A REACTION-DIFFUSION MODEL OF
LEUKEMIA TREATMENT BY
CHEMOTHERAPY
WENXIANG LIU AND H. I. FREEDMAN
ABSTRACT. A model of leukemia treatment by chemother-apy techniques is proposed utilizing a system of reaction diffu-sion equations representing the change in densities of normal cells, competing cancer cells, and chemotherapy in a given or-gan or area. We view the interactions between normal and cancer cells as being competitive for available resources, and we think of the chemotherapy agent as a predator on both normal and leukemia cells. The existence, uniqueness, and bounded-ness of the solutions are established by means of a comparison principle and a monotonicity method. We analyze the con-stant solutions and their stabilities. The main method used in studying the stability is the spectral analysis of the linearized operators. Persistence criteria for the normal cells and cancer cells are also derived. The analysis is carried out both analyti-cally and numerianalyti-cally.
1 Introduction Leukemia can be described as the disorganization of the hematopoietic system in which a malignant clone of cells acts to impede the growth of normal hemopoietic tissue. When leukemia develops, the body produces large numbers of abnormal blood cells. In most types of leukemias, the abnormal cells are white blood cells, and they usually look different from normal blood cells, and do not function properly. Leukemia is either acute or chronic. In acute leukemia, the abnormal blood cells are blasts that remain very immature and cannot carry out their normal functions. The number of blasts increases rapidly, and the disease gets worse quickly. In chronic leukemia, some blast cells are present, but in general, these cells are more mature and can carry out some of their normal functions. Also, the number of blasts increases less rapidly than in acute leukemia. As a result, chronic leukemia gets Keywords: Competition models, chemotherapy, persistence, lower and upper solutions.
Copyright cApplied Mathematics Institute, University of Alberta. 249
worse gradually. However, without treatment, even the chronic disorders may be fatal [1, 2].
It has been postulated [3–6] that the origins of acute leukemia can be found in pluripotent stem cells. Therefore, it is suggestive that in the acute leukemia state a pluripotent stem cell in the bone marrow becomes malignant, proliferates and displaces normal cells in the marrow. These abnormal cells fill the blood and the marrow and produce a malfunction of the body’s immune system. Also, leukemia cells inhibit the colony-forming capabilities of the normal proliferative cells. Consequently, the body’s hematopoietic system becomes disorganized.
In cancer treatment today, four types of treatment are most com-monly used in efforts to obtain long-term periods of disease-free re-mission. These include surgery, radiotherapy, chemotherapy and im-munotherapy. Cancer chemotherapy has demonstrated a definite capac-ity for controlling disseminated metastatic cancer and is therefore widely used (Dorr and Von Hoff [7], Frei [8], Liotta [9], Perry [10]). In cancer chemotherapy, anti-neoplastic drugs are designed to selectively destroy or inhibit the proliferative activity of cancer cells while the normal cells are affected to a lesser extent (Dorr and Von Hoff [7], Frei [8]).
The ultimate role of mathematical modelling in cancer chemother-apy is to provide a more rational basis for experimental design of the anti-cancer drugs and to make qualitative predictions with regard to the dynamic evolution of the disease based on the cytokinetic parameters of the patient and the drug parametric configuration. The fact that it is reasonable to view the interaction between normal cells and cancer cells as competitive is justified in Freedman [11] and Nani and Freedman [12, 13]. In [12], a model of cancer treatment by chemotherapy was pre-sented and conditions for the boundedness of solutions were analyzed. The equilibria and their stabilities, and conditions for the existence of small amplitude periodic solutions were also discussed. Persistence and extinction criteria of the normal cells and cancer cells were also derived. It is well known that the distributions of populations in general, being heterogeneous, depend not only on time, but also on the spatial po-sitions in the habitat. So it is natural and more precise to study the corresponding P.D.E. problem as suggested by the authors in [12]. For a detailed explanation of the ecological background of the problem, the reader is referred to [12]. Motivated by the conception of persistence in [14, 15], we introduce it into this paper and establish the existence, uniqueness, boundedness and persistence of the solutions for the Neum-man boundary condition problem by means of a comparison principle and a monotonicity method, (see e.g., [17, 18]). The main method used
in studying the stability of constant solutions is the spectral analysis of the linearized operators.
The idea of modelling cancer interactions with healthy tissue as a competition process was first proposed by Gatenby [21]. However, his paper did not consider treatment. The first paper to incorporate chemother-apy treatment was Nani and Freedman [12]. An extension to modelling cancer at several sites (metastasis) with chemotherapy treatment was carried out in [13].
This paper utilizes a special case (logistic) of the model developed in [12] (see [12] for the derivation of the model), and extends it to the diffusion case to model the spread of cancer within a site (such as leukemia in the bone marrow).
The organization of this paper is as follows. In Section 2 we describe the model. Section 3 deals with the no diffusion case, whereas in Sec-tion 4, we analyze the diffusion case. We conclude with a final secSec-tion which contains numerical examples to illustrate our results and a short discussion.
2 The model We take as our model of leukemia treatment by chemotherapy a system of reaction diffusion equations where u1(x, t) represents the density of normal cells, u2(x, t) the density of leukemia cells, andv(x, t) the density of chemotherapy agents in the affected re-gion at time t ≥ 0. We view u1(x, t) and u2(x, t) as competing for nutrient, oxygen, etc. and we think of v(x, t) as a predator capable of destroying both u1(x, t) and u2(x, t), but selectively is more lethal to
u2(x, t). The model then takes the form
(1) ∂u1 ∂t =D1 ∂2u 1 ∂x2 +α1u1 1− u1 K1 −q1u1u2− p1u1v a1+u1 , ∂u2 ∂t =D2 ∂2u 2 ∂x2 +α2u2 1− u2 K2 −q2u1u2− p2u2v a2+u2 , ∂v ∂t =D3 ∂2v ∂x2 + ∆− ξ+ c1u1 a1+u1 + c2u2 a2+u2 v , with initial conditions
u1(x,0) =u01(x), u2(x,0) =u02(x), v(x,0) =v0(x) in Ω, and Neumann homogeneous boundary conditions
∂u1 ∂n = ∂u2 ∂n = ∂v ∂n= 0 on∂Ω,
where Ω is an open boundary region with smooth boundary ∂Ω,∂/∂n denotes the derivative along the outward normal,u0
1(x), u02(x), v0(x), are smooth functions on Ω.
The constants in system(1) may be interpreted as follows:
αi, i= 1,2,are the specific birth rates of the normal and cancer cells
for small densities.
Ki, i= 1,2,are the respective carrying capacities.
qi, i = 1,2, are the competition coefficients between u1(x, t) and
u2(x, t).
pi, i= 1,2,are the predation coefficients ofv(x, t) onui(x, t).
ai, i= 1,2,determine the speeds at which ui(x, t), in the absence of
competition and predation, reaches carrying capacity.
∆ is the infusion rate of the chemotherapy to the specific region. ξ is the loss of poison for the chemotherapy agent within this region. ci, i= 1,2 are the combination rates of the chemotherapy agent with
the cells. Hence they are proportional topi,i= 1,2.
All constants are positive. To make this model more realistic, we impose certain inequalities among the parameters. It is well known that leukemia cells grow at a much faster rate than normal cells. Further, if no treatment is offered, most of time leukemia cells out-compete the normal cells independent of initial conditions. Furthermore, the chemotherapy agent must be considerably more effective in killing leukemia than in killing normal cells in order for the treatment to be effective. These lead to the following sets of inequalities:
α2> α1, p2>> p1.
In addition, there are other inequalities which we will list in the next section since they depend on homogeneous steady states values.
3 The no diffusion case When there is no spatial variation we obtain a system of ordinary differential equations of the form.
(2) du1 dt =α1u1 1− u1 K1 −q1u1u2− p1u1v a1+u1 , du2 dt =α2u2 1− u2 K2 −q2u1u2− p2u2v a2+u2, dv dt = ∆− ξ+ c1u1 a1+u1 + c2u2 a2+u2 v,
with initial conditions
u1(0) =u01>0, u2(0) =u02≥0, v(0) =v0>0.
At this point we will establish two important properties of solutions to system (2).
1. All solutions with positive initial values remain positive.
Proof. By uniqueness of solutions, since u1 ≡ 0 is a solution of the
first equation of (2), no solution with u1(t)>0 at any timet ≥0 can become zero in finite time. Similarly, the same is true foru2(t). Since
v(0) = ∆ > 0, no solution v(t) of (2) with v(t) >0 can become zero.
2. System (2)is dissipative.
Proof. Since the initial conditions are nonnegative, then so are the
solutions. From (2), we have du1(t) dt ≤α1u1(t)(1− u1(t) K1 ), du2(t) dt ≤α2u2(t)(1− u2(t) K2 ). From standard comparison theory, we obtain
lim sup t→∞ u1(t)≤K1, lim sup t→∞ u2(t)≤K2. Similarly,dv(t)/dt≤∆−ξv(t) gives lim sup t→∞ v(t)≤ξ−1∆.
Hence, the region < ={(u1, u2, v) ∈ R+3 : 0 ≤u1 ≤ K1,0 ≤u2 ≤
K2,0≤v ≤ξ−1∆}is an attracting invariant region proving the prop-erty.
Definition.In analytical terms persistence means that lim inft→∞x(t)>
0 for each populationx(t); in geometric terms, that each trajectory of the modelling system of differential equations is eventually bounded away from the coordinate planes.
3.1 The homogeneous steady states These equilibria are: E0(0,0, ξ−1∆), E1(ˆu1,0,ˆv), E2(0,u˜2,v˜), E3(u∗1, u ∗ 2, v ∗ ). Note that the trivial steady state E0(0,0, ξ−1∆) always exists, and
E1(ˆu1,0,ˆv),E2(0,u˜2,˜v),E3(u∗1, u
∗
2, v
∗
) may or may not exist. In particu-larE1(ˆu1,0,ˆv) represents the steady state where leukemia is eliminated, which is the most desirable state. E2(0,u˜2,v˜) represents the case where the leukemia has completely taken over the site. E3(u∗1, u
∗
2, v
∗) means
coexistence which is only desirable for small numbers of leukemia cells. The equilibriumE1(ˆu1,0,ˆv) exists provided that the algebraic system
(3) α1 1− u1 K1 − p1v a1+u1 = 0, ∆− ξ+ c1u1 a1+u1 v= 0 has a positive solution (ˆu1,ˆv).
This system has a unique positive solution provided
(4) P1∆< α1a1ξ .
Necessary and sufficient conditions for (3) to have two positive solu-tions are (5) ξa1< K1(ξ+c1), ξa1< p1∆ α1 < α1[a1ξ−K1(ξ+c1)] 2 4K1(ξ+c1) .
Analogously,E2(0,u˜2,v˜) exists provided that the algebraic system
(6) α2 1− u2 K2 − p2v a2+u2 = 0, ∆− ξ+ c2u2 a2+u2 v= 0 has a positive solution (˜u2,˜v).
Similar to the analysis ofE1, system (6) has a unique positive solution provided
(7) P2∆< α2a2ξ,
and exactly two positive solutions if
(8) ξa2< K2(ξ+c2), ξa2< p2∆ α2 < α2[a2ξ−K2(ξ+c2)]2 4K2(ξ+c2) . As a result, we have the following theorems.
Theorem 1. If(4)holds, thenE1(ˆu1,0,ˆv)exists uniquely. If(5)holds,
then there exists two distinct equilibria of typeE1(ˆu1,0,vˆ).
Theorem 2. If(7)holds, thenE2(0,u˜2,v˜)exists uniquely. If(8)holds,
there exists two distinct equilibria of type E2(0,u˜2,˜v).
In the next sections, we will analyze both cases when model (2) has only one or more than one solution of type E1(ˆu1,0,ˆv), E2(0,u˜2,˜v). Here we have their coordinates. In the case of only one solution of each type, we consider just the first solution with the following coordinates.
ˆ u1= α1[K1(ξ+c1)−a1ξ] 2α1(ξ+c1) ±{α 2 1[a1ξ−K1(ξ+c1)]2−4α1K1(ξ+c1)(p1∆−α1a1ξ)} 1 2 2α1(ξ+c1) , ˆ v= ∆(α1+ ˆu1) ξa1+ (ξ+c1)ˆu1 , ˜ u1= α2[K2(ξ+c2)−a2ξ] 2α2(ξ+c2) ±{α 2 2[a2ξ−K2(ξ+c2)]2−4α2K2(ξ+c2)(p2∆−α2a2ξ)} 1 2 2α2(ξ+c2) , ˜ v= ∆(α2+ ˜u2) ξa2+ (ξ+c2)˜u2 .
3.2 Local stability In order to compute the stability of the various equilibria of system (2), we let M be the Jacobian matrix about the point (u1, u2, v). Then M = α1−2α1u1 K1 −q1u2− a1p1v (a1+u1)2 −q1u1 − p1u1 a1+u1 −q2u2 α2− 2α2u2 K2 −q2u1− a2p2v (a2+u2)2 − p2u2 a2+u2 − a1c1v (a1+u1)2 a2c2v (a2+u2)2 − ξ+ c1u1 a1+u1 + c2u2 a2+u2 . ComputingM atE0(0,0, ξ−1∆), we get M0= α1−p1∆ a1ξ 0 0 0 α2− p2∆ a2ξ 0 −c1∆ a1ξ −c2∆ a2ξ −ξ ,
and the eigenvalues are λ1=α1− p1∆ a1ξ , λ2=α2− p2∆ a2ξ , λ3=−ξ . As a result, we have: Theorem 3. If P1∆ < α1a1ξ or P2∆ < α2a2ξ, then E0(0,0, ξ−1∆)
is a hyperbolic saddle point. If P1∆ > α1a1ξ and P2∆ > α2a2ξ, then
It is easy to see that one or both ofE1(ˆu1,0,ˆv),E2(0,u˜2,v˜) will exist ifE0(0,0, ξ−1∆) is a hyperbolic saddle point, and neither of them will exist ifE0(0,0, ξ−1∆) is stable. ComputingM atE1(ˆu1,0,vˆ), we obtain M1= α1 1−2ˆu1 K1 − a1p1vˆ (a1+ ˆu1)2 −q1uˆ1 − p1uˆ1 a1+ ˆu1 0 α2−q2uˆ1− p2vˆ a2 0 − c1∆ (a1+ ˆu1)2 −c2ˆv a2 − ξ+ c1uˆ1 a1+ ˆu1 .
Hence the eigenvalues are
λ2=α2−q2uˆ1−a−21p2v ,ˆ σ(A) ={λiλ2 −Tr (A)λ+ det(A) = 0, i= 1,3}, where A= α1 1−2ˆu1 K1 − a1p1vˆ (a1+ ˆu1)2 − p1uˆ1 a1+ ˆu1 − a1c1vˆ (a1+ ˆu1)2 − ξ+ c1uˆ1 a1+ ˆu1 .
By the Routh-Hurwitz criteria, if Tr (A) <0 and det(A) >0, then the eigenvalues ofAhave negative real parts.
If ˆu1> K1/2, then Tr (A) =α1− 2α1 K1 ˆ u1− a1p1vˆ (a1+ ˆu1)2 − ξ+ c1uˆ1 a1+ ˆu1 <0
and det(A) = α1 1−2ˆu1 K1 − a1p1vˆ (a1+ ˆu1)2 ξ+ c1uˆ1 a1+ ˆu1 − a1c1p1uˆ1vˆ (a1+ ˆu1)2 =−α1 1−2ˆu1 K1 ξ+ c1uˆ1 a1+ ˆu1 + p1α1∆ξ (a1+ ˆu1)[a1ξ+ (ξ+a1)ˆu1] >0.
As a result, we have the following theorem.
Theorem 4. Suppose that uˆ1 > K1/2 and α2 6= q2uˆ1 +a−21p2vˆ. If
α2 > q2uˆ1+a−21p2vˆ, then E1(ˆu1,0,vˆ) is a hyperbolic saddle point. If
α2< q2uˆ1+a−21p2vˆ, thenE1(ˆu1,0,ˆv)is locally asymptotically stable.
0 10 20 30 40 50 60
800 1000 1200 1400
1600 Trajectories of Populations in the Case of No Diffusion.
u 1 (t) 0 10 20 30 40 50 60 −1 0 1 2 u2 (t) 0 10 20 30 40 50 60 90 95 100 time t v(t)
FIGURE 1: Solutions for model (2) with α1 = 1.5, α2 = 10.0, K1 = 1460.0, K2 = 2100.0, q1 = 0.0075, q2 = 0.005, p1 = 0.0008, p2 = 0.08, a1 = 1.0, a2 = 1.0, c1 = 0.0024, c2 = 0.6,∆ = 2000.0, ξ = 20;u1(0) = 800.0, u2(0) = 0.1, v(0) = 90.0. Here the boundary equi-libriumE1(1460.0,0,100.0) is locally stable.
ComputingM atE2(0,u˜2,v˜), one obtains M2= α1−q1u˜2− p1˜v a1 0 0 −q2u˜2 α2− 2α2 K2 ˜ u2− a2p1v˜ (a2+ ˜u2)2 − p2u˜2 a2+ ˜u2 −c1v˜ a1 − a2c2˜v (a2+ ˜u2)2 − ξ+ c2u˜2 a2+ ˜u2 .
Hence the eigenvalues are
λ1=α1−q1u˜2−a−11p1v ,˜ σ(B) ={λi|λ2−Tr (B)λ+ det(B) = 0, i= 2,3}, where B= α2 1−2˜u2 K2 − a2p2˜v (a2+ ˆu2)2 − p2u˜2 a2+ ˜u2 − a1c1˜v (a2+ ˜u2)2 − ξ+ c2u˜2 a2+ ˜u2 .
Similar to the analysis ofE1, we have the following lemma.
Lemma 1 If u˜2 > K2/2, then the real parts of eigenvalues λ2, λ3 are
negative.
Based on Lemma 1, we have the following theorem.
Theorem 5. Suppose that u˜2 > K2/2 and α1 6= q1u˜2 +a−11p1˜v. If
α1 > q1u˜2+a−11p1v˜, then E2(0,u˜2,v˜) is a hyperbolic saddle point. If
α1< q1u˜2+a−11p1v˜, thenE2(0,u˜2,˜v)is locally asymptotically stable. Now we wish to examine criteria for there to be no limit cycles in the u1−vplane,u2−v plane, and theu1−u2plane.
In theu1−v plane: du1 dt =α1u1 1− u1 K1 − p1u1v a1+u1 dv dt = ∆− ξ+ c1u1 a1+u1 v .
0 1 2 3 4 5 6 7 8 9 10 −500
0 500 1000
1500 Trajectories of Populations in the Case of No Diffusion.
u 1 (t) 0 1 2 3 4 5 6 7 8 9 10 0 1000 2000 3000 u 2 (t) 0 1 2 3 4 5 6 7 8 9 10 90 92 94 96 98 time t v(t)
FIGURE 2: Solutions for model (2) approach E2(0,2100,97.6) if the treatment intensity is weak (small values ofa2−1p2 and/or ξ−1∆) with a large initialu2(0) = 1.0. Other parameters and initial conditions are the same as in Figure 1 exceptp2= 0.04.
Using Dulac’s negative criterion, we define D(u1, v) = ∂ ∂u1 a 1+u1 p1u1 α1u1 1− u1 K1 − p1u1 a1+u1 v + ∂ ∂v a 1+u1 p1u1 ∆− ξ+ c1u1 a1+u1 v =α1(K1−a1) p1K1 −2α1u1 p1K1 −ξ(a1+u1) p1u1 − c1 p1 .
Clearly,D(u1, v)<0 foru1, v >0 ifK1< a1. Therefore, ifK1< a1, then there are no periodic solutions in theu1−v plane forD(u1, v).
A similar statement holds for the corresponding system in theu2−v plane, that is, if K2 < a2, then there are no periodic solutions in the
0 1 2 3 4 5 6 7 8 9 10 80
90 100 110
120 Trajectories of Populations in the Case of No Diffusion.
u1 (t) 0 1 2 3 4 5 6 7 8 9 10 0 50 100 150 200 u2 (t) 0 1 2 3 4 5 6 7 8 9 10 920 940 960 980 1000 time t v(t)
FIGURE 3: A solution for model (2) with α1 = 13.9, α2 = 37.0, K1 = 125.0, K2 = 167.0, q1 = 0.08, q2 = 0.08, p1 = 0.0005, p2 = 18.0, a1 = 20.0, a2 = 801.0, c1 = 0.001, c2 = 36.0,∆ = 50000.0, ξ = 50;u1(0) = 120.0, u2(0) = 160.0, v(0) = 1000.0. Here system (2) is uniformly persis-tent and the interior equilibriumE3(90.0,40.0,960.0) is locally stable. Since there is no equilibrium in theu1−u2plane, there are no periodic solutions in this plane.
Based on the above results, we may address the question of an interior equilibrium in u1−u2−v space by using the techniques in Freedman and Waltman [15, 16], and the results in Butler et al. [17], and obtain the following theorem.
Theorem 6. Suppose thatK1≤min{a1,2ˆu1}andK2≤min{a2,2˜u2}.
If α1 > q1u˜2+a−11p1v˜ and α2 > q2uˆ1+a2−1p2ˆv, then system (2) is
uniformly persistent, and henceE3(u∗1, u∗2, v∗)exists.
4 The diffusion case To study the effects of spatial variations, we first note that the non-uniform diffusive steady state produces equations that can not be solved in closed form. We therefore consider the effects of small space - time perturbations of the uniform steady states,E0,E1,
E2,E3. Let us assume thatx=sfor the case of one spatial dimension with 0≤s≤a, whereais a constant. This spatial dimension could be measured over a section of the bone marrow or over the total space of distribution where leukemia activity may be significant. Before we study the stabilities of these steady states, we first establish the existence and uniqueness of solutions of system (1).
4.1 Preliminaries In this section we introduce the concept of upper and lower solutions as well as an existence-comparison theorem, which will be very useful to us in establishing the existence, uniqueness, and boundedness, and even in studying the asymptotic behavior (in some sense) of the solutions.
We first consider the more general system
(9) ∂u1 ∂t −L1u1=f1(u1, u2, u3), ∂u2 ∂t −L2u2=f2(u1, u2, u3), ∂u3 ∂t −L3u3=f3(u1, u2, u3), with boundary condition
Bi[ui] =αi(x)ui+βi(x)∂ui
∂n =hi(x), i= 1,2,3 on∂Ω×R
+,
and initial condition
ui(x,0) =u0i(x), i= 1,2,3 in Ω,
whereLi is a uniformly elliptic operator in Ω,i= 1,2,3.
We assume thatαi,βiandu0i are smooth nonnegative functions with
u0
i 6= 0 , αi+βi > 0 and that fi is continuously differentiable with
respect to its variables foruk≥0,i, k= 1,2,3. In addition, we require
that f = (f1, f2, f3) is a quasi-monotone function, i.e.:
∂f1 ∂u2 ≤0, ∂f1 ∂u3 ≤0, ∂f2 ∂u1 ≤0, ∂f2 ∂u3 ≤0, ∂f3 ∂u1 ≤0, ∂f3 ∂u2 ≤0
forui≥0,i= 1,2,3.
Now, we give the definition of upper and lower solutions.
Definition 4.1 Ordered smooth functions ¯u = (¯u1,u¯2,u¯3) and u = (u1, u2, u3) in ΩT are called upper and lower solutions of (9) respectively,
if they satisfy the following inequalities
(¯u1)t−L1u¯1−f1(¯u1, u2, u3)≥(u1)t−L1u1−f1(u1,u¯2,¯u3), (¯u2)t−L2u¯2−f2(u1,u¯2, u3)≥(u2)t−L2u2−f2(¯u1, u2,¯u3), (¯u3)t−L3u¯3−f3(u1, u2,u¯3)≥(u3)t−L3u3−f3(¯u1,u¯2, u3) in ΩT. Bi[¯ui]≥hi(x)≥Bi[ui], i= 1,2,3 onST, ¯ ui(x,0)≥u0i(x)≥ui(x,0), i= 1,2,3 on Ω,
where ΩT = Ω×(0, T], ST = ∂Ω×(0, T], and T < ∞ but can be
arbitrarily large.
Suppose ¯uanduexist. Denote
Σ ={(u1, u2, u3)∈ <3:ρi≤ui≤ρ¯i, i= 1,2,3}, Mi= sup P ∂fi ∂ui , i= 1,2,3,
whereρi= inf(x,t)∈ΩTui(x, t), ¯ρi= sup(x,t)∈ΩT u¯i(x, t), i= 1,2,3.
We construct the sequences {u¯(k)} and {u(k)} with ¯u(0) = ¯u and
u(0)=uas follows: (¯u(1k))t−L1u¯1(k)+M1u¯1(k)=M1u¯(k −1) 1 +f1(¯u(k −1) 1 , u (k−1) 2 , u (k−1) 3 ), (¯u(2k))t−L2u¯2(k)+M2u¯2(k)=M2u¯(k −1) 2 +f2(u(k −1) 1 ,¯u (k−1) 2 , u (k−1) 3 ), (¯u(3k))t−L3u¯3(k)+M3u¯3(k)=M3u¯(k −1) 3 +f3(u(k −1) 1 , u (k−1) 2 ,u¯ (k−1) 3 ), (u(1k))t−L1u1(k)+M1u1(k)=M1u(k −1) 1 +f1(u(k −1) 1 ,¯u (k−1) 2 ,u¯ (k−1) 3 ), (u(2k))t−L2u2(k)+M2u2(k)=M2u(k −1) 2 +f2(¯u(k −1) 1 , u (k−1) 2 ,u¯ (k−1) 3 ), (u(3k))t−L3u3(k)+M3u3(k)=M3u(k −1) 3 +f3(¯u(k −1) 1 ,¯u (k−1) 2 , u (k−1) 3 ),
and Bi[¯u(ik)] =hi(x) =Bi[ui(k)], i= 1,2,3, (x, t)∈ST, ¯ u(ik)(x,0) =u0i(x) =u (k) i (x,0), i= 1,2,3, x∈Ω.
By using standard techniques (e.g., C. V. Pao [18]), we can establish the following existence-comparison theorem.
Theorem 7. Suppose thatf = (f1, f2, f3)is a quasi-monotone function
and there exists a pair of upper and lower solutionsu¯= (¯u1,u¯2,¯u3)and
u= (u1, u2, u3)satisfyingui≤u¯i,i= 1,2,3. Then the sequences{u¯(k)}
and {u(k)} obtained as above converge monotonically from above and
below, respectively, to a unique solutionu= (u1, u2, u3)of(9)such that
ui(x, t)≤ui(x, t)≤u¯i(x, t), i= 1,2,3, (x, t)∈ΩT.
In view of Theorem 7, to obtain the existence and uniqueness of solutions of (1), we need only to find a pair of upper and lower solutions of (1). We do this as follows by using an appropriate O.D.E. problem to find upper and lower solutions.
For an upper solution, we study the O.D.E. system du1 dt =α1u1 1− u1 K1 , du2 dt =α2u2 1− u2 K2 , dv dt = ∆−ξv, with initial conditions
ui(0) = ˜ui≡sup Ω u0i(x)>0, i= 1,2,3, v(0) = ˜v≡sup Ω v0(x)>0.
Then we have ¯ u1(t) =K1 1 + K1−u˜1 ˜ u1 e −α1t −1 , ¯ u2(t) =K2 1 + K2−u˜2 ˜ u2 e−α2t −1 , ¯ v(t) = ∆ξ−1+ (˜v−∆ξ−1)e−tξ.
Clearly, (0,0,0) and (¯u1(t),u¯2(t),v¯(t)) are a pair of lower and upper solutions of (1). Hence we can use Theorem 7 for anyT >0 and obtain:
Theorem 8. There exists a unique solution(u1(x, t), u2(x, t), v(x, t))to
system(1)satisfying
0≤ui(x, t)≤u¯i(t), i= 1,2,
0≤v(x, t)≤v¯(t).
We have established the global existence and uniqueness of the tions of (1). Now, we will prove the global boundedness of these solu-tions.
From the above, it is easy to see that
0≤u1(x, t)≤u¯1(t)≤max{K1,u˜1}, 0≤u2(x, t)≤u¯2(t)≤max{K2,u˜2}, 0≤v(x, t)≤v¯(t)≤max{∆ξ−1,v˜}.
Hence, all solutions of (1) are uniformly bounded for (x, t)∈Ω×R+. Next, we analyze the asymptotic behavior of the three populations.
Theorem 9. Suppose thatα1−q1K2−a−11ξ−1p1∆ >0, α2−q2K1−
a−1
2 ξ−1p2∆>0. Then system(1)is persistent.
the property: du¯1(t) dt =α1u¯1 1− u¯1 K1 −q1u¯1u2, du¯2(t) dt =α2u¯2 1− u¯2 K2 −q2u1u¯2, dv¯(t) dt = ∆−vξ ,¯ and du1(t) dt =α1u1 1− u1 K1 −q1K2u1−a −1 1 p1u1v,¯ du2(t) dt =α2u2 1− u2 K2 −q2K1u2−a −1 2 p2u2v,¯ dv(t) dt = ∆−(ξ+a −1 1 c1K1+a−21c2K2)v , with initial conditions
¯ ui(0) = ˜ui≡sup Ω u0i(x)>0, i= 1,2, ¯ v(0) = ˜v0≡sup Ω v0(x)>0, and ui(0) =ui≡inf Ω u 0 i(x)>0, i= 1,2, v(0) =v0≡inf Ω v 0(x)>0.
Here, we takeKi= max{Ki,u˜i},i= 1,2 for convenience.
Obviously, (¯u1,u¯2,v¯) and (u1, u2, v) are a pair of lower and upper solutions of (1); moreover, we have
¯ v(t) = ∆ξ−1+ (˜v0−∆ξ−1)e−tξ, v(t) = ∆(ξ+a−1 1 c1K1+a2−1c2K2)−1 + [˜v0−∆(ξ+a−1 1 c1K1a−21c2K2)−1]e−t(ξ+a −1 1 c1K1+a−21c2K2).
Therefore, lim inf t→∞ v(t) = ∆(ξ+a −1 1 c1K1+a−21c2K2)−1>0, lim inf t→∞ v(x, t)≥lim inft→∞ v(t) = ∆(ξ+a −1 1 c1K1+a−21c2K2)−1>0. Again, u1(t) = (α1−q1K2−a−11p1v¯)ce(α1−q1K2−a −1 1 p1¯v)t 1 +α1K1−1ce(α1−q1K2−a −1 1 p1v¯)t ,
wherec is a constant. Hence
lim inf t→∞ u1(x, t)≥lim inft→∞ u1(x, t) = α1−q1K2−a−11ξ −1p 1∆ α1K1−1 >0. Similarly, we have lim inf t→∞ u2(x, t)≥lim inft→∞ u2(x, t) = α2−q2K1−a2−1ξ−1p2∆ α2K2−1 >0.
The proof is complete.
Now we will analyze the stability of all possible equilibria of sys-tem (1), E0(0,0, ξ−1∆), E1(ˆu1,0,ˆv), E2(0,u˜2,v˜), E3(u∗1, u ∗ 2, v ∗ ). 4.2 The analysis of E0(0,0, ξ− 1
∆) To examine the stability of the uniform steady stateE0(0,0, ξ−1∆) to perturbations, we write
(10)
u1(x, t) = 0 +ε1(x, t),
u2(x, t) = 0 +ε2(x, t),
By substituting (10) into (1), and linearizing the equations, we obtain (11) ∂ε1 ∂t =D1 ∂2ε 1 ∂x2 + (α1− p1∆ a1ξ)ε1+ 0ε2+ 0η, ∂ε2 ∂t =D2 ∂2ε 2 ∂x2 + 0ε1+ (α2− p2∆ a2ξ )ε2+ 0η, ∂η ∂t =D3 ∂2η ∂x2 − c1∆ a1ξ ε1− c2∆ a1ξ ε2−ξη, ∂ε1 ∂x x=0 =∂ε2 ∂x x=0 = ∂η ∂x x=0 = 0, ∂ε1 ∂x x=a = ∂ε2 ∂x x=a = ∂η ∂x x=a = 0.
For an examination of linear stability, it is sufficient to assume solutions of (11) are in the form
ε1∝etλcoskx,
ε2∝etλcoskx,
η∝etλcoskx ,
where λ and k are the frequency and wave number respectively. The eigenvalue equation then reads
λ+D1k2 +a−1 1 ξ −1p 1∆−α1 0 0 0 λ+D2k 2 +a−1 2 ξ−1p2∆−α2 0 a−1 1 ξ −1c 2∆ a−21ξc2∆ λ+D3k2+ξ = 0. Hence λ1=α1−a−11ξ −1p 1∆−D1k2, λ2=α2−a−21ξ −1p 2∆−D2k2, λ3=−ξ−D3k2.
Note that the diffusion has no effect on the stability ofE0(0,0, ξ−1∆) in this case. Now by applying the boundary conditions (11) we obtain k=nπ/a,wherenis an integral constant. Ifais very small, thenDik2
becomes very large. Within this frame work, it can be realized that the steady state would be stable to small space-time perturbations for all time, even though it is unstable without diffusion, which means leukemia can not win the competition within the small space-time perturbations and this is not true in the case where there is no diffusion whenPi∆<
αiaiξ,i= 1,2. Therefore, the steady stateE0(0,0, ξ−1∆) is unstable in the homogeneous case, as is shown by the analysis of system (2), but is stable in the presence of spatial variations.
4.3 The analysis of EEE111(ˆ(ˆ(ˆuuu111,,,000,,,vˆvˆˆv))) Again, to examine the stability of the uniform steady stateE1to perturbations, we write
(12)
u1(x, t) = ˆu1+ε1(x, t),
u2(x, t) = 0 +ε2(x, t),
v(x, t) = ˆv+η(x, t).
By substituting (12) into (1) and linearizing the equations, we obtain
(13) ∂ε1 ∂t =D1 ∂2ε 1 ∂x2 +a11ε1+a12ε2+a13η, ∂ε2 ∂t =D2 ∂2ε 2 ∂x2 +a21ε1+a22ε2+a23η, ∂η ∂t =D3 ∂2η ∂x2 +a31ε1+a32ε2+a33η, and (14) ∂ε1 ∂x x=0 = ∂ε2 ∂x x=0 = ∂η ∂x x=0 = 0, ∂ε1 ∂x x=a = ∂ε2 ∂x x=a =∂η ∂x x=a = 0,
where a11=α1 1−2ˆu1 K1 − a1p1vˆ (a1+ ˆu1)2 , a12=−q1uˆ1, a13=− p1uˆ1 a1+ ˆu1 , a21= 0, a22=α2−q2uˆ1−a−21p2v,ˆ a23= 0, a31=− a1c1vˆ (a1+ ˆu1)2 , a32=−a2−1c2ˆv, a33=− ξ+ c1uˆ1 a1+ ˆu1 . Let ε1∝etλcoskx, ε2∝etλcoskx η∝etλcoskx,
where λ and k are the frequency and wave-number respectively. The eigenvalue equation then reads
λ+D1k2−a11 −a12 −a13 0 λ+D2k2−a22 0 −a13 −a23 λ+D3k2−a33 = 0. Hence λ2=a22−D2k2, σ(A) ={λi|λ2−Tr (A)λ+ det(A) = 0, i= 1,3}, where A= a11−D1k2 a13 a13 a33−D3k2 .
The condition k = 0 corresponds to neglecting diffusion and the constant solution E1(ˆu1,0,vˆ) is unstable with α2 > q2uˆ1+a−21p2ˆv, ˆ
u1> K1/2. Now we will discuss the stability with the casek6= 0.
Case 1: ˆuuuˆˆ111> K> K> K111///222.
Tr (A) =a11+a33−(D1+D3)k2<0,
det(A) =D1D3k4−(a11D3+a33D1)k2+ (a11a33−a13a31)>0, and so, a diffusion-driven instability can be immediately excluded with the case,α2< q2uˆ1+a−21p2ˆv.
0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 800 900 1000 1100 1200 1300 1400 1500 Distance x Evolution of Normal Cells in the Space and Time.
Time t u1
(x,t)
FIGURE 4: A solution for model (1) with the same parameters and initial conditions as in Figure 1 along with, D1 = 1, D2 = 2, D3 = 3. Here the uniform steady sateu1(x, t) = ˆu1is locally stable and diffusion has no effects on the stability.
Theorem 10. Suppose thatuˆ1> K1/2, andα26=q2uˆ1+a−21p2ˆv. Then
the diffusion has little effect on the stability of the uniform steady state
compared with the case where there is no diffusion. That is, if α2 >
q2ˆu1+a−21p2vˆ+D2k2, then E1(ˆu1,0,vˆ) is unstable. If α2 < q2uˆ1+
a−1
2 p2vˆ+D2k2, thenE1(ˆu1,0,vˆ)is locally asymptotically stable. From the boundary conditions (14), we can see thatk =nπ/a. Ifa is sufficiently small, then α2 > q2uˆ1+a2−1p2ˆv+D2k2 could be easily violated, and then leukemia will eventually be driven to extinction. Ifa is large enough, i.e. the diffusion region is large enough, the chemother-apy agent is not effective anymore and leukemia will eventually kill the normal cells and will win the competition.
Letδ=α2−q2uˆ1−a−21p2v >ˆ 0,as=nπ
p
D2/δ. Then we obtain the stable region, [0, as]. Within this region, the leukemia will be eventually
0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 20 0.02 0.04 0.06 0.08 0.1 Distance x Evolution of Abnormal Cells in the Space and Time.
Time t
u2
(x,t)
FIGURE 5: A solution for model (1) with the same parameters and initial conditions as in Figure 4. Here cancer cells decay with time and diffusion has no effects on the stability.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 800
1000 1200 1400
1600 Solution Prolifes at a Selection of Distances.
u1 (x) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.05 0.1 u2 (x) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 90 95 100 Time t v(x)
FIGURE 6: A collection of solution profiles for model (1) with a selection of distances in both Figure 4 and 5, which demonstrates that the uniform steady steadyE1 is locally stable in the diffusion case.
Case 2: µµµ111<<<uuuˆˆˆ111< µ< µ< µ222(((< K< K< K111///2)2)2) . In this case,
Tr (A) =a11+a33−(D1+D3)k2,
det(A) =D1D3k4−(a11D3+a33D1)k2+ (a11a33−a13a31). By the Routh-Hurwitz criterion, if Tr (A)<0 and det(A)>0, then the eigenvalues ofAhave negative real parts.
Let k2m= max a 11+a33 D1+D2 , a11D3+a33D1+{(a11D3+a33D1)2+ 4D1D3(a13a31−a11a33)}1/2 2D1D3 . Then, Tr (A)<0, det(A)>0 ifk > km.
Theorem 11. Supposeµ1<uˆ1< µ2(< K1/2)andα26=q2uˆ1+a−21p2vˆ.
If α2 > q2ˆu1+a−21p2ˆv+D2k2, then E1(ˆu1,0,ˆv) is unstable. If α2 <
q2ˆu1+a2−1p2vˆ+D2k2 andk > km, thenE1(ˆu1,0,ˆv)is locally
asymptot-ically stable.
By applying the boundary conditions, we obtain the stable region [0, nπ/km]. Within this region, the leukemia will be eventually driven to
extinction. The conditionα2> q2uˆ1+a2−1p2vˆ+D2k2, which guarantees that leukemia wins could be easily violated again when ais sufficiently small and the other parameters are fixed. That is, when the diffusion region is small and falls into [0, nπ/km], the chemotherapy agent is very
effective and will kill all leukemia eventually. However, when the dif-fusion region grows out of the stable region, the chemotherapy agent is not effective enough to kill the leukemia and the normal cells finally lose the competition.
4.4 The analysis ofEEE222(0(0(0,,,uuu˜˜˜222,,,v˜vv˜˜))) To examine the stability of the uni-form steady stateE2 to perturbations, we write
(15)
u1(x, t) = 0 +ε1(x, t),
u2(x, t) = ˜u2+ε2(x, t),
By substituting (15) into (1) and linearizing the equations, we obtain (16) ∂ε1 ∂t =D1 ∂2ε 1 ∂x2 +b11ε1+b12ε2+b13η, ∂ε2 ∂t =D2 ∂2ε 2 ∂x2 +b21ε1+b22ε2+b23η, ∂η ∂t =D3 ∂2η ∂x2 +b31ε1+b32ε2+b33η, and (17) ∂ε1 ∂x x=0 = ∂ε2 ∂x x=0 = ∂η ∂x x=0 = 0, ∂ε1 ∂x x=a = ∂ε2 ∂x x=a =∂η ∂x x=a = 0, where b11=α1−q1u˜2−a1−1p1v,˜ b12= 0, b13= 0, b21=−q2u˜2, b22=α2 1−2˜u2 K2 − a2p2˜v (a2+ ˜u2)2 , b23=− p2u˜2 a2+ ˜u2 , b31=−a−11c1v,˜ b32=− a2c2v˜ (a2+ ˜u2)2, b33=− ξ+ c2u˜2 a2+ ˜u2 . Letting ε1∝etλcoskx, ε2∝etλcoskx, η∝etλcoskx,
where λ and k are the frequency and wave-number respectively; thus, we get the eigenvalue equation
λ+D1k2−b11 0 0 −b21 λ+D2k2−b22 −b23 −b13 −b23 λ+D3k2−b33 = 0.
Hence λ1=b1−D1k2, σ(B) ={λi λ2 −Tr (B)λ+ det(B) = 0, i= 2,3}, where (18) B= " b22−D2k2 b23 b23 b33−D3k2 # .
The condition k = 0 corresponds to neglecting diffusion and the constant solution E2(0,u˜2,v˜) is unstable with α1 > q1u˜2+a−11p1˜v, ˜
u2 > K2/2. Again, we will try to work on the stability with the case
k6= 0.
Case 1: ˜uuu˜˜222> K> K> K222///222. Here we have
Tr (B) =b22+b33−(D2+D3)k2<0,
det(B) =D2D3k4−(b22D3+b33D2)k2+ (b22b33−b32b23)>0.
Theorem 12. Suppose thatu˜2> K2/2, andα16=q1u˜2+a−11p1v˜+D1k2.
If α1 > q1˜u2+a−11p1˜v+D1k2, then E2(0,u˜2,˜v) is unstable. If α1 <
q1˜u2+a−11p1v˜+D1k2, thenE2(0,u˜2,˜v)is locally asymptotically stable. From the boundary conditions (13), we can see thatk =nπ/a. Ifa is sufficiently small, then α1 > q1u˜2+a1−1p1˜v+D1k2 could be easily violated, and then the leukemia will eventually be eliminated. If a is large enough, i.e. the diffusion region is large enough, the chemother-apy agent is not effective anymore and leukemia will eventually kill the normal cells and will win the competition.
Letσ =α1−q1u˜2−a1−1p1v >˜ 0,au =nπpD1/σ. Then we obtain the unstable region, [au,∞). Within this region, the leukemia would be
eventually eliminated and the normal cells win the competition.
Case 2: τττ111<<<uuu˜˜˜222< τ< τ< τ222(((< K< K< K222///2)2)2). Then
Tr (B) =b22+b33−(D2+D3)k2,
By the Routh-Hurwitz criterion, if Tr (B)<0 and det(B)>0, then the eigenvalues ofB have negative real parts. Let
k2 m= max b 22+b33 D2+D3 , b22D3+b33D2+{(b22D3+b33D2)2+ 4D2D3(b23b32−b22b33)}1/2 2D2D3 . Then, Tr (B)<0, det(B)>0 ifk > km.
As a result, we have the following theorem.
Theorem 13. Supposeτ1<u˜2< τ2(< K2/2)andα16=q1u˜2+a−11p1v˜+
D1k2. If α1 > q1u˜2+a−11p1v˜+D1k2, thenE2(0,u˜2,˜v) is unstable. If
α2 < q2u˜1+a2−1p2v˜+D2k2 and k > km, then E2(0,u˜2,v˜) is locally
asymptotically stable.
By applying the boundary conditions, we obtain the stable region [0, nπ/km]. Within this region, normal cells will be eventually extinct.
5 Numerical results and discussion In order to perform the nu-merical simulations of system (1) and (2), we impose some conditions based both on the analytical results and on some physiological argu-ments:
a) α2> α1 (cancer cells grow faster than normal cells).
b) K1≤K2(the carrying capacity of cancer cells may be greater than that of normal cells).
c) p2 >> p1 (the drug is more potent against the cancer cells than against the normal cells).
d) c2>> c1 (a consequence of the above item).
e) the hypotheses of Theorems 1 and 2 which guarantee the existence ofE1 andE2.
Also the initial condition is such that: u01> u02 (in general).
The three most important steady states of our model from a phys-iological point of view are E1, E2 and E3. E1 represents a cancer free state, and of course the most desirable result would be to haveE1 glob-ally asymptoticglob-ally stable. E2 represents a state which is exclusively
cancerous and is a lethal state for the individual. E3 represents a state where normal and cancer cells exist simultaneously.
We are able to obtain analytic criteria for E1 to be locally asymp-totic stable, but not globally stable. Numerically, we can show that if the initial cancer value is sufficiently small, solutions may approachE1 (Figure 1). However, if u0
2 is large, solutions may approachE2 if the treatment intensity is weak (small values of a−1
2 p2 and/orξ−1∆), and may approachE3 (Figure 3) if the treatment intensity is strong (higher values ofa−1
2 p2and/orξ−1∆). This demonstrates an equi-asymptotical stability in the large forE3. Also the region of stability ofE1is found nu-merically to increase with higher treatment levels. On the other hand, we show that it is possible to choose parameters and initial values so that solutions of system (1) approach a positive (physiologically) state E3 with the cancer level small. In these parameters we note thatξ−1∆ is significantly higher than in the previous cases. In the small space region, we find numerically that diffusion has no effect on the stability of constant solutions if the treatment is strong enough and solutions of system (1) approach the uniform steady stateE1 (Figures 4, 5 and 6). However, for a large initial cancer valueu0
2, solutions would not approach the uniform steady state E2 under a reasonable weak treatment inten-sity, but they eventually approach the uniform steady stateE3exhibiting coexistence of three populations with a small cancer lever.(Figures 7, 8 and 9). When we increase the cancer initial value a little more under the same treatment condition, solutions still approach the uniform interior steady sate but with a large cancer level (Figures 10–12). This implies that there may be accumulations of cells at certain sites and depletions at other sites and through such processes leukemic cells may occupy sites of normal cells as the propagation of spatial heterogeneities occur. As a result, it may be suggested that the positions occupied by the leukemic cells as they expand, may be very fertile areas that are rich in nutrients needed for hematopoiesis. This is because those positions used to be occupied by the displaced normal cells. Thus, leukemic cell numbers may increase very rapidly. Also, upon introduction of more leukemic cells, existing normal cell colonies go through a process of shrinkage (Figure 10)as their positions are invaded by emerging colonies of abnor-mal cells. The resulting leukemic dominance may cause damage to and disturb the colony-forming capabilities of the normal cells.
Proceeding from the facts above, it is appropriate to suggest that through certain diffusive processes and mechanisms, the normal cells are displaced from their positions by colonies of leukemic cells over a wide region of space and are eventually driven to extinction. Essentially the
leukemic colonies display a tendency to invade the spaces designated for normal cell growth. Also, the rapid increase in the leukemic population over a period of time may result in a high leukemic cell density. This could lead to a migration of leukemic cells, possibly through a diffusive process as we described in this article to regions of lower cell density and nutrient availability. This could account for the reasons why other organs of the body become clogged with masses of abnormal cells, as is noted in [19]. It is important to mention that the predictions of the model may hold for some acute leukemias but not for the chronic leukemias in which there is a more gradual procession towards leukemic dominance [20].
Finally, we point out how our results may be useful to leukemia treat-ment. For those situations in which our models may apply, the theorems identify which combination of coefficients, i.e., sufficiently high or low values would lead to low levels of cancer, or elimination altogether. It is known [1] that different leukemias can be treated with various suc-cesses (or failures), and our results may be used to help in improving the successes (i.e., the rate of success or length of remission time).
0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 800 900 1000 1100 1200 1300 1400 Distance x Evolution of Normal Cells in the Space and Time.
Time t
u1
(x,t)
FIGURE 7: A solution for model (1) with the same parameters and initial conditions as in Figure 4 exceptp2 = 0.04(a weak intensive treat-ment, i.e. smalla−12 p2) and a small initial cancer numberu02 = 0.1. Here the normal cells eventually approach a uniform interior steady state.
0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 Distance x Evolution of Abnormal Cells in the Space and Time.
Time t
u2
(x,t)
FIGURE 8: A solution for model (1) with the same parameters and initial conditions as in Figure 7. Here the abnormal population survives the treatment and eventually approaches an interior steady stateu2 = 2.5 under the weak treatment intensity.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 800
1000 1200
1400 Evolution of Solution Profiles at a Selection of Distance.
u1 (x) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 u2 (x) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 90 95 100 Time t v(x)
FIGURE 9: A collection of solution profiles for model (1) with a selection of distances in both Figure 7 and 8, which shows that solutions of model (1) approach an interior uniform steady state with a small cancer lever under a weak treatment intensity.
0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 0 200 400 600 800 1000 1200 Distance x Evolution of Normal Cells in the Space and Time.
Time t
u1
(x,t)
FIGURE 10: A solution for model (1) with the same parameters and initial conditions as in Figure 7 except a large initial cancer numberu02= 0.6. Here the normal cells begin to go through a process of shrinkage.
0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 0 200 400 600 800 1000 1200 1400 Distance x Evolution of Abnormal Cells in the Space and Time.
Time t
u2
(x,t)
FIGURE 11: A solution for model (1) with the same parameters and initial conditions as in Figure 10. Here the abnormal population survives the treatment and eventually dominates the populations.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 500 1000 1500 u1 (x)
Evolution of Solution Profiles at a Selection of Distances.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 500 1000 1500 u2 (x) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 90 95 100 Time t v(x)
FIGURE 12: A collection of solution profiles for model (1) with a selec-tion of distances in both Figure 10 and 11, which shows that soluselec-tions of model (1) approach an interior uniform steady state with a high cancer level under a weak treatment intensity.
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Department of Mathematical & Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
E-mail address: wliu@math.ualberta.ca