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

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 [1–9] 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 [10–20] 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:𝜕𝜎Δ_𝑟𝜎=Γ₁𝜎,0<𝑟<𝑅(𝑡),𝑡>0,(1.1)𝜕𝑟(0,𝑡)=0,𝜎(𝑅(𝑡),𝑡)=𝜎^∞Δ,𝑡>0,(1.2)_𝑟𝜕𝛽𝛽=Γ₂𝛽,0<𝑟<𝑅(𝑡),𝑡>0,(1.3)𝜕𝑟(0,𝑡)=0,𝛽(𝑅(𝑡),𝑡)=𝛽^∞𝑑,𝑡>0,(1.4)𝑑𝑡4𝜋𝑅³(𝑡)3=4𝜋^{𝑅(𝑡−𝜏1)0}𝜆𝜎𝑟,𝑅𝑡−(𝑡)𝜏₁=𝜑𝑟²(𝑑𝑡𝑟),−4𝜋−^{𝑅(𝑡−𝜏2)0}𝜏≤𝑡≤𝜆0𝜎,+𝜇𝛽𝑟,𝑡−𝜏₂𝑟²𝑑𝑟,𝑡>0,((11..56)) where Γ₁, Γ₂, 𝜆, 𝜇, 𝜎_∞, 𝛽_∞, 𝜎, 𝜏₁, and 𝜏₂ are positive constants, 𝜑 is a given positive function. Δ_𝑟=(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) is𝜎(𝑟,𝑡)=𝜎^∞𝑅(𝑡)sinh𝑅(𝑡)sin𝑟h𝑟,𝛽(𝑟,𝑡)=𝛽^∞𝑅(𝑡)sinh(𝜃𝑅(𝑡))sinh𝑟(𝜃𝑟).(1.7) Substituting (1.7) into (1.6) we obtain𝑅̇(𝑡)=𝑅(𝑡)⎡⎢⎢⎣𝜆𝜎_∞𝑝𝑅𝑡−𝜏₁𝑅𝑡−𝜏¹𝑅(𝑡)³−𝜇𝛽_∞𝑝𝑅𝑡−𝜏₂+13𝜆𝜎𝑅𝑡−𝜏²𝑅(𝑡)³⎤⎥⎥⎦;(1.8) here 𝑝(𝑥)=(𝑥coth𝑥−1)/𝑥². Set 𝜔(𝑡)=𝑅³(𝑡), and we have𝜔̇(𝑡)=3𝜆𝜎_∞𝑝𝜔^1/3𝑡−𝜏₁𝜔𝑡−𝜏₁−3𝜇𝛽_∞𝑝𝜔^1/3𝑡−𝜏₂+𝜆𝜎𝜔𝑡−𝜏₂.(1.9) Using the step method (see, e.g., [21]), we can easily show that if there exists a solution for 𝑡∈[(𝑛−1)𝜏₃,𝑛𝜏₃], then the solution for 𝑡∈[𝑛𝜏₃,(𝑛+1)𝜏₃], where 𝑛∈𝑁, 𝜏₃=min(𝜏₁,𝜏₂), is defined by the formula𝜔(𝑡)=𝜔𝑛𝜏₃+^𝑡𝑛𝜏₃3𝜆𝜎_∞𝑝𝜔^1/3𝑠−𝜏₁𝜔𝑠−𝜏₁−3𝜇𝛽_∞𝑝𝜔^1/3𝑠−𝜏₂+𝜆𝜎𝜔𝑠−𝜏₂𝑑(𝑠1..10) Clearly the step method gives the existence of unique solution to (1.9) because of 𝑠−𝜏₁,𝑠−𝜏₂∈[(𝑛−1)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 equation3𝜆𝜎_∞𝑝𝑥^1/3−3𝜇𝛽_∞𝑝𝜃𝑥^1/3−𝜆𝜎𝑥=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, andlim_𝑥→0+𝑝(𝑥)=13,lim^𝑥→∞𝑝(𝑥)=0.(2.2) Let 𝑔(𝑥)=3𝜆𝜎_∞𝑝(𝑥^1/3)−3𝜇𝛽_∞𝑝(𝜃𝑥^1/3)−𝜆𝜎, 𝑥>0. Then𝑔lim_𝑥→0+𝑔(𝑥)=𝜆𝜎_∞−𝜇𝛽_∞−𝜆𝜎,_(𝑥)=𝑥^−2/3𝜆𝜎_∞𝑝^𝑥^1/3−𝜃𝜇𝛽_∞𝑝^𝜃𝑥^1/3=−𝜃𝜇𝛽_∞𝑥^−2/3𝑝^𝑥^1/3𝑝^𝜃𝑥^1/3𝑝^𝑥^1/3−𝜃𝜆𝜇𝜎𝛽^∞_∞.(2.3) By [12], we know that 𝑝^(𝜃𝑦)/𝑝^(𝑦) is strictly monotone increasing (resp., decreasing) if 0<𝜃<1 (resp., 𝜃>1) andlim_𝑦→0+𝑝^(𝜃𝑦)𝑝^(𝑦)=𝜃,lim^𝑦→∞𝑝^(𝜃𝑦)𝑝^(𝑦)=𝜃1².(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 𝛽_∞≥𝜆𝜎_∞/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 𝜎≥𝜎_∞−𝜇𝛽_∞/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<𝜎_∞−𝜇𝛽_∞/𝜆, there exists a unique positive solution 𝑥=𝜔_𝑠 for (2.1), that is, the problem (1.9) has a unique positive stationary solution 𝑥=𝜔_𝑠. Moreover 𝑔^(𝜔_𝑠)<0.Assume that 𝜃>1. Then the following assertions hold. (3) If 𝛽_∞≥𝜃𝜆𝜎_∞/𝜇 then there exist no positive solutions for (2.1), that is, the problem (1.9) has no positive stationary solutions. (4)If 𝛽_∞≤𝜆𝜎_∞/𝜃²𝜇, then in the case 𝜎≥𝜎_∞−𝜇𝛽_∞/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<𝜎_∞−𝜇𝛽_∞/𝜆, there exists a unique positive solution 𝑥=𝜔_𝑠 for (2.1), that is, the problem (1.9) has a unique positive stationary solution. Moreover 𝑔^(𝜔_𝑠)<0.(5)If  𝜆𝜎_∞/𝜃²𝜇<𝛽_∞<𝜃𝜆𝜎_∞/𝜇, then there exists a unique 𝑥^∗>0 such that𝑝^𝜃𝑦^∗𝑝^(𝑦^∗)=𝜃𝜆𝜇𝜎𝛽^∞_∞;(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<𝜎≤𝜎_∞−𝜇𝛽_∞/𝜆 there exists a unique positive solution 𝑥=𝜔_𝑠 for (2.1) which satisfy 𝑔^(𝜔_𝑠)<0, that is, the problem (1.9) has a unique positive stationary solution satisfying 𝑔^(𝜔_𝑠)<0. If 𝜎_∞−𝜇𝛽_∞/𝜆<𝜎<3𝑀 there exist two positive solutions 𝑥₁=𝜔_𝑠1<𝑥₁=𝜔_𝑠2 for (2.1) which satisfy 𝑔^(𝜔_𝑠1)>0, 𝑔^(𝜔_𝑠2)<0, that is, the problem (1.9) has two positive stationary solutions satisfying 𝑔^(𝜔_𝑠1)>0, 𝑔^(𝜔_𝑠2)<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 obtain𝜔̇(𝑡)=−𝐴₁𝜔𝑡−𝜏₁−𝐴₂𝜔𝑡−𝜏₂,(2.6) where 𝐴₁=−𝜆𝜎_∞[𝜔^𝑠1/3𝑝^(𝜔^𝑠1/3)+3𝑝(𝜔^𝑠1/3)], 𝐴₂=𝜇𝛽_∞(𝜃𝜔^𝑠1/3𝑝^(𝜃𝜔^𝑠1/3)+3𝑝(𝜃𝜔^𝑠1/3))+𝜆𝜎.

Similarly linearizing (1.9) at the trivial stationary solution we get𝜔̇(𝑡)=−𝐵₁𝜔𝑡−𝜏₁−𝐵₂𝜔𝑡−𝜏₂,(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,𝑑𝑦³𝑝(𝑦)𝑑𝑦>0⇔𝑦²(𝑦𝑝̇(𝑦)+3𝑝(𝑦))>0⇔𝑦𝑝̇(𝑦)+3𝑝(𝑦)>0.(2.8) This readily implies that 𝐴₁=−𝜆𝜎_∞(𝑦𝑝(̇𝑦)+3𝑝(𝑦))|_𝑦=𝜔^𝑠1/3<0. Immediately, 𝐴₂=𝜇𝛽_∞(𝑦𝑝(̇𝑦)+3𝑝(𝑦))|_{𝑦=𝜃𝜔}^𝑠1/3+𝜎>0. Hence the claim is true.

The characteristic of (2.6) is as follows:𝑧=−𝐴₁𝑒^−𝑧𝜏1−𝐴₂𝑒^−𝑧𝜏2.(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 𝑥̇(𝑡)=𝑓𝑥𝑡−𝜏₁,𝑥𝑡−𝜏₂,(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: 𝑥̇(𝑡)=−𝐵₁𝑥𝑡−𝜏₁−𝐵₂𝑥𝑡−𝜏₂.(2.11) Then (1)if 𝐵₁<0, 𝐵₂>|𝐵₁|, and 𝜏₁∈(0,𝜋/2𝐵²²−𝐵²¹], 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 𝑥̇(𝑡)=𝑓𝑥𝑡−𝜏₁,𝑥𝑡−𝜏₂,(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=𝑥_𝑠, that is, 𝑓(𝑥_𝑠,𝑥_𝑠)=0. Let the linearized equation around the positive stationary solution of (2.10) be as follows: 𝑥̇(𝑡)=−𝐴₁𝑥𝑡−𝜏₁−𝐴₂𝑥𝑡−𝜏₂.(2.13) Then (1)if 𝐴₁<0, 𝐴₂>|𝐴₁|, and 𝜏₁∈(0,𝜋/2𝐴²²−𝐴²¹], 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_∞[𝜔^𝑠1/3𝑝^(𝜔^𝑠1/3)+3𝑝(𝜔^𝑠1/3)], 𝐴₂=𝜇𝛽_∞(𝜃𝜔^𝑠1/3𝑝^(𝜃𝜔^𝑠1/3)+3𝑝(𝜃𝜔^𝑠1/3))+𝜎 and 𝜔_𝑠>0 satisfy (2.1), by direct computation, we have𝐴₂−||𝐴₁||=𝐴₂+𝐴₁=−𝜆𝜎_∞𝑝^𝜔^𝑠1/3+𝜃𝜇𝛽_∞𝑝^𝜃𝜔^𝑠1/3.(2.14) Since 𝑝(𝑥) is strictly monotone decreasing for 𝑥>0, we are readily get𝐴₂>||𝐴₁||⇔𝑝^𝜃𝜔^𝑠1/3𝑝^𝜔^𝑠1/3<𝜃𝜆𝜇𝜎𝛽^∞_∞𝐴⇔𝑔^𝜔_𝑠<0,₂<||𝐴₁||⇔𝑝^𝜃𝜔^𝑠1/3𝑝^𝜔^𝑠1/3>𝜃𝜆𝜇𝜎𝛽^∞_∞⇔𝑔^𝜔_𝑠>0.(2.15) Clearly 𝐵₁<0, by direct computation, we obtain𝐵₂>||𝐵₁||⇔𝜎>𝜎_∞−𝜇𝛽^∞𝐵₂𝜆,<||𝐵₁||⇔𝜎<𝜎_∞−𝜇𝛽^∞𝜆.(2.16)

Noticing𝛽_∞<𝜆𝜇𝜎^∞−𝜎⇔𝜎>𝜎_∞−𝜇𝛽^∞𝜆⇔𝐵²𝛽_∞>||𝐵₁||,>𝜆𝜇𝜎^∞−𝜎⇔𝜎<𝜎_∞−𝜇𝛽^∞𝜆⇔𝐵²<||𝐵₁||,(2.17) by Lemma 2.2, we can conclude the following. (i)Assume that 𝛽_∞<(𝜆/𝜇)(𝜎_∞−𝜎) and 𝜏₁∈(0,𝜋/2𝐵²²−𝐵²¹] 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)(𝜎_∞−𝜎) 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_∞<𝜆𝜇𝜎^∞−𝜎,𝛽_∞<𝜆𝜎^∞𝜇𝜃²0<𝜃<1,𝛽_∞⇔<𝜆𝜎^∞𝜇𝜃²,𝜎<𝜎_∞−𝜇𝛽^∞𝜆or𝜃>1,𝛽^∞<𝜆𝜎^∞𝜇𝜃²,𝜎<𝜎_∞−𝜇𝛽^∞𝜆(2.18)⇒ (1.9) has a positive stationary solution 𝜔_𝑠 and 𝑔^(𝜔_𝑠)<0⇔𝐴₂>|𝐴₁|. Then by Corollary 2.3, we have the following. Assume that 𝛽_∞<(𝜆/𝜇)(𝜎_∞−𝜎) and 𝛽_∞<𝜆𝜎_∞/𝜇𝜃² hold, then for 𝜏₁∈(0,𝜋/2𝐴²²−𝐴²¹] 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)(𝜎_∞−𝜎)<𝛽_∞<𝜆𝜎_∞/𝜇𝜃²⇔𝛽_∞<𝜆𝜎_∞/𝜇𝜃², 𝜎>𝜎_∞−𝜇𝛽_∞/𝜆⇒𝐵₂>|𝐵₁|. From Lemma 2.1 we know that if 0<𝜃<1 and (𝜆/𝜇)(𝜎_∞−𝜎)<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)(𝜎_∞−𝜎)<𝛽_∞<𝜆𝜎_∞/𝜇𝜃² hold, then for 𝜏₁∈(0,𝜋/2𝐵²²−𝐵²¹] 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)(𝜎_∞−𝜎)⇒𝐵₂>|𝐵₁|;

if (𝜆/𝜇)(𝜎_∞−𝜎)<𝛽_∞<𝜆𝜎_∞/𝜇𝜃²⇒𝜎>𝜎_∞−𝜇𝛽_∞/𝜆⇔𝐵₂>|𝐵₁|;

if 𝜆𝜎_∞/𝜇𝜃²<𝛽_∞<𝜃𝜆𝜎_∞/𝜇 and if 𝜎>3𝑀⇒𝜎<𝜎_∞−𝜇𝛽_∞/𝜆⇔𝐵₂<|𝐵₁|; if 𝜎<𝜎_∞−𝜇𝛽_∞/𝜆⇒𝐵₂>|𝐵₁|; if 𝜎_∞−𝜇𝛽_∞/𝜆<𝜎<3𝑀⇒𝑔^(𝜔_𝑠2)<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)(𝜎_∞−𝜎)<𝛽_∞<𝜆𝜎_∞/𝜇𝜃² holds, then for 𝜏₁∈(0,𝜋/2𝐵²²−𝐵²¹] 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)(𝜎_∞−𝜎)<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<𝜎_∞−𝜇𝛽_∞/𝜆, then for 𝜏₁∈(0,𝜋/2𝐵²²−𝐵²¹] 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 𝜎_∞−𝜇𝛽_∞/𝜆<𝜎<3𝑀, then for 𝜏₁∈(0,𝜋/2𝐴²²−𝐴²¹] there exists 𝜏⁰²>0 such that for 𝜏₂∈[0,𝜏⁰²) the positive stationary solution 𝜔_𝑠2 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)(𝜎_∞−𝜎) and 𝜏₁∈(0,𝜋/2𝐵²²−𝐵²¹] 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)(𝜎_∞−𝜎) 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)(𝜎_∞−𝜎) and 𝛽_∞<𝜆𝜎_∞/𝜇𝜃² hold, then for 𝜏₁∈(0,𝜋/2𝐴²²−𝐴²¹] 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)(𝜎_∞−𝜎)<𝛽_∞<𝜆𝜎_∞/𝜇𝜃² hold, then for 𝜏₁∈(0,𝜋/2𝐵²²−𝐵²¹] 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)(𝜎_∞−𝜎)<𝛽_∞<𝜆𝜎_∞/𝜇𝜃² holds, then for 𝜏₁∈(0,𝜋/2𝐵²²−𝐵²¹] 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<𝜎_∞−𝜇𝛽_∞/𝜆, then for 𝜏₁∈(0,𝜋/2𝐵²²−𝐵²¹] 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 𝜎_∞−𝜇𝛽_∞/𝜆<𝜎<3𝑀, then for 𝜏₁∈(0,𝜋/2𝐴²²−𝐴²¹] there exists 𝜏⁰²>0 such that for 𝜏₂∈[0,𝜏⁰²) the positive stationary solution 𝜔_𝑠2 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 form𝜔̇(𝑡)=𝑓𝜔𝑡−𝜏₁,𝜔𝑡−𝜏₂,(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 equation𝜔̇(𝑡)=𝑓(𝜔(𝑡),𝜔(𝑡)),𝑥(0)=𝑥₀>0,(3.2) or retarded differential equation with only one delay of the form𝜔̇(𝑡)=𝑓(𝜔(𝑡−𝜏),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).