# Logistic Growth Model in Human Birth Rate

Introduction

The study discusses how the logistic equation, which is a mathematical model of population growth, can be used to forecast the human birth rate concerning available resources. Most of the world’s resources are nonrenewable. While this is a widely accepted fact, the 21st century presents unique challenges in terms of balancing demand for resources such as water, fish, and land with sustainability requirements. Seo (18) states that the effects of climate change have only made the situation worse, especially as rainfall patterns are becoming increasingly unpredictable, water sources are becoming depleted, forest cover is declining, the acreage of arable land is shrinking, and fish populations are dwindling. While all this is happening, the global community continues to grow, raising serious questions about the planet’s ability to support itself.

Is there a way that the correlation between human birth rate and resource utilization can be analyzed, predicted, and modeled to enable better population planning? At what point can available resources become incapable of supporting the human birth rate? By using the logistic growth model, the relationship between the human birth rate and resource availability can be modeled using the following differential equation: dP/dt=kP(1-P/N)-λ. In this formula, the population is represented by P(t), the birth rate is denoted by k, N symbolizes rate of resource consumption, and the carrying capacity is signified by λ. An examination of the answers to this equation for different resource consumption levels confirms that a critical resource consumption level, also referred to as a bifurcation value, is a reality.

The model reveals that if the resource consumption level surpasses this bifurcation value by the smallest degree, then the birth rate will dramatically reduce (Schiesser 49). The logic behind the equation is that when a region, country, society, or community is approaching the bifurcated value, the slightest increase in resource utilization can have devastating effects on birth rate (Schiesser 68).

A community that grazes its cattle on a public piece of land, for instance, may not take any action to moderate the intensity of grazing that the cattle are allowed to engage in. After some time, pastures on the land will become extinct and unable to sustain the population of the community, leading to low birth rates. Overutilization of resources leading to the collapse or extinction of whole societies or communities also referred to as overharvesting, is not a new phenomenon. Indeed, there are historical accounts that prove it should be treated as a matter of urgency. According to Seo (29), a good example is the deterioration of the community on Easter Island, home to the Rapa Nui people. Credible documents show that a key ingredient in society’s disintegration was the total depletion of forest cover on the island, made possible by human deforestation.

Today, many would wonder how naive the Rapa Nui were to decimate all trees in their habitat. How did no one realize that the forest cover was rapidly disappearing? Why did no one intervene to arrest the situation? These questions are easy to pose now, but back then the Rapa Nui were oblivious to the level of destruction they were causing to themselves and their environment. Seo (57) adds that ironically, while it is easy for current generations to feel arrogantly smarter than the Rapa Nui, it is likely that future generations will ask the same questions regarding actions that are presently causing the extinction of species and overconsumption of food sources.

Growth models

Estimating the world population and the carrying capacity limit can help the world to plan on how to use its natural resources sparingly to sustain the world for a longer period, without total destruction. Several models have been put forward to predict the population growth rate. However, some of these models have limitation. One such model is exponential growth rate model.  Malthus proposed exponential growth model in 1798, and hence often referred to as the Malthusian Growth Model. The logistic growth model was proposed by Verhulst in 1845. Both models emanated from the observations of biological reproduction process (Balakrishnan 14).  But in relation to human population, the constant rate of growth can be observed in uncertainty. Humanbeings reproduce sexually and are intelligent creatures with consciences. Therefore, it is uncertain to establish the condition in which a person can be reproduced.  The logistic model is an improvement of exponential growth since it takes into consideration carrying capacity(Balakrishnan 14). However, whether it can be used to estimate the population growth rate of humanbeings is an issue under studies.  Since the objective of the paper was to come up with a useful model that can be used to predict the maximum limit of people that the world can sustain, based on the available resources.  Logistic growth model will be tested by population data of a country to show whether it effective in providing important information for government planning

Logistic growth Model

Usually exponential growth model is applicable in cases where the available natural resources are infinite, and this is not practical in the real world.  In an attempt to model the limited resources reality, logistic growth model can be applied.

Carrying Capacity and the logistic model

It is not possible for the world population to grow infinitely in the real world characterized by the scarce resources which continue to reduce by each year.  Exponential growth is only possible in a given environment in which there are few people with vast resources. However, when the number of people increases, the resources will ultimately be depleted slowing the rate of population growth. Ultimately, the rate of growth will level off or plateau.  The maximum population size that a given environment can support is known as carrying capacity denoted by symbol K.

At carrying capacity, the deaths balance births, thus

Therefore when

As the population size (N) to approach K, the resources, as well as other forces, start to limit growth. The limit is often imposed by the increases in d (Deaths and decreases in b (births).  The logistic model assumes that every person in a given population have equal access to resources and therefore, have equal survival chances.  For plants, the amount of sunlight, water, nutrients as well as space to grow is key resources, whereas, in animals, key resources include, mates, nesting space, shelter, water, and food. In the real world, the phenotypes variation among people in a population implies that some people will better adapt to their environment than another will.

An environment that  has limited population assumes that the population growth rate reduces as we approach the limiting population. The logistic model is described as

Where:

, is the maximum load of the environment ( carrying capacity)

is the unknown function based on time given time

is the proportionality constant

K  represents the population size that a given environment  can sustain in the long term

Hence this differential equation in a continuous equation is given by

Such population net rate is denoted by  this shows the rate at which the population increase in (N) size as (t) time progresses (Frauenthal 540).  The equation gives a sigmoid growth curve that approaches a stable carrying capacity (K). as in figure 1.

Fig 1:  The logistic growth model graph

Implementation of the outcomes

This part, the proposed model is utilized on the actual population of Jordan and is used to predict the country population up to 2100

 Year Actual Population 1955 645, 724 1960 888,632 1965 1,119,798 1970 1,654,769 1975 1,985,121 1980 2,280,670 1985 2,782,885 1990 3,358,453 1995 4,320, 158 2000 4,767, 476 2005 5,332, 982 2010 6,517, 912 2015 7,594, 547 2016 7,737,800

Table 1: Population of  Jordan from (1955-2016) based on the world statistics (Worldometers 1)

The Country population is forecasted to be 11, 716, 525 by 2050 with about 1.2% increase per year. The logistic growth model can be used to predict the actual population and the comparison made.

For the Year 1955,

The rate of growth (r) is a constant that can be calculated from the Jordan population between 1955 and 1960.

With the actual population of Jordan in 1960, and  during this year, the rate of growth can be calculated.

When  and

5α in the exponent is since the original data are a 5-years interval.

The general Logistic growth model for this case becomes

Thus, Logistic population growth model for Jordan becomes,

We thus attain the estimation of Jordan population from 1970 onwards by substituting the time values and population of Jordan in Table 1.  The results are as shown in figure 2 below.

The green line shows the population of Jordan using a logistic growth model while the Blue bars are the actual population data of the country.

Fig 2: Logistic growth model showing the population of Jordan

The limitation of the Logistic growth model is that we only used two historic years in creating the models. Besides, the value of limiting capacity should be definite, and this model cannot predict it (Frauenthal 540).

Using regression tools based on the logistic growth model to calculate the population of Jordan

Jordan’s population estimation by Logistic model

Logistic growth model only works best and can provide accurate results when the population growth rate  (α)as well as carrying capacity (k) is accurately obtained. The  CFTOOL of MATLAB toolbox is utilized to calculate the value of α and k.  Fig 3 is cftool screenshot, and table 2, indicates the final values of α and k.

Fig 3: Using MATLABCFTOOL to fit the logistic growth model

MatlabCftool leads to the following equation:

was found to be 0.0456 with confidence interval level of 95% between 0.04326 and 0.04795.  The carrying capacity was 17.84 with the level of confidence interval at 95% of (13.30, 22.390), having determination coefficient of 99.67-percent showing the model can fit the current data appropriately. This method is applicable to finding the value of carrying capacity using the current data.

Projection of Jordan’s Population using logistic, growth models

 Year Actual population (in millions) (Statista 1). Logistic Growth Model estimate (in millions). 1955 0.646 0.804 1960 0.889 0.998 1965 1.120 1.236 1970 1.655 1.525 1975 1.985 1.875 1980 2.281 2.293 1985 2.783 2.788 1990 3.358 3.368 1995 4.320 4.035 2000 4.767 4.791 2005 5.333 5.631 2010 6.518 6.544 2015 7.595 7.514 2016 7.748 7.713 2020 8.519 2025 9.535 2030 10.534 2035 11.494 2040 12.393 2045 13.215 2050 13.953 2055 14.601 2060 15.163 2065 15.641 2070 16.044 2075 16.381 2080 16.658 2085 16.886 2090 17.072 2095 17.224 2100 17.346

Table 2: Jordan Population projection using the logistic model

Conclusion

From table 2, the Jordan population was about 3.36 million, and it gradually grew until 1995, in which it hit a 4.32 million mark. The population increase was caused by the first Gulf war since many migrants from Gulf nations who settled in Jordan.  In 2010-2016, the population also inflated from 6.518 million to 7.738 million in 2016. The rise is because of the crisis in Iraq and Syria. Based on the logistic growth model, predicted the population of Jordan to be about 17.346 million by 20100. This model accuracy mainly depends on carrying capacity hence its application in real human population is limited due to the complexity of the various population dynamics, which makes it hard to come with an accurate value of k for a particular habitat. Besides, k is not constant and is prone to often changes based on various conditions. Therefore, this model is not accurate in providing population forecasts. More research needs to be conducted to find the best model to calculate the value of k. with the real value of k; the logistic growth model can be used to predict population growth rate.

Work Cited

Balakrishnan, Narayanaswamy. Handbook Of The Logistic Distribution. Dekker, 2013.

Frauenthal, James C. Introduction to Population Modeling. Boston, MA: Birkhäuser Boston, 2013. Internet resource.

Schiesser, William E. Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R. New York, NY: Wiley, 2014. Print .and

Seo, Niggol S. Natural and Man-Made Catastrophes: Theories, Economics, and Policy Designs.

Statista. “Jordan – Average Age Of The Population 1950-2050 | Statistic”. Statista, 2019, https://www.statista.com/statistics/383922/average-age-of-the-population-in-jordan/.

Worldometers. “Jordan Population (2019) – Worldometers”. Worldometers.Info, 2019, http://www.worldometers.info/world-population/jordan-population/.

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