 † ÿ D1 …x1 † ÿ x1 x2 q1 …x1 ; x2 †
) x_ 1 < B1 …x1 † ÿ D1 …x1 †:
But there exists K1 such that B1 …K1 † ˆ D1 …K1 † by hypothesis. Thus
x1 …t† 6 max…K1 ; x10 †:
Note that x_ 1 < 0 for x1 > K1 and hence
lim sup x1 …t† 6 K1 :
t!1

For
x_ 2 ˆ B2 …x2 † ÿ D2 …x2 † ÿ x1 x2 q2 …x1 ; x2 † ÿ h…x2 ; w†;
a similar analysis gives
x2 …t† 6 max…K2 ; x20 †;
and
lim sup x2 …t† 6 K2 :
t!1

Now consider
w_ ˆ Q1 ÿ a1 e1 …w† ‡ f …w; z† ÿ bh…x2 ; w†;
) w_ < Q1 ‡ f …w; z† ÿ a1 e1 …w†;
) w_ < Q1 ‡ w max fe …w; z† ÿ wa1 min ee 1 …w†;
w;z2B

w2B

F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

165

where
f …w; z† ˆ w fe …w; z†;
e1 …w† ˆ we
e 1 …w†:

Now


w_ < Q1 ‡ w max max fe …w; z† ÿ a1 min e
e 1 …w† :
w;z2B

Let

w;z2B

w;z2B



e 1 …w† :
d1 ˆ max max fe …w; z† ÿ a1 min e
w;z2B

…4:2†

w;z2B

w;z2B

We shall henceforth assume that
max fe …w; z† < a1 min ee 1 …w†;
w;z2B

w;z2B

and consequently d1 < 0. Then w 6 ÿ Q1 =d1 ‡ w0 ed1 t . Thus


Q1
w 6 max ÿ ; w0 ;
d1
Q1
) lim sup w 6 ÿ ; d1 < 0; w0 P 0:
d1
t!1

…4:3†

Similarly we consider the z equation
z_ ˆ Q2 ÿ a2 e2 …z† ÿ gf …w; z†
) z_ < Q2 ÿ a2 e2 …z†
< Q2 ÿ a2 z
min e
e 2 …z†;
w;z2B

where e2 …z† ˆ z
e
e 2 …z†.
We shall henceforth assume that
d2 ˆ a2 min e
e 2 …z† > 0:

…4:4†

w;z2B

Then z 6 Q2 =d2 ‡ z0 eÿd2 t . Thus


Q2
Q2
z 6 max
; z0
:
and lim sup z 6
t!1
d2
d2



5. The equilibria: existence and local stability
The equilibria of system (3.1) are obtained by solving the system of isocline equations

…4:5†

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F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

B1 …x1 † ÿ D1 …x1 † ÿ x1 x2 q1 …x1 ; x2 † ˆ 0;
B2 …x2 † ÿ D2 …x2 † ÿ x1 x2 q2 …x1 ; x2 † ˆ 0;

…5:1†

Q1 ÿ a1 e1 …w† ‡ f …w; z† ÿ bh…x2 ; w† ˆ 0;
Q2 ÿ a2 e2 …z† ÿ gf …w; z† ˆ 0;
subject to the hypotheses P1 ±P6 , (4.2), (4.4) and (4.5).
The possible equilibria are of the form
 

…i† E0 …0; 0; w; z†;
 z†;
…ii† E1 …x1 ; 0; w;
b ; bz †;
…iii† E2 …0; b
x2; w

…5:2†

…iv† E3 …x1 ; x2 ; w ; z †:

The existence and local stability of the prospective equilibria are analysed as follows.
 
‡ :
Existence and local stability of E0 …0; 0; w; z†: The system of equation (3.1) is restricted to R
wz
This leads to the system
w_ ˆ Q1 ‡ f …w; z† ÿ a1 e1 …w†;
z_ ˆ Q2 ÿ a2 e2 …z† ÿ gf …w; z†;
w…0† ˆ w0 P 0;

…5:3†

z…0† ˆ z0 P 0:

Theorem 5.1. Let
L1 ˆ max f …w; z† > 0;
w;z2B


L2 ˆ min a1 mine
e 1 …w†; a2 min e
e 2 …z† > 0:
w;z2B

w2B

…5:4†

z2B

Then

…w ‡ z† 6

Q1 ‡ Q2 ÿ …g ÿ 1†L1
‡ …w0 ‡ z0 † eÿL2 t ;
L2

and
lim sup …w ‡ z† 6

t!1

Q1 ‡ Q2 ÿ …g ÿ 1†L1
:
L2

…5:5†

Proof. Using the system of equations (5.3) we obtain the di€erential equation
…w ‡ z†0 ˆ Q1 ‡ Q2 ‡ f …w; z† ÿ gf …w; z† ÿ a1 e1 …w† ÿ a2 e2 …z†
6 Q1 ‡ Q2 ‡ …1 ÿ g†max f …w; z† ÿ a1 wmin e
e 2 …z†
e 1 …w† ÿ a2 zmin e
z2B
w2B
w;z2B


6 Q1 ‡ Q2 ÿ …g ÿ 1†L1 ÿ …w ‡ z†min a1 min e
e 1 …w†; a2 min e
e 2 …z† ;
w;z2B

w2B

z2B

F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

167

and
…w ‡ z† 6

Q1 ‡ Q2 ÿ …g ÿ 1†L1
‡ …w0 ‡ z0 † eÿL2 t ;
L2

and hence
lim sup…w ‡ z† 6

t!1

Q1 ‡ Q2 ÿ …g ÿ 1†L1
:
L2

 
 ‡ such that
Lemma 5.2. Suppose there exists …w; z† 2 R
wz



Q1 ÿ a1 e1 …w† ‡


1

Q2 ÿ a2 e2 …z† ˆ 0
g
 

as t ! 1. Then E0 …0; 0; w; z† exists.

Proof. By equating the right-hand side of system (5.3) to zero, we have the two surfaces
C1 :

Q1 ÿ a1 e1 …w† ‡ f …w; z† ˆ 0;

C2 :

Q2 ÿ a2 e2 …z† ÿ gf …w; z† ˆ 0:

We have shown by Theorem 5.1 that system (5.3) is dissipative under the stated conditions of the
theorem. Now
a1 e1 …w† ÿ Q1 ˆ f …w; z†;
1
…Q2 ÿ a2 e2 …z†† ˆ f …w; z†:
g
Then

1
a1 e1 …w† ÿ Q1 ˆ …Q2 ÿ a2 e2 …z††;
g
1
() a1 e1 …w† ÿ Q1 ÿ …Q2 ÿ a2 e2 …z†† ˆ 0;
g
1
() Q1 ÿ a1 e1 …w† ‡ …Q2 ÿ a2 e2 …z†† ˆ 0:
g

The lemma now follows immediately.



We now discuss the (local) linearized stability of system (3.1) restricted to a neighborhood of
 
the equilibrium E0 …0; 0; w; z†:
The Jacobian matrix due to the linearization of (3.1) about an arbitrary equilibrium
 4 is given by
E…x1 ; x2 ; w; z† 2 R
‡

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F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

0

JE…x1 ;x2 ;w;t†

B01 …x1 † ÿ D01 …x1 †
ÿx2 q1 …x1 ; x2 †
ÿx1 x2 q1;x1 …x1 ; x2 †

B
B
B
B
B
B
B ÿx2 q2 …x1 ; x2 †
B
ˆB
B ÿx1 x2 q2;x1 …x1 ; x2 †
B
B
B
B
0
B
B
@
0

ÿx1 q1 …x1 ; x2 †
ÿx1 x2 q1;x2 …x1 ; x2 †

0

B02 …x2 † ÿ D02 …x2 †
ÿx1 q2 …x1 ; x2 †
ÿx2 x1 q2;x2 …x1 ; x2 †
ÿhx2 …x2 ; w†

ÿ hw …x2 ; w†

ÿ bhx2 …x2 ; w†

fw …w; z† ÿ bhw …x2 ; w†
ÿa1 e01 …w†

0

ÿ gfw …w; z†

1

C
C
C
C
C
C
C
C
0
C:
C
C
C
C
fz …w; z† C
C
C
0
ÿa2 e2 …z† A
ÿgfz …w; z†
0

…5:6†

Using hypotheses P1 ±P6 , the Jacobian matrix due to linearization of (3.1) about the rest point
 
E0 …0; 0; w; z† is given by the expression
1
0
B01 …0†
0
0
0
C
B ÿD0 …0†
C
B
1
0
0
C
B
…0†
…0†
ÿ
D
B
2
2
C
B
0
0
0

C
B
ÿhx2 …0; w†
C
B
C:
B
 
…5:7†
JE0 …0;0;w ;z† ˆ B
C



f
…w
;
z
†
w
C
B
…w
;
z
†
f
0
ÿ
bh
x2 …0; w†
z

C
B
ÿa1 e01 …w†
C
B

C
B
 
ÿa2 e02 …z† A
@
0
0
ÿ gfw …w; z†
 
ÿgfz …w; z†
Henceforth we let M22 de®ne the matrix
 

M22 ˆ



fw …w; z† ÿ a1 e01 …w†
 
ÿgfw …w; z†

!
 
fz …w; z†
:
 

ÿ a2 e02 …z† ÿ gfz …w; z†

…5:8†

The eigenvalues of JE0 …0;0;w ;z† are given by
k1 ˆ B01 …0† ÿ D01 …0†;



k2 ˆ B02 …0† ÿ D02 …0† ÿ hx2 …0; w†;

…5:9†

and the eigenvalues of M22 which are given by
r…M22 † ˆ fki j det…M22 ÿ kI† ˆ 0; i ˆ 3; 4g
ˆ fki j k2 ÿ …trace M22 †k ‡ det M22 ˆ 0; i ˆ 3; 4g:

…5:10†

By the Routh±Hurwitz criteria, the eigenvalues of M22 have negative real parts, i.e.,
Re r…M22 † < 0, i.e., if ÿTrace M22 > 0, and det M22 > 0.
Theorem 5.3. If
(i) B01 …0† ÿ D01 …0† < 0,

(ii) B02 …0† ÿ D02 …0† ÿ hx2 …0; w† < 0, and
 
(iii) Trace M22 < 0 with det M22 > 0, then the rest point E0 …0; 0; w; z† is locally asymptotically stable.

169

F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199
 

Proof. The proof is by inspection of the eigenvalues of the Jacobian matrix for E0 …0; 0; w; z† and
the qualitative theory of di€erential equations (cf. [21±24]). 
Theorem 5.4. Suppose
(i) B01 …0† ÿ D01 …0† > 0,

(ii) B02 …0† ÿ D02 …0† ÿ hx2 …0; w† > 0; and
(iii) Trace M22 < 0 with det M22 > 0.
 
Then the rest point E0 …0; 0; w; z† is a hyperbolic saddle and is repelling in both x1 and x2 directions
locally. In particular, the dimensions of the stable manifold W ‡ and unstable manifold W ÿ are given,
respectively, by


 
Dim W ‡ E0 …0; 0; w; z† ˆ 2;



 
Dim W ÿ E0 …0; 0; w; z† ˆ 2:

Proof. This result follows directly from inspection of the eigenvalues of the Jacobian matrix for
 
E0 …0; 0; w; z† and examples from Freedman and Mathsen [22]. 
 

Remark. Clinically the rest point E0 …0; 0; w; z† is not therapeutically feasible since it has neither
normal nor cancer cells. It is also highly unstable.
 z]
5.1. Existence and local stability analysis of E1 [x1 , 0, w,
 ‡ as represented by
Consider system (3.1) restricted to R
x1 wz
x_ 1 ˆ B1 …x1 † ÿ D1 …x1 †;
w_ ˆ Q1 ÿ a1 e1 …w† ‡ f …w; z†;
z_ ˆ Q2 ÿ a2 e2 …z† ÿ gf …w; z†;
x1 …0† ˆ x10 P 0;

z…0† ˆ z0 P 0;

…5:11†
w…0† ˆ w0 P 0:

 ‡ are E
e 1 ‰0; w;
e 1 ‰x1 ; w;
 zŠ and E
 zŠ: In particular, the existence of
The possible equilibria in R
x1 wz
~
 zŠ (which will be shown by persistence analysis) will imply the existence of E1 ‰x1 ; 0; w;
 zŠ.
E1 ‰x1 ; w;
 ‡ if
 zŠ exists in R
Using methods similar to the previous section, we can conclude that E~1 ‰0; w;
x1 wz
 z such that
there exists w;

1
 ‡ Q2 ÿ a2 e2 …z† ˆ 0:
Q1 ÿ a1 e1 …w†
g
e 1 ‰0; w;
 zŠ. This procedure leads to the
We now linearize system (5.11) in the neighborhood of E
result


e 1 …0; w;
 z† n;
…5:12†
n_ ˆ DF E

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F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

e 1 …0; w;
 z†† is the Jacobian matrix of the linearization and is given by
where DF … E
2 0
3
B1 …0† ÿ D01 …0†
0
0


5:
e 1 …0; w;
 z† ÿ a1 e01 …w†

 z†
0
fw …w;
fz …w;
 z† ˆ 4
DF E
0
 z†
 z†
ÿ a2 e2 …z† ÿ gfz …w;
0
ÿ gfw …w;

…5:13†

e 1 …0; w;
 z†† are given by
The eigenvalues of DF … E
k1 ˆ B01 …0† ÿ D01 …0†;

and

r…M22 † ˆ fki j det…M22 ÿ kI† ˆ 0; i ˆ 2; 3g;
 

 z†. Note that Re k2 < 0 and Re k3 < 0 if
where M22 is de®ned as in (5.8), with …w; z† replaced by …w;
Trace M22 < 0 and det M22 > 0.
 ‡ is
e 1 ‰0; w;
 zŠ 2 R
Theorem 5.5. The rest point E
x1 wz
(i) a hyperbolic saddle if
k1 ˆ B01 …0† ÿ D01 …0† > 0;

e 1 ‰0; w;
 zŠ is repelling in the x1 -direction,
and Trace M22 < 0 with det M22 > 0. In particular E
(ii) a hyperbolic source if
k1 ˆ B01 …0† ÿ D01 …0† > 0 and Re ki > 0
(iii) asymptotically stable (sink) if
k1 ˆ B01 …0† ÿ D01 …0† < 0 and

for i ˆ 2; 3;

Trace M22 < 0

Proof. Similar to those of the previous subsection.

with det M22 > 0:


De®nition 5.6. A set A  S is a strong attractor with respect to S if
lim sup q…u…t†; A† ˆ 0;

…5:14†

t!1

where u…t† is an orbit such that u…t0 †  S and q is the Euclidean distance function.
Lemma 5.7. The invariance box




Q1
Q2
‡ 
;
A1 ˆ …x1 ; w; z† 2 Rx1 wz 0 6 x1 6 K1 ; 0 6 w 6 ÿ ; 0 6 z 6
d1
d2
where


d1 ˆ max max f~…w; z† ÿ a1 min e
e 1 …w† < 0;
w;z

w;z

…5:15†

w

d2 ˆ a2 min e~2 …z† > 0;
z

is a strong attractor set with respect to R‡
w1 wz .
Proof. The proof is done using standard comparison theorems as in the previous sections.



F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

171

Remark. Since A1 is a strong attractor, it implies that all solutions of (5.11) with initial conditions
 ‡ are dissipative, uniformly bounded, and eventually enter the region A1 .
in R
x1 wz
 zŠ. Suppose
Theorem 5.8. Existence of E1 ‰x1 ; 0; w;
(i) Lemma 5.2 holds.
 ‡ and repelling locally in the x1 -direction (see
 z† is a unique hyperbolic rest point in R
(ii) E~1 …0; w;
x1 wz
Theorem 5.5 (i)).
R
 ‡ ; … T …B0 …0† ÿ D0 …0†† dt > 0†:
(iii) No periodic nor homoclinic orbits exist in the planes of R
1
1
x1 wz
0
Then
lim inf x1 …t† > 0;
t!1

lim inf w…t† > 0;
t!1

lim inf z…t† > 0:
t!1

 ‡ exhibits uniform persistence and consequently, the rest point
In particular, the subsystem in R
x1 wz
 zŠ exists.
E1 ‰x1 ; 0; w;
Proof. The proof follows from the de®nition of uniform persistence by Butler et al. [25], Freedman
and Rai [21,24]. 
 zŠ and using hyThe Jacobian matrix due to linearization around the rest point E1 ‰x1 ; 0; w;
potheses H1 ±H4 , and P1 ±P6 and expression (5.6) is given by
3
2
0
7
6 B1 …x1 †
0
0
ÿ x1 q1 …x1 ; 0†
7
6 ÿD0 …x1 †
1
7
6
0
0
7
6
B2 …0† ÿ D2 …0†
7
6
7
6
ÿx1 q2 …x1 ; 0†
0
0
0
7
6
7:
6
…5:16†
JE1 ‰x1 ;0;w;

ÿhx2 …0; w†
 zŠ ˆ 6
7
7
6

…
w;

z
†
f
w
6
 z† 7

0
ÿ bhx2 …0; w†
fz …w;
7
6

ÿa1 e01 …w†
7
6
0
7
6
ÿa
e
…
z
†
2 2
5
4
 z†
0
0
ÿ gfw …w;
 z†
ÿgfz …w;
 zŠ are given by
The corresponding eigenvalues of the Jacobian matrix for E1 ‰x1 ; 0; w;
k1 ˆ B01 …x1 † ÿ D01 …x1 †;

k2 ˆ B02 …0† ÿ D02 …0† ÿ x1 q2 …x1 ; 0† ÿ hx2 …0; w†;
and k3 ; k4 which belong to the set
r…M22 ˆ fki j k2 ÿ …Trace M22 †k ‡ det M22 ˆ 0; i ˆ 3; 4; g
where M22 is de®ned by (5.8).

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F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

Theorem 5.9. Let
 >0
(i) B02 …0† ÿ D02 …0† ÿ xq2 …x; 0† ÿ hx2 …0; w†
(ii) B01 …x† ÿ D01 …x† < 0
(iii) Trace M22 < 0 and det M22 > 0.
 zŠ is a hyperbolic saddle point which is repelling in the x2 -direction
Then the equilibrium E1 ‰x1 ; 0; w;
 z†† is the x1 ÿ w ÿ z space and the unstable
locally. In particular, the stable manifold, W ‡ …E1 …x1 ; 0; w;
 z†† is the x2 -direction, with Dim W ÿ …E1 † ˆ 1:
manifold W ÿ …E1 …x1 ; 0; w;
 zŠ is locally asymptotically stable (hyperbolic sink)
Theorem 5.10. The rest point E1 ‰x1 ; 0; w;
if B02 …0† ÿ D02 …0† ÿ x1 q2 …x1 ; 0† ÿ hx2 …0; w† < 0; B01 …x1 † ÿ D0 …x1 † < 0 and Trace M22 < 0 with
det M22 > 0.
The proofs of Theorems 5.8 and 5.10 follow from an inspection of the Jacobian matrix of
 zŠ and using the Routh±Hurwitz criteria.
linearization in the neighborhood of E1 ‰x1 ; 0; w;

b ; bz ]
5.2. The existence and stability of E2 [0; b
x2; w

b ; bz Š:
We now establish criteria for the existence and stability of the rest point E2 ‰0; b
x2; w
 ‡ ; we obtain the following subsystem:
When system (3.1) is restricted to R
x2 wz
x_ 2 ˆ B2 …x2 † ÿ D2 …x2 † ÿ h…x2 ; w†;

w_ ˆ Q1 ÿ a1 e1 …w† ‡ f …w; z† ÿ bh…x2 ; w†;
z_ ˆ Q2 ÿ a2 e2 …z† ÿ gf …w; z†;
x2 …0† ˆ x20 P 0;

w…0† ˆ w0 P 0;

…5:17†
z…0† ˆ z0 P 0:

‡
The possible equilibria corresponding to system (5.7) are in R
x2 wz
e
b ; bz Š and
(i) E 2 ‰0; w
e 2 ‰b
b ; bz Š:
(ii) E
x2; w
e 2 ‰0; w
b ; bz Š exists if there
Using the arguments from Lemma 5.2 we can conclude that the rest point E
b ; bz † such that
exist … w
1
b † ‡ …Q2 ÿ a2 e2 …bz †† ˆ 0
Q1 ÿ a 1 e 1 … w
g

as t ! 1.
e 2 ‰b
b ; bz Š and hence E2 ‰0; b
b ; bz Š will be established similar to the
x2; w
x2; w
The existence of E
previous section using persistence analyses.
e 2 ‰0; w
b ; bz Š† due to linearization of (5.17) in the neighborhood of
The Jacobian matrix DF … E
‡

e 2 ‰0; w
b ; bz Š in R
E
satis®es
the
ordinary
di€erential equation
x wz
 2

e 2 ‰0; w
b ; bz Š g;
g_ ˆ DF E
…5:18†

F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

where

2

B02 …0† ÿ D02 …0†
6 ÿhx …0; w
b†
2
6
6
e 2 ‰0; w
b ; bz Š† ˆ 6
DF … E
b†
6 ÿbhx2 …0; w
6
4
0

0
b†
ÿa1 e01 … w

b ; bz †
‡fw … w

b ; bz †
ÿ gfw … w

0

173

3

7
7
7
b ; bz † 7
fz … w
7:
7
ÿa2 e02 …bz † 5
b ; bz †
ÿgfz … w

…5:19†

The hypotheses H1±H4 and P1 ±P6 are again used in the computation of the entries of the Jacobian
matrix together with expression (5.6).
e 2 ‰0; w
b ; bz Š† are given by
The eigenvalues of DF … E
k1 ˆ B02 …0† ÿ D02 …0† ÿ hx2 …0; w†

and k2 ; k3 2 r…M22 †:

e 2 ‰0; w
b ; bz Š of (5.17) is such that
Theorem 5.11. The rest point E
e
b ; bz Š is a hyperbolic saddle point (repelling, in the x2 -direction) if
(i) E 2 ‰0; w
b † > 0 and Trace M22 < 0 with det M22 > 0:
B02 …0† ÿ D02 …0† ÿ hx2 …0; w
e 2 ‰0; w
b ; bz Š is a hyperbolic source if B02 …0† ÿ D02 …0† ÿ hx2 …0; w† > 0 and Re ki > 0; i ˆ 2; 3.
(ii) E
e 2 ‰0; w
b ; bz Š is a hyperbolic sink and hence locally asymptotically stable if
(iii) E
0
0
b † < 0 and Trace M22 < 0 with det M22 > 0.
B2 …0† ÿ D2 …0† ÿ hx2 …0; w

Proof. These results follow immediately from inspection of the Jacobian matrix due to linearie 2 ‰0; w
b ; bz Š and applying the qualitative theory of ordinary di€erential
zation of (5.17) around E
equations. 
Lemma 5.12. The (non-negatively) invariant set


Q1
Q2
‡

;
A2 ˆ …x2 ; w; z† 2 Rx2 wz j 0 6 x2 6 K2 ; 0 6 w 6 ÿ ; 0 6 z 6
d1
d2

…5:20†

where d1 and d2 are defined as in (4.2) and (4.4), respectively, is a strong attractor with respect to
solutions initiating from int R‡
x2 wz with non-negative initial conditions.
Proof. Similar to the previous section's proof for the invariant set A1 :



Remark. Since the compact set A2 is a strong attractor, it therefore means that, all solutions of
 ‡ are dissipative, uniformly bounded, and eventually enter
(5.17) with initial conditions in int R
x2 wz
the region A2 .
b ; bz Š. Suppose
Theorem 5.13. Existence of E2 ‰0; b
x2; w
(i) Lemma 5.11 holds.
 ‡ (see Theorem 3.7 (i)).
e 2 ‰0; w
b ; bz Š is a unique hyperbolic saddle repelling in the x2 direction of R
(ii) E
x2 wz
‡
(iii) There are no periodic nor homo/hetero-clinic trajectories in the planes of R
x2 wz
Z T


 0
B2 …0† ÿ D02 …0† ÿ hx2 …0; w† dt > 0 :
0

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F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

e x2; w
b ; bz Š exists
Then the subsystem (5.17) exhibits uniform persistence and the interior equilibrium E‰b
‡

b ; bz Š exists.
x2; w
in Rx2 wz and consequently E2 ‰0; b


 zŠ in Section 3.3.2.
Proof. The proof is similar to that for the existence of E1 ‰x1 ; 0; w;

b ; bz Š. The Jacobian
We now perform the linearized stability analyses for the rest point E2 ‰0; b
x2; w
b ; bz Š is given by the
x2; w
matrix due to linearization of system (5.17) in the neighborhood of E2 ‰0; b
expression
2

B01 …0† ÿ D01 …0†
6 ÿb
x2†
6 x 2 q1 …0; b
6
6 ÿb
x2†
6 x 2 q2 …0; b
6
JE ‰0; bx ; bw ; bz Š ˆ 6
2
2
6
0
6
6
4
0

0

0
D02 …bx 2 †

x2†
B02 …b

ÿ
b†
ÿhx2 …b
x2; w

b†
x2; w
ÿ hw …b

b†
b ; bz † ÿ a1 e01 … w
fw … w
b†
ÿbhw …b
x2; w

b†
ÿ bhx2 …b
x2; w

b ; bz †
ÿ gfw … w

0

0

3

7
7
7
7
0
7
7:
7
b ; bz † 7
fz … w
7
7
ÿa2 e02 …bz † 5
b ; bz †
ÿgfz … w

…5:21†

The eigenvalues of JE ‰0; bx ; bw ; bzŠ are given by
2
2
x 2 q1 …0; bx 2 †
k1 ˆ B01 …0† ÿ D01 …0† ÿ b

and k2 ; k3 ; k4 2 r…M33 †:

In particular M33 is the matrix de®ned by
0
x 2 † ÿ D02 …b
x2†
B0 …b
b†
B 2
ÿhw …bx 2 ; w
B ÿhx2 …b
b†
x2; w
B
B
b ; bz † ÿ a1 e01 … w
b†
fw … w
B
M33 ˆ B ÿbhx2 …b
b†
x2; w
B
b†
ÿbhw …b
x2; w
B
B
@
b ; bz †
0
ÿgfw … w
0

b 11
w
B
b 21
ˆ: @ m

Now

b 31
m

b 12
m
b 22
m

b 32
m

1
b 13
m
C
b 23 A:
m

0

1

C
C
C
C
C
b ; bz †
fz … w
C
C
C
ÿa2 e02 …bz † C
A
b ; bz †
ÿgfz … w

…5:22†

b 33
m

r…M33 † ˆ p…k; M33 † ˆ det‰M33 ÿ kI3 Š
a 1 k2 ‡ b
a2k ‡ b
a 3 ˆ 0; i ˆ 2; 3; 4g;
ˆ fki j k3 ‡ b

…5:23†

F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

175

where
b 11 ‡ m
b 22 ‡ m
b †;
b
a 1 ˆ ÿ …Trace M33 † ˆ ÿ… m

 33 




m

m
b 11 m
b 22 m
b 23 
b 13 
m
 b 11
b
‡
det
‡
det
a 2 ˆ det 
m
m
b 32 m
b 33 
b 31 m
b 33 
b 21
m
b
a 3 ˆ ÿ det M33 :


b 12 
m
;
b 22 
m

…5:24†

Lemma 5.14. The eigenvalues of M33 have negative real parts if
b
a 1 > 0;

b
a3 > 0

and

b
a1b
a2 > b
a3:

Proof. The proof uses the Routh±Hurwitz criterion.



Theorem 5.15. Let
(i) B01 …0† ÿ D01 …0† ÿ bx 2 q1 …0; b
x 2 † > 0,
(ii) b
a 1 > 0; b
a 3 > 0 and b
a1b
a3.
a2 > b
b ; bz Š is a hyperbolic saddle point and repelling in the x1 -direction. In particular, the
Then E2 ‰0; b
x2; w
stable manifold W ‡ …E2 † is the x2 ÿ w ÿ z space and the unstable manifold W ÿ …E2 † is the x1 -direction,
such that Dim W ‡ …E2 † ˆ 3 and Dim W ÿ …E2 † ˆ 1.
b ; bz Š is locally asymptotically stable (hyperbolic sink) if
Theorem 5.16. E2 ‰0; b
x2; w
x 2 † < 0, and
(i) B01 …0† ÿ D01 …0† ÿ bx 2 q1 …0; b
(ii) b
a 1 > 0; b
a 3 > 0; b
a1 b
a 3 hold concurrently.
a2 > b

The proofs of Theorems 5.15 and 5.16 follow directly from linearized stability analysis and
application of the Routh±Hurwitz criteria.
b ; bz Š corresponds to the scenario in which the normal cells in the
Remark. The equilibrium E2 ‰0; b
x2; w
cancer-a€ected tissue or organ are all destroyed. This will eventually lead to the demise of the
b ; bz Š is highly
x2; w
cancer patient unless a transplant of a new organ is implemented. Thus E2 ‰0; b
clinically unstable.
5.3. Existence of E3 [x1 ; x2 ; w ; z ]

In this section, we shall establish sucient conditions for the existence of a positive interior
equilibrium E3 ‰x1 ; x2 ; w ; z Š. This will be done by showing that system (3.1) is uniformly persistent
(see [21,24,30]).
‡
To show uniform persistence in R
x1 x2 wz we must assume or verify the following hypotheses for
system (3.1).
‡ .
H0 : All dynamics are trivial on oR
x1 x2 wz
H1 : All invariant sets (equilibria/rest points) are hyperbolic and isolated.
‡
H2 : No invariant sets on oR
x1 x2 wz are asymptotically stable.
 ‡ , it must be
H3 : If an equilibrium exists in the interior of any 3-dimensional subspace of R
x1 x2 wz
globally asymptotically stable with respect to orbits initiating in that interior.

176

F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

 ‡ , and W ‡ …M† is its strong stable manifold, then
H4 : If M is an invariant set on oR
x1 x2 wz
W  …M† \ int Rx1 x2 wz ˆ ;:

…5:25†

H5 : The given system of di€erential equations is dissipative and eventually uniformly bounded
for t 2 R‡ with respect to a strong (compact) attractor set.
H6 : All invariant sets are acyclic.
Remark. H1 ±H5 gives persistence and H6 is required for uniform persistence.

6. Global stability of subspace equilibria
 x]
6.1. Global asymptotic stability of E1 ,[x1 , w,
In this section, we derive criteria for the global stability hypothesis H3 to be valid. First criteria
 zŠ with respect to solutions initiating in int R‡
for the global asymptotic stability of E1 ‰x1 ; 0; w;
x1 wz
will be established.
In R‡
x1 wz we choose the Liapunov function,
V …x1 ; w; z† ˆ x1 ÿ x1 ÿ x1 ln

x1 1
 2 ‡ 12k2 …z ÿ z†2 ;
‡ 2k1 …w ÿ w†
x1

…6:1†

where ki 2 R‡ for i ˆ 1; 2.
 ‡ is given by the expression
The derivative of (6.1) along the solution curves of (5.11) in R
x1 wz
 ‰Q1 ÿ a1 e1 …w† ‡ f …w; z†Š ‡ k2 …z ÿ z†‰Q2 ÿ a2 e2 …z† ÿ gf …w; z†Š;
V_ ˆ …x1 ÿ x1 †g1 …x1 † ‡ k1 …w ÿ w†
…6:2†
where we set
Bi …xi † ÿ Di …xi †  xi gi …xi †;

i ˆ 1; 2:

…6:3†

Thus
V_ ˆ …x1 ÿ x1 †g1 …x1 †
 1 ‰e1 …w†
 ÿ e1 …w†Š ‡ k1 …w ÿ w†‰f
 …w; z† ÿ f …w;
 z†Š
‡ k1 …w ÿ w†a
 z† ÿ f …w; z†Š:
‡ k2 …z ÿ z†a2 ‰e2 …z† ÿ e2 …z†Š ‡ k2 …z ÿ z†g‰ f …w;
Let
0

1
v1
v1 ˆ x1 ÿ x1

X ˆ @ v2 A such that v2 ˆ w ÿ w
v3
v3 ˆ z ÿ z

…6:4†

F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

177

and set
g1 …x1 †
; a12 ˆ 0; a13 ˆ 0;
x1 ÿ x1

‰e1 …w† ÿ e1 …w†Š
;
a22 ˆ ÿ k1 a1

…w ÿ w†
 z†Š
 z†Š
‰f …w; z† ÿ f …w;
‰f …w; z† ÿ f …w;
a23 ˆ k1
ÿ gk2
;

…z ÿ z†
…w ÿ w†
‰e2 …z† ÿ e2 …z†Š
a33 ˆ ÿ k2 a2
:
…z ÿ z†
Thus
a11 ˆ

…6:5†

V_ ˆ a11 v21 ‡ a22 v22 ‡ a23 v2 v3 ‡ a33 v23
ˆ a11 v21 ‡ 12a12 v1 v2 ‡ 12a13 v1 v3 ‡ 12a12 v1 v2 ‡ a22 v2 v2 ‡ 12a23 v2 v3 ‡ 12a13 v1 v3 ‡ 12a23 v2 v3 ‡ a33 v23 ;
…6:6†
where aij ˆ aji with a12 ˆ a13 ˆ 0. But
V_ ˆ X T AX ˆ X T A
X ˆ hAX ; X i;
where

0

a11

B
B1
AˆB
B 2 a12
@
1
a
2 13

1
a
2 12

1
a
2 13

a22

1
a
2 23

1
a
2 23

a33

1

C
C
C:
C
A

…6:7†

…6:8†

In particular, A is symmetric and real such that A ˆ 12…A ‡ At † where t denotes transpose.
Lemma 6.1. Negative Definiteness of V_ .
(i) V_ is negative if X T AX is negative definite.
(ii) X T AX is negative if A is negative definite.
(iii) A is negative definite if the (eigenvalues) zeros of the polynomial
p…k; A† ˆ det…A ÿ kIn † ˆ 0
have negative real parts.
A complete discussion and proofs of the lemma can be found in Refs. [26,27].
Lemma 6.2 (Frobenius 1876). Let
0 1
x1
B x2 C
B C
B C
X ˆ B x3 C; X T ˆ …x1 ; x2 ; x3 ; . . . ; xn † 2 Rn :
B .. C
@ . A
xn

Let A be a symmetric n  n matrix over R. Then the real quadratic form X T AX is negative definite if
A is negative definite. In particular, a necessary and sufficient condition for the real, symmetric

178

F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

matrix A to be negative definite is that the principal minors of A, starting with that of the first-order,
be alternatively negative and positive.
The discussion of Lemma 6.2 is found in Howard Eves book (see [26]). We now state additional
hypotheses.
Q: Let V_ ˆ X T AX ; A ˆ faij gnn , where A is a real symmetric n  n matrix. Then the aij 's are
such that
(i) aij 2 C 0 …R‡  R‡  R‡ ; R†,
(ii) limx!x aij exist as a ®nite number, where x is rest point,
(iii) the aij are bounded.
Let the matrix A be given as in (6.8). Then
p…k; A† ˆ det…A ÿ kI†
 1 k2 ‡ m
2k ‡ m
 3 ˆ 0;
ˆ k3 ‡ m

…6:9†

where
 1 ˆ ÿ trace A ˆ ÿ…a11 ‡ a22 ‡ a23 †;
m





 a11 1 a12 
 a11 1 a13 
 a22





2
2
 2 ˆ det 
‡ det 
‡ det 
m


 1 a12 a22 
 1 a13 a33 
 1 a23
2
2
2

 3 ˆ ÿ det A:
m

1
a
2 23

a33




;


These reduce with a12 ˆ a13 ˆ 0 to
 1 ˆ ÿ …a11 ‡ a22 ‡ a33 †;
m
 2 ˆ a11 a22 ‡ a11 a33 ‡ a22 a33 ÿ 14a223 ;
m

…6:10†

 3 ˆ a11 …a22 a33 ÿ 14 a223 †:
m
Hence by the Routh±Hurwitz criterion and Lemma 6.1 (iii), the matrix A is negative de®nite if
 1 > 0;
m

 3 > 0 and m
1m
2 > m
 3:
m

…6:11†

A re®nement of the criteria (6.11) leads to the following theorem.
 ‡ is globally asymptotically stable with respect to
e 1 ‰x1 ; w;
 zŠ 2 R
Theorem 6.3. The rest point E
x1 wz
 ‡ if
solution trajectories initiating from int R
x1 wz
(i) a11 < 0; a22 < 0; a33 < 0, and
(ii) a22 a33 ÿ 14 a223 > 0.
In an alternative approach, using Frobenius theorem, we see that the leading principal minors
of A are


 a11 1 a12 
2
 and det A:
a11 ; det  1
a
a22 
2 12

F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

Thus A is negative de®nite if


 a11 1 a12 
2
 > 0;

a11 < 0; det  1
a
a22 
2 12

179

…6:12†

and det A < 0, by Lemma 6.2.
Since a12 ˆ a13 ˆ 0, we arrive at re®ned criteria for the negative de®niteness of A as
…i† a11 < 0;
…ii† a22 a33 ÿ

a22 < 0;
1 2
a
4 23

a33 < 0

and

> 0:

…6:13†

This agrees with Theorem 6.3.
e 2 [b
b ; bz ]
6.2. Global asymptotic stability of E
x2; w

In this section, criteria for global asymptotic stability of the 3-dimensional equilibrium
e
b ; bz Š or equivalently E2 ‰0; b
b ; bz Š with respect to solutions initiating from int R‡
E 2 ‰b
x2; w
x2; w
x2 wz will be
established.
We consider the subsystem (5.17) and choose the Liapunov function
1
b †2 ‡ 12 k3 …z ÿ bz †2 :
x2 ÿ b
x 2 ln x2 ‡ k2 …w ÿ w
…6:14†
V ˆ x2 ÿ b
bx 2 2
Let
h…x2 ; w† ˆ x2 h1 …x2 ; w†

and h1 …x2 ; w† ˆ wh2 …x2 ; w†:

…6:15†

Then using (3.33) and (3.43) we have
V_ ˆ …x2 ÿ b
x 2 †‰g2 …x2 † ÿ h1 …x2 ; w†Š
b †‰Q1 ÿ a1 e1 …w† ‡ f …w; z† ÿ bh…x2 ; w†Š
‡ k2 …w ÿ w
‡ k3 …z ÿ bz †‰Q2 ÿ a2 e2 …z† ÿ gf …w; z†Š:

…6:16†

Simplifying (3.44) leads to

b h2 …x2 ; w
b †Š ÿ …x2 ÿ w
b 2†w
b h2 …x2 ; w
b†
V_ ˆ …x2 ÿ b
x 2 †g2 …x2 † ÿ …x2 ÿ b
x 2 †‰wh2 …x2 ; w† ÿ w
b †‰a1 …e1 … w
b † ÿ e1 …w††Š ‡ k2 …w ÿ w
b †‰f …w; z† ÿ f … w
b ; bz †Š
‡ k2 …w ÿ w
b †‰h…b
b † ÿ h…x2 ; w†Š
‡ bk2 …w ÿ w
x2; w

b ; bz † ÿ f …w; bz †Š:
‡ k3 …z ÿ bz †‰a2 …e2 …bz † ÿ e2 …z††Š ‡ k3 g…z ÿ bz †‰f … w

…6:17†

We now set V_ ˆ X T BX with
0 1 0
1
x2 ÿ b
x2
v1
b A;
X ˆ @ v2 A ˆ @ w ÿ w
v3
z ÿ bz

where

0

b11
1
@
B ˆ 2 b12
1
b
2 13

1
b
2 12

b22
1
b
2 23

1
b
2 13
1
b
2 23

b33

1

A with b13 ˆ b31 ˆ 0:

…6:18†

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F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

Note that bij ˆ bji . Thus B is a real and symmetric 3  3 matrix, such that
1
B ˆ …B ‡ Bt †:
2
In particular the bij 's are de®ned as
b h2 …x2 ; w
b †;
b11 ˆ g2 …x2 † ÿ w
b h2 …x2 ; w
b†
b † ÿ h…x2 ; w†Š
wh2 …x2 ; w† ÿ w
‰h…b
x2; w
‡ bk2
;
b12 ˆ b21 ˆ
b†
…w ÿ w
…x2 ÿ b
x2†
b13 ˆ b31 ˆ 0;

b † ÿ e1 …w†Š
‰e1 … w
b22 ˆ k2 a1
;
b†
…w ÿ w
‰e2 …bz † ÿ e2 …z†Š
:
b33 ˆ k3 a2
…z ÿ bz †

…6:19†

The leading principal minors of B are


 b11 1 b12 
2
 and det B:

b11 ; det  1
b
b22 
2 12

By Frobenius' theorem, B will be negative de®nite if


 b11 1 b12 
2
 > 0 and det B < 0:

b11 < 0; det  1
b
b22 
2 12

…6:20†

But b13 ˆ b31 ˆ 0 and hence (3.50) simpli®es the criteria
…i† b11 < 0;

b22 < 0;

b33 < 0;

b12 < 0;

…ii† b11 b22 ÿ 14b212 > 0;
…iii† b22 b33 ÿ

1 2
b
4 23

…6:21†

> 0:

This leads to the following theorem.
 ‡ is globally asymptotically stable with respect to
e 2 ‰b
b ; bz Š 2 R
Theorem 6.4. The rest point E
x2; w
x2 wz
‡

solution trajectories initiating from int Rx2 wz if
(i) b11 < 0; b22 < 0; b33 < 0; b12 < 0, and
(ii) b11 b22 ÿ 14 b212 > 0,
(iii) b22 b33 ÿ 14 b223 > 0.
 z] in R‡
6.3. Global asymptotic stability of E0 [0; 0,w,
wz
 ‡ as depicted by (5.11).
Consider system (3.1) restricted to R
wz
e 0 ‰w ; z Š and consequently E0 ‰0; 0; w ; zŠ exists
We have shown that the 2-dimensional equilibrium E
if Lemma 5.2 holds. In this section, we shall establish criteria for the global asymptotic stability of
 
‡ .
E0 ‰0; 0; w; zŠ with respect to solutions emanating from the interior of R
wz

F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

181

 ‡ . We choose the Liapunov function V such
Let G be a neighborhood of any point in R
wz
that
 2

 2

V ˆ 12 c1 …w ÿ w† ‡ 12c2 …z ÿ z† :

…6:22†

Note that
 
‡ .
(i) V is positive de®nite with respect to E0 ‰0; 0; w; zŠ in R
wz
2
2
(ii) V ! 1 as w ‡ z ! 1.
(iii) V is a Liapunov function for (5.11) in G.
(iv) V 2 C 0 …R2‡ ; R† and is bounded below.
Now


V_ ˆ c1 …w ÿ w†w_ ‡ c2 ‰z ÿ zŠ_z

…6:23†

along the solution trajectories of (3.2). From (6.23) we obtain the expression

V_ ˆ c1 …w ÿ w†‰Q1 ÿ a1 e1 …w† ‡ f …w; z†Š


‡ c2 …z ÿ z†‰Q2 ÿ a2 e2 …z† ÿ gf …w; z†Š;

…6:24†

or


 
V_ ˆ c1 …w ÿ w†‰a1 e1 …w† ÿ f …w; z† ÿ a1 e1 …w† ‡ f …w; z†Š


 



‡ c2 …z ÿ z†‰a2 e2 …z† ‡ gf …w; z† ÿ a2 e2 …z† ÿ gf …w; z†Š






 

ˆ a1 c1 …w ÿ w†‰e1 …w† ÿ e1 …w†Š ‡ c1 …w ÿ w†‰f …w; z† ÿ f …w; z†Š




 

‡ a2 c2 …z ÿ z†‰e2 …_z† ÿ e2 …z†Š ‡ gc2 …z ÿ z†‰f …w; z† ÿ f …w; z†Š:
In particular,
V_ ˆ X T CX ;
where
Cˆ



where X ˆ

1
c
2 12

c11
1
c
2 12

c22



v1
v2



ˆ



…6:25†

 
wÿw ;

zÿz


;

…6:26†

and


c11 ˆ ÿ a1 c1

‰e1 …w† ÿ e1 …w†Š


…w ÿ w†

;

 

c12 ˆ c21 ˆ c1

‰f …w; z† ÿ f …w; z†Š


…z ÿ z†


c22 ˆ ÿ a2 c2

‰e2 …z† ÿ e2 …z†Š


…z ÿ z†

:

 

ÿ gc2

‰f …w; z† ÿ f …w; z†Š


…w ÿ w†

;

…6:27†

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F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

De®ne
A0 ˆ






Q1
Q2
‡

…w; z† 2 Rwz 0 6 w 6 ÿ ; 0 6 z 6 ; d1 < 0; d2 > 0 ;
d1
d2

where d1 and d2 are as de®ned by expressions (4.2) and (4.4).
We now de®ne the sets




S1 ˆ …w; z† 2 A0 \ int R‡
wz V …w; z† ˆ 0 ;
o
n

w ˆ w ; z ˆ z :
S2 ˆ …w; z† 2 int R‡
wz

…6:28†

…6:29†
…6:30†

By inspection, we see immediately that
S1  S2 :

Now de®ne the set E as follows:
n
o

V_ ˆ 0 \ G:

E ˆ …w; z† 2 R‡
wz

…6:31†
 

Then the largest invariant set in E is E0 …0; 0; w; z† restricted to R‡
wz .
e 0 ‰w ; zŠ or consequently
Hence by LaSalle's Invariant Principle, cf. [27±29], we conclude that E
 
E0 ‰0; 0; w; zŠ is globally asymptotically stable with respect to solutions initiating from int R‡
wz if the
matrix C is negative de®nite.
 
 ‡ is globally asymptotically stable with respect to
Theorem 6.5. The equilibrium E0 …0; 0; w; z† 2 R
wz
‡

solution trajectories emanating from int Rwz if
(i) c11 < 0; c22 < 0, and
(ii) c11 c22 ÿ 14 c212 > 0.

Proof. The proof follows from computing the leading principal minors of (6.26) and using the
Frobenius theorem, see Lemma 6.2 or alternatively by means of Lemma 6.1. 
7. Persistence, uniform persistence and existence of E3 [x1 ; x2 ; w ; z ]
In this section, we shall present results on persistence, uniform persistence and ®nally give
sucient criteria for the existence of a positive interior equilibrium E3 ‰x1 ; x2 ; w ; z Š.
Theorem 7.1. Assume system (3.1) is such that
 
(i) E0 …0; 0; w; z† is a hyperbolic saddle point and is repelling in the x1 and x2 -directions locally (see
Theorem 5.4)
 z† is a hyperbolic saddle point and is repelling in the x2 -direction locally (see
(ii) E1 …x1 ; 0; w;
Theorem 5.8)
b ; bz † is a hyperbolic saddle point and is repelling in the x1 -direction locally (see
x2; w
(iii) E2 …0; b
Theorem 5.10)
‡
(iv) system (3.1) is dissipative and solutions initiating in int R
x1 x2 wz are eventually uniformly
bounded (see Theorem 5.1)

F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

183

 

 z† and E2 ‰0; b
b ; bz Š are globally asymptotically stable
(v) the equilibria E0 …0; 0; w; z†, E1 …x1 ; 0; w;
x2; w
‡ ‡
‡


with respect to Rwz , Rx1 wz and Rx2 wz , respectively, (see Theorems 6.3±6.5).
Then system (3.1) exhibits (robust) persistence.
Proof. The proof will be done using the Butler±McGehee Lemma (cf. [30]). Let




Q1
Q2
‡

 R4‡ ;
B ˆ …x1 ; x2 ; w; z† 2 Rx1 x2 wz 0 6 x1 6 K1 ; 0 6 x2 6 K2 ; 0 6 w 6 ÿ ; 0 6 z 6
d1
d2

where d1 , and d2 are as de®ned by (4.2) and (4.4).We have shown in Theorem 5.1 that B is
positively invariant and any solution of system (3.1) initiating at a point in B  R4‡ is eventually
 
 z† and E2 ˆ E2 ‰0; b
b ; bz Š are the only
bounded. However E0 ˆ E0 …0; 0; w; z†, E1 ˆ E1 …x1 ; 0; w;
x2; w
compact invariant sets on oR4‡ . Let M ˆ E3 ‰x1 ; x2 ; w ; z Š be such that M 2 int R4‡ .
The proof is completed by showing that no point Qi 2 oR4‡ belongs to X…M†. The proof is
divided into ®ve steps.
Step 1. We show that
E0 62 X…M†:

Suppose E0 2 X…M†. Since E0 is hyperbolic, E0 6ˆ X…M†. By the Butler±McGehee lemma, there
4
‡
‡
‡
exists a point Q‡
0 2 W …E0 †nfE0 g† such that Q0 2 X…M†. But W …E0 † \ …R‡nfE0 g† ˆ ;. This
4
contradicts the positive invariance property of B  R‡ . Thus E0 62 X…M†.
Step 2. We show that
E1 62 X…M†:
‡
‡
If E1 2 X…M†; then there exists a point Q‡
1 2 W …E1 †nfE1 g such that Q1 2 X…M† by the Butler±
 zŠ is globally asymptotically stable
McGehee lemma. But W ‡ …E1 † \ …int R4‡ † ˆ ; and E1 ‰x1 ; 0; w;
‡
‡
with respect to Rx1 wz . This implies that the closure of the orbit O…Q‡
1 † through Q1 either contains
E0 or is unbounded. This is a contradiction. Hence E1 62 X…M†:
Step 3. We show that

E2 62 X…M†:
The proof is similar to Step 2.
Step 4. We show that
oR4‡ \ X…M† ˆ ;:
Suppose oR4‡ \ X…M† ˆ
6 ;. Let Q 2 oR4‡ and Q 2 X…M†. Then, the closure of the orbit through Q,
i.e., O…Q† must either contain E0 ; E1 ; E2 or is unbounded.
This gives a contradiction.
Step 5. Thus we see that if E0 is unstable then
W ‡ …E0 † \ …R4‡ j fE0 g† ˆ ;:
Also, we deduce that if E1 is unstable, then
W ‡ …E1 † \ …int R4‡ † ˆ ;;
6 ;:
W ÿ …E1 † \ …R4 nR4‡ † ˆ

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F. Nani, H.I. Freedman / Mathematical Biosciences 163 (2000) 159±199

Similarly if E2 is unstable, then
W ‡ …E2 † \ …int R4‡ † ˆ ;;
6 ;;
W ÿ …E1 † \ …R4 nR4‡ † ˆ
and the persistence result follows since X…M† must be in int R4‡ .