Multivariate Calculus
&
Ordinary Differential Equations
Lecture 7
Population, 1950 - 2022
Source: ourworldindata.org
Population Growth
Source: ourworldindata.org
1 Ordinary Differential Equations
1.5.2 Malthus: "... doomed to misery and vice"
Without inhibiting factors the rate at which the human population grows is proportional to its existing population; if $P$ is the population at time $t$ then $$\dif{P}{t}=rP,$$ where $r$ is the growth rate (per human per year, say).
1 Ordinary Differential Equations
1.5.2 Malthus: "... doomed to misery and vice"
Without inhibiting factors the rate at which the human population grows is proportional to its existing population; if $P$ is the population at time $t$ then $$\dif{P}{t}=rP,$$ where $r$ is the growth rate (per human per year, say).
The world population is currently growing at a rate of approximately $r=0.0125$. If we take as initial condition the year 2000 with roughly 6 billion people on the planet, and ignore global warming so that we may take the total area of land on Earth to be constant at 150 million square kilometres, find in which year each human will have, on average, one square metre of land left to stand on.
1 Ordinary Differential Equations
1.5.2 Malthus: "... doomed to misery and vice"
$\ds \dif{P}{t}=rP,$
The world population is currently growing at a rate of approximately $r=0.0125$. If we take as initial condition the year 2000 with roughly 6 billion people on the planet, and ignore global warming so that we may take the total area of land on Earth to be constant at 150 million square kilometres, find in which year each human will have, on average, one square metre of land left to stand on.
Here we have $\, \dfrac{dP}{dt}= 0.0125P,\,$ $\, P(2000)=6,\,$ $t$ is in years and $P$ is in billions.
1 Ordinary Differential Equations
1.5.2 Malthus: "... doomed to misery and vice"
[...] If we take as initial condition the year 2000 with roughly 6 billion people on the planet, [...]
Since $\, \dfrac{dP}{dt}= 0.0125P,\,$ then $\, \ds \int\frac{dP}{P} =\int 0.0125~dt \,$
$\Ra \ln \abs{P} = 0.0125 t + C$ $\;\,\Ra\, P(t) =Ae^{0.0125t}$
We need $\,P(2000) = 6$ $\;\,\Ra\, 6 = A e^{0.0125(2000)}$
$\Ra A = 6e^{-0.0125(2000)}$
$\Ra P(t) = 6e^{-0.0125(2000)} e^{0.0125t}$ $= 6 e^{0.0125(t-2000)}$
1 Ordinary Differential Equations
1.5.2 Malthus: "... doomed to misery and vice"
[...] and ignore global warming so that we may take the total area of land on Earth to be constant at 150 million square kilometres, find in which year each human will have, on average, one square metre of land left to stand on.
👉 $\; P(t) = 6 e^{0.0125(t-2000)}$
Now we want to find the year in which there will be 150,000 billion people.
So we have to solve for $t$:
$\qquad \qquad 150,000 = 6 e^{0.0125(t-2000)}$
$\qquad \qquad \qquad \;\,\Ra t\approx 2810.$
1 Ordinary Differential Equations
1.5.3 Verhulst to the rescue; down with the Malthusian law
The problem with the Malthusian population model is that it does not contain a "damping" factor reflecting issues such as over-population, limited food or water supply, etc. The logistic model of Verhulst is the simplest model that has such built-in damping:
$\ds \dif{P}{t}=rP\Bigl(1-\frac{P}{\theta}\Bigr),$
where $\theta$ is some very large constant, known as the carrying capacity.
1 Ordinary Differential Equations
1.5.3 Verhulst to the rescue; down with the Malthusian law
👉 $\ds \dif{P}{t}=rP\Bigl(1-\frac{P}{\theta}\Bigr)$
Note that when $P$ is small(ish) then $P/\theta$ is small and we approximately have
$\ds \dif{P}{t}\approx rP$
as before. When the population becomes close to the carrying capacity (i.e., $P/\theta$ very close to 1 so that not just $1-P/\theta\approx 0$ but also $P(1-P/\theta)\approx 0$) then growth comes almost to a standstill.
1 Ordinary Differential Equations
1.5.3 Verhulst to the rescue; down with the Malthusian law
👉 $\ds \dif{P}{t}=rP\Bigl(1-\frac{P}{\theta}\Bigr)$
If we introduce new variables $y=P/\theta$ and $\tau=rt,$ then the ODE becomes
$\ds\frac{\dup y}{\dup \tau} = y(1-y).$
We have already seen that if $\,y(0)=y_0$ then this has solution
$\ds y = \frac{y_0 e^\tau}{1-y_0+y_0e^\tau}.$
1 Ordinary Differential Equations
1.5.3 Verhulst to the rescue; down with the Malthusian law
👉 $\ds \dif{P}{t}=rP\Bigl(1-\frac{P}{\theta}\Bigr)\; \;$ and $\;\;\ds\frac{\dup y}{\dup \tau} = y(1-y).$
We have already seen that if $\,y(0)=y_0$ then this has solution
$\ds y = \frac{y_0 e^\tau}{1-y_0+y_0e^\tau}.$
If $P(0)=P_0$ in our original variables then we have
$\ds P = \frac{\theta P_0 e^{rt}}{\theta-P_0+P_0e^{rt}}.$
1 Ordinary Differential Equations
1.5.3 Verhulst to the rescue; down with the Malthusian law
Example: Identify the equilibrium (or singular) solutions to the logistic population model, and interpret these solutions in terms of population growth. Include a diagram.
$\ds\frac{dP}{dt} = rP\left(1-\frac{P}{\theta}\right)$ Then equilibrium solution: $P=0, \theta$
1 Ordinary Differential Equations
$\ds\frac{dP}{dt} = rP\left(1-\frac{P}{\theta}\right)$
1 Ordinary Differential Equations
1.5.3 Verhulst to the rescue; down with the Malthusian law
Example: The Pacific Halibut Fishery uses a logistic model plus an extra term to take into account their harvesting, which is at a rate proportional to the existing population.
If $E$ is the constant of proportionality for harvesting, their model is
$\ds \dif{P}{t} = rP\Bigl(1-\frac{P}{\theta}\Bigr)-EP.$
📝 = Exercise!
1 Ordinary Differential Equations
1.5.4 Extra reading: numerically solving ODEs (Not assessable! 😃)
Matlab has some more sophisticated numerical methods for solving ODEs than Euler's method. Matlab's ode45 uses a Runge-Kutta fourth-order integration technique. To use it you need to write two little programs. The first one, say logistic.m, defines the differential equation
function dy = logistic(t, y)
dy = y * (1 - y);
The second program will numerically solve the ODE and plot your solution.
y0 = 0.1;
tfinal = 6;
[t, y] = ode45('logistic', tfinal, y0);
plot(t, y);
1 Ordinary Differential Equations
By the way, you can also write in the MATLAB live editor:
logistic = @(t, y) y * (1 - y); % Anonymous func.
y0 = 0.1;
tspan = [0, 6];
[t, y] = ode45(logistic, tspan, y0);
plot(t, y);
1 Ordinary Differential Equations
1.5.5 Main points
- You should be able to apply Newton's law of cooling and solve the resulting IVP.
- You should know how to model population growth and solve the resulting IVP.
- You should be able to interpret the obtained solutions.