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A nonlinear dynamic simulative model has been discussed with variable cycles of entire world per capita ecological footprint taken from 1961 to 2003. The model was further classified and decomposed and extracted by the empirical mode decomposition (EMD) method. To deal with the problems proposed in the
*Living Planet Report* 2006, three ecological footprint scenarios are presented. Simulative numerical values of the three global per capita (GPC) ecological footprint scenarios are also analyzed based on the simulative model. The results show that: 1) The clear varying cycle of global per capita EF growth is 4.6 years, 9.5 years, 19.5 years and 41 years over the last 42 years; 2) According to the business-as-usual scenario, if the global per capita increases positively with the constant growth, it is expected that GPC EF would be 3.262 gha in 2050. Assuming global per capita biocapacity (BC) to be 1.236 gha, global per capita ecological deficit (ED) would increase from 0.4 gha in 2003 to 2.026 gha in 2050; 3) The slow-shift scenario shows global per capita EF would decrease from 2.23 gha in 2003 to 1.619 gha in 2080 and 1.406 gha in 2100, if the negative annual change rate of it is 0.447 percent. Global per capita ED would decrease from 0.4 gha in 2003 to 0.222 gha in 2080 and 0.038 gha in 2100, if global per capita BC is 1.397 gha; 4) The rapid-reduction scenario depicts global per capita EF would decrease to 1.414 gha in 2050, if the negative annual change rate is 0.842% from 2003 to 2050. Assuming global per capita BC to be 1.461 gha, global per capita ecological reserve (ER) would be 0.047 gha, and overshoot would be eliminated in 2050. Global per capita EF would decrease to 1.054 gha in 2100, if the negative annual change rate is 0.438% from 2050 to 2100. Assuming global per capita BC to be 1.474 gha, GPC ER would be 0.420 gha. Then, wild species of the planet will be allocated nearly 28.5% of the planet’s biological productivity, which coincides with the results of
*Living Planet Report* 2006.

Natural capital is essential core sustainability [

The conventional spectrum analysis method of time series is mostly based on Fourier spectral analysis. The resolution rate of Fourier spectral analysis can be very high in frequency, but extremely low in time [

The Hilbert-Huang Transform (HHT) was presented by N. E. Huang in 1998 [

The EF analysis method has three primary indices: Ecological Footprint (EF), Biocapacity (BC) and Ecological Deficit or Reserve (ED/ER). EF is defined as summative places of land and water which are occupied by individuals, and the individual uses all the resources and also reproduce them. The participants in that economy have capabilities to absorb all waste and manage it using prevailing technology [

The latest available data mentioned in Living Planet Report 2006 indicate that humanity’s EF, in the earth, the impact of the living species is increasing and now reaches three times since 1961 [

There is an ecological reserve in the non-EU, Latin America, the Caribbean, and Africa. The average footprint per capita changed little in those low-and-medium-income countries, while, nearly about eighteen percent increase in the footprints are noted for high income countries. In conclusion, in the last 40 years, the countries with low income have yielded 0.8 global hectares per capita.

Using the envelops, the local maximum and minimum criteria are separately used by the help of decomposition method. Cube spline is used to measure upper and lower envelops, when extreme values are identified.

But, as the

A specific criterion is used to stop the sifting process by repeating the process again and again. A value of 0.2 or 0.30 is set as values of SD.

From the data,

In Equation (4), components that contain information for more extended periods are denoted by. It is considered as a newly generated data and endangered to the same sifting process.

Dealt is treated as the new data and as described above. Equation (5) is a result of repetition of all

i.e.

Thus, classified IMFs are achieved, and a residual,

There is no doubt that EMD is a compelling method. However, there are some problems associated with EDM i.e. endpoint treatment is strongly influenced by this method. We employed cube spline to calculate envelopes. But endpoints make the spline very sensitive. It is imperative to take measures to avoid from the propagation of end effects into the interior solution. Hence, the current problem will be solved by the extrema extending method [

The Boundary Processing exists not only in designing of the numeric filter and transforming wavelet, but also in the EMD method. The boundary processing method of stretching symmetrically mirror image is adopted in this paper [

The global time series from 1961 to 2003 in this paper is obtained by considering simplicity and authority. Our assessment and analysis are based on the series data of Living Planet Report by WWF [

In this section, the EMD method is presented to analyse the fluctuation and causes of global per capita EF at multi-time scales. From

that the data points are classified into one residential and four IMFs. _{1} ponderance expresses the 4.6-years-period fluctuation, while IMF_{2}, IMF_{3}, and IMF_{4} express 9.5 years period, 19.5 years, and 41 years, respectively (

Three ecological footprint scenarios of the world were proposed in Living Planet Report 2006 (

IMF_{i} ponderances | IMF_{1} | IMF_{2} | IMF_{3} | IMF_{4} | RES |
---|---|---|---|---|---|

Periods (T_{i}, Years) | 4.6 | 9.5 | 19.5 | 41 | ∞ |

Variance contribution (k_{i}, %) | 1.08 | 1.76 | 2.22 | 15.22 | 79.72 |

The dynamic simulation model is constructed based on the above analysis as follow:

Then:

_{i} is the period.

Taking T_{1}, T_{2}, T_{3}, and T_{4} in

Formula (9) is the nonlinear dynamic simulative model with periodic fluctuation for global per capita EF.

The predictive results of global per capita EF from 1961 to 2003 are shown in

Year | Real Values | Predictive Values | Error (%) |
---|---|---|---|

1961 | 1.692 | 1.692 | 0 |

1965 | 1.874 | 1.832 | −2.235 |

1970 | 2.117 | 1.894 | −10.514 |

1975 | 2.165 | 1.914 | −11.578 |

1980 | 2.262 | 1.965 | −13.124 |

1985 | 2.188 | 2.019 | −7.741 |

1990 | 2.243 | 2.018 | −10.045 |

1995 | 2.198 | 2.029 | −7.669 |

2000 | 2.196 | 2.234 | 1.728 |

2001 | 2.190 | 2.278 | 4.007 |

2002 | 2.2 | 2.330 | 5.909 |

2003 | 2.23 | 2.381 | 6.771 |

The simulative results of global per capita EF from 2003 to 2050 are shown in

_{2} emissions [_{2} ecological footprints would increase by only about 60% in 2050. However, we think it could be hardly achieved because of our present energy consumption structure and the difficulty in implementing the Kyoto Protocol.

The per capita ecological deficit would increase from 0.45 global hectares in 2003 to 2.026 global hectares in 2050, if the average BC per person reduces from 1.78 global hectares in 2003 to 1.236 global hectares in 2050, as predicted by WWF in Living Planet Report 2006. In the current condition of the ecological deficit, ecological assets exhaustion and large-scale ecosystem collapse will become increasingly possible before 2050 [

The slow-shift scenario in Living Planet Report 2006 shows that total human

Year | 2004 | 2005 | 2010 | 2015 | 2020 | 2025 |
---|---|---|---|---|---|---|

Per Capita EF | 2.304 | 2.352 | 2.453 | 2.518 | 2.495 | 2.629 |

Year | 2030 | 2035 | 2040 | 2045 | 2050 | |

Per Capita EF | 2.599 | 2.642 | 2.766 | 3.099 | 3.262 |

ecological footprint in 2100 would be 15% less than that in 2003. If biocapacity gains are sustained with an increase of 20% by 2100 and inhabitants are growing moderately, the situation will arise in which ecological footprint per capita would decrease from 2.2 global hectares to around 1.5 global hectares. That is human EF would fall to 1.505 gha in 2080 and 1.263 gha in 2100.

The annual decreasing rate of global per capita EF should be 0.447% (r = −0.00447), if it drops from 2.23 gha in 2003 to 1.263 gha in 2100.

Only when the average annual decreasing rate of global per capita EF reaches 0.62%, could the global per capita EF fall to 1.396 gha in 2080 and 1.176 gha in 2100 by repeated adjustment with Model 9. In this way, global ecological overshoot would end in 2080; in result, the allocation portion for the wild species will

Year | 2004 | 2005 | 2010 | 2015 | 2020 | 2025 | 2030 | 2035 | 2040 | 2045 |
---|---|---|---|---|---|---|---|---|---|---|

Per Capita EF | 2.184 | 2.156 | 2.099 | 2.064 | 2.077 | 2.008 | 2.023 | 2.002 | 1.944 | 1.806 |

Year | 2050 | 2055 | 2060 | 2065 | 2070 | 2075 | 2080 | 2085 | 2090 | 2100 |

Per Capita EF | 1.751 | 1.730 | 1.704 | 1.674 | 1.679 | 1.696 | 1.619 | 1.537 | 1.471 | 1.406 |

be about 14% of the planet’s biological productivity. This is only possible, if the global per capita biocapacity is 1.397 gha in 2080 and 1.368 gha in 2100.

The rapid-reduction scenario in Living Planet Report 2006 shows that humanity’s footprint in 2100 would be 40% less than that in 2003. It depicts that, by the mid-century, Earth’s biological productivity will be eight years ahead of ecological debt. So that 30% of biocapacity would be allowed for the use of wild species by then. According to [

The annual decreasing rate of global per capita EF should be 0.842% (r = −0.00842) if it drops from 2.23 gha in 2003 to 1.348 gha in 2050. The simulative results of global per capita EF from 2003 to 2050 are shown in

Year | 2004 | 2005 | 2010 | 2015 | 2020 | 2025 | 2030 |
---|---|---|---|---|---|---|---|

Per Capita EF | 2.145 | 2.029 | 1.990 | 1.928 | 1.950 | 1.831 | 1.856 |

Year | 2035 | 2040 | 2045 | 2050 | |||

Per Capita EF | 1.820 | 1.723 | 1.504 | 1.414 |

2050 if its average annual decreasing rate would be 0.842%. By then global ecological overshoot would have been concluded if global per capita biocapacity is 1.461 gha. While global ecological surplus would be 0.047 gha in 2050.

The annual decreasing rate of global per capita EF should be 0.438% (r = −0.00438), if it drops from 1.348 gha in 2050 to 1.053 gha in 2100.

A nonlinear dynamic simulative model has been discussed with variable cycles of entire world per capita ecological footprint taken from 1961 to 2003. The model was further classified and decomposed and extracted by the empirical

Year | 2050 | 2055 | 2060 | 2065 | 2070 | 2075 | 2080 | 2090 | 2095 | 2100 |
---|---|---|---|---|---|---|---|---|---|---|

Per Capita EF | 1.348 | 1.275 | 1.249 | 1.252 | 1.221 | 1.218 | 1.214 | 1.128 | 1.086 | 1.054 |

mode decomposition (EMD) method. Simulative numerical values of the three global per capita ecological footprint scenarios are analyzed based on the simulative model. The results show that:

1) The obvious undulation cycle of global per capita EF growth is 4.6 years, 9.5 years, 19.5 years and 41 years respectively over the last 42 years.

2) Global per capita EF would be 3.262 gha in 2050, if the business-as-usual scenario and the positive annual change rate remain constant. Assuming the global per capita biocapacity (BC) to be 1.236 gha, the global per capita ecological deficit (ED) would increase from 0.4 gha in 2003 to 2.026 gha in 2050.

3) The slow-shift scenario shows global per capita EF would fall from 2.23 gha in 2003 to 1.619 gha in 2080 and 1.406 gha in 2100 if the negative annual change rate of it is 0.447% constantly. Assuming the global per capita BC to be 1.397 gha, the global per capita ED would decrease from 0.4 gha in 2003 to 0.222 gha in 2080 and 0.038 gha in 2100. Overshoot would end in 2080 and the allocation portion for the wild species will be about 14% of the planet’s biological productivity. We think it needs more deliberations.

4) The rapid-reduction scenario depicts global per capita EF would fall to 1.414 gha in 2050, if the negative annual change rate remains 0.842% from 2003 to 2050. Assuming the global per capita BC to be 1.461 gha, then the global per capita ecological reserve (ER) would be 0.047 gha, and overshoot would be eliminated in 2050. The global per capita EF would fall to 1.054 gha in 2100, if the negative annual change rate is 0.438% from 2050 to 2100. Assuming the global per capita BC to be 1.474 gha, global per capita ER would be 0.420 gha, and about 28.5% of the planet’s biological productivity could be allocated for the use of wild species, which coincides with the results of Living Planet Report 2006.

The nonlinear dynamic simulative model proposed with the cycles and multiple scenarios simulation is analyzed in this paper. The purpose is to offer access to the prediction study on the Ecological Footprint Method. It is meaningful in reducing humanity’s EF for policy-makers by simulating the parameter. The variation of EF could be simulated if the values of r and T are appropriate. Theoretical references could be offered for those policy-makers from the public, government, and environment departments. Of course, the multiple scenarios simulation of this paper needs to be improved since their values are calculated at various hypothesis premises (e.g., the annual change rate of global per capita EF would be consistent). In fact, the variation of EF is very complicated as it is influenced by social, economic and natural factors such as population, consumption, land use, climate, technology, management and etc. Solutions to these questions will provide further perception into Ecological Footprint Method prediction.

This paper is supported by the Doctoral Research Foundation of Zaozhuang University (2017BS03). We would also like to appreciate Yun-Tao Zhao and Shi-Guan Zhuang of WWF China, and Audrey Peller of WWF, for their significant assistance to this paper.

The authors declare no conflicts of interest regarding the publication of this paper.

Xiang, L.H. and Chen, C.Z. (2019) Multiple Scenarios Simulation of Global Ecological Footprint Based on Empirical Mode Decomposition Method. Open Journal of Ecology, 9, 506-520. https://doi.org/10.4236/oje.2019.911033