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Mathematical Modeling in Current Trends of Human Population Growth

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Mathematical Modeling in Current Trends of Human Population Growth International Journal of Research, Science, Technology & Management ISSN Online: 2455-2240, Volume 23 Issue I, April 2021 Mathematical Modeling in Current Trends of Human Population Growth 1 Rakesh Verma, 2 Renu Kushwah 1 Associate Professor, Applied Sciences Department 2 Assistant Professor, Applied Sciences Department 1,2 Sagar Institute of Research & Technology Excellence, Bhopal, M.P. differential equations govern the growth of various species. Abstract: Bhopal is an overpopulated and the most densely Hence, in this paper, mathematical modeling for global populated city in M.P.. In present scenario, sustainability is the population growth based on exponential and logistic growth ability of a system to exist constantly at a cost in a universe. In model but we can study to change the major part of population the 21 century, it refers generally to the capacity for biosphere through differential equation. Today the main critical problem and human civilization to coexist. As we know that human of world is eerie growth of the population. According to population is increasing day by day as well as human demands situation of the world, more than thousand people are is also increasing like water, daily routine things, trees etc. The increasing every day. And each need like land, water, shelter, role of mathematical models are very useful for the behaviour food etc is sufficient for good life. Mathematical modeling is a of system depend on physics, chemistry, biology etc. broad interdisciplinary science that uses mathematical and Importance of mathematical models to effectively predict computational techniques to model and manifest the economic, social system including the population growth. occurrence arising in life sciences. It involves the following Some works are going on growth of population problem processes: through exponential and logistic growth model but the differential equation with which we can study the change of i. The construction of mathematical model for a real major parts of population through time, which is helpful for world problem. predict of population of a certain place at certain time.IJRSTM ii. The analysis of the emerging mathematical problem. Keywords: Mathematical modeling, Population growth, iii. The explanation of the mathematical results in the Logistic growth, Exponential growth, Differential equations. reference of the original situation. These models can be in different shapes, sizes and styles. INTRODUCTION: At first glance, modeling the growth of a species would seem to be impossible since the population of any species The projection of future population is normally based on always changes by integer amounts. The projection of present population and reliable growth rate and the first order future populations is normally based on present population 7 www.ijrstm.net International Journal of Research, Science, Technology & Management ISSN Online: 2455-2240, Volume 23 Issue I, April 2021 and reliable growth rate and the first order differential In practice N(t) is a non-negative integer. We assume that N(t) equations govern the growth of various species. Hence in is continuously differentiable. The growth rate of population is this paper, mathematical modeling for global population the rate of at which population changes [5]. growth based on exponential and logistic growth models is presented. The use of these growth models is widely Suppose that we know the population N0 at some given time established in many fields of modeling and forecasting. t t0 , and we are interested in protecting the population N, at some future time t t , i.e. to find the population function METHODOLOGY: 1 N(t) for t0 t t1 that satisfies N(t0 ) N 0 . Methodology is more effective than the simple methods in which one intends to use for collecting data, analysis, and Then the growth model is described with initial value problem interpretation of data. In this paper, secondary population of the Bhopal (M.P.) from 1960 to 2010 (during gas tragedy in dN r.N; t0 t t1; N(t0 ) N0 (1) 1984) was collected as par the Madhya Pradesh government dt data. The exponential and logistic growth mathematical model was used to compute the predicted population values. On integrating by variable of separable method of equation (i), MALTHUSIAN GROWTH MODEL: dN r.dt A Malthusian growth model, sometimes called a simple N exponential growth model, is essentially exponential growth ln N r.t c based on the idea of the function being proportional to the r.tc speed to which the function grows. The model is named after N(t) e r.t c Thomas Robert Malthus. The Malthusian growth model is also N(t) e .e IJRSTM r.t known as simple exponential growth model [7]. Malthusian N(t) C.e model is single species population model. This is a reasonable assumption for a population of a bacteria or animal under ideal Equation (ii) is the general solution of equation (i). conditions. In exponential growth, a population’s per And now for the initial value condition: N(t ) N then we individual growth rate stays the same regardless of population 0 0 size, making the population grow faster and faster as it gets have larger. 8 www.ijrstm.net International Journal of Research, Science, Technology & Management ISSN Online: 2455-2240, Volume 23 Issue I, April 2021 N(t) C.er.t population over time, or a positive growth rate. However, most r.t populations cannot continue to grow forever because they will N(t ) C.e 0 0 eventually run out of water, food, sunlight, space or other N C.er.t0 0 resources. As these resources begin to run out, population r.t0 N 0 .e C growth will start to slow down. When the growth rate of a population decreases as the number of individuals increases, Then by equation (ii), this is called logistic population growth. r.t0 r.t N(t) N 0.e .e Logistic model is a modification of Malthusian model. The r(tt0 ) Logistic model is a model of population growth first published N(t) N 0.e (2) by a Belgian mathematical biologist Pierre Verhulst [6]. He Where ‘r’ is a constant known as the Malthus factor, is the found that the population growth not only depends on the multiple that deter mines the growth rate. And equation (iii) is population size but also on how far this size is from its carrying known as Malthusian law of growth [5]. The growth rate capacity [11]. He modified Malthusian model to make a increases as the population increases. Equation (i) is population size proportional to both the previous population appropriate for modeling population growth under ideal and a new term [14] conditions, thus we have to recognize that a more realistic must a bN(t) reflect the fact a given environment has a limited resource. (3) a LOGISTIC GROWTH MODEL: Where ‘a’ and ‘b’ are the vital coefficients of the population. In logistic growth model, a population’s per capita growth rate a Now as the population value gets closer to this new term gets smaller and smaller as population size approaches a b maximum imposed by limited resources in the environmentIJRSTM will become very small and tends to zero, which gives the right known as the carrying capacity. feedback to limit the population growth [15]. Thus, the second term models the competition for available resources, which A group of individuals of the same species living in the same tends to limit the population growth. So, the modified equation area is called a population. The measurement of how the size of using this new term is a population changes over time is called the population growth rate, and it depends upon the population size, birth rate and dN a.N(t)[a bN(t)] death rate. As long as there are enough resources available, (4) dt a there will be an increase in the number of individuals in a 9 www.ijrstm.net International Journal of Research, Science, Technology & Management ISSN Online: 2455-2240, Volume 23 Issue I, April 2021 The equation (4) is known as the logistic law of population growth model. a Solving the equation (4) and applying the initial condition N b (9) N(t ) N then equation (4) become a 0 0 1 b 1.eat N 0 dN 2 aN bN dt If we take the limit of equation (9) as t , we get dN dt (5) N.(a bN ) a N max lim N (10) t b On using the separation of variables, Taking t=0, t=1 and t=2, then the value of N are N1 and N 2 1 1 b respectively then we get equation (9), dN dt (6) a N a bN a Now on integrating and we get, b a 1 e (1 e ) (i) a N1 N0 (11) 1 b 1 e2a log N log(a bN) t c (7) (1 e2a ) (ii) a a N 2 N 0 At t=0, we get N(t0 ) N0 b Dividing the (11)(ii) by (11)(i) and eliminate then we get, a 1 1 e2a c log N 0 log(a bN0 ) (8) a IJRSTM N N 1 ea 2 0 Now equation (7) becomes 1 ea N1 N 0 1 1 log N log9a.bN) t log N log(a bN ) So that, a a 0 0 a N 0 (N 2 N1 ) e (12) N 2 (N1 N 0 ) Putting this value into the first equation of (1) then we get 10 www.ijrstm.net International Journal of Research, Science, Technology & Management ISSN Online: 2455-2240, Volume 23 Issue I, April 2021 292130.10 103100 2019963.13 b N 2 N N 1 0 2 (13) 7 0 a N (N N 2N N N N ) 1 0 1 0 2 1 2 299776.59 142800 Thus the limiting value of P is given by 4282321.834 1 0 a N (N N 2N N N N ) 302702.08 185400 1 0 1 0 2 1 2 9078522.281 N max lim N (14) t b N 2 N N 1 0 1 0 2 MAPE: 303875.92 19246467.22 3 The absolute percentage errors shown on the following 304378.49 worksheet are rounded to the nearest whole percentage.
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