Transitions in a Logistic Growth Model Induced by Noise Coupling and Noise Color

Transcription

1 Commun. Theor. Phys. (Beijing, China) 46 (2006) pp c International Academic Publishers Vol. 46, No. 1, July 15, 2006 Transitions in a Logistic Growth Model Induced by Noise Coupling and Noise Color SHI Jin and ZHU Shi-Qun School of Physical Science and Technology, Suzhou University, Suzhou , China (Received October 25, 2005) Abstract With unified colored noise approximation, the logistic growth model is used to analyze cancer cell population when colored noise is included. It is found that both the coupling between noise terms and the noise color can induce continuous first-order-like and re-entrance-like phase transitions in the system. The coupling and the noise color can also increase tumor cell growth for small number of cell mass and repress tumor cell growth for large number of cell mass. It is shown that the approximate analytic expressions are consistent with the numerical simulations. PACS numbers: e, a, r Key words: logistic growth model, first-order-like phase transition, re-entrance-like phase transition, colored noise 1 Introduction Recently, nonlinear stochastic systems with noise terms have attracted extensive attention. [1] There are wide applications of noise-induced transitions in the fields of physics, chemistry, and biology. [2 5] The intensity fluctuations of a laser system have been investigated when both multiplicative and additive noise terms are included. [6,7] The existence of the first-order-like phase transition in the most probable intensity of the steady state probability distribution shows very interesting behavior. [8,9] The re-entrance-like phase transition has been found in Markovian systems of infinite dimensions [10,11] and in non-markovian systems. [12,13] The non-markovian systems can be analyzed by Markovian approaches to zerodimensional system, but there are also studies for extended systems. [14 16] For a non-markovian system including colored noise, the analytical methods of unified colored noise approximation (UCNA) and functional analysis can be applied. [17 25] The colored noise plays an important role in some physical systems. The observed magnetic resonance line shapes induced by thermal fluctuations can be explained by Kubo s model with colored noise in the frequency. [19] The experimental measurements of statistical properties in dye laser system offer a solid evidence of colored noise with finite noise correlation time. The experimental data show that the fluctuations of the pump parameter in dye lasers can induce multiplicative colored noise, while the statistics of photon emission in dye lasers can induce additive white noise. [20 25] In nonlinear stochastic systems, the logistic model has also been used in many cases, such as a model of cell growth, particularly, tumor cell growth. [26 32] Some methods are employed to model the effects of chemotherapy or radiotherapy with the logistic growth model. One of them is to assume that the drug kills instantly, thus giving a pulse-type action. Another method is to assume that the chemotherapeutic or radiotherapeutic effects are modelled by continuous or piecewise-continuous periodic functions. These periodic functions alternate the growth rate between the presence of drug and the recovery stage in the logistic growth model. The effects of drug and environment are also assumed to generate noise terms in a logistic model. [28 32] In most of the previous analysis, much attention has been paid for the white noise. However, the colored nature of the noise in a logistic growth system needs to be considered. In this paper, UCNA is applied to investigate the logistic model with colored noise. In Sec. 2, the theoretical logistic model with coupling between two noise terms is presented. Analytic expressions of the steady state probability distribution is derived. In Sec. 3, numerical simulations is presented to check the validity of the approximation method. In Sec. 4, the parameter planes of the first-order-like and the re-entrance-like phase transitions are calculated when the coupling between noise terms are varied. In Sec. 5, the effects of noise color on the phase diagram and the steady state probability distribution are studied when the noise correlation time is varied. A discussion concludes the paper. 2 Theoretical Model Some external factors, such as temperature, drugs, and radiotheraphy, can alter the cell growth rate and generate a multiplicative noise. Meanwhile, these factors can refrain the cell number and give a negative white noise. Since the two kinds of fluctuations have a common origin, The project supported by the Natural Science Foundation of Jiangsu Province of China under Grant No. BK Corresponding author, szhu@suda.edu.cn

2 176 SHI Jin and ZHU Shi-Qun Vol. 46 they may be coupled with each other. When these influences are included, the tumor cell growth system can be describe by a logistic stochastic equation, [32] dx dt = ax bx2 + xε(t) Γ(t). (1) The multiplicative noise can enhance the cell growth, while the additive noise can refrain the cell growth. Since the two noise terms are from the same origin, the minus sign before the additive noise may be used to reflect the opposite effect of the additive noise. In Eq. (1), x is the tumor cell mass, a is the tumor cell growth rate, b is the tumor cell decay rate, ε(t) is Gaussian colored noise and Γ(t) is Gaussian white noise. These noise terms have the following properties, ε(t) = Γ(t) = 0, (2) ε(t)ε(t ) = D [ τ exp 1 ] τ t t, (3) Γ(t)Γ(t ) = 2αδ(t t ), (4) ε(t)γ(t ) = 2λ αdδ(t t ), (5) where D and α are the intensities of colored and white noise terms, τ is the correlation time of the colored noise ε(t), λ denotes the strength of the coupling between noise terms of ε(t) and Γ(t) with 1 λ 1. Applying the unified colored noise approximation (UCNA), [1,17,18] the non-markovin process can be transformed to a Morkovin process. The Langevin equation of Eqs. (1) (5) can be approximately reduced to dx dt = τbx (ax bx2 ) τbx [ Dxξ(t) αη(t)], (6) where ξ(t) and η(t) are Gaussian white noise terms and are given by ξ(t) = η(t) = 0, (7) ξ(t)ξ(t ) = η(t)η(t ) = 2δ(t t ), (8) ξ(t)η(t ) = 2λδ(t t ). (9) It should be mentioned that the boundary condition of the approximate method used in deriving Eq. (6) is (1 + τbx) > 0. [1,17,18] Since both of the quantities τ and b are greater than zero, and the cell mass x is always greater than zero, the boundary condition of (1 + τbx) > 0 is always satisfied. From mathematical point of view, the additive noise term will always introduce a finite probability for x to be negative. Physically speaking, the tumor cell mass x cannot be negative. Thus the cell mass x needs to be greater than zero in Eq. (1). Then the corresponding Fokker Planck equation of the probability density function P (x, t) can be written as [33 35] P (x, t) t A(x)P (x) = + 2 B(x)P (x, t) x x 2, (10) where P (x, t) is the probability density function of the cell mass. The drift coefficient A(x) and the diffusion coefficient B(x) are given by A(x) = ax bx2 1 + τbx + Dx λ Dα τbα + λτbx Dα (1 + τbx) 3, (11) B(x) = Dx2 2λ Dαx + α (1 + τbx) 2. (12) In Eqs. (10) (12), there are natural boundary conditions that need to be fulfilled. These natural boundary conditions include P (0, t) and P (, t) 0. When t, the probability distribution function P (x, t) tends to a steady state. After a straightforward calculation, the steady state probability distribution function P st (x) can be obtained, [36,37] P st = N(1 + τbx)(dx 2 2λx Dα + α) γ exp [ τb2 2D x2 + where N is the normalization constant and β = 1 Dα(1 λ2 ) γ = 1 [a + ατb2 2D D + 2λ [ (2λ 2 1) α D α D (τab 2λτb2 α/d b) D x + β arctg (Dx λ Dα) ], (13) Dα(1 λ2 ) α ) α (τab 2λτb 2 D b + aλ D + ατb2 λ α ], (14) D D (τab 2λτb 2 α D b )] 1 2. (15) To determine the maxima of the steady state probability distribution function P st (x), one has dp st (x)/dt = 0. Then the equation of dp st (x)/dt = 0 can be easily written as τ 2 b 3 x 4 + (2τb 2 τ 2 ab 2 )x 3 + (b 2τab)x 2 + (D a + τbλ Dα)x (τbα + λ Dα) = 0. (16) According to the combination of Sturm s theorem and Descarte s rule of signs, [38] the relations between the tumor cell growth rate a, the coupling strength λ, the noise correlation time τ, and the strength D of colored noise can be derived from Eq. (16).

3 No. 1 Transitions in a Logistic Growth Model Induced by Noise Coupling and Noise Color Numerical Simulation In order to check the validity of the approximate method employed in the derivation, it is necessary to perform numerical simulation. The coupling between two noise terms can be separated. The colored noise terms in Eq. (2) can be described by a differential equation of Ornstein Uhlenbeck (OU) noise with constant intensity. The differential equation of the colored noise can be written as dε(t) = ε(t) + ξ(t), (17) dt τ τ where ξ(t) is Gaussian white noise with zero mean and correlation ξ(t)ξ(t ) = 2Dδ(t t ). (18) The numerical simulations are performed by integrating the dynamical equations (1), (17), and (18). Gaussian white noise is generated using the Box Müller algorithm and a pseudo-random number generator. [39] The numerical data of time series are obtained using the second order Runge Kutta algorithm with a time step of t = Then the data are saved over 500 different trajectories. Each single trajectory evolves over 10 6 periods. The numerical results of the steady state probability distributions are plotted in the corresponding figures using the symbols of,,, and. 4 Coupling Between Noise Terms To see the effects of coupling λ between noise terms, the relation of colored noise strength D and the cell growth rate a needs to be calculated when the coupling λ is varied from 1.0 to The phase diagram between D and a is plotted in Fig. 1 when λ is varied. Fig. 1(a) is a plot of the parameter plane (a, D) for 0.0 λ 1.0 and τ = 1.0. The parameter plane is divided into two regions. In region I, there is a single peak in P st (x). In region II, there are three extremes of two peaks and one valley in P st (x). Regions I and II are separated by two critical curves. The shape of region II is like a horn. The apex of the horn is shifted to large values of both D and a when λ is increased. The upper left part of the boundary line between regions I and II is almost the same no matter whether λ is 0 or 1.0. However, the right part of the boundary line is raised when λ is increased. That is, region II is reduced as λ is increased. From Fig. 1(a), it is noted that the re-entrance-like phase transition phenomenon appears. [10 13,40] For suitable values of a, the curve of P st (x) is changed from one peak to two peaks and one valley, and then to one peak again by increasing D. Similarly, for suitable value of D, the same re-entrance-like phase transition phenomenon occurs by increasing a. Figure 1(b) is a plot of the parameter plane (a, D) for 1.0 λ < 0.0 and τ = 1.0. It is clear that the parameter plane is divided into three regions. In region I, there is no extreme in P st (x). The curve of P st (x) is a monotonically decreasing function of x. In region II, there are two extremes of one peak and one valley in P st (x). In region III, there is a single peak in P st (x). The critical line between regions I, II, and III is the horizontal line. The horizontal line is increased as the coupling λ is increased. That is, region III is expanded as λ is increased. The lower part of the boundary between regions I and II is shifted to larger value of D and a as λ is increased. For the upper part of the boundary, three lines are almost combined to one. That is, for very large values of D and a, the critical line between regions I and II are almost the same even if λ is varied. It is clear that the first-order-like phase transition phenomenon appears. [8,9] Fig. 1 Parameter plane of (a, D) when λ is varied. The parameters are chosen as τ = 1.0, b = 0.1, α = 3.0. (a) 0.0 λ 1.0 with λ = 0.0 (solid line), 0.5 (dashed line), 1.0 (dash-dotted line); (b) 1.0 λ < 0.0 with 1.0 (solid line), 0.5 (dashed line), λ = 0.3 (dash-dotted line). The analytic expressions and the numerical simulations of P st (x) are plotted in Fig. 2 as a function of x. Figure 2(a) is a plot of P st (x) for λ = 0.5 with different values of a and fixed value of D. The parameters of curves

4 178 SHI Jin and ZHU Shi-Qun Vol. 46 a, b, and c are chosen as those marked by a, b, and c in Fig. 1(a). It is clear that the curve of P st (x) is changed from one peak to two peaks and one valley, and then to one peak again. It shows clearly the re-entrance-like phase transition phenomenon. [10 13,40] From Fig. 2(a), it is seen that the height of P st (x) is decreased while the position of the peak in P st (x) is moved to larger value of x when a is increased and D is fixed. This means that the distribution of tumor cell population can be suppressed by increasing the growth rate a. The curve a shows a sharp and very high peak in P st (x) at small values of x. Then the curve of P st (x) drops very fast and finally decreases to zero. This means that the probability distribution of tumor cell population is very high at small tumor cell mass x but decreases to zero very fast at large values of x when the tumor cell growth rate is small. The curves b and c in P st (x) shows quite a low height with broad distribution. This means that the probability distribution of tumor cell population is small but it covers from small to large values of tumor cell mass x. The probability of tumor cell population is decreased to zero very slowly when the tumor cell growth rate is large. Figure 2(b) is a plot of P st (x) for λ = 0.5 and τ = 1.0 when parameters a and D are changed. The parameters of curves d, e, and f are chosen as those marked by d, e, and f in Fig. 1(b). It is clearly seen that the curve of P st (x) undergoes monotonically decreasing function to one peak and one valley, and then to one peak. It shows clearly the first-order-like phase transition phenomenon. [8,9] Since the intensity D of colored noise is fixed in curves d and e, the comparison of curves d and e can reveal the effect of the cell growth rate a. The curve d shows that there is no peak in P st (x) and P st (x) is a monotonically decreasing function of x. This means that the probability distribution of tumor cell population is decreased as the tumor cell mass x is increased when the tumor cell growth rate is small. The curve e in P st (x) shows quite a low height with a broad distribution. This means that the probability distribution of tumor cell population is small but it covers the range from small to medium values of tumor cell mass x when the tumor cell growth rate is increased. Since the cell growth rate a is fixed in curves e and f, the comparison of curves e and f can reveal the effect of the intensity D of the colored noise. From curve f, it is seen that the peak in P st (x) is very high for small values of D and large values of a. This means that the distribution of tumor cell population can be increased by decreasing the multiplicative colored noise and increasing the cell growth rate. From Fig. 2, it is seen that the analytic expressions are consistent with the numerical simulations. Fig. 2 The steady state probability distribution P st(x) as a function of x. The analytic expressions are shown by solid, broken, and dotted lines while the numerical simulations are plotted by,, and. The parameters are chosen as τ = 1.0, b = 0.1, α = 3.0. (a) λ = 0.5, D = 5.0. The parameters are a = 0.5 (curve a: ); a = 2.5 (curve b: ); and a = 4.5 (curve c: ); (b) λ = 0.5. The parameters are a = 0.5, D = 2.0 (curve d: ); a = 2.0, D = 2.0 (curve e: ); and a = 2.0, D = 0.25 (curve f: ). To see the effects of the coupling λ, the analytic and numerical results of P st (x) are plotted in Fig. 3 when λ is varied. The parameters are chosen as those marked by A in Fig. 1(a) and B in Fig. 1(b) for τ = 1.0. From Fig. 3(a), it is seen that the curve P st (x) is changed from one peak to two peaks and one valley, and then to one peak again when λ is increased. For λ 0, there is continuous changes from region I to region II, and then to region I again by varying λ. The re-entrance-like phase transition phenomenon appears by continuously varying λ. From Fig. 1(a), it is also clear that the parameter region of appearance of re-entrance-like phase transition phenomenon is extremely small. From Fig. 3(b), it is clear that the curve P st (x) is changed from monotonically decreasing function to one peak and one valley,

5 No. 1 Transitions in a Logistic Growth Model Induced by Noise Coupling and Noise Color 179 and then to single peak as λ is increased. For λ < 0, the curve of P st (x) is also continuously changed from region I to region II, and then to region III by varying λ. The first-order-like phase transition occurs by varying λ continuously. From Fig. 3(a), it is seen that the approximate analytic results are in good agreement with the numerical simulations for λ = 0. For λ = 0.5 and 1, the curve of the approximate analytic results are shifted to smaller values of x compared to that of the numerical simulations. The deviations are noticeable. However, there is reasonably good agreement between the two results. While for λ < 0 in Fig. 3(b), the approximate analytic results are in good agreement with the numerical simulations. Fig. 3 The steady state probability distribution P st(x) as a function of x when λ is varied. The analytic expressions are shown by solid, broken, and dotted lines while the numerical simulations are plotted by,, and. The parameters are chosen as τ = 1.0, b = 0.1, α = 3.0. (a) 1.0 λ 0, a = 1.64, D = 2.8. The parameters are chosen as λ = 0 : ; λ = 0.5 : ; λ = 0.9 : ; (b) 1.0 λ < 0, a = 0.4, D = The parameters are chosen as λ = 0.3 : ; λ = 0.5 : ; λ = 1 :. 5 Effects of Noise Color To see the effects of noise color, the relation of the noise strength D and the cell growth rate a needs to be calculated when the noise correlation time τ is varied from 0.5 to 1.5. Fig. 4 Parameter plane of (a, D) when τ is varied. The parameters are chosen as b = 0.1, α = 3.0. The curves are plotted as τ = 0.5 (dashed line), 1.0 (solid line), 1.5 (dash-dotted line). (a) λ = 0.5; (b) λ = 0.5. The parameter plane of (a, D) is plotted in Fig. 4 when τ is varied. Figure 4(a) is a plot of the parameter plane (a, D) for λ = 0.5. Similar result as that in Fig. 1(a) is obtained. The apex of the curves of region II is shifted to smaller values of a and D when τ is increased from 0.5 to 1.5. The upper part of the curve is expanded, i.e., region II is expanded as τ is increased. From Fig. 4(a), it is seen that the re-entrance-like phase transition phenomenon appears again by increasing either D or a for suitable values of the parameters. [10 13,40] Figure 4(b) is a plot of the parameter plane (a, D) for λ = 0.5. Similar to Fig. 1(b), the parameter plane is also divided into three regions. The upper part of the critical line between regions I and II is shifted to smaller values of

6 180 SHI Jin and ZHU Shi-Qun Vol. 46 a when the noise correlation time τ is increased. That is, region II is expanded as τ is increased. It is clear that the first-order-like phase transition phenomenon appears again. [8,9] The approximate analytic results and numerical computations of P st (x) are plotted in Fig. 5 as a function of x. Figure 5(a) is a plot of P st (x) for λ = 0.5 and τ = 1.0 with different values of D and fixed value of a. The parameters of curves a, b, and c are chosen as those marked by a, b, and c in Fig. 4(a). It shows clearly the re-entrance-like phase transition phenomenon as D is increased. [10 13,40] The height of the peak is increased and shifted to small values of x. This means that the peak distribution of tumor cell population can be enhanced by increasing the multiplicative noise. Meanwhile, the position of the most probability distribution of tumor cell population can be reduced to smaller value of x by increasing D. At small values of tumor cell mass x, the distribution of tumor cell can be increased by increasing D. Comparing Figs. 2(a) and 5(a), it is shown that increasing the tumor cell growth rate a can reduce the height of the probability distribution of tumor cell population anomalously if the colored noise intensity D is fixed. If the tumor cell growth rate a is fixed, the height of the probability distribution of the tumor cell population can only be increased slightly by increasing the colored noise intensity D. Figure 5(b) is a plot of P st (x) for λ = 0.5 and τ = 1.0 when parameters a and D are changed. The parameters of curves d, e, and f are chosen as those marked by d, e, and f in Fig. 4(b). It is clearly seen that the first-order-like phase transition occurs again. [8,9] Comparing Figs. 2(b) and 5(b), it is seen that similar results are obtained. Fig. 5 The steady state probability distribution P st(x) as a function of x. The analytic expressions are shown by solid, broken, and dotted lines while the numerical simulations are plotted by,, and. The parameters are chosen as b = 0.1, α = 3.0. (a) λ = 0.5, a = 2.0. The parameters are D = 1.5 (curve a: ); D = 3.5 (curve b: ); and D = 5.5 (curve c: ); (b) λ = 0.5. The parameters are a = 0.5, D = 2.0 (curve d: ); a = 1.5, D = 2.0 (curve e: ); and a = 1.5, D = 0.1 (curve f: ). Fig. 6 The steady state probability distribution P st(x) as a function of x. The analytic expressions are shown by solid, broken, and dotted lines while the numerical simulations are plotted by,, and. The parameters are chosen as those marked by A and B in Fig. 4 with b = 0.1, α = 3.0. (a) λ = 0.5, a = 3.0, D = 5.5. τ = 0.5 : ; τ = 1.0 : ; τ = 1.5 : ; (b) λ = 0.5, a = 0.4, D = τ = 0.5 : ; τ = 1.0 : ; τ = 1.5 :.

7 No. 1 Transitions in a Logistic Growth Model Induced by Noise Coupling and Noise Color 181 From Fig. 5, it is clear that the analytic results are in good agreement with the numerical simulations. To see the effects of noise color on the transition, the approximate analytic results and numerical simulations of P st (x) are plotted in Fig. 6 when the noise correlation time τ is varied. The parameters are chosen as those marked by A in Fig. 4(a) for λ = 0.5 and B in Fig. 4(b) for λ = 0.5. From Fig. 6(a), it is seen that P st (x) is changed from a single peak to two peaks and one valley, and then to single peak again when τ is increased from 0.5 to 1.5. For λ 0, the re-entrance-like phase transition phenomenon appears by varying τ continuously. The peak in P st (x) is decreased, then increased and moved to larger values of x as τ is increased. From Fig. 6(b), it is clear that the curve of P st (x) is changed from monotonically decreasing function to one peak and one valley, and then to single peak as τ is increased from 0.5 to 1.5. For λ < 0, the first-order-like phase transition phenomenon appears by varying τ continuously. Comparing Figs. 3 and 6, it is clear that similar results are obtained. Figure 6 shows that the approximate results are consistent with the numerical simulations. It is very interesting to note that the noise color can induce continuous first-order-like phase transition and reentrance-like phase transition phenomena in the logistic growth model. [8 13,40] The approximate results and numerical computations of the steady state probability distribution P st (x) are plotted in Fig. 7 when the multiplicative colored noise correlation time τ is varied. Figure 7(a) is a plot of P st (x) when the coupling strength is λ = 0.5. From Fig. 7(a), it is seen that the height of the peak is increased and the position of the peak is moved to large values of x when τ is increased. Meanwhile, when τ is increased, the broad distribution of P st (x) with a long tail is changed to a sharp distribution. That is, the most probability distribution of tumor cell population is increased and moved to large values of tumor cell mass x. The distribution of tumor cell population is also concentrated to a small range of tumor cell mass with medium values of x. For small and large values of tumor cell mass x, the distribution of tumor cell population is almost zero for large noise correlation time τ. Figure 7(b) is a plot of P st (x) when the coupling strength is λ = 0.5. Similar results as those in Fig. 7(a) are obtained. It is clear that the height of the peak in P st (x) for λ = 0.5 is lower than that for λ = 0.5. From Fig. 7, it is seen that the height of the distribution of tumor cell population is increased and concentrated to medium value of tumor cell mass x when the noise correlation time τ is increased. When τ is increased, the peak of the approximate result is higher than that of the numerical simulation. For τ = 20, the deviation is obvious. However, the analytic results are reasonably in good agreement with the numerical computations. Fig. 7 The steady state probability distribution P st(x) as a function of x when τ is varied. The analytic expressions are shown by solid, broken, dotted and broken dotted lines while the numerical simulations are plotted by,, and. The parameters are chosen as a = 1.0, b = 0.1, D = 0.3, α = 3.0. (a) λ = 0.5. τ = 0 : ; τ = 0.5 : ; τ = 5.0 : ; τ = 20 : ; (b) λ = 0.5. τ = 0 : ; τ = 0.5 : ; τ = 5.0 : ; τ = 20 :. 6 Discussion The effects of the environmental fluctuation on tumor cell growth are investigated through the coupling between noise terms and noise color in logistic growth model. It is seen that the re-entrance-like phase transition phenomenon occurs when the value of λ is positive. The re-entrance-like phase transition phenomenon can be achieved either by varying λ or by varying τ continuously. The first-order-like phase transition phenomenon occurs when the value of λ is negative. There is no re-entrance-like phase transition phenomenon if λ is negative. There is continuous change of first-order-like phase transition phenomenon when either λ or τ is changed. It is clear that either the coupling λ

8 182 SHI Jin and ZHU Shi-Qun Vol. 46 between noise terms or the noise color τ can induce re-entrance-like phase transition and first-order-like phase transition continuously. That is, both the coupling between noise terms and noise color can enhance tumor cell growth when the number of tumor cell is very small and repress the tumor cell growth when the number of tumor cell is very large. When the noise correlation time τ is increased, the distribution of tumor cell population is concentrated and sharply peaked to a small range of medium value of tumor cell mass. The phenomenon of the noise induced transitions in the logistic growth model can provide the effects of the external factors on the probability distribution function of the cancer cell mass x and give some suggestions of the medical treatment. It is found that the approximate theoretical results are in good agreement with the numerical simulations. Acknowledgments It is a pleasure to thank Yin-Sheng Ling, Xiao-Qin Luo, and Dan Wu for many valuable suggestions and numerical simulations. References [1] P. Hanggi and P. Jung, Adv. Chem. Phys. 89 (1995) 239. [2] W. Horsthemke and R. Lefever, Noise-Induced Transitions, Springer-Verlag, Berlin (1984). [3] A. Fulinski and T. Telejko, Phys. Lett. A 152 (1991) 11. [4] B.Q. Ai, X.J. Wang, L.G. Liu, M. Nakano, and H. Matsuura, Chin. Phys. Lett. 19 (2002) 137. [5] S.I. Denisov, A.N. Vitrenko, and W. Horsthemke, Phys. Rev. E 68 (2003) [6] S. Zhu, Phys. Rev. A 47 (1993) [7] Y. Ja and J. Li, Phys. Rev. E 53 (1996) [8] L. Lin, L. Cao, and D. Wu, Phys. Rev. A 48 (1993) 739. [9] L. Cao, D. Wu, and L. Lin, Phys. Rev. A 49 (1994) 506. [10] C. Van den Broeck, J.M.R. Parrondo, and R. Toral, Phys. Rev. Lett. 73 (1994) [11] J.H. Li and Z.Q. Huang, Phys. Rev. E 53 (1996) [12] Y. Jia and J.R. Li, Phys. Rev. Lett. 78 (1997) 994. [13] F. Castro, A.D. Sanchez, and H.S. Wio, Phys. Rev. Lett. 75 (1995) [14] S.E. Mangioni, R.R. Deza, H.S. Wio, and R. Toral, Phys. Rev. Lett. 79 (1997) [15] S.E. Mangioni, R.R. Deza, R. Toral, and H.S. Wio, Phys. Rev. E 61 (2000) 223. [16] S.E. Mangioni, R.R. Deza, and H.S. Wio, Phys. Rev. E 66 (2002) [17] P. Jung and P. Hanggi, Phys. Rev. A 35 (1987) [18] P. Jung and P. Hanggi, J. Opt. Soc. Am. B 5 (1988) 979. [19] R. Kubo, A stochastic theory of a line-shape and relaxation, in Fluctuation, Relaxation and Resonance in Magnetic Systems, ed. by D. Ter Haar, Oliver and Boyd, Edinburgh, Scotland (1962) pp [20] P. Lett, R. Short, and L. Mandel, Phys. Rev. Lett. 52 (1984) 341. [21] S.N. Dixit and P.S. Sahni, Phys. Rev. Lett. 50 (1983) [22] S. Zhu, A.W. Yu, and R. Roy, Phys. Rev. A 34 (1986) [23] M. San Miguel, L. Pesquera, M.A. Rodriguez, and A. Hernandez-Machado, Phys. Rev. A 35 (1987) 208. [24] R.F. Fox and R. Roy, Phys. Rev. A 35 (1987) [25] X. Luo and S. Zhu, Eur. Phys. J. D 19 (2002) 111. [26] M. Eisen, Mathematical Models in Cell Biology and Cancer Chemotherapy, Lecture Notes in Biomathematics, Vol. 30, Springer-Verlag, New York (1979). [27] G.W. Swan, Optimization of Human Cancer Radiotherapy, Lecture Notes in Biomathematics, Vol. 42, Springer- Verlag, New York (1981). [28] A. Lipowski and D. Lipowska, Physica A 276 (2000) 456. [29] R. Kemkemer, S. Schrank, W. Vogel, H. Gruler, and D. Kaufmann, Proc. Nat. Acad. Sci. 99 (2002) [30] S. Michelson and J.T. Leith, J. Theor. Biol. 169 (1994) 327. [31] J.C. Panetta, Appl. Math. Lett. 8 (1995) 83. [32] B.Q. Ai, X.J. Wang, G.T. Liu, and L.G. Liu, Phys. Rev. E 67 (2003) [33] D.J. Wu, L. Cao, and S.Z. Ke, Phys. Rev. E 50 (1994) [34] C.W. Gardiner, Handbook of Stochastic Methods, 2nd ed., Springer-Verlag, Berlin (1990). [35] H. Risken, The Fokker Planck Equation, 2nd ed., Springer-Verlag, Berlin (1990). [36] S.I. Denisov and W. Horsthemke, Phys. Rev. E 65 (2002) [37] S.I. Denisov and W. Horsthemke, Phys. Rev. E 65 (2002) [38] J. Nathan, Basic Algebra 1, Freeman, New York (1985). [39] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in FORTRAN, 2nd ed., Cambridge University Press, Cambridge, England (1992). [40] L. Cao and D.J. Wu, Phys. Lett. A 260 (1999) 126.