# Population Growth

Population growth is a dynamic process that can be effectively described using differential equations. We consider here a few models of population growth proposed by economists and physicists.

## Malthusian Growth Model

The simplest model was proposed still in $$1798$$ by British scientist Thomas Robert Malthus. This model reflects exponential growth of population and can be described by the differential equation

$\frac{{dN}}{{dt}} = aN,$

where $$a$$ is the growth rate (Malthusian Parameter). Solution of this equation is the exponential function

$N\left( t \right) = {N_0}{e^{at}},$

where $${N_0}$$ is the initial population.

The given simple model properly describes the initial phase of growth when population is far from its limits. However, the accuracy of the exponential model drops at a later stage due to saturation or other nonlinear effects (Figure $$1$$).

## Logistic Model

This kind of population models was proposed by French mathematician Pierre Francois Verhulst in $$1838.$$ This model is also called the logistic model and is written in the form of differential equation:

$\frac{{dN}}{{dt}} = aN\left( {1 - \frac{N}{M}} \right),$

where $$M$$ is the maximum size of the population.

The right side of this equation can be represented as

$aN - \frac{{a{N^2}}}{M},$

where the first term is responsible for growth of population and the second term limits this growth due to lack of available resources or other reasons (Figures $$2,3$$).

The logistic differential model has exact solution, which we derive below.

$\frac{{dN}}{{dt}} = aN\left( {1 - \frac{N}{M}} \right),\;\; \Rightarrow \int {\frac{{dN}}{{N\left( {1 - \frac{N}{M}} \right)}}} = \int {adt} .$

The integrand in the left integral can be found using the partial fraction decomposition method:

$\frac{1}{{N\left( {1 - \frac{N}{M}} \right)}} = \frac{A}{N} + \frac{B}{{1 - \frac{N}{M}}},\;\; \Rightarrow \frac{1}{{N\left( {1 - \frac{N}{M}} \right)}} = \frac{{A\left( {1 - \frac{N}{M}} \right) + BN}}{{N\left( {1 - \frac{N}{M}} \right)}},\;\; \Rightarrow 1 \equiv A - A\frac{N}{M} + BN,\;\; \Rightarrow \left\{ {\begin{array}{*{20}{l}} {A = 1}\\ {B = \frac{1}{M}} \end{array}} \right..$

Then the integral in the left side is

$\int {\frac{{dN}}{{N\left( {1 - \frac{N}{M}} \right)}}} = \int {\left( {\frac{1}{N} + \frac{{\frac{1}{M}}}{{1 - \frac{N}{M}}}} \right)dN} = \int {\frac{{dN}}{N}} + \int {\frac{{d\left( {\frac{N}{M}} \right)}}{{1 - \frac{N}{M}}}} = \ln \left| N \right| - \ln \left| {1 - \frac{N}{M}} \right| = \ln \left| {\frac{N}{{1 - \frac{N}{M}}}} \right| = \ln \frac{N}{{1 - \frac{N}{M}}}.$

Thus, the general solution of the logistic differential equation is given by

$\ln \frac{N}{{1 - \frac{N}{M}}} = at + \ln C,\;\; \Rightarrow \ln \frac{N}{{1 - \frac{N}{M}}} = \ln {e^{at}} + \ln C,\;\; \Rightarrow \ln \frac{N}{{1 - \frac{N}{M}}} = \ln C{e^{at}},\;\; \Rightarrow \frac{N}{{1 - \frac{N}{M}}} = C{e^{at}}.$

The last algebraic equation can be solved for $$N:$$

$N = C{e^{at}} - \frac{N}{M}C{e^{at}},\;\; \Rightarrow N\left( {1 + \frac{1}{M}C{e^{at}}} \right) = C{e^{at}},\;\; \Rightarrow N = \frac{{C{e^{at}}}}{{1 + \frac{1}{M}C{e^{at}}}} = \frac{{CM{e^{at}}}}{{M + C{e^{at}}}}.$

The constant $$C$$ can be determined from the initial condition $$N\left( {t = 0} \right) = {N_0},$$ so that

${N_0} = \frac{{CM \cdot 1}}{{M + C}},\;\; \Rightarrow CM = {N_0}M + C{N_0},\;\; \Rightarrow C = \frac{{{N_0}M}}{{M - {N_0}}}.$

Substituting this value for $$C$$ into the general solution, we obtain:

$N\left( t \right) = \frac{{\frac{{{N_0}{M^2}{e^{at}}}}{{M - {N_0}}}}}{{M + \frac{{{N_0}M{e^{at}}}}{{M - {N_0}}}}} = \frac{{{N_0}{M^2}{e^{at}}}}{{{M^2} - {N_0}M + {N_0}M{e^{at}}}} = \frac{{{N_0}M{e^{at}}}}{{M - {N_0} + {N_0}{e^{at}}}} = \frac{{{N_0}M}}{{{N_0} + \left( {M - {N_0}} \right){e^{ - at}}}}.$

The graph of the logistic function (see above) has a nice view. Figure $$2$$ shows a few logistic curves at different values of $${N_0},$$ and Figure $$3$$ shows how the shape of the curve changes depending on the growth rate $$a.$$

We see that the family of logistic curves on the segment $$t \gt 0$$ can describe nonlinear population growth with saturation, when the maximum allowed value has a limit.

## Hyperbolic Growth Models

The models we considered above are useful in the analysis of demographic processes on a scale of centuries. If consider population growth for several thousand years (Figure $$4$$), it can be seen that the main explosive growth from $$2$$ to $$7$$ billion people occured on the past $$50$$ years. This type of dependency is similar to the hyperbolic curve. A simple hyperbolic growth model was suggested by several researchers (von Forster $$\left( {1960} \right),$$ von Hoster $$\left( {1975} \right),$$ and Shklovskii $$\left( {1980} \right)$$) in the following form:

$N\left( t \right) = \frac{C}{{{T_1} - t}} = \frac{{200}}{{2025 - t}}\,\left( \text{bln.} \right)$

As it follows from this model the world population goes off to infinity as the year $$2025$$ approaches.

However, the real growth dynamics demonstrates that the so-called demographic transition follows after the explosive growth phase. This new state is characterized by declining fertility and mortality. Such a transition has already occurred in many developed countries. As a result of the demographic transition, the population growth ceases and may even fall. The global world population had just entered the phase of demographic transition in the beginning of $$21$$st century.

It turns out that such a complex population dynamics can be also well described using differential equations! A model of this type was recently (in $$1997$$) developed by Russian physicist Sergey Kapitsa. Kapitsa proposed to describe the explosive growth using the following equation:

$\frac{{dN}}{{dt}} = \frac{C}{{{{\left( {{T_0} - t} \right)}^2} + {\tau ^2}}},$

where $${{T_0}}, C$$ and $$\tau$$ are certain approximation parameters. This differential equation has the exact solution as a function:

$N\left( t \right) = \frac{C}{\tau }\text{arccot}\, \frac{{{T_0} - t}}{\tau }.$

The given function describes the explosive population growth remarkably well at the following values of the parameters: $$C = 1.86 \times {10^{11}},$$ $${T_0} = 2007,$$ $$\tau = 42.$$ Besides that, the model covers the demographic transition phase when the population growth reaches saturation (Figure $$4$$).

According to this model, the global world population will reach about $$12$$ billion in $$2200-2300.$$