# Differential Equations

## First 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 $$1$$).

### 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 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 $$2,3$$). Figure 2. Figure 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:

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

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 $$2200-2300.$$