Spatial and Temporal Taylor's Law in 1-Dim Chaotic Maps
aa r X i v : . [ n li n . C D ] S e p S PATIAL AND T EMPORAL T AYLOR ’ S L AW IN IM C HAOTIC M APS
Hiroki Kojima
University of Tokyo [email protected]
Yuzuru Mitsui
University of Tokyo [email protected]
Takashi Ikegami
University of Tokyo [email protected]
September 29, 2020 A BSTRACT
By using low-dimensional chaos maps, the power law relationship established between the samplemean and variance called Taylor’s Law (TL) is studied. In particular, we aim to clarify the rela-tionship between TL from the spatial ensemble (STL) and the temporal ensemble (TTL). Since thespatial ensemble corresponds to independent sampling from a stationary distribution, we confirmthat STL is explained by the skewness of the distribution. The difference between TTL and STL isshown to be originated in the temporal correlation of a dynamics. In case of logistic and tent maps,the quadratic relationship in the mean and variance, called Bartlett’s law, is found analytically. Onthe other hand, TTL in the Hassell model can be well explained by the chunk structure of the trajec-tory, whereas the TTL of the Ricker model have a different mechanism originated from the specificform of the map.
We challenged the unexplained widespread power-law behavior between the mean and variance, called Taylor’s law(TL) [1], with one-dimensional (1-D) chaotic maps. TL was originally found in the field of population ecology andhas since been reported more widely, ranging from demographic ecology to prime number distribution [2], complexnetworks [3], and so on [4, 5]. TL expresses that a power-law scaling relationship holds between the mean ( M ) andvariance ( V ) as V = αM β . When the mean and variance are calculated by randomly sampling from a stationary state,it is called spatial TL (STL), and, when it is sampled from a time series, it is called temporal TL (TTL).STL and TTL have been analyzed in several dynamical systems of ecological models [6, 7, 8, 9]. For example,Ballantyne [7] showed that the exponent of TTL becomes 2 when the solution of the map is linearly scaled by changesin the parameters, and Kilpatrick and Ives [6] used the noisy Ricker model of multiple species to show TTL betweenthe mean and variance. Perry [8] found STL in the chaotic regions of the Hassell model controlled by noise, but theyprovided no analytical explanations. Cohen showed that the exponential growth model satisfies STL with the exponent β = 2 in the limit of large time [9].In this letter, we investigate both STL and TTL in the chaotic regime of 1-D maps, discussing a possible mechanism ofboth STL and TTL. There are only a few studies that have investigated the relationship of STL and TTL [10, 11, 12].They use empirical data with a probabilistic model, but no theoretical explanation was provided. We discuss that spatialTL is explained by the skewed distribution function, while temporal TL is dependent on the temporal correlations ina time series. Zhao et al., [12] argued that the skewness of the population abundance was an important factor of theexponent β . Although they recognized that the autocorrelation of the time series was important, their main numericalsimulations did not consider it. Here, we will show that autocorrelation greatly influenced TL. PREPRINT - S
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29, 2020Figure 1: The chaotic time series of maps used here. (a) Logistic map with r = 3 . . (b) Tent map with µ = 1 . . (c)Hassell model with κ = 10 , ν = 12 , λ = 100 . (d) Ricker model with r = 5 . We examined four discrete one-dimensional chaotic map systems that basically adopted ecological modelingsi.e.,logistic map, tent map, Hassell model [13, 14] and Ricker model [15]which have been widely used as populationmodels [16, 17]. x n +1 = rx n (1 − x n ) , (1) x n +1 = µ − µ (cid:12)(cid:12)(cid:12)(cid:12) x n − (cid:12)(cid:12)(cid:12)(cid:12) , (2) x n +1 = λx n (1 + κx n ) ν , (3) x n +1 = x n exp { r (1 − x n ) } . (4)The typical time series generated from these maps are shown in FIG. 1.We calculated the multiple trajectories by varying the initial states with fixed parameters. Each trajectory starts fromdifferent initial values randomly sampled from a uniform distribution. The length of each trajectory is set at 20,000time steps. We eliminated the initial transients (discarded the first 10,000 steps) and analyzed the last 10,000 steps( T = 10000 ). By denoting the n th value of the i th trajectory as x in , we analyzed the ensemble of N samples;computing M n = (1 /N ) P Ni =1 x in and V n = (1 /N ) P Ni =1 ( x in − M n ) for STL. As for computing TTL, we compute M i = (1 /T ) P Tn =1 x in and V i = (1 /T ) P Tn =1 ( x in − M i ) (FIG. 2).The calculation settings of STL and TTL here can be interpreted as the observation of population density at indepen-dent multiple places of the equal environmental conditions (e.g., [8, 18, 19]).We study the TL relationship in terms of four values as follows: b = cov ( M, V ) / var ( M ) , var ( M ) , and var ( V ) . Thefirst quantity b corresponds to the slope of the linear regression line on the M − V plane. When the linear regressionis valid, this exponent b approximates the exponent β of TL as β ≃ MV b ( M and V are the sample mean and variancecalculated over all trajectories, respectively) [20].These results are summarized in the ”Numerical” columns in TABLE 1. In the following sections, we will theoretically analyze and reproduce the above arguments.2
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29, 2020Figure 2: Results of the numerical simulations of 10,000 sets of ( M, V ) on the M − V plane. The data points arerepresented as blue dots, and the linear regression lines are shown as red lines. (a) STL, logistic map with r = 3 . . (b)TTL, logistic map with r = 3 . . (c) STL, tent map with µ = 1 . . (d) TTL, tent map with µ = 1 . . (e) STL, Hassellmodel with κ = 10 , ν = 12 , λ = 100 . (f) TTL, Hassell model with κ = 10 , ν = 12 , λ = 100 . (g) STL, Ricker modelwith r = 5 . (h) TTL, Ricker model with r = 5 . The spatial ensemble is organized by randomly sampling from a stationary distribution of a given map. The followingquantities are given in [21, 22, 20]: var ( M ) = µ N , (5)var ( V ) = 1 N (cid:18) µ − N − N − µ (cid:19) , (6)cov ( M, V ) = µ N , (7)where N is the sample size. µ is equal to the variance (or the second central moment) of the stationary distribution. µ denotes the skewness of the stationary distribution (or the third central moment), and µ denotes fourth momentsof the stationary distribution. 3 PREPRINT - S
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29, 2020The covariance of the sample mean and variance are proportional to the skewness of the distribution. Cohen et al. [20]argued that it is one of the possible origins of TL.In order to confirm that STL is explained from the skewness of the stationary distribution, we computed a stationarydistribution from the 10,000 trajectories to calculate µ , µ and µ . The predicted value matches the value of thenumerical calculation, as shown in TABLE 1. TL relationships have different appearances in spatial and temporal ensembles. If each trajectory is independent,spatial TL only depends on the stationary distribution, as stated above, but temporal TL also depends on the temporalstructure of the trajectories. We first show how temporal correlation in trajectories relates to temporal TL; next, weestimate the temporal TL of each map based on the characteristics of each of the temporal structures.
Eq.5-7 only hold when the variables are sampled independently. This holds true for the spatial ensembles but not forthe temporal ensembles, as each system has its own characteristic memory length.The following generalized equations are applicable when the sample size N is large enough (see Appendix):var ( M ) = µ N + 1 N N − X τ =1 N − τ ) R ( τ ) var ( V ) ≃ N (cid:0) µ − µ (cid:1) + 1 N N − X τ =1 N − τ ) R ( τ ) cov ( M, V ) ≃ µ N + 1 N N − X τ =1 ( N − τ )( R ( τ ) + R ( − τ )) , where the first term is equal to Eq.5-7 when the system size is large. The second order terms are given by the followingterms. y i = x i − E [ x ] R ( τ ) = E [ y i y i + τ ] ,R ( τ ) = E (cid:2) ( y i − µ )( y i + τ − µ ) (cid:3) ,R ( τ ) = E (cid:2) y i y i + τ (cid:3) , The second order terms will vanish when the variables are uncorrelated. Specifically, the difference between thevar( M ) values calculated from the spatial ensembles and temporal ensembles is proportional to the sum of the auto-correlation function.We can calculate the TTL if we know R ( τ ) , R ( τ ) and R ( τ ) , but, typically, we need to actually sample thetrajectories and directly calculate from them to acquire the functions. Below, we estimate the TTL of each map onlyusing map and conspicuous temporal structure information. The mean and variance of the temporal ensembles from the logistic map are strongly correlated (FIG. 2 (b)). We foundthat this is a direct consequence of the quadratic form of the logistic map (Eq.1). By taking the temporal summationof Eq.1, we analytically derived the following equation (see Appendix): V = (1 − r ) M − M . (8)In general, a dynamic system containing a quadratic function ( x n +1 = cx n + dx n ) shows this relationship in thetemporal ensemble as follows (see Appendix): 4 PREPRINT - S
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29, 2020 V = 1 − dc M − M . (9)This quadratic component is evident when we plot the temporal mean and variance calculated from short trajectoriesas shown in FIG. 3.In order to check whether it reproduces the results in TABLE 1, we calculated the exponent b using dVdM (cid:12)(cid:12)(cid:12)(cid:12) M = M ( M :sample mean of the all values in the 10,000 × M − V relationship in the logistic map. Here, r = 4 , the number of time steps tocalculate the temporal mean and variance is 100, and the number of plots is 2,000. The red and green lines representthe linear and Bartlett’s law approximations, respectively. The TTL of the tent map can be also analytically derived from the equation of the map. V = µµ + 1 M − M . (10)This quadratic relation is clearly seen when we plot the temporal mean and variance calculated from short trajectoriesas well as in the logistic map (FIG. 4).We calculated b using this equation in the same way as for the logistic map (see Appendix), and the result is shown inthe column ”Prediction” in TABLE 1, which well-reproduced the value from the ”Numerical column.5 PREPRINT - S
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29, 2020Figure 4: The typical temporal M − V relationship in the tent map. Here, µ = 2 , the number of time steps to calculatethe temporal mean and variance is 100, and the number of plots is 2,000. The red and green lines represent the linearand Bartlett’s law approximations, respectively.The quadratic relationship between the sample mean and variance in general was proposed by Bartlett [23], anddescribed as follows, V = pM + qM . Sometimes, this relationship has been compared to TL [24, 25], and, recently, it was called Bartlett’s law [26].
The trajectory of the Hassell model in a given parameter region has a chunk structure, in which the value monotonicallyincreases, and, finally, it abruptly decreased to ∼ to start the next chunk (FIG. 5). We assume that this chunkstructures is responsible for the M − V relationship in the Hassell model. In order to check the importance of thechunk structures, we reshuffled the temporal ensembles, retaining the chunk structure to build a surrogate ensemble(see Appendix). The result from the surrogate is shown in the column ”Prediction” in TABLE 1. In Table I, theprediction values are well-reproduced by the ”Numerical” values. The time trajectory of the Ricker model in this parameter region also has a chunk structure, but, unlike the Hassellmodel, the temporal ensemble did not show a correlation in M − V at all, and var ( M ) is much smaller than the spatialensemble.First, we constructed the surrogate ensemble in the same way as the Hassell model. The result is shown in the upperrow in the column ”Prediction” in TABLE 1. This did not well-reproduce the result from the ”Numerical” column,except for var ( V ) .We found that the discrepancy in var ( M ) is the direct consequence of the form of the time evolution function, and weanalytically showed that the time average depends only on the first term and the ( N + 1) th term; M = 1 N r (log x − log x N +1 ) + 1 , (11)which caused the decorrelation between the mean and variance. This relationship can be generalized to the multi-variable stochastic Ricker model used in [6] (see Appedix).Based on this, we constructed the surrogate ensemble by random sampling two values from the stationary distributionand applying Eq.11. The result is shown in the lower row in the column ”Prediction” in TABLE 1, and it well-reproduced the values in the ”Numerical” column. 6 PREPRINT - S
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29, 2020Figure 5: Chunk structures of the orbits of the Hassell model and Ricker model. (a) Hassell model with κ = 10 , ν = 12 , λ = 100 . (b) Ricker model with r = 5 . b = cov ( M, V ) / var ( M ) var ( M ) var ( V ) Model Type Prediction Numerical Prediction Numerical Prediction NumericalLogistic( r = 3 . ) Spatial -0.056207(-0.056260-0.056149) -0.056188(-0.059249-0.053120) . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − )Temporal -0.44142(-0.44147-0.44136) -0.44132(-0.44151-0.44113) . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − )Tent( µ = 1 . ) Spatial -0.027104(-0.027151-0.027060) -0.026905(-0.030327-0.023764) . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − )Temporal -0.45362(-0.45367-0.45357) -0.45348(-0.45367-0.45331) . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − )Hassell( κ = 10 , ν = 12 , λ = 100 ) Spatial 0.14187(0.141860.14189) 0.14186(0.140680.14285) . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − )Temporal 0.43951(0.436900.44229) 0.58675(0.585000.58821) . ∗ − ( . ∗ − . ∗ − ) . ∗ − (4 . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − )Ricker( r = 5 ) Spatial 7.2162(7.21517.2172) 7.2193(7.17197.2508) . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − ) 0.038743(0.0387250.038759) 0.038667(0.0373030.040184)Temporal Chunk30.484(29.09931.681)Formula7.5120(4.152511.040) 7.3799(3.087411.399) Chunk . ∗ − ( . ∗ − . ∗ − )Formula . ∗ − ( . ∗ − . ∗ − ) . ∗ − ( . ∗ − . ∗ − ) Chunk0.0050676(0.00492900.0052013) 0.0065328(0.00637320.0067021) Table 1: The results from the numerical simulations (Numerical) and the predicted values from the theoretical models(Prediction). In the ”Numerical” column, we calculated the indices from 10,000 distinct trajectories. In the ”Predic-tion” column, we calculated the indices from the theoretical models of each map. These results are described with amedian and 95% CI calculated from 200 repeated calculations (95% CI is given below the associated median value).7
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In summary, we have investigated STL and TTL in several chaotic dynamical systems and analyzed the mechanisms ofTaylor’s law. STL originated from the skewness of the stationary distribution, which is the same mechanism discussedin [20], and the difference between STL and TTL was derived from the temporal correlation of the state variables.In our case, each trajectory was independent of each other, and the relationship between STL and TTL only dependson the temporal correlation in each trajectory. So, our results were complementary to past research [11, 12], whichconsidered the case of dependent trajectories and argued how the dependence between the trajectories, such as theenvironmental synchrony and density-dependent dispersal, affected the exponent, although the possibility that thetemporal correlation might affect the exponent of TTL was pointed out in [12].We also showed that the TTL was well-approximated based on the characteristic temporal structure of each map. Inthe Hassell model, the temporal structure over several time steps, chunk structures, mostly contributed to the M − V relationships, while, in the logistic map and the tent map, the relationships between the present value and the valueof the next time step, which are expressed in the time-evolution equations, strongly influenced the mean-variancerelationships. Interestingly, while the trajectories from the Ricker model have chunk structures similar to those of theHassell model, the TTL cannot be well approximated by the chunk structure due to the presence of the relationship inthe time-evolution equation, as found in the logistic map and the tent map.The quadratic relation obtained for the TTL of the logistic map and tent map is an example of the relationship calledBartlett’s law [23]. Our explanation, based on the relationship between the values of consecutive time steps, canprovide another explanation for the quadratic relations. On the other hand, the mechanism of TL in high-dimensionalsystems may be different, and further research is needed. Acknowledgments
The authors would like to deeply thank Joel E. Cohen and Yuzuru Sato for valuable discussions.
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A Appendix
A.1 Calculation Details of Prediction and Numerical
We evaluated the mean-variance relationship in two ways,
P rediction and
N umerical . In
N umerical , we made10,000 distinct trajectories starting from different initial conditions randomly chosen from a uniform distribution be-tween 0 and 1, and the length of the trajectories was 20,000. The first 10,000 steps were discarded to eliminate theeffects of transient dynamics. Then, we got 10,000 × ( M, V ) , var ( M ) , and var ( V ) . On the other hand, wecalculated 10,000 time average and variance to test temporal TL. From this data, we calculated cov ( M, V ) , var ( M ) ,and var ( V ) . In P rediction , to confirm the
N umerical results in the spatial format, we made the stationary distribu-tions from the 10,000 × ( M, V ) , var ( M ) , and var ( V ) using Eq.5-7. To confirmthe N umerical results in the temporal format, we performed different analyses for each map in
P rediction . Forthe logistic map and tent map, we used Eq.8 and Eq.10. First, we calculated 10,000 pairs of ( M, V ) in the spa-tial format and calculated the average of these M and V ( M and V ). Next, we calculated the straight lines whoseslopes were dVdM (cid:12)(cid:12)(cid:12)(cid:12) M = M (this quantity corresponds to b ) passing through ( M , V ) . For the Hassell model, the 10,000 × b = cov ( M, V ) / var ( M ) , var ( M ) , and var ( V ) . For the Ricker model, in addition to using the same analysis as for theHassell model, another analysis was also performed using Eq.11. var ( M ) was predicted by randomly extracting twovalues from the stationary distribution and applying the formula Eq.11. We repeated the above procedure 200 times tocreate a 95 % confidence interval. A.2 Relationship between temporal TL and auto-correlation
When we set y i = x i − E [ x i ] , then the following equation holds. E " N X i =1 y i = 0 . (12) V = 1 N − N X i =1 x i − N N X j =1 x j = 1 N − N X i =1 x i − N x i N X j =1 x j + N N X j =1 x j = 1 N − N X i =1 x i − N N X i =1 x i N X j =1 x j + N X i =1 N N X j =1 x j = 1 N − N X i =1 x i − N N X i,j =1 x i x j (13)10 PREPRINT - S
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29, 2020var ( M ) = E N N X i =1 x i − E " N N X i =1 x i = 1 N E N X i =1 ( x i − E [ x i ]) ! = 1 N E N X i =1 y i ! = 1 N E N X i =1 y i + N X i = j y i y j = µ N + 1 N N − X τ =1 N − τ ) R ( τ ) var ( V ) = E N − N X i =1 x i − N N X j =1 x j − E N − N X i =1 x i − N N X j =1 x j = E N − N X i =1 y i − N N X j =1 y j − E N − N X i =1 y i − N N X j =1 y j = E N − N X i =1 y i − N N X i,j =1 y i y j − E N − N X i =1 y i − N N X i,j =1 y i y j ( ∵ Eq. ≃ E N N X i =1 (cid:0) y i − E [ y i ] (cid:1)! = 1 N N X i,j =1 E (cid:2)(cid:0) y i − E [ y i ] (cid:1) (cid:0) y j − E [ y j ] (cid:1)(cid:3) = 1 N N X i =1 E h(cid:0) y i − E [ y i ] (cid:1) i + N X i = j E (cid:2)(cid:0) y i − E [ y i ] (cid:1) (cid:0) y j − E [ y j ] (cid:1)(cid:3) = 1 N (cid:0) µ − µ (cid:1) + 1 N N − X τ =1 N − τ ) R ( τ ) PREPRINT - S
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29, 2020cov ( M, V ) = E N N X i =1 x i − E " N N X i =1 x i N − N X i =1 x i − N N X j =1 x j − E N − N X i =1 x i − N N X j =1 x j = 1 N ( N − E N X i =1 y i ! N X i =1 y i − N N X j =1 y j − E N X i =1 y i − N N X j =1 y j = 1 N ( N − E N X i =1 y i N X i =1 y i − N N X j =1 y j ( ∵ Eq. N ( N − E N X i =1 y i N X i =1 y i − N N X i =1 y i N X j =1 y j ( ∵ Eq. N ( N − E N X i,j =1 y i y j y j − N N X k =1 y k ! ≃ N E N X i,j =1 y i y j = 1 N E N X i =1 y i + N X i = j y i y j = µ N + 1 N N − X τ =1 ( N − τ )( R ( τ ) + R ( − τ )) A.3 Bartlett’s law of quadratic dynamical systems
Dynamical systems described in quadratic form: x n +1 = cx n + dx n , (14)where c and d are constants, satisfy Bartlett’s law. We define the sample mean ( M ) and the sample variance ( V ) asfollows: M = 1 N N X n =1 x n ,V = 1 N − N X n =1 ( x n − M ) . Bartlett’s law is described as follows: V = pM + qM . PREPRINT - S
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29, 2020Proof. x n +1 = cx n + dx n , = c (cid:18) x n + d c (cid:19) − d c , = c (cid:26) ( x n − M ) + (cid:18) M + d c (cid:19)(cid:27) − d c , = c ( x n − M ) + c (cid:18) M + d c (cid:19) + 2 c (cid:18) M + d c (cid:19) ( x n − M ) − d c . Taking the sum and dividing by N on both sides of the equation, and if N is large enough:(Left side) = 1 N N X n =1 x n +1 , ≃ N N X n =1 x n , = M. On the other hand:(Right side) = 1 N N X n =1 ( c ( x n − M ) + c (cid:18) M + d c (cid:19) ) + 2 c (cid:18) M + d c (cid:19) N N X n =1 ( x n − M ) − d c , ≃ cV + c (cid:18) M + d c (cid:19) − d c . Therefore, we obtain: M = cV + c (cid:18) M + d c (cid:19) − d c , ⇔ V = 1 − dc M − M . A.4 Bartlett’s law of tent map
Tent map (Eq.15) satisfies Bartlett’s law with p = µµ + 1 , q = − . x n +1 = µ − µ (cid:12)(cid:12)(cid:12)(cid:12) x n − (cid:12)(cid:12)(cid:12)(cid:12) . (15)Proof. x n +1 = µ − µ (cid:12)(cid:12)(cid:12)(cid:12) x n − (cid:12)(cid:12)(cid:12)(cid:12) , ⇔ (cid:12)(cid:12)(cid:12)(cid:12) x n − (cid:12)(cid:12)(cid:12)(cid:12) = 12 − µ x n +1 . By squaring both sides, we obtain, (cid:18) x n − (cid:19) = (cid:18) − µ x n +1 (cid:19) . (Left side) = (cid:26) ( x n − M ) + (cid:18) M − (cid:19)(cid:27) , = ( x n − M ) + (cid:18) M − (cid:19) + 2( x n − M ) (cid:18) M − (cid:19) . PREPRINT - S
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29, 2020(Right side) = 1 µ n ( x n +1 − M ) + (cid:16) M − µ (cid:17)o , = 1 µ (cid:26) ( x n +1 − M ) + (cid:16) M − µ (cid:17) + 2 (cid:16) M − µ (cid:17) ( x n +1 − M ) (cid:27) . Taking the sum of both sides of the equation and dividing both sides by N , we obtain the following equations:(Left side) = 1 N N X n =1 ( ( x n − M ) + (cid:18) M − (cid:19) + 2( x n − M ) (cid:18) M − (cid:19)) , ≃ V + (cid:18) M − (cid:19) . (Right side) = 1 µ N N X n =1 ( x n +1 − M ) + 1 µ N N X n =1 (cid:16) M − µ (cid:17) + 2 µ (cid:16) M − µ (cid:17) N N X n =1 ( x n +1 − M ) . If N is large enough, N N X n =1 ( x n +1 − M ) ≃ N − N X n =1 ( x n − M ) , = V, N N X n =1 ( x n +1 − M ) ≃ N N X n =1 ( x n − M ) , = 0 , we obtain: V + (cid:18) M − (cid:19) = 1 µ V + 1 µ (cid:16) M − µ (cid:17) , ⇔ V = µµ + 1 M − M . (16) A.5 Temporal Mean of Ricker model
We applied log to each side of x n +1 = x n exp { r (1 − x n ) } and obtained the following equation. log x n +1 = log x n + r (1 − x n ) (17)Using this equation iteratively, we obtain: log( x N +1 ) = log( x N ) + r (1 − x N )= log( x N − ) + r (1 − x N − ) + r (1 − x N ) ... = log( x ) + N X n =1 r (1 − x n ) , ⇔ M = 1 N r (log( x ) − log( x N +1 )) + 1 . In a similar way, we can obtain the temporal mean of the multi-variable stochastic Ricker model: x in +1 = x in exp ( r i − x in + P i = j α ij x jn K i ! + ǫ in ) . (18)14 PREPRINT - S
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29, 2020Considering the log of the each side of Eq.18, we get: log( x in +1 ) = log( x in ) + r i − x in + P i = j α ij x jn K i ! + ǫ in . (19)Using this equation iteratively, we obtain: log( x n +1 ) = log( x ) + N r − r i K i A N X n =1 x n + N X n =1 ǫ n , ⇔ r i K i A N X n =1 x n = log( x ) − log( x n +1 ) + N r + N X n =1 ǫ n , ,where ( x n ) i = x in , ( ǫ n ) i = ǫ in , ( A ) ij = α ij ( i = j ) , ( A ) ii = 1 ,and r i = r i . Therefore, we can get the followingformula: M i = 1 N N X n =1 x in = K i N r i A − ( log x − log x N +1 + N r + N X n =1 ǫ n ) i15