### Year : 2018 Volume : 14 Issue : 28

**Türkiye Nüfus Büyümesi ve Tahminleri: Matematiksel Büyüme Modelleri ve İstatistiksel Analiz İle Kuramsal ve Uygulamalı Bir Yaklaşım
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#### Abstract

Bu çalışmanın amacı, Verhulst ve Gompertz tarafından ilk tanımlamaları yapılan ve nüfus analizlerinde geniş uygulama imkânları bulan matematiksel büyüme fonksiyonlarıyla, 1925-2015 dönemi için yeterli veri arz eden Türkiye nüfusunun büyüme eğilimini ve özelliklerini analiz gelecekte ulaşacağı maksimum seviyeyi tespit etmektir. İlaveten bu dönem zarfında sağlanan mutlak ve nispi büyüme oranlarını yine bu eğriler üzerinden hesaplamaktır. Geliştirdiğimiz matematiksel analiz ve istatistiksel uygulamayla Türkiye nüfus verilerinin soyut düzeyde temsiliyeti hedeflenmiş, yapılan ileri istatistiksel çalışmayla keyfiyet test edilmiştir. Daha sonra söz konusu fonksiyonlarla geleceğe dönük tahmin çalışmaları yapılmıştır. Bu şekilde Türkiye nüfusu hakkında matematiksel büyüme modelleri, istatistiksel analiz ve geleceğe dönük tahminler ile kuramsal bir çerçeve tanımlanmıştır. Türkiye’nin 1925- 2015 dönemi toplam nüfus sayımı istatistikleri çalışmamızda kullanılmıştır. Lojistik fonksiyon ve Gompertz fonksiyonunda alt ve üst asimptot arasında nüfusun gelişimi önce süratle artan sonra azalarak artan bir seyir halinde olacağı varsayımı ile hareket edilir. Nüfusun gelişimi üst sınır olan taşıma kapasitesi ile sınırlıdır. Büyüme fonksiyonlarında nüfus bağımsız değişkene göre sonsuz büyümez. Fonksiyonlarının birinci türevleri yıllık mutlak büyüme rakamlarının hesabında ve ortalama yıllık büyümenin hesaplanmasında, ikinci türevler ise değişimlerdeki değişimin ve fonksiyon dönüm noktalarının hesabında kullanılmıştır. Çalışmamızda SAS bilgisayar yazılımı kullanılmıştır. Gompertz fonksiyonu üzerinden yapılan ilave çalışmalara ve Amerikalı bilim adamlarının nüfus çalışmasına da ayrıntılı olarak değinilmiştir.

#### Keywords

Türkiye-Nüfusu Matematiksel-Büyüme-Fonksiyonları Lojistik-Fonksiyon Gompertz-Fonksiyonu Nüfus-tahminleri#### Corresponding Author

Cemil İskender#### Authors

Cemil İskender#### References

- Allen, R. G. D. (1969). Mathematical analysis for economists, Macmillan and Co. Ltd.
- Berger, R. D. (1981). Comparison of the Gompertz and Logistic Equations to describe plant disease progress. Phytopathology, 71, 716‒719.
- Burley, H. T. (1996). Growth rate tables. Cambridge University Press.
- Carey, E. (2009). Using Calculus to Model the Growth of L. Plantarum Bacteria. Undergraduate Journal of Mathematical Modeling: One + Two, 1(2), 1‒11. http://dx.doi.org/10.5038/2326-3652.1.2.2
- Chukwu, A. U. & Oyamakin, S. O. (2015). On Hyperbolic Gompertz Growth Model. World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 9(3), 189‒193.
- Fekedulegn D. B. & Colbert, J. J. (1999). Parameter Estimation of Nonlinear Growth Models in Forestry, Silva Fennica 33(4), 327‒336.
- Gebremariam, B. (2014). Is Nonlinear Regression Throwing you a curve? New diagnos

#### Details

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**DOI
** 10.26650/ekoist.2018.14.28.0004

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**Submission :
** 24 May 2018

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**Early Viewed :
** 20 Eyl 2018

**Turkish Population Growth and Estimates
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#### Abstract

This study aims to analyze the population growth trend and properties of Turkey between 1925 and 2015 and estimate the maximum level to be reached in the future by using mathematical growth functions of Verhulst and Gompertz. The study used SAS software. Additionally, the study calculates and charts absolute and relative growth rates of population through the curves. The representation of Turkish population data at an abstract level was targeted with mathematical analysis and statistical application. After the successful results were taken from the mathematical representation and statistical proof stages, predictions for the future were made. Thus, a theoretical framework with mathematical growth models, statistical analyses, and predictions related to the future of Turkish population is defined. The necessity of using logistic and Gompertz growth functions for population analysis is explained in detail. These functions analyze growth with the assumption that population rapidly increases first, then the rate of increase slows down, reaching a maximum level at upper asymptote (carrying capacity). There is a limit to growth in these functions based on the geography and resources of the country. The first derivatives of the growth functions are used for calculating annual absolute growth and average annual growth rates of population. The second derivatives have been used for calculating change amount of absolute growth figures and reflection points of functions. Additionally, some thoughts on Gompertz function and American experience on population are explained in detail.

#### Abstract Extended

This study aims to analyze the population growth trend and properties of Turkey for the period 1925–2015 and estimate the maximum level to be reached in the future by using commonly accepted mathematical growth functions of which the first definitions were developed by Verhulst (Belgium) and Gompertz (England) in the nineteenth century. Additionally, the study calculates and charts the absolute and relative growth rates of population studied during this period through the curves. Turkish population statistics of Turkish Statistical Institute were analyzed by using logistic growth functions, two-stage logistic growth function and Gompertz growth functions, and on the basis of this purpose, non-linear function solutions were used in statistical applications. Representation of the Turkish population data at an abstract level was targeted with mathematical analysis and statistical application, and all population data were processed with advanced statistical tests. After the successful results were taken from the mathematical representation and statistical proof stages, predictions were made with the aforementioned functions and the results obtained are discussed. Thus, a theoretical framework is defined with mathematical growth models, statistical analyses, and predictions related to the future of Turkish population with data of the 1925–2015 period. Pierre François Verhulst (1804–1849, Belgium) anticipated that population cannot continue to grow all the time according to the exponential function of the Malthusian model; on the contrary, after a rapid increase in speed, growth will decrease to nearly zero, which adds a restrictive variable to the function. This variable is carrying capacity–the limiting factor of growth. In 1825, the mathematician and actuarial and insurance specialist Benjamin Gompertz (1779–1865) wrote his famous article about human mortality tables, putting forward a non-linear growth function for this purpose. Although Gompertz developed this function for calculating human mortality tables, today’s intended framework has already been surpassed and the relevant function has been used in many areas of sciences as a growth function. The concept of “carrying capacity” is already included in Gompertz function, as in the logistic function, as a limiting variable. We can calculate growth by simply dividing figures with each other in discrete units of time (by years) or by means of continuity of a mathematical function. Simple and compound growth formulas are used for short periods of time. For studying longterm data, functional analysis with growth functions is required. According to our knowledge about the data at hand, we can choose what best suits them. This also applies to the human population statistics. It is preferable to apply statistical methods to the data at hand by adopting growth functions and calculating absolute and relative growth rates of population for the long term.An increase in the number of available mass of population is based on the existence of resources at hand, which is a restrictive variable. In this case, it is useful to apply logistic growth function to data over a long period of time with the assumption that nothing will permanently increase or decrease and there will be increasing, slowing, and decreasing period of time for the variable at hand. When applied with non-linear statistical techniques, growth functions and curves with the flexibility to provide predictions of multiple parameters on the same function at a time draw a close model to actual data. Initially, there is exponential growth in the logistic function. Following the exponential growth period, an inflection point is reached. This is the beginning of second phase growth, which slows down and reaches an upper limit. Logistic models have upper and lower asymptotes. The turning point is right in the center of the curve, which is symmetric here. Symmetry is an element restricting the structure of the curve. But the growth curve developed by Gompertz overcomes the symmetry restriction and provides us with a better functional analysis. In the analysis with the logistic function, the independent variable is time and the dependent variable is population. It is useful to work with this function when the variable starts from a point close to the horizontal axis with a slow acceleration, reaches the inflection point rapidly, and then slows down toward the upper asymptote. This is the restrictive point where carrying capacity—the region where the population reaches the limits in place—comes in. Moving the carrying capacity to an upper level is only possible with the introduction of new resources and geography. Because human population is closely related to the economy, it will be more meaningful to add the national income to carrying capacity. Apart from geography, natural resources, energy, etc., national income is also an effective variable for carrying capacity; this is the point where economics comes in to the function. There are more than one unknown parameters in the logistic function to be estimated, and ordinary least squares methods of estimation are not sufficient for statistical tests. There is one equation but more than one parameter to be estimated. Specially developed least squares methods based on minimization of the error sum of squares are required for solving and determining parameters of non-linear functions. Fortunately, techniques developed by theoretical statisticians in the last fifty years have made the job easier in terms of solving non-linear functions. All these methods are used in our statistical analysis. The Turkish census statistics from 1927 to 2015 were used in our study. Preliminary statistical studies have been made to prepare a series for 5-year periods and to provide estimated data for missing years. In these studies, estimations have been made for 1925, 1930, and 1995, which have no census results, with the help of simple curve fitting and regression analysis. Together with census statistics of Turkish Statistical Institute and estimations for missing years, 19 observations have been obtainedfor basic statistical analysis and non-linear curve fitting. Additionally, qualitative analysis has been done over this period, and this analysis has been included in the quantitative analyses as a variable. The development of the Turkish population over the aforementioned period has been divided into three sub-phases: stagnation period (1925–1950), development and growth period (1950–1985), and growth and maturity period (1990–2015). The qualitative aspects of these periods are given in detail in the main article. These three phases were indexed as 1, 2, and 3 and used in twophase logistic function (James H. Ricketts/Geoffrey A. Head function) and Gompertz function as dummy variables. Thus, it includes a qualitative variable in addition to the time variable. More successful results can be obtained with this approach. Two groups of functions used in the statistical analysis are logistic function and Gompertz function. In addition to basic logistic function, two-phase logistic function—successfully developed by authors James H. Ricketts and Geoffrey A. Head—has been extensively used in our study. Furthermore, adding inverse hyperbolic sine variable to this function gave more fruitful results. The two-phase logistic function has given far better results than the original logistic function. We added the allometric variable (v) of Richards’ logistic function to obtain accurate, meaningful statistical tests. The first and second derivatives of the logistic function have been used for calculations of annual absolute growth figures, average annual growth rates, and changing amount of the first derivative in absolute growth figures. Inflection points were calculated from second-degree derivatives. Re-parametrization of variables to be determined is calculated for curves in order to find more linear equations. Otherwise, meaningless statistical results were obtained from calculations of the curves. Gompertz functions, which give more flexible results than the logistic functions used in the study, have confirmed same upper asymptote levels (115 million people) for the Turkish population, along with the two-phase logistic function. Hyperbolic variable is also included in the Gompertz function for better results. Logistic and Gompertz growth functions do not have a linear structure. A nonlinear function has a non-linear relation between parameters. Therefore, least squares estimation methods used in the estimation of parameters of ordinary linear equations cannot be used here. Non-linear least squares methods developed for non-liner functions are used for the estimation of parameters of the non-linear functions. We have also followed this method in our work. On the contrary, it is impossible to make these calculations manually or with entry-level computer software. In this case, Microsoft Excel can be used only to prepare the data. There are no standard nonlinear regression models in Excel; therefore, we used Excel as a data-preparation tool. The software “SAS University Edition NLIN Procedure” was used in this studyfor statistical analysis of non-linear functions. In this software, convergence criteria/ iterative phase, minimization of error sum of squares, compliance diagnosis charts, F-tests, standard errors, 95% confidence limits and bootstrap confidence limits, Box bias criteria and Hougaard skewness criteria, global non-linearity measures, etc. were used for statistical tests of the functions. In order to find the required linearity of tested growth functions, parametrization and calibration of the parameters of functions to be estimated have been necessary in many functions, and this has been done in the SAS software. Detailed explanations are given in the main article for the aforementioned criteria followed and applied in the SAS NLIN procedure. Logistic functions have passed all the advanced statistical tests in the SAS NLIN procedure. The lower asymptote and upper asymptote (carrying capacity) figures of logistic function have been found as 9.5 and 94 million people, respectively. Turkey has a long way to go if we consider that Turkey’s population as of 2015 is 78.1 million people. The intrinsic growth rate of the logistic function has been found to be 4.93%. This is not an annual average growth rate of population that we analyze. Intrinsic growth rate is a general growth rate for the function, and it is valid throughout the period under study. A higher intrinsic growth rate means a sharper increase of the curve during the reach to the upper asymptote. We mentioned that the logistic function is symmetric around the reflection point. In order for the curve to follow this symmetry, carrying capacity is lower than expected, as will be confirmed by Gompertz functions having asymmetrical structures. The carrying capacity is 115 million people in Gomperz functions, which is more realistic than the logistic function’s 94 million people. The error sum of squares as a percentage of the total sum of squares of logistic function is 0.0121%, which is negligible. Data representation of the function is high and acceptable. But the Ricketts/Head logistic function has produced far better results than a simple logistic function from an error sum of squares point of view. The reflection point of the curve is 51.7 million people at the year 1986.38. Top level in relative growth was in 1965, with 2.55% growth rate. This figure has reached below 1% circa 2015. A two-phase asymmetric logistic function developed by James H. Ricketts and the Geoffrey A. Head for medical purposes, was used in our population analysis with some modifications. The Richards parameter was added to the function and, with the other arrangements made, contributed to the success of the function in representing the Turkish population. Intrinsic growth rate of the function is 12.82%; the lower asymptote and upper asymptotes are 7 and 115 million people, respectively. This is a sharper curve upwards than that of the logistic function. Hundred and fifteen million people is a more realistic figure and can also be confirmed by Gompertz function. The error sum of squares has been found as 0.006318% of the total sum of squares.e have achieved successful statistical results with the Gompertz functions as well. The intrinsic rate of growth is 2.71% and the upper asymptote is 113.8 million people for the first Gompertz function. The inflection point was the highest in 1984, with a population of 50.4 million people. The highest relative rate of growth was calculated as 2.7% in 1960. We have dealt with two properties of the functions. The first is the ability to represent the data to which they apply. The second is whether they are powerful enough for predictions. The method that we used to prove the ability of functions at representing the Turkish population was to get the test results of non-linear equations—part of an advanced stage of statistics—with the relevant computer software. I believe that we have achieved this goal in the study. Verhulst’s logistics function, a two-stage Ricketts/Head function that is a continuation of logistic function and Gompertz function and their mathematical properties, was successful and gave the expected results for our Turkish population analysis. Nineteen observations over 90 years between 1925 and 2015 comprise enough data for our population analysis. After the mathematical representation and statistical proof, successful results were obtained for the future population discussed. It is expected that Turkey’s population will level off at 115–120 million people. Additionally, some thoughts on Gompertz function and interesting experiences by American scientists on population are detailed in our study. Detailed statistical outputs of the growth functions can be found in the appendices.

#### Keywords

Turkish-Population Logistic-Growth-Function Mathematical-Growth-Functions Gompertz-Growth-Function Population-Estimates#### Corresponding Author

Cemil İskender#### Authors

Cemil İskender#### References

- Allen, R. G. D. (1969). Mathematical analysis for economists, Macmillan and Co. Ltd.
- Berger, R. D. (1981). Comparison of the Gompertz and Logistic Equations to describe plant disease progress. Phytopathology, 71, 716‒719.
- Burley, H. T. (1996). Growth rate tables. Cambridge University Press.
- Carey, E. (2009). Using Calculus to Model the Growth of L. Plantarum Bacteria. Undergraduate Journal of Mathematical Modeling: One + Two, 1(2), 1‒11. http://dx.doi.org/10.5038/2326-3652.1.2.2
- Chukwu, A. U. & Oyamakin, S. O. (2015). On Hyperbolic Gompertz Growth Model. World Academy of Science, Engineering and Technology International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 9(3), 189‒193.
- Fekedulegn D. B. & Colbert, J. J. (1999). Parameter Estimation of Nonlinear Growth Models in Forestry, Silva Fennica 33(4), 327‒336.
- Gebremariam, B. (2014). Is Nonlinear Regression Throwing you a curve? New diagnos

#### Details

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**DOI
** 10.26650/ekoist.2018.14.28.0004

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**Submission :
** 24 May 2018

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**Early Viewed :
** 20 Eyl 2018