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Abstract and Applied Analysis
Volume 2011, Article ID 980686, 10 pages
http://dx.doi.org/10.1155/2011/980686
Research Article

Hopf Bifurcation of a Mathematical Model for Growth of Tumors with an Action of Inhibitor and Two Time Delays

1Institute of Systems Science and Mathematics, Naval Aeronautical and Astronautical University, Yantai, Shandong 264001, China
2Department of Mathematics, Zhaoqing University, Zhaoqing, Guangdong 526061, China

Received 3 November 2010; Accepted 25 January 2011

Academic Editor: Allan C. Peterson

Copyright © 2011 Bao Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A mathematical model for growth of tumors with two discrete delays is studied. The delays, respectively, represent the time taken for cells to undergo mitosis and the time taken for the cell to modify the rate of cell loss due to apoptosis and kill of cells by the inhibitor. We show the influence of time delays on the Hopf bifurcation when one of delays is used as a bifurcation parameter.

1. Introduction

Within last four decades, an increasing number of partial differential equation models for tumor growth or therapy have been developed; compare [19] and references cited therein. Most of those models are in form of free boundary problems. Rigorous mathematical analysis of such free boundary problems has drawn great interest, and many interesting results have been established; compare [1020] and references cited therein. Analysis of such free boundary problems not only provides a sound theoretical basis for tumor medicine, but also greatly enriches the understanding of deferential equations.

In this paper, we study a mathematical model for growth of tumors with two discrete delays. The delays, respectively, represent the time taken for cells to undergo mitosis and the time taken for the cell to modify the rate of cell loss due to apoptosis and kill of cells by the inhibitor. The model is as follows:Δ𝑟𝜎=Γ1𝜎,0<𝑟<𝑅(𝑡),𝑡>0,(1.1)𝜕𝜎𝜕𝑟(0,𝑡)=0,𝜎(𝑅(𝑡),𝑡)=𝜎Δ,𝑡>0,(1.2)𝑟𝛽=Γ2𝛽,0<𝑟<𝑅(𝑡),𝑡>0,(1.3)𝜕𝛽𝜕𝑟(0,𝑡)=0,𝛽(𝑅(𝑡),𝑡)=𝛽𝑑,𝑡>0,(1.4)𝑑𝑡4𝜋𝑅3(𝑡)3=4𝜋𝑅(𝑡𝜏1)0𝜆𝜎𝑟,𝑡𝜏1𝑟2𝑑𝑟4𝜋𝑅(𝑡𝜏2)0𝜆𝜎+𝜇𝛽𝑟,𝑡𝜏2𝑟2𝑑𝑟,𝑡>0,(1.5)𝑅(𝑡)=𝜑(𝑡),𝜏𝑡0,(1.6) where Γ1, Γ2, 𝜆, 𝜇, 𝜎, 𝛽, 𝜎, 𝜏1, and 𝜏2 are positive constants, 𝜑 is a given positive function. Δ𝑟=(1/𝑟2)(𝜕/𝜕𝑟)(𝑟2(𝜕./𝜕𝑟)). The term Γ1𝜎 in (1.1) is the consumption rate of nutrient in a unit volume; Γ2𝛽 in (1.3) is the consumption rate of inhibitor in a unit volume; 𝜎 reflects constant supply of nutrient that the tumor receives from its surface; 𝛽 reflects constant supply of inhibitor that the tumor receives from its surface. 𝜏1 represents the time taken for cells to undergo mitosis, and 𝜏2 represents the time taken for the cell to modify the rate of cell loss due to apoptosis and kill of cells by the inhibitor. The two terms on the right hand side of (1.5) are explained as follows: the first term is the total volume increase in unit time interval induced by cell proliferation; 𝜆𝜎 is the cell proliferation rate in unit volume. The second term is total volume shrinkage in unit time interval caused by cell apoptosis (cell death due to aging) and the kills of cells by the inhibitor; the cell apoptosis rate is assumed to be constant; 𝜆𝜎 does not depend on either 𝜎 or 𝛽.

The study of effects of time delay in growth of tumors by using the method of mathematical models was initiated by Byrne [1]. Recently this study has drawn attention of some other researchers; compare Bodnar and Foryś [10], Fory and Bodnar [15], Foryś and Kolev [16], Sarkar and Banerjee [7] with one delay; and compare Piotrowska [18], Xu [19] with two delays. This mathematical model is established by modifying the model of Byrne [2] by considering two independent time delays effect as in [18, 19]. The modifications are based on biological considerations, see Cui and Xu [14] for details. In [14], the authors studied the problem (1.1)–(1.6) with only one delay in proliferation, that is, 𝜏2=0, and showed that the dynamical behavior of solutions of the model with a delay in proliferation is similar to that of solutions for corresponding nondelayed problem. The aim of this paper is to investigate the influence of time delays on the Hopf bifurcation when 𝜏2 is used as a bifurcation parameter.

Denote 𝜃=(Γ2/Γ1). By rescaling the space variable, we may assume that Γ1=1. Accordingly, we have 𝜃=Γ2. The solution of (1.1)–(1.4) is𝜎𝜎(𝑟,𝑡)=𝑅(𝑡)sinh𝑅(𝑡)sinh𝑟𝑟𝛽,𝛽(𝑟,𝑡)=𝑅(𝑡)sinh(𝜃𝑅(𝑡))sinh(𝜃𝑟)𝑟.(1.7) Substituting (1.7) into (1.6) we obtaiṅ𝑅(𝑡)=𝑅(𝑡)𝜆𝜎𝑝𝑅𝑡𝜏1𝑅𝑡𝜏1𝑅(𝑡)3𝜇𝛽𝑝𝑅𝑡𝜏2+13𝑅𝜆𝜎𝑡𝜏2𝑅(𝑡)3;(1.8) here 𝑝(𝑥)=(𝑥coth𝑥1)/𝑥2. Set 𝜔(𝑡)=𝑅3(𝑡), and we havė𝜔(𝑡)=3𝜆𝜎𝑝𝜔1/3𝑡𝜏1𝜔𝑡𝜏13𝜇𝛽𝑝𝜔1/3𝑡𝜏2𝜔+𝜆𝜎𝑡𝜏2.(1.9) Using the step method (see, e.g., [21]), we can easily show that if there exists a solution for 𝑡[(𝑛1)𝜏3,𝑛𝜏3], then the solution for 𝑡[𝑛𝜏3,(𝑛+1)𝜏3], where 𝑛𝑁, 𝜏3=min(𝜏1,𝜏2), is defined by the formula𝜔(𝑡)=𝜔𝑛𝜏3+𝑡𝑛𝜏33𝜆𝜎𝑝𝜔1/3𝑠𝜏1𝜔𝑠𝜏13𝜇𝛽𝑝𝜔1/3𝑠𝜏2𝜔+𝜆𝜎𝑠𝜏2𝑑𝑠.(1.10) Clearly the step method gives the existence of unique solution to (1.9) because of 𝑠𝜏1,𝑠𝜏2[(𝑛1)𝜏3,𝑛𝜏3].

Using Theorem 1.2 from [22], we can get nonnegative initial condition 𝜔0, and the solution of (1.9) can become negative in a finite time. Therefore, through the rest of the paper we assume that a positive solution of (1.9) with initial function 𝜔0 exists for every 𝑡>0.

2. Stability of the Stationary Solutions and Existence of Local Hopf Bifurcation

In this section, we will study stability of the stationary solutions and existence of local Hopf bifurcation.

The first step is to find stationary solutions. Stationary solutions to (1.9) satisfy the equation3𝜆𝜎𝑝𝑥1/33𝜇𝛽𝑝𝜃𝑥1/3𝜆𝜎𝑥=0.(2.1) Clearly, (2.1) has the trivial solution 𝑥=0. Next, we consider the positive solutions to (2.1). From [17] we know that 𝑝(𝑥) is strictly monotone decreasing for 𝑥>0, andlim𝑥0+1𝑝(𝑥)=3,lim𝑥𝑝(𝑥)=0.(2.2) Let 𝑔(𝑥)=3𝜆𝜎𝑝(𝑥1/3)3𝜇𝛽𝑝(𝜃𝑥1/3)𝜆𝜎, 𝑥>0. Thenlim𝑥0+𝑔(𝑥)=𝜆𝜎𝜇𝛽𝑔𝜆𝜎,(𝑥)=𝑥2/3𝜆𝜎𝑝𝑥1/3𝜃𝜇𝛽𝑝𝜃𝑥1/3=𝜃𝜇𝛽𝑥2/3𝑝𝑥1/3𝑝𝜃𝑥1/3𝑝𝑥1/3𝜆𝜎𝜃𝜇𝛽.(2.3) By [12], we know that 𝑝(𝜃𝑦)/𝑝(𝑦) is strictly monotone increasing (resp., decreasing) if 0<𝜃<1 (resp., 𝜃>1) andlim𝑦0+𝑝(𝜃𝑦)𝑝(𝑦)=𝜃,lim𝑦𝑝(𝜃𝑦)𝑝(=1𝑦)𝜃2.(2.4) Using these results, we can easily prove the following lemma (see [11] or [14]).

Lemma 2.1. Assume that 0<𝜃<1. Then the following assertions hold. (1)If 𝛽𝜆𝜎/𝜃2𝜇 then there exist no positive solutions for (2.1), that is, the problem (1.9) has no positive stationary solutions.(2)If 𝛽<𝜆𝜎/𝜃2𝜇, then in the case 𝜎𝜎𝜇𝛽/𝜆 there exist no positive solutions for (2.1), that is, the problem (1.9) has no positive stationary solutions, and in the opposite case 𝜎<𝜎𝜇𝛽/𝜆, there exists a unique positive solution 𝑥=𝜔𝑠 for (2.1), that is, the problem (1.9) has a unique positive stationary solution 𝑥=𝜔𝑠. Moreover 𝑔(𝜔𝑠)<0.Assume that 𝜃>1. Then the following assertions hold. (3) If 𝛽𝜃𝜆𝜎/𝜇 then there exist no positive solutions for (2.1), that is, the problem (1.9) has no positive stationary solutions. (4)If 𝛽𝜆𝜎/𝜃2𝜇, then in the case 𝜎𝜎𝜇𝛽/𝜆 there exist no positive solutions for (2.1), that is, the problem (1.9) has no positive stationary solution, and in the opposite case 𝜎<𝜎𝜇𝛽/𝜆, there exists a unique positive solution 𝑥=𝜔𝑠 for (2.1), that is, the problem (1.9) has a unique positive stationary solution. Moreover 𝑔(𝜔𝑠)<0.(5)If  𝜆𝜎/𝜃2𝜇<𝛽<𝜃𝜆𝜎/𝜇, then there exists a unique 𝑥>0 such that𝑝𝜃𝑦𝑝(𝑦)=𝜆𝜎𝜃𝜇𝛽;(2.5) here 𝑦=(𝑥)1/3, and 𝑥 is the maximum point of 𝑔(𝑥). Denote 𝑔(𝑥)=𝑀. Then if 𝜎>3𝑀 there exist no positive solutions for (2.1), that is, the problem (1.9) has no positive stationary solutions. If 0<𝜎𝜎𝜇𝛽/𝜆 there exists a unique positive solution 𝑥=𝜔𝑠 for (2.1) which satisfy 𝑔(𝜔𝑠)<0, that is, the problem (1.9) has a unique positive stationary solution satisfying 𝑔(𝜔𝑠)<0. If 𝜎𝜇𝛽/𝜆<𝜎<3𝑀 there exist two positive solutions 𝑥1=𝜔𝑠1<𝑥1=𝜔𝑠2 for (2.1) which satisfy 𝑔(𝜔𝑠1)>0, 𝑔(𝜔𝑠2)<0, that is, the problem (1.9) has two positive stationary solutions satisfying 𝑔(𝜔𝑠1)>0, 𝑔(𝜔𝑠2)<0, respectively.

The next step is to study the stability and the Hopf bifurcation of (1.9). Linearizing (1.9) at positive stationary solutions, we obtaiṅ𝜔(𝑡)=𝐴1𝜔𝑡𝜏1𝐴2𝜔𝑡𝜏2,(2.6) where 𝐴1=𝜆𝜎[𝜔𝑠1/3𝑝(𝜔𝑠1/3)+3𝑝(𝜔𝑠1/3)], 𝐴2=𝜇𝛽(𝜃𝜔𝑠1/3𝑝(𝜃𝜔𝑠1/3)+3𝑝(𝜃𝜔𝑠1/3))+𝜆𝜎.

Similarly linearizing (1.9) at the trivial stationary solution we geṫ𝜔(𝑡)=𝐵1𝜔𝑡𝜏1𝐵2𝜔𝑡𝜏2,(2.7) where 𝐵1=𝜆𝜎, 𝐵2=𝜇𝛽+𝜆𝜎.

We claim 𝐴1<0, 𝐴2>0. Actually, for 𝑦>0, 𝑦3𝑝(𝑦) is strictly monotone increasing in 𝑦 (see [14]), that is, for 𝑦>0,𝑑𝑦3𝑝(𝑦)𝑑𝑦>0𝑦2(𝑦̇𝑝(𝑦)+3𝑝(𝑦))>0𝑦̇𝑝(𝑦)+3𝑝(𝑦)>0.(2.8) This readily implies that 𝐴1=𝜆𝜎(𝑦̇𝑝(𝑦)+3𝑝(𝑦))|𝑦=𝜔𝑠1/3<0. Immediately, 𝐴2=𝜇𝛽(𝑦̇𝑝(𝑦)+3𝑝(𝑦))|𝑦=𝜃𝜔𝑠1/3+𝜎>0. Hence the claim is true.

The characteristic of (2.6) is as follows:𝑧=𝐴1𝑒𝑧𝜏1𝐴2𝑒𝑧𝜏2.(2.9)

From [14] we know that if 𝜏2=0 then for arbitrary 𝜏1>0, the dynamical behavior of solutions of problem (1.9) with nonnegative initial function is similar to that of solutions for corresponding nondelayed problem. By continuity, for sufficiently small 𝜏2>0, the dynamical behavior of solutions of problem (1.9) with nonnegative initial function is also similar to that of solutions for corresponding nonretarded problem.

In the following, we will study stability of the stationary solutions and existence of local Hopf bifurcation. From biological point of view it is reasonable to take 𝜏2 as bifurcation parameter, for detail see [18] and the references therein.

The case when 𝐵1<0, 𝐵2>0 was studied in [23, 24], and the proof of the following lemma can be found in it.

Lemma 2.2. Consider the equation 𝑥̇𝑥(𝑡)=𝑓𝑡𝜏1,𝑥𝑡𝜏2,(2.10) with a nonnegative initial continuous function 𝜑[𝜏,0]𝑅+, where 𝜏1, 𝜏2 are the positive constants, 𝜏=max(𝜏1,𝜏2), and 𝑓 is a continuously differentiable nonlinear function. Assume that (2.10) has the trivial stationary solution, that is, 𝑓(0,0)=0. Let the linearized equation around the trivial solution of (2.10) be as follows: ̇𝑥(𝑡)=𝐵1𝑥𝑡𝜏1𝐵2𝑥𝑡𝜏2.(2.11) Then (1)if 𝐵1<0, 𝐵2>|𝐵1|, and 𝜏1(0,𝜋/2𝐵22𝐵21], then there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the trivial solution to (2.10) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs;(2)if 𝐵1<0, 0<𝐵2<|𝐵1|, the trivial solution to (2.10) is unstable independently on the values of both delays, and there is no Hopf bifurcation.

Use Lemma 2.2, we easily have the following.

Corollary 2.3. Consider the equation 𝑥̇𝑥(𝑡)=𝑓𝑡𝜏1,𝑥𝑡𝜏2,(2.12) with a nonnegative initial continuous function 𝜑[𝜏,0]𝑅+, where 𝜏1, 𝜏2 are the positive constants, 𝜏=max(𝜏1,𝜏2), 𝑓 is a continuously differentiable nonlinear function. Assume that (2.12) has the positive stationary solution 𝑥=𝑥𝑠, that is, 𝑓(𝑥𝑠,𝑥𝑠)=0. Let the linearized equation around the positive stationary solution of (2.10) be as follows: ̇𝑥(𝑡)=𝐴1𝑥𝑡𝜏1𝐴2𝑥𝑡𝜏2.(2.13) Then (1)if 𝐴1<0, 𝐴2>|𝐴1|, and 𝜏1(0,𝜋/2𝐴22𝐴21], then there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the positive stationary solution to (2.10) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs;(2)if 𝐴1<0, 0<𝐴2<|𝐴1|, the positive stationary solution to (2.10) is unstable independently on the values of both delays, and there is no Hopf bifurcation.

Noticing 𝐴1=𝜆𝜎[𝜔𝑠1/3𝑝(𝜔𝑠1/3)+3𝑝(𝜔𝑠1/3)], 𝐴2=𝜇𝛽(𝜃𝜔𝑠1/3𝑝(𝜃𝜔𝑠1/3)+3𝑝(𝜃𝜔𝑠1/3))+𝜎 and 𝜔𝑠>0 satisfy (2.1), by direct computation, we have𝐴2||𝐴1||=𝐴2+𝐴1=𝜆𝜎𝑝𝜔𝑠1/3+𝜃𝜇𝛽𝑝𝜃𝜔𝑠1/3.(2.14) Since 𝑝(𝑥) is strictly monotone decreasing for 𝑥>0, we are readily get𝐴2>||𝐴1||𝑝𝜃𝜔𝑠1/3𝑝𝜔𝑠1/3<𝜆𝜎𝜃𝜇𝛽𝑔𝜔𝑠𝐴<0,2<||𝐴1||𝑝𝜃𝜔𝑠1/3𝑝𝜔𝑠1/3>𝜆𝜎𝜃𝜇𝛽𝑔𝜔𝑠>0.(2.15) Clearly 𝐵1<0, by direct computation, we obtain𝐵2>||𝐵1||𝜎>𝜎𝜇𝛽𝜆,𝐵2<||𝐵1||𝜎<𝜎𝜇𝛽𝜆.(2.16)

Noticing𝛽<𝜆𝜇𝜎𝜎𝜎>𝜎𝜇𝛽𝜆𝐵2>||𝐵1||,𝛽>𝜆𝜇𝜎𝜎𝜎<𝜎𝜇𝛽𝜆𝐵2<||𝐵1||,(2.17) by Lemma 2.2, we can conclude the following. (i)Assume that 𝛽<(𝜆/𝜇)(𝜎𝜎) and 𝜏1(0,𝜋/2𝐵22𝐵21] hold, then there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the trivial solution to (1.9) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs.(ii)Assume that 𝛽>(𝜆/𝜇)(𝜎𝜎) holds, the trivial solution to (1.9) is unstable independently on the values of both delays, and there is no Hopf bifurcation.

By simple computation, we have𝛽<𝜆𝜇𝜎𝜎,𝛽<𝜆𝜎𝜇𝜃20<𝜃<1,𝛽<𝜆𝜎𝜇𝜃2,𝜎<𝜎𝜇𝛽𝜆or𝜃>1,𝛽<𝜆𝜎𝜇𝜃2,𝜎<𝜎𝜇𝛽𝜆(2.18) (1.9) has a positive stationary solution 𝜔𝑠 and 𝑔(𝜔𝑠)<0𝐴2>|𝐴1|. Then by Corollary 2.3, we have the following. Assume that 𝛽<(𝜆/𝜇)(𝜎𝜎) and 𝛽<𝜆𝜎/𝜇𝜃2 hold, then for 𝜏1(0,𝜋/2𝐴22𝐴21] there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the positive stationary solution to (1.9) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs.

Since 0<𝜃<1 and (𝜆/𝜇)(𝜎𝜎)<𝛽<𝜆𝜎/𝜇𝜃2𝛽<𝜆𝜎/𝜇𝜃2, 𝜎>𝜎𝜇𝛽/𝜆𝐵2>|𝐵1|. From Lemma 2.1 we know that if 0<𝜃<1 and (𝜆/𝜇)(𝜎𝜎)<𝛽<𝜆𝜎/𝜇𝜃2 hold, then (1.9) has no positive stationary solution. By Lemma 2.2, we readily have the following. Assume that 0<𝜃<1 and (𝜆/𝜇)(𝜎𝜎)<𝛽<𝜆𝜎/𝜇𝜃2 hold, then for 𝜏1(0,𝜋/2𝐵22𝐵21] there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the trivial solution to (2.10) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs.

Assume that 𝜃>1, then we have the following.

If 𝛽>(𝜃𝜆𝜎/𝜇)(>𝜆𝜎/𝜇𝜃2)𝛽<(𝜆/𝜇)(𝜎𝜎)𝐵2>|𝐵1|;

if (𝜆/𝜇)(𝜎𝜎)<𝛽<𝜆𝜎/𝜇𝜃2𝜎>𝜎𝜇𝛽/𝜆𝐵2>|𝐵1|;

if 𝜆𝜎/𝜇𝜃2<𝛽<𝜃𝜆𝜎/𝜇 and if 𝜎>3𝑀𝜎<𝜎𝜇𝛽/𝜆𝐵2<|𝐵1|; if 𝜎<𝜎𝜇𝛽/𝜆𝐵2>|𝐵1|; if 𝜎𝜇𝛽/𝜆<𝜎<3𝑀𝑔(𝜔𝑠2)<0 and then by Lemma 2.2 and Corollary 2.3, we have the following. (1)Assume that 𝜃>1, 𝛽>𝜃𝜆𝜎/𝜇 or 𝜃>1, (𝜆/𝜇)(𝜎𝜎)<𝛽<𝜆𝜎/𝜇𝜃2 holds, then for 𝜏1(0,𝜋/2𝐵22𝐵21] there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the trivial solution to (2.10) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs.(2)Assume that 𝜃>1, (𝜆/𝜇)(𝜎𝜎)<𝛽<𝜆𝜎/𝜇𝜃2 hold. Then if 𝜎>3𝑀, the trivial solution to (1.9) is unstable independently on the values of both delays, and there is no Hopf bifurcation; if 𝜎<𝜎𝜇𝛽/𝜆, then for 𝜏1(0,𝜋/2𝐵22𝐵21] there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the trivial solution to (2.10) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs; if 𝜎𝜇𝛽/𝜆<𝜎<3𝑀, then for 𝜏1(0,𝜋/2𝐴22𝐴21] there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the positive stationary solution 𝜔𝑠2 to (1.9) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs.

We summarize as follows.

Theorem 2.4. (i) Assume that 𝛽<(𝜆/𝜇)(𝜎𝜎) and 𝜏1(0,𝜋/2𝐵22𝐵21] hold, then there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the trivial solution to (1.9) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs.
(ii) Assume that 𝛽>(𝜆/𝜇)(𝜎𝜎) holds, the trivial solution to (1.9) is unstable independently on the values of both delays, and there is no Hopf bifurcation.
(iii) Assume that 𝛽<(𝜆/𝜇)(𝜎𝜎) and 𝛽<𝜆𝜎/𝜇𝜃2 hold, then for 𝜏1(0,𝜋/2𝐴22𝐴21] there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the positive stationary solution to (1.9) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs.
(iv) Assume that 0<𝜃<1 and (𝜆/𝜇)(𝜎𝜎)<𝛽<𝜆𝜎/𝜇𝜃2 hold, then for 𝜏1(0,𝜋/2𝐵22𝐵21] there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the trivial solution to (2.10) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs.
(v) Assume that 𝜃>1, 𝛽>𝜃𝜆𝜎/𝜇 or 𝜃>1, (𝜆/𝜇)(𝜎𝜎)<𝛽<𝜆𝜎/𝜇𝜃2 holds, then for 𝜏1(0,𝜋/2𝐵22𝐵21] there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the trivial solution to (2.10) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs.
(vi) Assume that 𝜃>1, 𝜆𝜎/𝜇𝜃2<𝛽<𝜃𝜆𝜎/𝜇 hold. Then if 𝜎>3𝑀, the trivial solution to (1.9) is unstable independently on the values of both delays, and there is no Hopf bifurcation; if 𝜎<𝜎𝜇𝛽/𝜆, then for 𝜏1(0,𝜋/2𝐵22𝐵21] there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the trivial solution to (2.10) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs; if 𝜎𝜇𝛽/𝜆<𝜎<3𝑀, then for 𝜏1(0,𝜋/2𝐴22𝐴21] there exists 𝜏02>0 such that for 𝜏2[0,𝜏02) the positive stationary solution 𝜔𝑠2 to (1.9) is asymptotically stable and for 𝜏2=𝜏02 the Hopf bifurcation occurs.

3. Conclusion

In this paper, we study a mathematical model for growth of tumors with two discrete delays. The delays, respectively, represent the time taken for cells to undergo mitosis and the time taken for the cell to modify the rate of cell loss due to apoptosis and kill of cells by the inhibitor. Final mathematical formulation of the model is retarded differential equation of the form𝜔̇𝜔(𝑡)=𝑓𝑡𝜏1,𝜔𝑡𝜏2,(3.1) with a nonnegative initial continuous function 𝜑[𝜏,0]𝑅+, where 𝜏1, 𝜏2, and 𝜏=max(𝜏1,𝜏2) are the positive constants, 𝑓 is a continuously differentiable function. The results show that the two independent delays control the dynamics of the solution of the problem (3.1) and the dynamic behavior is different to the corresponding nonretarded ordinary equatioṅ𝜔(𝑡)=𝑓(𝜔(𝑡),𝜔(𝑡)),𝑥(0)=𝑥0>0,(3.2) or retarded differential equation with only one delay of the forṁ𝜔(𝑡)=𝑓(𝜔(𝑡𝜏),𝜔(𝑡)),(3.3) with a nonnegative initial continuous function 𝜑[𝜏,0]𝑅+. However the dynamic behaviors of the problem (3.2) and (3.3) are similar, see [14].

Acknowledgments

The work of the third author is partially supported by NNSF (10926128), NSF, and YMF of Guangdong Province (9251064101000015, LYM10133).

References

  1. H. M. Byrne, “The effect of time delays on the dynamics of avascular tumor growth,” Mathematical Biosciences, vol. 144, no. 2, pp. 83–117, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. H. M. Byrne, “Growth of nonnecrotic tumors in the presence and absence of inhibitors,” Mathematical Biosciences, vol. 130, no. 2, pp. 151–181, 1995. View at Publisher · View at Google Scholar
  3. H. M. Byrne and M. A. J. Chaplain, “Growth of necrotic tumors in the presence and absence of inhibitors,” Mathematical Biosciences, vol. 135, no. 2, pp. 187–216, 1996. View at Publisher · View at Google Scholar
  4. M. J. Dorie, R. F. Kallman, D. F. Rapacchietta et al., “Migration and internalization of cells and polystyrene microspheres in tumor cell spheroids,” Experimental Cell Research, vol. 141, no. 1, pp. 201–209, 1982. View at Google Scholar
  5. H. Greenspan, “Models for the growth of solid tumor by diffusion,” Studies in Applied Mathematics, vol. 51, pp. 317–340, 1972. View at Google Scholar
  6. H. P. Greenspan, “On the growth and stability of cell cultures and solid tumors,” Journal of Theoretical Biology, vol. 56, no. 1, pp. 229–242, 1976. View at Publisher · View at Google Scholar
  7. R. R. Sarkar and S. Banerjee, “A time delay model for control of malignant tumor growth,” in National Conference on Nonlinear Systems and Dynamics, pp. 1–4, 2006.
  8. K. Thompson and H. Byrne, “Modelling the internalisation of labelled cells in tumor spheroids,” Bulletin of Mathematical Biology, vol. 61, pp. 601–623, 1999. View at Google Scholar
  9. J. Ward and J. King, “Mathematical modelling of avascular-tumor growth II: modelling grwoth saturation,” IMA Journal of Mathematics Applied in Medicine and Biology, vol. 15, pp. 1–42, 1998. View at Google Scholar
  10. M. Bodnar and U. Foryś, “Time delay in necrotic core formation,” Mathematical Biosciences and Engineering, vol. 2, no. 3, pp. 461–472, 2005. View at Google Scholar · View at Zentralblatt MATH
  11. S. Cui, “Analysis of a mathematical model for the growth of tumors under the action of external inhibitors,” Journal of Mathematical Biology, vol. 44, no. 5, pp. 395–426, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. S. Cui and A. Friedman, “Analysis of a mathematical model of the effect of inhibitors on the growth of tumors,” Mathematical Biosciences, vol. 164, no. 2, pp. 103–137, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. S. Cui and A. Friedman, “Analysis of a mathematical model of the growth of necrotic tumors,” Journal of Mathematical Analysis and Applications, vol. 255, no. 2, pp. 636–677, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. S. Cui and S. Xu, “Analysis of mathematical models for the growth of tumors with time delays in cell proliferation,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 523–541, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. U. Fory and M. Bodnar, “Time delays in proliferation process for solid avascular tumour,” Mathematical and Computer Modelling, vol. 37, no. 11, pp. 1201–1209, 2003. View at Publisher · View at Google Scholar
  16. U. Foryś and M. Kolev, “Time delays in proliferation and apoptosis for solid avascular tumour,” in Mathematical Modelling of Population Dynamics, vol. 63 of Banach Center Publications, pp. 187–196, Polish Academy of Sciences, Warsaw, Poland, 2004. View at Google Scholar · View at Zentralblatt MATH
  17. A. Friedman and F. Reitich, “Analysis of a mathematical model for the growth of tumors,” Journal of Mathematical Biology, vol. 38, no. 3, pp. 262–284, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. M. J. Piotrowska, “Hopf bifurcation in a solid avascular tumour growth model with two discrete delays,” Mathematical and Computer Modelling, vol. 47, no. 5-6, pp. 597–603, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. S. Xu, “Hopf bifurcation of a free boundary problem modeling tumor growth with two time delays,” Chaos, Solitons & Fractals, vol. 41, no. 5, pp. 2491–2494, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. S. Xu, “Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation,” Nonlinear Analysis: Real World Applications B, vol. 11, no. 1, pp. 401–406, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977, Applied Mathematical Sciences.
  22. M. Bodnar, “The nonnegativity of solutions of delay differential equations,” Applied Mathematics Letters, vol. 13, no. 6, pp. 91–95, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. X. Li, S. Ruan, and J. Wei, “Stability and bifurcation in delay-differential equations with two delays,” Journal of Mathematical Analysis and Applications, vol. 236, no. 2, pp. 254–280, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. M. J. Piotrowska, “A remark on the ODE with two discrete delays,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 664–676, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH