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

Authors: Bao Shi; Fangwei Zhang; Shihe Xu
Journal: Abstract and Applied Analysis
 (2011)

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## 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.


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## Body

## 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:(1.1)Δrσ=Γ1σ,0<r<R(t),t>0,(1.2)∂σ∂r(0,t)=0,σ(R(t),t)=σ∞,t>0,(1.3)Δrβ=Γ2β,0<r<R(t),t>0,(1.4)∂β∂r(0,t)=0,β(R(t),t)=β∞,t>0,(1.5)ddt4πR3(t)3=4π∫0R(t-τ1)λσ(r,t-τ1)r2dr-4π∫0R(t-τ2)[λσ̃+μβ(r,t-τ2)]r2dr,t>0,(1.6)R(t)=φ(t),-τ≤t≤0,
where Γ1, Γ2, λ, μ, σ∞, β∞, σ̃, 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; λσ̃ 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.Denoteθ=(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) is(1.7)σ(r,t)=σ∞R(t)sinhR(t)sinhrr,β(r,t)=β∞R(t)sinh(θR(t))sinh(θr)r.
Substituting (1.7) into (1.6) we obtain(1.8)Ṙ(t)=R(t)[λσ∞p(R(t-τ1))(R(t-τ1)R(t))3-(μβ∞p(R(t-τ2))+13λσ̃)(R(t-τ2)R(t))3];
here p(x)=(xcothx-1)/x2. Set ω(t)=R3(t), and we have(1.9)ω̇(t)=3λσ∞p(ω1/3(t-τ1))ω(t-τ1)-(3μβ∞p(ω1/3(t-τ2))+λσ̃)ω(t-τ2).
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 formula(1.10)ω(t)=ω(nτ3)+∫nτ3t3λσ∞p(ω1/3(s-τ1))ω(s-τ1)-(3μβ∞p(ω1/3(s-τ2))+λσ̃)ω(s-τ2)ds.
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(2.1)(3λσ∞p(x1/3)-3μβ∞p(θx1/3)-λσ̃)x=0.
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, and(2.2)limx→0+p(x)=13,limx→∞p(x)=0.
Let g(x)=3λσ∞p(x1/3)-3μβ∞p(θx1/3)-λσ̃, x>0. Then(2.3)limx→0+g(x)=λσ∞-μβ∞-λσ̃,g′(x)=x-2/3(λσ∞p′(x1/3)-θμβ∞p′(θx1/3))=-θμβ∞x-2/3p′(x1/3)(p′(θx1/3)p′(x1/3)-λσ∞θμβ∞).
By [12], we know that p′(θy)/p′(y) is strictly monotone increasing (resp., decreasing) if 0<θ<1 (resp., θ>1) and(2.4)limy→0+p′(θy)p′(y)=θ,limy→∞p′(θy)p′(y)=1θ2.
Using these results, we can easily prove the following lemma (see [11] or [14]).Lemma 2.1.
Assume that0<θ<1. Then the following assertions hold. (1)
Ifβ∞≥λσ∞/θ2μ then there exist no positive solutions for (2.1), that is, the problem (1.9) has no positive stationary solutions.(2)
Ifβ∞<λσ∞/θ2μ, then in the case σ̃≥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 σ̃<σ∞-μβ∞/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)
Ifβ∞≥θλσ∞/μ then there exist no positive solutions for (2.1), that is, the problem (1.9) has no positive stationary solutions.(4)
Ifβ∞≤λσ∞/θ2μ, then in the case σ̃≥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 σ̃<σ∞-μβ∞/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)
Ifλσ∞/θ2μ<β∞<θλσ∞/μ, then there exists a unique x*>0 such that(2.5)p'(θy*)p'(y*)=λσ∞θμβ∞;
here y*=(x*)1/3, and x* is the maximum point of g(x). Denote g(x*)=M. Then if σ̃>3M there exist no positive solutions for (2.1), that is, the problem (1.9) has no positive stationary solutions. If 0<σ̃≤σ∞-μβ∞/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 σ∞-μβ∞/λ<σ̃<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 obtain(2.6)ω̇(t)=-A1ω(t-τ1)-A2ω(t-τ2),
where A1=-λσ∞[ωs1/3p′(ωs1/3)+3p(ωs1/3)], A2=μβ∞(θωs1/3p′(θωs1/3)+3p(θωs1/3))+λσ̃.Similarly linearizing (1.9) at the trivial stationary solution we get(2.7)ω̇(t)=-B1ω(t-τ1)-B2ω(t-τ2),
where B1=-λσ∞, B2=μβ∞+λσ̃.We claimA1<0, A2>0. Actually, for y>0, y3p(y) is strictly monotone increasing in y (see [14]), that is, for y>0,(2.8)d(y3p(y))dy>0⇔y2(yṗ(y)+3p(y))>0⇔yṗ(y)+3p(y)>0.
This readily implies that A1=-λσ∞(yṗ(y)+3p(y))|y=ωs1/3<0. Immediately, A2=μβ∞(yṗ(y)+3p(y))|y=θωs1/3+σ̃>0. Hence the claim is true.The characteristic of (2.6) is as follows:(2.9)z=-A1e-zτ1-A2e-zτ2.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 takeτ2 as bifurcation parameter, for detail see [18] and the references therein.The case whenB1<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(2.10)ẋ(t)=f(x(t-τ1),x(t-τ2)),
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:
(2.11)ẋ(t)=-B1x(t-τ1)-B2x(t-τ2).
Then(1)
ifB1<0, B2>|B1|, and τ1∈(0,π/2B22-B12], then there exists τ20>0 such that for τ2∈[0,τ20) the trivial solution to (2.10) is asymptotically stable and for τ2=τ20 the Hopf bifurcation occurs;(2)
ifB1<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 Lemma2.2, we easily have the following.Corollary 2.3.
Consider the equation(2.12)ẋ(t)=f(x(t-τ1),x(t-τ2)),
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:
(2.13)ẋ(t)=-A1x(t-τ1)-A2x(t-τ2).
Then(1)
ifA1<0, A2>|A1|, and τ1∈(0,π/2A22-A12], then there exists τ20>0 such that for τ2∈[0,τ20) the positive stationary solution to (2.10) is asymptotically stable and for τ2=τ20 the Hopf bifurcation occurs;(2)
ifA1<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.NoticingA1=-λσ∞[ωs1/3p′(ωs1/3)+3p(ωs1/3)], A2=μβ∞(θωs1/3p′(θωs1/3)+3p(θωs1/3))+σ̃ and ωs>0 satisfy (2.1), by direct computation, we have(2.14)A2-|A1|=A2+A1=-λσ∞p′(ωs1/3)+θμβ∞p′(θωs1/3).
Since p(x) is strictly monotone decreasing for x>0, we are readily get(2.15)A2>|A1|⇔p′(θωs1/3)p′(ωs1/3)<λσ∞θμβ∞⇔g′(ωs)<0,A2<|A1|⇔p′(θωs1/3)p′(ωs1/3)>λσ∞θμβ∞⇔g′(ωs)>0.
Clearly B1<0, by direct computation, we obtain(2.16)B2>|B1|⇔σ̃>σ∞-μβ∞λ,B2<|B1|⇔σ̃<σ∞-μβ∞λ.Noticing(2.17)β∞<λμ(σ∞-σ̃)⇔σ̃>σ∞-μβ∞λ⇔B2>|B1|,β∞>λμ(σ∞-σ̃)⇔σ̃<σ∞-μβ∞λ⇔B2<|B1|,
by Lemma 2.2, we can conclude the following. (i)
Assume thatβ∞<(λ/μ)(σ∞-σ̃) and τ1∈(0,π/2B22-B12] hold, then there exists τ20>0 such that for τ2∈[0,τ20) the trivial solution to (1.9) is asymptotically stable and for τ2=τ20 the Hopf bifurcation occurs.(ii)
Assume thatβ∞>(λ/μ)(σ∞-σ̃) 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 have(2.18)β∞<λμ(σ∞-σ̃),β∞<λσ∞μθ2⇔0<θ<1,β∞<λσ∞μθ2,σ̃<σ∞-μβ∞λorθ>1,β∞<λσ∞μθ2,σ̃<σ∞-μβ∞λ⇒ (1.9) has a positive stationary solution ωs and g′(ωs)<0⇔A2>|A1|. Then by Corollary 2.3, we have the following. Assume that β∞<(λ/μ)(σ∞-σ̃) and β∞<λσ∞/μθ2 hold, then for τ1∈(0,π/2A22-A12] there exists τ20>0 such that for τ2∈[0,τ20) the positive stationary solution to (1.9) is asymptotically stable and for τ2=τ20 the Hopf bifurcation occurs.Since0<θ<1 and (λ/μ)(σ∞-σ̃)<β∞<λσ∞/μθ2⇔β∞<λσ∞/μθ2, σ̃>σ∞-μβ∞/λ⇒B2>|B1|. From Lemma 2.1 we know that if 0<θ<1 and (λ/μ)(σ∞-σ̃)<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∞-σ̃)<β∞<λσ∞/μθ2 hold, then for τ1∈(0,π/2B22-B12] there exists τ20>0 such that for τ2∈[0,τ20) the trivial solution to (2.10) is asymptotically stable and for τ2=τ20 the Hopf bifurcation occurs.Assume thatθ>1, then we have the following.Ifβ∞>(θλσ∞/μ)(>λσ∞/μθ2)⇒β∞<(λ/μ)(σ∞-σ̃)⇒B2>|B1|;if(λ/μ)(σ∞-σ̃)<β∞<λσ∞/μθ2⇒σ̃>σ∞-μβ∞/λ⇔B2>|B1|;ifλσ∞/μθ2<β∞<θλσ∞/μ and if σ̃>3M⇒σ̃<σ∞-μβ∞/λ⇔B2<|B1|; if σ̃<σ∞-μβ∞/λ⇒B2>|B1|; if σ∞-μβ∞/λ<σ̃<3M⇒g′(ωs2)<0 and then by Lemma 2.2 and Corollary 2.3, we have the following. (1)
Assume thatθ>1, β∞>θλσ∞/μ or θ>1, (λ/μ)(σ∞-σ̃)<β∞<λσ∞/μθ2 holds, then for τ1∈(0,π/2B22-B12] there exists τ20>0 such that for τ2∈[0,τ20) the trivial solution to (2.10) is asymptotically stable and for τ2=τ20 the Hopf bifurcation occurs.(2)
Assume thatθ>1, (λ/μ)(σ∞-σ̃)<β∞<λσ∞/μθ2 hold. Then if σ̃>3M, the trivial solution to (1.9) is unstable independently on the values of both delays, and there is no Hopf bifurcation; if σ̃<σ∞-μβ∞/λ, then for τ1∈(0,π/2B22-B12] there exists τ20>0 such that for τ2∈[0,τ20) the trivial solution to (2.10) is asymptotically stable and for τ2=τ20 the Hopf bifurcation occurs; if σ∞-μβ∞/λ<σ̃<3M, then for τ1∈(0,π/2A22-A12] there exists τ20>0 such that for τ2∈[0,τ20) the positive stationary solution ωs2 to (1.9) is asymptotically stable and for τ2=τ20 the Hopf bifurcation occurs.We summarize as follows.Theorem 2.4.
(i) Assume thatβ∞<(λ/μ)(σ∞-σ̃) and τ1∈(0,π/2B22-B12] hold, then there exists τ20>0 such that for τ2∈[0,τ20) the trivial solution to (1.9) is asymptotically stable and for τ2=τ20 the Hopf bifurcation occurs.
(ii) Assume thatβ∞>(λ/μ)(σ∞-σ̃) holds, the trivial solution to (1.9) is unstable independently on the values of both delays, and there is no Hopf bifurcation.
(iii) Assume thatβ∞<(λ/μ)(σ∞-σ̃) and β∞<λσ∞/μθ2 hold, then for τ1∈(0,π/2A22-A12] there exists τ20>0 such that for τ2∈[0,τ20) the positive stationary solution to (1.9) is asymptotically stable and for τ2=τ20 the Hopf bifurcation occurs.
(iv)  Assume that0<θ<1 and (λ/μ)(σ∞-σ̃)<β∞<λσ∞/μθ2 hold, then for τ1∈(0,π/2B22-B12] there exists τ20>0 such that for τ2∈[0,τ20) the trivial solution to (2.10) is asymptotically stable and for τ2=τ20 the Hopf bifurcation occurs.
(v) Assume thatθ>1, β∞>θλσ∞/μ or θ>1, (λ/μ)(σ∞-σ̃)<β∞<λσ∞/μθ2 holds, then for τ1∈(0,π/2B22-B12] there exists τ20>0 such that for τ2∈[0,τ20) the trivial solution to (2.10) is asymptotically stable and for τ2=τ20 the Hopf bifurcation occurs.
(vi) Assume thatθ>1, λσ∞/μθ2<β∞<θλσ∞/μ hold. Then if σ̃>3M, the trivial solution to (1.9) is unstable independently on the values of both delays, and there is no Hopf bifurcation; if σ̃<σ∞-μβ∞/λ, then for τ1∈(0,π/2B22-B12] there exists τ20>0 such that for τ2∈[0,τ20) the trivial solution to (2.10) is asymptotically stable and for τ2=τ20 the Hopf bifurcation occurs; if σ∞-μβ∞/λ<σ̃<3M, then for τ1∈(0,π/2A22-A12] there exists τ20>0 such that for τ2∈[0,τ20) the positive stationary solution ωs2 to (1.9) is asymptotically stable and for τ2=τ20 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 form(3.1)ω̇(t)=f(ω(t-τ1),ω(t-τ2)),
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 equation(3.2)ω̇(t)=f(ω(t),ω(t)),x(0)=x0>0,
or retarded differential equation with only one delay of the form(3.3)ω̇(t)=f(ω(t-τ),ω(t)),
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].



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