Time Delay Induced Stochastic Resonance in One Species Competition Ecosystem without a Periodic Signal

Transcription

1 Commun. Theor. Phys. 57 (2012) Vol. 57, No. 4, April 15, 2012 Time Delay Induced Stochastic Resonance in One Species Competition Ecosystem without a Periodic Signal WANG Xiu-Hua ( ), 1 BAI Li (Ü ), 1 ZHOU Zhong-Rao ( ), 1 NIE Lin-Ru (Ñ ), 1, and MEI Dong-Cheng (Öý ) 2 1 Faculty of Science, Kunming University of Science and Technology, Kunming , China 2 Department of Physics, Yunnan University, Kunming , China (Received July 11, 2011; revised manuscript received November 21, 2011) Abstract One-species competition ecosystem with noise and time delay was investigated as not driven by a periodic force. The results show that the time delay is responsible for stochastic resonance of the system as delay time is smaller than critical point of the Hopf bifurcation. PACS numbers: a, n Key words: one-species competition ecosystem, time delay, stochastic resonance 1 Introduction In recent years, much attention has been paid to stochastic resonance (SR) [1 3] in various fields such as physics, [4 6] biology, [7] laser, [8] ecology, [9 10] etc. SR is a seemingly counterintuitive phenomenon, caused by the interaction among nonlinear system, noise, and small periodic signal. As SR takes place, variables of the system oscillate more harmonically with the periodic signal and the signal is magnified, and the output signal-to-noise ratio (SNR) exhibits a maximum with respect to noise intensity. In fact, the small periodic force is not the necessary condition for SR. Reference [11] investigated a predatorprey system without a periodic force, equally found an SR phenomenon. But it did not consider time delay. In the absence of an external signal, it has been realized that noise also can enhance regular dynamics in nonlinear systems, when an internal time scale is present in the system. This phenomenon was initially considered as SR in an autonomous system and later named coherence resonance (CR). [12 13] As CR occurs, time evolution of the system exhibits maximal regularity at an optimal noise intensity. The regularity can be reflected by some statistical quantities (e.g., characteristic correlation time, coefficient of variation). As we check whether there exists SR or not in a system, however, a general method is to simulate the system s power spectra whose quality depends on both time series regularity and amplitude size of state variable, and lets us to determine an optimal noise intensity at which the SNR attains a maximum. In realistic systems, however, an inclusion of time delay is more natural. Several authors have investigated effect of time delay, and have found the consequent dynamic phenomena such as multistability, [14 15] Hopf bifurcation, [16 17] clustering, [14] amplitude death, [18] anticipated synchronization, [19] and stability switch, [20] For the system with noise and time delay, their combinations usually make it more complex and richer dynamic behaviors. Reference [21] studied mutualism ecosystems, and found that noise and time delay can suppress population explosion. In a symmetric two-species competition system subjected to noise, coherent resonance-like with respect to delay time was reported. [22] The purpose of this paper is to discuss effect of time delay on the SR in one-species competition ecosystem without a periodic signal. The paper is structured as follows. In Sec. 2, the Hopf bifurcation about delay time in the onespecies competition system without noise is presented. In Sec. 3, the effect of time delay on the SR of the system is analysed. In Sec. 4, conclusions are made. 2 Hopf Bifurcation In the deterministic case, the classical Lotka Volterra model of one-species system can be described by the following differential equation [23 24] = x(t)[r + ax(t)], (1) where x(t) represents the species density at time t, r is the growth rate of the species, and a is the intraspecies interaction parameter. If a > 0, the system is symbiotic, otherwise competitive. Here we consider the case of a < 0. If time delay is further considered, Eq. (1) can be Supported by the Yunnan Provincial Foundation of China under Grant Nos. 2009CD036 and 08Z0015, and the National Natural Science Foundations of China under Grant No lrnie@163.com c 2011 Chinese Physical Society and IOP Publishing Ltd

2 620 Communications in Theoretical Physics Vol. 57 rewritten into [25 26] = x(t)[r + ax(t τ)], (2) where τ is the delay time. τ = 0 means that all members of the species survive to the same age, and the egg is instantaneously converted into an adult. In this model, it is assumed that the birth rate coefficient is diminished by the population of the preceding generation, τ being the generation time (the time required in going from an egg stage to the adult stage). Fig. 1 Time evolution of species density x at different delay times: (a) τ = 0.1; (b) τ = ; (c) τ = ; and (d) τ = 0.2. The values of parameters are a = 1 and r = 10. The initial values are x(0) = 9.9, and x(t τ) = 9.9 as t < τ, with time step Fig. 2 Hopf bifurcation of x with respect to delay time τ. r = 10, a = 1, 2, 5, 10 for (a); a = 1, r = 2, 3,6, 10 for (b). By means of Eq. (2), the time evolution of state variable x is numerically calculated at different delay times, and the results are shown in Fig. 1. It can be seen from Fig. 1 that there is a critical phenomenon about delay time in the system. x decays vibrationally to zero with time as τ is smaller than about , while in the case of τ > the state variable x oscillates periodically at a constant amplitude. Namely, the time delay can cause the Hopf bifurcation of the system with respect to delay time. The Hopf bifurcations of the state variable as a function of delay time under different values of a and r are depicted in Figs. 2(a) and 2(b), respectively. From Fig. 2 we can see that critical points of the bifurcations are dependent on the value of r and independent of the value of a. With

3 No. 4 Communications in Theoretical Physics 621 the decrement of r, the critical value of τ increases. As r = 10, the critical value τ c = Analytically we will demonstrate the Hopf bifurcation caused by time delay. In Eq. (2) there is a nontrivial equilibrium point x eq = r/a. Linear stability of the system near the equilibrium point can be analysed by means of the following method. [27 28] Substituting x(t) = x eq +y(t) into Eq. (2), we obtain the equation the deviant variable y(t) satisfies dy(t) = ry(t τ). (3) The solution of Eq. (3) has the form y(t) = y(0)e λt, where λ is the characteristic root λ = r e λτ. (4) It can be seen from Eq. (4) that as τ = 0, λ = r < 0, namely, the system is stable. With the increment of τ = 0, the solution of Eq. (4) about λ gradually becomes an imaginary number. As the system transits from stable region to oscillatory region, λ is an imaginary number. Substituting λ = iz into Eq. (4), we have cos(zτ) = 0, (5) r sin(zτ) = z. (6) Eliminating z from Eqs. (5) and (6), the critical value of τ can be given by τ c = π 2r. (7) have the following statistical properties: ξ(t) = 0 and ξ(t)ξ(t ) = 2Dδ(t t ), with D being the noise intensity. Then Eq. (2) becomes [21,25] = x(t)[r + ax(t τ)] + x(t) 2 ξ(t). (8) Only the external noise is considered here. There is also internal reaction noise that should be quite relevant, but is neglected in the present study. Also, the assumed form of white noise for the proliferation rate r represents a somewhat arbitrary choice, and some additional motivations could be useful. The delay Fokker Planck equation related to Eq. (8) reads [29] P(x, t) = [x(r + ax τ ) 2Dx 3 ] t x P(x, t; x τ, t τ)dx τ 2 + D x 2 x4 P(x, t; x τ, t τ)dx τ, (9) where x τ denotes x(t τ), P(x, t; x τ, t τ) denotes the joint probability density. Some interesting dynamic behaviors of a system often take place near a critical point. In what follows, we mainly emphasize on the stochastic behaviors of x near the Hopf bifurcation point τ c. Fig. 3 The parameter plane of τ-r. Equation (7) indicates that the critical value τ c only depends r. According to Eq. (7), we plotted the parameter plane of τ-r describing the system s stability, see Fig Stochastic Resonance An ecosystem is always influenced by external environments, such as temperature and climate change. This will give rise to a fluctuation of intraspecies interaction parameter. We describe the fluctuation by inducing a stochastic term a a + ξ(t), in which ξ(t) is the stochastic force of the Gaussian white noise, and assumed to Fig. 4 Time evolution of species density x at delay time τ = at different levels of noise: (a) D = 10 6 ; (b) D = 10 3 ; and (c) D = The other parameter values are the same as in Fig. 1. In the case of small delay time, Eq. (9) can be approximately solved. But for the delay time τ = near the Hopf bifurcation point, stochastic simulation is a feasible method to deal with the system. Using Eq. (8) to simulate stochastically the time evolution of x at different noise intensities in the time delay (τ = ), the results are shown in Fig. 4. Figure 4 indicates that in the case of time delay the noise causes the state variable x make periodic oscillations with time, but in the deterministic case x decays periodically to zero (see Fig. 1(b)). This means

4 622 Communications in Theoretical Physics Vol. 57 that the noise can activate the potential vibratory behaviors of x in the deterministic case. Furthermore, for the smaller noise intensity (e.g., D = 10 6 ), the oscillation of x is well-ordered, but its amplitude is relatively smaller. With the increment of noise intensity (e.g., D = 10 3 ), the amplitude of x is amplified to a great extent although its periodicity is somewhat damaged. For the greater noise intensity (e.g., D = ), x oscillates acutely with time at a greater amplitude, yet its time series is the most irregular. be seen from Fig. 6 that β as a function of noise intensity exhibits a non-monotonic behavior, and there exists an optimal noise intensity which makes β be maximum, namely, SR phenomenon. Fig. 6 The quality factor β as a function of noise intensity. The other parameter values are the same as in Fig. 4. Fig. 5 The power spectrums of x corresponding to Fig. 4. An interesting thing in the system not driven by a periodic force is that in the case of time delay the noise can cause its state variable to vibrate sinusoidally with time. It means that the noise stimulates the system to emit a periodic signal. The exploration of the signal depends on its signal-to-noise ratio. Thus we numerically simulate the power spectra at the three noise intensities corresponding to Fig. 4, and the results are plotted in Fig. 5. Figure 5 indicates that there is a peak at the frequency ω p = π in the power spectrum no matter what value the noise intensity takes, and both the height and the wih of the peak are affected by levels of the noise. With the increment of noise intensity, the peak of the power spectrum first becomes higher and higher then approaches to a saturation value, but its wih is greater and greater. The quality factor of the power spectrum can be measured by the quantity β, defined as [30] ( ω ) 1 β = h, (10) ω p where h is the peak height, ω p represents the peak frequency, and ω denotes the peak wih at the height h 1 = e 1/2 h. The peak height h increases with the signal power at ω = ω p. The wider the peak, the greater the noise power. So β corresponds to a signal-to-noise ratio in a sense. By means of Eq. (10) we numerically calculate the dependence of β on noise intensity, shown in Fig. 6. It can 4 Conclusions Up to now, we have investigated the SR behavior in the one-species competition ecosystem with noise and time delay. In the deterministic case the state variable decays to zero as the delay time is smaller than the critical point τ c. Yet in the stochastic case the noise can cause the state variable oscillate periodically. As the noise intensity takes an optimal value, the response of the system can be enhanced maximally. It can be seen easily from Fig. 7 that as τ = 0 the SR disappears. Therefore the counterintuitive phenomenon is induced by the time delay. The different place from conventional SRs is that the system is not driven by a periodic force. Fig. 7 Time evolution of species density x obtained by means of stochastic simulations of Eq. (8) at different delay times: (a) τ = 0; (b) τ = 0.08; and (c) τ = , in the case of noise intensity D = The other parameter values are the same as in Fig. 1.

5 No. 4 Communications in Theoretical Physics 623 References [1] L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70 (1998) 223. [2] J.H. Li, Commun. Theor. Phys. 50 (2008) [3] P.S. Burada, G. Schmid, D. Reguera, M.H. Vainstein, J.M. Rubi, and P. Hänggi, Phys. Rev. Lett. 101 (2008) [4] Y. Jia, X.P. Zheng, X.M. Hu, and J.R. Li, Phys. Rev. E 63 (2001) [5] J.H. Li and Y.X. Han, Phys. Rev. E 74 (2006) [6] J.H. Li, Commun. Theor. Phys. 53 (2010) 517. [7] W.R. Zhong, Y.Z. Shao, and Z.H. He, Phys. Rev. E 73 (2006) (R). [8] A. Joshi and M. Xiao, Phys. Rev. A 74 (2006) [9] José M.G. Vilar and Ricard V. Solé, Phys. Rev. Lett. 80 (1998) [10] L.R. Nie, J.H. Peng, and D.C. Mei, Commun. Theor. Phys. 55 (2011) 829. [11] R. Rai and H. Singh, Phys. Rev. E 62 (2000) [12] Y.M. Kang, Europhys. Lett. 94 (2011) [13] M.A. Arteaga, M. Valencia, M. Sciamanna, H. Thienpont, M. López-Amo, and K. Panajotov, Phys. Rev. Lett. 99 (2007) [14] S.H. Park, S. Kim, H.B. Pyo, and S. Lee, Phys. Rev. E 60 (1999) [15] S. Kim, S.H. Park, and H.B. Pyo, Phys. Rev. Lett. 82 (1999) [16] Z.H. Wang and H.Y. Hu, Acta Mech. Sin. 24 (2008) 449. [17] Z.H. Wang and H.Y. Hu, Int. J. Bifurcation and Chaos 17 (2007) [18] D.V. Ramana Reddy, A. Sen, and G.L. Johnston, Phys. Rev. Lett. 85 (2000) [19] C. Masoller, H.L.D. de S. Cavalcante, and J.R. Rios Leite, Phys. Rev. E 64 (2001) [20] Z.H. Wang and H.Y. Hu, J. Sound and Vibration 233 (2000) 215. [21] L.R. Nie and D.C. Mei, Europhys. Lett. 79 (2007) [22] L.R. Nie and D.C. Mei, Phys. Rev. E 77 (2008) [23] A.J. Lotka, Proc. Nat. Acad. Sci. USA 6 (1920) 410. [24] V. Volterra, Mem. Accad. Nazionale Lincei Ser. 6(2) (1926) 31. [25] N.S. Goel, S.C. Maitra, and E.W. Montroll, Rev. Mod. Phys. 43 (1971) 231. [26] J. Hofbauer and K. Zigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge (1988). [27] H.Y. Hu and Z.H. Wang, Chaos, Solitons and Fractals 40 (2009) 13. [28] H.Y. Hu and Z.H. Wang, J. Sound and Vibration 214 (1998) 213. [29] T.D. Frank, Phys. Rev. E 72 (2005) [30] G. Hu, T. Ditzinger, C.Z. Ning, and H. Haken, Phys. Rev. Lett. 71 (1993) 807.