In this video I go over an introduction to differential equations and explain a bit about how they are the most important applications of calculus to the real-world. Differential equations are simply equations that consist of a function and some of its derivatives. These types of equations model real-world applications very well because often times physical concepts change at a rate that is proportional to its current state.
One such example is in population growth in which the rate of growth depends on what the current population is. In this video I use differential equations to model population growth. The first differential equation for population growth that I go over is for ideal conditions and is simply stated as the rate of growth is proportional to the current population. But a more accurate model assumes that there is a maximum carrying capacity in which the population levels off. This latter model is known as the Logistic Differential Equation and was first proposed by the Dutch mathematical biologist Pierre-François Verhulst in the 1840s. I also go over a brief history lesson on Verhulst.
This is an extremely important video as it lays the foundation for my later videos on one of the most powerful mathematical concepts, differential equations, so make sure to watch this video!
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Modeling with Differential Equations: Population Growth
Perhaps the most important of all the applications of calculus is to differential equations.
When physical scientists or social scientists use calculus, more often than not it is to analyze a differential equation that has arisen in the process of modeling some phenomenon that they are studying.
Although it is often impossible to find an explicit formula for the solution of a differential equation, we will see that graphical and numerical approaches provide the needed information.
The process of modeling involves formulating a mathematical model of a real-world problem either through intuitive reasoning about the phenomenon or from a physical law based on evidence from experiments.
The mathematical model often takes the form of a differential equation, that is, an equation that contains an unknown function and some of its derivatives.
This is not surprising because in a real-world problem we often notice that changes occur and want to predict future behavior on the basis of how current values change.
One example of how differential equations arise when we model physical phenomena is in population growth.
Models of Population Growth
One model for the growth of a population is based on the assumption that the population grows at a rate proportional to the size of the population.
That is a reasonable assumption for a population of bacteria or animals under ideal conditions.
- Conditions such as unlimited environment, adequate nutrition, absence of predators, immunity from disease, etc.
Let's identify and name the variables in this model:
t = time (the independent variable)
P = population (the dependent variable)
The rate of growth of the population is the derivative dP/dt.
So our assumption that the rate of growth of the population is proportional to the population size is written as the equation:
Where:
- k is the proportionality constant.
This equation is our first model of population growth.
It is a differential equation because it contains an unknown function P and its derivative dP/dt.
Having formulated a model, let's look at its consequences.
If we rule out a population of 0, then P(t) > 0 for all t.
So, if k > 0, then this equation shows that P'(t) > 0 for all t.
This means that the population is always increasing.
In fact, as P(t) increases, the equation shows that dP/dt becomes larger.
In other words, the growth rate increases as the population increases.
Let's try to think of a solution of the equation.
This equation asks us to find a function whose derivative is a constant multiple of itself.
We know that exponential functions have that property.
In fact, if we let P(t) = Cekt, then:
Thus, any exponential function of the form P(t) = Cekt is a solution to our differential equation.
- In later videos I will show how there is in fact no other solution but the exponential function.
Allowing C to vary through all the real numbers, we get the family of solutions P(t) = Cekt whose graphs are shown in the figure below:
But populations have only positive values so we are interested only in the solutions of C > 0.
And we are probably concerned only with values of t greater than the initial time t = 0.
The figure below shows the physically meaningful solutions.
Putting t = 0 we get P(0) = Cek(0) = C.
Thus the constant C turns out to be the initial population, P(0).
The differential equation we used is appropriate for modeling population growth under ideal conditions, but we have to recognize that a more realistic model must reflect the fact that a given environment has limited resources.
Many populations start by increasing in an exponential manner, but the population levels off when it approaches the carry capacity K (or decreases toward K if it ever exceeds K).
For a model to take into account both trends, we make two assumptions:
(1) Initially, the growth rate is proportional to P.
(2) P decreases if it ever exceeds K.
A simple expression that incorporates both assumptions is given by the equation:
Notice that if P is small compared with K, then P/K is close to 0 and so dP/dt ≈ kP.
If P > K, then 1 - P/K is negative and so dP/dt < 0.
Brief Mainstream History
This equation is called the logistic differential equation and was proposed by the Dutch mathematical biologist Pierre-Francois Verhulst in the 1840s as a model for world population growth.
- Pierre was born in Brussels, Belgium and lived from 1804 to 1849.
- He was a mathematician and a doctor in number theory from the University of Ghent in 1825.
- He published the logistic equation in 1838 and its solution in 1845 but it was not until 1920 when this model was rediscovered by American biological researchers Raymond Pearl and Lowell Reed, who promoted its use.
I will go over techniques that enable us to find explicit solutions of the logistic equation in my later videos so stay tuned for those!
But for now, we can deduce qualitative characteristics of the solutions directly from the logistic equation.
We first observe that the constant functions P(t) = 0 and P(t) = K are solutions because, in either case, one of the factors on the right side of the logistic equation is zero.
- This certainly make physical sense:
- If the population is ever either 0 or at the carrying capacity, it stays that way
- These two constant solutions are called equilibrium solutions.
If the initial population P(0) lies between 0 and K, then the right side of the logistic equation is positive, so dP/dt > 0 and the population increases.
But if the population exceeds the carrying capacity (P > K), then 1 - P/K is negative, so dP/dt < 0 and the population decreases.
Notice that, in either case, if the population approaches the carrying capacity (P → K), then dP/dt → 0, which means the population levels off.
So we expect that the solutions of the logistic differential equation have graphs that look something like the ones in the figure below:
Notice that the graphs move away from the equilibrium solution P = 0 and move toward the equilibrium solution P = K.