Growth trajectories of medium-sized municipal corporations in Uttar Pradesh: Application of fourth order Runge-Kutta algorithm

Introduction

Population projections and estimates are essential inputs for planners and policymakers in preparing development plans for a country, state, or region. These projections are generally derived using: a) Mathematical models, and b) the component method of projections. The Mathematical models depend on the mathematical equations with assumptions that form quite suitable for small area population projections. The component method of projections derived from birth, death and migration data by age group with the available data sources such as census, surveys or vital registrations. The first non-synchronous census in India was conducted between 1865 and 1872, while the modern census began in 1901. Since then, censuses have been conducted every decade 1961, 1971, 1981, 1991, 2001, and 2011. The census provides comprehensive information on population size, distribution, literacy, occupation, housing, and other socio-economic characteristics at the state, district, and village levels. Based on population size, the Census of India (Registrar General 2011) [1,2] broadly divides the country into urban and rural administrative divisions. Urban areas are further classified into Class I to Class VI Municipal corporations/towns/cities, million-plus cities, and megacities. The growth of mega and metropolitan cities has largely been driven by industrial expansion, making them major centres of development. However, due to the heavy concentration of industries and infrastructure, many of these cities are approaching saturation in terms of population. Consequently, planners and policymakers are increasingly focusing on the development of medium-sized municipal corporations to ensure balanced population growth. In line with this perspective, the present study aims to examine the growth trajectories of medium-sized municipalities that are approaching a population of one million and illustrate with real data derived from the Census of India (Registrar General 2011) [1,2] , for the state of Uttar Pradesh.

Data and Methodology

To achieve the objectives, data were collected from the Census of India (Data sources 1-6) for Uttar Pradesh. According to 2011 census, Uttar Pradesh consists of 71 districts and Ghaziabad district has the largest urban population, with the minimum in Saravati. The number of cities having a population of 1 to 10 lakh is 64, whereas the number of cities having a population of 10 lakh or more is 7.

For the present study, 57 medium-sized Municipal Corporations (MC) having more than 1 lakh population and less than 10 lakh population are considered to estimate the time required to reach 10 lakhs and above. The list of medium-sized municipalities with population during the 1911, 2001 and 2011 census periods is listed in Table 1.

Mathematical models for Population projections

Basic mathematical model for Population projection begins with the exponential model postulated by Malthus, T.R. (1798) [10] , with an assumption that Population will grow at a geometric growth rate and resources in a linear pattern. Malthusian models have the following form: The above exponential model gives Population estimates but does not derive the population saturation level and period. Further Population growth trajectory depends on the past rate of growth (r). To achieve the time required to estimate the saturation level, the Logistic Model has been used and tested as the most suitable model.

Logistic Model:

The logistic growth model equation is:

Where: 𝐾 = carrying capacity (to be estimated), r = intrinsic growth rate. 𝐴 = constant determined by the initial population This model can be rewritten in the form of a differential equation as:

-------- (1) Where P(t) = Population at time t, r = Intrinsic rate of growth, K = carrying capacity of the environment to meet the population, and t = time in years. [8,9,12,13,17] . The logistic equation predicts future population based on the current growth rate and a predetermined carrying capacity (K value). The model assumes that population growth slows as it approaches the carrying capacity. The logistic model, initially proposed by Verhulst P.F. (1838) [16] , incorporates the concept of a carrying capacity, with which the model is able to estimate the saturation effect as the population approaches environmental limits. To solve this nonlinear ordinary differential equation, with a varying K value, the fourth-order Runge-Kutta (RK4) algorithm has been employed to derive accurate results (Eq. 1).

Fourth-Order Runge-Kutta (RK4) Method:

RK4 is a relatively simple numerical approximation; it provides a high degree of accuracy (Anido et [7,11,13,14,18] and is widely used to derive approximate solutions of Ordinary Differential Equations (ODEs). In this study, the fourth-order RK4 method was applied to project the population of Municipal Corporations (MCs) in Uttar Pradesh with populations ranging from 100,000 (one lakh) to 1,000,000 (10 lakh), and to estimate the time required to reach saturation. Since growth trajectories vary across municipalities, the K value was derived separately for each using the logistic model. The estimated values of r and K for the 57 selected municipalities are presented in Table 1. The RK4 model calculations were performed using STATA 16.1 software. The core commands are given in Table 1A.

RK4 Algorithm:

The RK4 algorithm approximates the value of P(t) iteratively over small intervals "h" rather than considering the same "r" value throughout the projected period. The general formula for the RK4 method is as follows:

Model Application:

The growth trajectory is estimated by applying the logistics model and Fourth order Runge-Kutta (RK4) method. Average intrinsic rate of growth = (r1 + r2)/2. The calculated r1, r2, r and K values for 57 municipalities/towns are given in Table 1.

Input data for the Fourth Order Runge-Kutta (RK4) method:  Initial Population P 0 = 587211  Intrinsic rate of growth r = 0.021616 (calculated for the period 1991-2001, 2001-2011 as above)  Carrying capacity K = 2714391 (upper bound calculated for each area)  Time step h=1 year  Numerical method: Fourth Order Runge-Kutta (RK4) method  Period of Projection: 1991-2031

The Projected Population by the RK4 method is listed in Table 2. The calculated values of MAPE, RPE and TU for 57 Municipal corporations/towns in Uttar Pradesh are listed in Table 3, and for visualisation, Fig. 1 is drawn between MAPE and TU.

Model

Results

The RK4-based projections for 57 municipalities in Uttar Pradesh ( It can be concluded that the fourth-order Runge-Kutta (RK4) method proves to be a suitable and robust approach for population projections in Uttar Pradesh, as it effectively models continuous and predictable demographic growth.

References

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