Differential Equations

1st Order Equations

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 \(1\)).

Exponential and actual growth of population

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.

Logistic model of population growth for different initial population levels

Figure 2.

Logistic model of population growth for different growth rates

Figure 3.

The right side of this equation can be presented 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 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:

\[{N\left( t \right) }={ \frac{C}{\tau }\text{arccot}\, \frac{{{T_0} – t}}{\tau }.}\]
Hyperbolic model of population growth

Figure 4.

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 (see Figure \(4\)). According to this model, the global world population will reach about \(12\) billion in \(2200-2300.\)