Fluctuation-enhanced stability of a metapopulation

Physics Letters A 371 (2007) 111–117 www.elsevier.com/locate/pla

Fluctuation-enhanced stability of a metapopulation Lin-Ru Nie a,b,∗ , Dong-Cheng Mei a a Department of Physics, Yunnan University, Kunming 650091, China b Science School, Kunming University of Science and Technology, Kunming 650051, China

Received 7 January 2007; received in revised form 23 June 2007; accepted 8 August 2007 Available online 10 August 2007 Communicated by A.R. Bishop

Abstract The simplified incidence function model with cross-correlated noises was employed to study the stability of a metapopulation perturbed by environments. Through numerically computing the stationary probability distribution function (PDF) and stochastically simulating the extinction time of a metapopulation, we found that: (i) The multiplicative noise intensity D inhibits the fluctuation of dynamic variable while the additive noise intensity α intensifies it, whether there is a correlation between the multiplicative noise and the additive noise; (ii) As the correlation strength (λ) between them is greater than zero, there is an optimal D in which the PDF curve deviates furthest from the extinction position, and another optimal D which maximally delays the extinction time of a metapopulation; (iii) For the constant D and α, the increment of λ not only upgrades the probability that patches are occupied by a metapopulation, but also delays the time that a metapopulation goes to extinction. © 2007 Elsevier B.V. All rights reserved. PACS: 05.40.-a; 87.23.-n Keywords: Metapopulation; Noise intensity; Correlated strength; Stability

1. Introduction Metapopulation theory used in ecology is a powerful tool to understand the relationship between metapopulations and survival environments [1]. The term metapopulation has been used in a variety of meanings [2]. Hanski et al. [3] defined a typical metapopulation as a system satisfying the following four standards: (i) a survivable niche exists in the form of scattering patches; (ii) the largest regional metapopulation is faced with the risk of extinction as the smallest one; (iii) patches are not too separated to be recolonized; (iv) all regional metapopulations are not likely to synchronize completely. In Moilanen and Hanski’s application a metapopulation is an assemblage of local populations inhabiting spatially distinct habitat patches [4]. Hastings and Harrison [5] described a metapopulation as a set of subpopulations of a species that are isolated from one another and in which extinctions and colonizations * Corresponding author at: Department of Physics, Yunnan University, Kunming 650091, China. Tel.: +86 0871 5031744; fax: +86 0871 5031744. E-mail address: [email protected] (L.-R. Nie).

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occur. The earliest mathematical model used to study it is the well-known one posed by Levins [6,7], which assumes an infinite number of identical patches, in which case colonization and local extinction were uncorrelated among patches, habitat quality of patches remained constant, and colonization of all empty patches was equally likely. But the Levins model does not include two important forms of population structure: the distribution of local population sizes and the spatial configuration of the landscape. So it has been expanded to include many types of spatially structured models [2,8]. Ovaskainen et al. [9] developed metapopulation models for extinction threshold in spatially correlated landscapes, assuming homogeneous space. A new metapopulation model using a well-known technique in population genetics was derived by Jordi Bascompte [10], in which spatial heterogeneities are captured by an aggregate statistical measure of spatial correlation. In all spatially structured models, the incidence function model (IFM) [11] based on a first-order, linear Markov chain model occupies a paramount position. The IFM links the landscape characteristics (interpatch distance and patch area) to colonization and extinction probabilities through partly descriptive (i.e., statistical) and

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partly mechanistic formulae. A notable feature of the IFM is that it uses spatial information on patch occupancy patterns for parameter estimation. As far as the spatial structures are concerned, there are two main types of metapopulation models: spatially implicit models (i.e., mean field approximation) which are analytically tractable but neglect spatial heterogeneities and spatially explicit models which are more realistic but too complex. As a matter of fact, the results are all approximate to a large extent whether the models discussed are based on spatially implicit or explicit structures, because determinant equations are incapable to reflect accurately the influences of some stochastic factors (such as climate fluctuations, interruptions by surrounding animals, etc.) on systems. Thus theoretical ecologists have developed various approaches to understand the roles that space and stochasticity play in ecological systems [12–14]. The asymptotically exact analysis of stochastic metapopulation dynamics with explicit spatial structure was made by Otso Ovaskainen and Stephen J. Cornell [15] by means of a method which is based on a systematic perturbation expansion around the deterministic, non-spatial mean-field theory, using the theory of distributions to account for space and the underlying stochastic differential equations to account for stochasticity. Johan Grasman and Reinier HilleRisLambers [16] introduced a one-dimensional Wiener process (or Brownian motion) into the model formulated by Grasman [17], and wrote the corresponding Fokker–Planck equation to study the expected time to extinction of metapopulations. To our knowledge, few efforts have been made to study the effects of correlated noises on the extinction time of metapopulations up to now. The Letter is devoted to exploring the roles of correlated noises in the IFM, and to investigate if there exist an optimal intensity to delay the extinction time and enhance the persistence of metapopulations. The Letter is arranged as follows. Section 2 provides a simplified deterministic IFM, which is analyzed through potential under different patch characteristics. Section 3 presents the metapopulation system with the correlated noise, whose stability is in detail investigated in terms of the stationary probability distribution function (PDF) and the extinction time of a metapopulation by means of noise parameters. Section 4 concludes. 2. Deterministic model The IFM modified by Niklas Wahlberg et al. [18] and Ovaskainen and Hanski [19] is termed as our study object. Denoting by xi (t) the probability that patch i is occupied at time t , it satisfies the following first-order differential equation:    dxi = Ci x(t) 1 − xi (t) − Ei xi (t), dt with Si2

Ci =

e Ei = b , Ai

, Si2 + 1/c  Aj e−αdij xi (t). Si = j =i

xi ∈ [0, 1],

(1)

Here Ci (x(t)) are the colonization rates of empty patches and Ei are the extinction rates of extant populations. Si denotes the population-dynamic connectivity of patch i, 1/α gives the average migration distance and dij is the distance between patches i and j . c and e represent the colonization and extinction rate parameters, respectively. Ai is the area of patch i and b is a parameter. In the following processes, we will simplify the above model in order to capture the kernels of metapopulation patch dynamics. To deal with the above IFM involves rather complex calculations if the spatially heterogeneous structures of patches are taken into account. Thus, after assuming theoretically that: (i) the spatial structures of patches satisfy the coupled map lattice [20], and (ii) these patches are possessed of identical structural characteristics and qualities (i.e., Ai = A, di,i±1 = d), their probabilities occupied at any time are equal to each other, xi (t) = x(t). Eqs. (1), (2) and (3) can be combined into the simple form: e x 2 (1 − x) dx − x, = 2 dt x + y 2 /A2 Ab

x ∈ [0, 1],

(4)

where c1/2 ), and f is the structure factor of patches, y = 1/(f −αd ij . f = j =i e According to the simplified IFM (i.e., Eq. (4)), the system of the metapopulation is determined by four parameters, and the increment of the structure factor f can compensate for the decrement of colonization rate parameter c. From a physical view of point, the stability of a particle in the nonlinear system is reflected by its potential U (x). After integrating    2 e x (1 − x) − x dx, x ∈ [0, 1], − (5) x 2 + y 2 /A2 Ab the potential reads:



1 y e A U (x) = 1 + b x 2 − x + arctan x 2 A A y

2 2 y y ln x 2 + 2 . − 2A2 A

(6)

From Eq. (6), we calculated the variants of the potential with x under different values of A and y. The results are plotted in Figs. 1(a) and (b), respectively. Fig. 1 shows that there is only a non-trivial equilibrium state in the metapopulation system. As the area of patch decreases or the parameter y increases, the equilibrium state gradually shifts to the origin of coordinates, namely, the metapopulation becomes less and less stable. The metapopulation will more easily go to extinction once it is perturbed by environments. So the increment of patch area A and the product of structure factor f and colonization rate parameter c is beneficial to the permanent survival of a metapopulation. 3. Metapopulations with noises

(2) (3)

From the determinant model, we can see that a metapopulation in a non-trivial equilibrium state will survive forever without the perturbations of environment. Factually it is not just the case. It is well known that the extinction of species from

L.-R. Nie, D.-C. Mei / Physics Letters A 371 (2007) 111–117

η(t)η(t  ) = 2Dδ(t − t  ),  ξ(t)ξ(t  ) = 2αδ(t − t  ), √  η(t)ξ(t  ) = 2λ Dαδ(t − t  ),

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(8)

D and α representing noise intensities and λ being a crosscorrelated strength. The Stratonovich interpretation of the Fokker–Planck equation for the PDF P (x, t) corresponding to Eqs. (7) and (8) has the following form [28,29]:   2 √ ∂P e ∂ x (1 − x) − x + Dx − λ Dα P =− ∂t ∂x x 2 + y 2 /A2 Ab √  ∂2  + 2 Dx 2 − 2λ Dαx + α P . ∂x

(9)

The stationary probability distribution of the system can be obtained from Eq. (9) Pst (x) =

N −φ(x)/D , e g(x)

(10)

2 where √ N is a normalization constant, g(x) = (Dx − 2λ Dαx + α)1/2 and φ(x) is the modified potential given by

x 

x 2 (1 − x) √ (x 2 + y 2 /A2 )(x 2 − 2λ α/Dx + α/D)  x e dx. − b 2 √ A x − 2λ α/Dx + α/D

φ(x) = −

Fig. 1. The potential as a function of x, e = 0.8 and b = 0.2. (a) corresponds to that under different values of A, and (b) to that under different values of y.

the biosphere is a common event at the geological time scale. A great deal of attention to explaining this phenomenon is paid by Bak and Sneppen [21] and Bennett [22] with interspecies competition or environmental fluctuations documented in the fossil record [23]. Solé et al. [24] submitted radically the extinction to intrinsic nonlinear dynamics under 1/f noise while De Blasio [25] to that under environmental colored noise. Here we mainly focus on the influence of Gaussian white noises on the metapopulation system. Thus, two types of noises are introduced into Eq. (4): one is additive and the other is multiplicative, which express respectively the influence of the internal fluctuation on the system and the effect of the external environmental fluctuation on the system. The external environmental fluctuation can influence the internal fluctuation. Because of the influence of the external environmental fluctuation on the internal fluctuation, additive and multiplicative noise are not independent (there is correlation between them) [26,27]. So the Langevin equation corresponding to Eq. (4) is expressed into: e x 2 (1 − x) dx − b x − xη(t) + ξ(t), = 2 2 2 dt A x + y /A

x ∈ [0, 1],

(7)

where η(t) and ξ(t) are the Gaussian white noises with zero means and following correlation functions:

(11)

In terms of whether a cross-correlated strength is equal to zero, the metapopulation system with noises is divided into two cases to further investigate. 3.1. Metapopulation with uncorrelated noises If the multiplicative noise is mutually independent of the additive one, the cross-correlated strength λ = 0. Eq. (10) having been applied to calculating the variances of the PDF with multiplicative and additive noise intensities, the results are shown in Figs. 2(a) and (b). As usual, the increment of a noise intensity will drive a system to fluctuate more largely and its PDF curve becomes flatter and flatter, as shown in Fig. 2(a). From Fig. 2(b), however, it can be seen that as the multiplicative noise intensity increases, the PDF curve undergoes a change of being narrower and narrower. In other words, the multiplicative noise can inhibit markedly the fluctuation caused by an additive one although they belong to independent noise sources. This is certainly a novel phenomenon taking place in the metapopulation system. In Figs. 2(a) and (b), there is a common place that the peak position of the PDF curve basically remains fixed, not affected by noise intensities. For a metapopulation, an important physical quantity to which ecologists [16,30] pay attention is the time it goes extinction. If the environmental change is too fast with respect to the time scales of mutation and migration, the population will not be able to follow the new conditions and the phenotypic

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Fig. 2. (a) and (b) are the stationary PDF of x vs additive and multiplicative intensities, respectively, with λ = 0, e = 0.8, b = 0.2, y = 0.01 and A = 5.

Fig. 3. (a) and (b) are the mean extinction time Tex as a function of additive and multiplicative noise intensities intensities at different values of A and y, respectively, with λ = 0, e = 0.8, b = 0.2.

characters of the individuals in the population will shift progressively out of the optimal range, leading the species to a risk of extinction. From a physical view of point, the environmental change is regarded as a stochastic force and the extinction time as the mean first passage time that it takes averagely for the metapopulation to jump from non-trivial equilibrium to extinction position (x = 0) by noises. After stochastically simulated via Eqs. (7) and (8), the mean extinction time Tex as a function of additive and multiplicative noise intensities is displayed in Fig. 3, Fig. 3(a) corresponding to A = 0.1, 5 and Fig. 3(b) to y = 0.01, 0.8. Figs. 3(a) and (b) all present a generally admitted fact that the dramatically changing noise, whether an additive or multiplicative one, can swiftly result in the extinction of a metapopulation. Yet the larger habitat area can delay to some extent the extinction time, which is explained for the enhancement of

adaptivity of a metapopulation. To reduce the distance d between patches intensifies the connectivity or structure factor of patches so that they work together to fight against atrocious environments. So choosing the smaller parameter y also can enhance the viability of a metapopulation to postpone the extinction. 3.2. Metapopulation with correlated noises For the nonlinear system with an additive and multiplicative noise, a correlation strength λ can affect greatly its statistical properties. Dongcheng Mei et al. [31] studied the bistable system with correlated noises and concluded that λ enhances the fluctuation decay of dynamical variable for D/α  1. The correlation strength also causes a switch in the gene transcriptional

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Fig. 5. The variance of the stationary PDF with a correlation strength λ. D = 1, α = 0.1, e = 0.8, b = 0.2, y = 0.01 and A = 5.

Fig. 4. The stationary PDF vs noise intensities in the correlated case. The parameter values are λ = 0.9, e = 0.8, b = 0.2, y = 0.01 and A = 5.

regulatory system [32]. Here we shall consider how λ, D and α influence the ecological properties of the metapopulation. Performing numerical computation of the stationary PDF as a function of additive and multiplicative noise intensities by means of Eq. (10) with a correlation strength λ = 0.9, the curves are plotted in Figs. 4(a) and (b), respectively. Compared with those in uncorrelated noises, The results in Figs. 4(a) and (b) exhibit two interesting phenomena. First, the multiplicative noise intensity kept a constant (i.e., D = 1), the peak of the PDF curve will go far from the origin of x coordinate as the additive one increases. It means that the intensification of an additive noise will upgrade the probability the metapopulation occupies more patches. But it is not to say that the additive noise can delay the extinction time of the metapopulation because it simultaneously widens the PDF curve. Even so, it is certainly deduced from Figs. 2(a) and 4(a) that in the case of identical additive

and multiplicative noise intensities, the metapopulation system with larger λ should be more stable than that with smaller λ. Second, α remaining a constant, there is an optimal D which corresponds to the furthest PDF curve from the extinction position x = 0 (see Fig. 4(b)). The metapopulation under the noise intensity possesses the largest probability that patches are occupied, and maybe the longest extinction time near the optimal point, namely, the correlated multiplicative noise can enhance the stability of the metapopulation. In order to make clear the contribution of λ to the system, we computed the variance of the PDF with λ, letting D = 1 and α = 0.1, and the results are shown in Fig. 5. A forniciform saddle is displayed in Fig. 5, which indicates that as λ changes from 0 to 1, the PDF curve experiences a process of becoming cuspidal → obtuse → cuspidal. In the limit of λ → 1, the PDF is very close to Dirac function. Besides, when λ is greater than about 0.6, it swerves to avoid the extinction position. Therefore the role that the correlation strength plays in the metapopulation system is embodied by: (i) λ first potentiates then suppresses the fluctuation of dynamic variable as it increases from 0 to 1; (ii) The larger λ is helpful to raise the probability that patches are occupied by a metapopulation. In addition, λ also impacts on the mean extinction time of the metapopulation. Stochastically simulating the variances of the mean extinction time Tex with additive and multiplicative noise intensities in different values of λ by means of Eq. (7), we arrange the results in Figs. 6(a), (b) and (c). Fig. 6 indicates that in the same noise intensities the Tex for the larger λ is obviously greater than that for the smaller λ, and that for the larger λ there is an optimal multiplicative noise intensity which maximally delays the extinction time of a metapopulation, but not an optimal additive one. Furthermore, the optimal multiplicative noise intensity has nothing to do with the additive one, completely dependent on the λ. So choosing an optimal multiplicative noise, which is positively correlated by the greatest

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Fig. 6. The mean extinction time Tex vs additive and multiplicative noise intensities under different values of λ: 0.9 in (a), 0.7 in (b) and −0.85 in (c). e = 0.8, b = 0.2, y = 0.01 and A = 5.

extent with the additive one, is a crucial strategy for a metapopulation to survive for a longer time. 4. Conclusions Above the simplified IFM with the cross-correlated Gaussian white noises was employed to investigate the metapopulation system perturbed by capricious environments. We derived the PDF of the system subjected to correlated noises. Through numerically computing the PDF and stochastically simulating the Langevin equation (7), we studied the stability and the extinction time of a metapopulation. The results show that the multiplicative noise intensity D, the additive noise intensity α and the cross-correlated strength λ play very important roles in determining the stability of a metapopulation. First, whether there is a correlation between the multiplicative noise and the additive noise or not, D and α all play opposite roles, namely,

D inhibits the fluctuation of x while α intensifies it. Second, in the case of correlated noises (λ > 0), there is an optimal D in which the PDF curve goes furthest from the extinction position x = 0, and another optimal D which maximally delays the extinction time, but there is not an optimal α. Finally, for the constant D and α, the increment of λ not only upgrades the probability that patches are occupied by a metapopulation, but also delays the time that a metapopulation goes to extinction. In conclusion, by tuning an optimal D and making it positively correlated with α as much as possible, we can improve the ability of a metapopulation to withstand and exploit the capricious change of environments to enhance the stability of itself. Acknowledgements This work was supported by the Nation Nature Science Foundation of China (Grant No. 10363001).

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