Influence of Noise on Stability of the Ecosystem

Transcription

1 Commun. Theor. Phys. 60 (2013) Vol. 60, No. 4, October 15, 2013 Influence of Noise on Stability of the Ecosystem LIU Tong-Bo ( ) and TANG Jun (» ) College of Science, China University of Mining and Technology, Xuzhou , China (Received December 19, 2012; revised manuscript received March 7, 2013) Abstract Based on a simplified predator-prey model, the influence of noise on the ecosystem has been studied. The results show the following facts. (i) For all parameter values, the existence of noise maintains the oscillatory state of the ecosystem, and enough strong noise can destroy the ecosystem, which means the annihilation of the species. (ii) Comparing to oscillation with small amplitude, while the ecosystem oscillates explosively with large amplitude, it is more likely to lose balance. In addition, the small-amplitude oscillation takes on higher level of regularity. All the numerical results are reasonable comparing to the general knowledge about ecosystem. PACS numbers: Ac, Cb, a Key words: noise, predator-preg model, stability The relationship between predator and prey populations is the most essential factor contributing to ecosystem conservation. So, much theoretical attention has been paid on the predator-prey population dynamics. Since the pioneer work of Lotka [1] and Vortella, [2] many different mathematical models have been presented to mimic different types of ecosystems under different environmental conditions. For example, based on the experimental results, Holling et al. have introduced different model equations for different ecosystems. [3 4] Chattopadhyay et al. [5] have studied the ecosystem system with diseased species by using a three-variable model. In recent years, the migration of species has attracted much attention, which is simulated mathematically as a form of diffusion in the 2D ecosystem. The research results show that fruitful pattern formation appears in the ecosystem with migrating species, such as spiral pattern, Turing pattern, etc. [6] Single-armed, even multi-armed spiral pattern are observed in an ecosystem which contains three species with cyclic competition, and the competition rate is a important fact that determines the stability of spiral waves and the emergence of biodiversity, i.e., the change of competition rate makes the ecosystem transit from a state of coexistence of species to a state of extinction of one species. [7 8] In a four-species ecological system with cyclic dominance, phase transition from the coexistence of all four species to the existence of only two neutral species emerges through the changing of parameter across the threshold value. [9] In addition, some research work focus on the noise [10] and time-delay. [11] Based on Levins model with only one species, Wang [10] et al. find multiplicative and additive noise can influence the stability of the meta-population. Based on the classical Lotka Volterra model of one-species, the Stochastic Resonance (SR), which is a common phenomena in nonlinear noisy system, is theoretically and numerically observed. [12 13] However, the model with one species can not explicitly contain the relationship between predator and prey population. Noise originates from the fluctuation in a variety of physical, chemical or biological systems, and the source of noise is variable for different systems. For instance, the main noise source in the thermodynamic system is the fluctuation in environmental temperature. In the biochemical systems, the external noise originates from the random variations of one or more of the externally set control parameters, such as the rate constants associated with a given set of reactions, the internal noise comes from discrete nature of biochemical events such as transcription, translation, multimerization, and protein/mrna decay processes. Noise is often perceived as being undesirable and unpredictable, however, more and more advantages of noise are found in recent decades. For example, it has been documented that noise can induce sub-threshold oscillation in nonlinear systems, and improve the regularity of the oscillation. [14 16] Noise helps to achieve synchronized spikes in coupled neurons, [17 20] and can initiate spiral wave in the excitable system. [21] Our previous work show that an intermediate amount of noise plays a constructive role in persisting memory through noise-induced changing from monostable to bistable region. [22] Ecosystem is embedded inherently in some environment with different climate or atmosphere conditions, the fluctuation in which can bring noise to the system. In last decades, pollution badly disrupts the environment, different kinds of extreme climate appear more frequently, and ecosystems are destroyed intensively. Thereby, the influ- Supported by the National Natural Science Foundation of China under Grant No , and the Fundamental Research Funds for the Central Universities of China under Grant No. 2010QNA36 Corresponding author, tjuns1979@126.com c 2013 Chinese Physical Society and IOP Publishing Ltd

2 No. 4 Communications in Theoretical Physics 511 ence of noise on the ecosystem is a theoretical problem relate closely to the reality. In this paper, a simplified Rosenzweig MacArthur (RM) predator-prey model is used to mimic the ecosystem with two species. [23] The model equations are given by: du ( dt = αu 1 u v ) = f(u, v), 1 + βu dv dt = v ( βu 1 + βu γ ) = g(u, v), (1) where the two model variables u and v represent the population densities of prey and predator. The model parameter α is the ratio between growth rate of the prey and birth rate of the predator, β is the effective carrying capacity of the prey population, and γ is the ratio between death and growth rate of the predator. In this letter, we set α = 0.3, β = 7.5, and we focus on γ, the value of γ varies in the region (0 1). Fig. 1 The bifurcation diagram of the Presented predator-prey model, and γ is the bifurcation parameter. The dark thick lines indicate the stable fix points, while the thin lines represents unstable fix points or periodic orbits. The red lines indicate maximum and minimum of stable limit cycles. To investigate the dynamics of the system, the fix points of the deterministic model and the corresponding stability are calculated. In Fig. 1, we can see that, while γ decreases from 1.0, the annihilated state of predator (v = 0) becomes unstable through the bifurcation point (BP), at which γ = In addition, a new stable fix point appears through the BP, for which v 0. This stable fix point corresponds to the state at which the populations of prey and predator maintain on constant values. Then, the stable fix point becomes unstable again and a periodic orbit appears through a hopf bifurcation point (HB), at which γ = Although the bifurcation diagram is depicted only for γ > 0.4, the fix points for γ < 0.4 are obvious because no other bifurcation point is found. Furthermore, we can inferred from Fig. 1 that although the model is always oscillatory for γ < , the amplitude is changed with γ. While the value of γ is close to the HB, the amplitude is small and the minimum is not zero, but for smaller γ, the amplitude is large and the minimum becomes zero. We find the large-amplitude oscillation is similar to the excitable system, such as electrical activity in neuron system, which here will be named as explosive oscillation. As mentioned above, the fluctuation in the environmental condition can influence the ecosystem. Theoretically, the influence is usually studied by allowing the fluctuation of the model parameter. In this paper, we suppose parameter γ is subjected to additive random fluctuation, i.e., γ γ + ξ(t). The additive fluctuation gives rise to a multiplicative noise term in Eqs. (1), and the stochastic version of Eqs. (1) is given by du dv = f(u, v), = g(u, v) vξ(t), (2) dt dt where the multiplicative noise are interpreted in Stratonovich sense, ξ(t) is the Gaussian white noise. The statistical properties are given by ξ(t) = 0, ξ(t)ξ(t ) = Dδ(t t ), (3) where D is the corresponding noise intensity and δ is the Kronecker symbol. We use a forward Euler integration scheme with a time step 0.01 time unit. Simulations verify further time step reduction does not significantly improve accuracy. The numerical algorithm presented by Sancho et al. [24] will be used to simulate the noise term. To quantitatively describe the regularity of the oscillations, we employ the normalized auto correlation function of v time-series [25] ṽ(t)ṽ(t + τ) C(τ) = ṽ 2, ṽ = v v, (4) Then the characteristic correlation time can be integrated as follows: τ c = 0 C 2 (t)dt. (5) Firstly, the influence of noise in the parameter region < γ < (between the two bifurcation points) will be studied. As an example, we let γ = It is found that the oscillatory dynamics is dependent on the noise intensity. For small intensity of noise, predator population v undergoes small-amplitude phase oscillation [see top panel in Fig. 2 for D = ] with high-level regularity. While D increases, the dynamics of v becomes explosive oscillation (see center panel in Fig. 2 for D = 0.05). Additionally, it is obvious that the regularity of the explosive oscillation is much lower than that of small-amplitude phase oscillation. Further increasing of D lead the predator population vanishing, which may correspond to annihilation of the species, and the ecosystem is totally corrupted (see bottom panel in Fig. 2 for D = 0.11). Obviously, large fluctuation in climate brings the emergence of extreme climate conditions. It has been estimated more extreme climate conditions are easier to result in population decline and extinction, [26] and this is accord with our numerical result. Sequentially, it can be concluded that

3 512 Communications in Theoretical Physics the explosive oscillation is closer to the ecosystem corruption than the small-amplitude phase oscillation. Furthermore, our simulations show that although the noise intensity is extremely small (for example, D < ), the small-amplitude oscillation can be found yet. It tells us that although for the ideal environment (corresponding to small fluctuation), the ecosystem prefers oscillating with small-amplitude rather than being in a balance state with constant species population, i.e., the oscillation of ecosystem is robust. Vol. 60 τc of the v time-series for the oscillatory region in Fig. 3. Figure 4 shows the dependence of τc on noise intensity for three selected γ value. Obviously, with the increasing of noise intensity, τc decreases, i.e., the regularity of the oscillation decreases. In fact, this decreasing should be ascribed to the transiting from the small-amplitude phase oscillation to the explosive oscillation. Furthermore, the configuration of the lines in Fig. 4 can be divided into two regions. While D > 0.01 approximately, τc is very small for all γ value, and it corresponds to the explosive oscillation, i.e., the regularity of explosive oscillation is always low-level. On the contrary, for D < 0.01, the ecosystem undergoes small-amplitude phase oscillation, the destroying of noise on the regularity can be depicted explicitly. In addition, for a given weaker noise, τc decreases with parameter γ (see the inset in Fig. 4), i.e., while the system is closer to the annihilated state, the regularity of the oscillation is easier to be destroyed by the noise. Fig. 2 Time series of v for γ = Top: D = ; Center: D = 0.05; Bottom: D = We have also calculated the evolution of species population for other value of γ in the parameter region < γ < , similar results has been obtained. As mentioned above, enough large D leads the predator population vanishing, and corrupts the ecosystem. The critical value of D, for which the ecosystem can be corrupted, is calculated and shown in Fig. 3. The critical value Dc decreases linearly with the value of γ. As shown in Fig. 1, larger γ is closer to the annihilated state (γ > ). So, we can conclude that the ecosystem closer to the annihilated state, the ecosystem is easier to be corrupted. Fig. 3 The critical noise intensity Dc, across which the ecosystem transits from explosive oscillation to annihilation. To show the character of the oscillation more explicitly, we have calculated the characteristic correlation time Fig. 4 The dependence of the characteristic correlation time τc on noise intensity D for three selected γ value. Inset: the dependence of τc on γ for a given noise intensity D = Furthermore, in Fig. 4, we can see that τc maintain on a low level for all values of noise intensity for γ = To understand this result, the noisy time series of v for different γ value are compared in Fig. 5. While γ = 0.77, the ecosystem possesses small-amplitude phase oscillation. It is because that γ = 0.77 is close to the HB, and noise can induce the subthreshold oscillation. As the increasing of γ, the value of the parameter is far from the HB, and then instead of noise-induced subthreshold oscillation, the ecosystem undergoes small-amplitude fluctuation around the steady state. So, we can conclude that while the value of γ is far from the HB, the subthreshold phase oscillation will not appear, and arbitrary intensity of noise makes the system undergoes fluctuating around the steady state,

4 No. 4 Communications in Theoretical Physics which makes the regularity of the time series maintain on a low-level for all noise intensity. 513 the transiting from small-amplitude phase oscillation to explosive oscillation. While D increases across a critical value, the population of predator will annihilate after a period of explosive oscillation (see the bottom panel in Fig. 6). Comparing Figs. 2 and 6, we conclude that for all parameter regions, the system always undergoes the same successive transition with the increasing of noise intensity. The sequence is, small-amplitude phase oscillation, explosive oscillation, and annihilation. We also calculate the critical value Dc in the parameter region for phase oscillation, as done for the region < γ < The results (see Fig. 7) are similar to those shown in Fig. 3. The critical value Dc decreases linearly with the value of γ. Fig. 5 Time series of v for different value of γ (D = ). Above results are obtained in the parameter region for which the deterministic model being in the steady state. In what follows, we will focus on the parameter region for oscillation, i.e., γ < In Fig. 1, the oscillating region can be divided into two segments. For γ > 0.7, the ecosystem possesses phase oscillation, while γ < 0.7, the oscillation becomes explosive one, for which the minimum of v is close to zero. Intuitively, in a real ecosystem, the population of species should not come to zero, although the system is oscillating. So, our following attention will be paid on the phase oscillating region (0.7 < γ < ). Fig. 6 Time series of v for different value of D (γ = 0.74). As an example, the time series of v for γ = 0.74 is compared in Fig. 6. Apart from the disturbing of the noise (D = 0), the system undergoes small-amplitude phase oscillation with ideal regularity. While the noise intensity D increases to 0.01, the regularity of the oscillation is destroyed. Further increasing of noise intensity induces Fig. 7 The critical noise intensity Dc, across which the ecosystem transits from explosive oscillation to annihilation. In summary, the influence of noise on the ecosystem is studied for different parameter region, including phase oscillatory region and region for steady states. Noise leads the ecosystem oscillating for all parameter values. The amplitude and style of the oscillation are dependent on the intensity of noise. For weak noise, the ecosystem possesses small-amplitude phase oscillation, and the system exhibits explosive oscillation for stronger noise, while the noise intensity increases across a critical value Dc, the ecosystem will lose the balance, which means the annihilation of the species. It tells us that the ecosystem is easier to be destroy while the system possessing explosive oscillation. It has ever been found this kind of annihilation can be induced by the change of the parameter value in other ecosystems.[7 9] On the other hand, it is also been found that the regularity of the small-amplitude oscillation is better than explosive oscillation, and the level of the regularity decreases with the noise intensity. In addition, we find the critical value Dc change with model parameter γ linearly. As discussed above, our numerical results accord with the general knowledge about ecosystem. The deteriorating ecological environment destroy the balance between species in the ecosystems, which makes our theoretical study has great realistic significance.

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