5. Growth – checked and not

[status: just starting]

5.1. Motivation, prerequisites, plan

5.1.1. Motivation

I am preparing this mini-course on 2020-03-12 as New Mexico has just entered a state of emergency over the coronavirus epidemic. It seems topical and a nice opportunity to introduce our students to some aspects of the mathematical equations that describe growth of species.

5.1.2. Prerequisites

The 10-hour “serious programming” course.
The “Data files and first plots” mini-course in Section 2
The “A tour of functions” mini-course in Section 4

5.1.3. Plan

There is an attractive and accessible video at:

https://www.youtube.com/watch?v=Kas0tIxDvrg

and:

https://www.youtube.com/watch?v=PUwmA3Q0_OE

which students can watch before the mini-course.

Then we go on to show how difference equations can lead to exponential growth for an unchecked population.

Then on to the logistic equation and how exponential growth actually works in the real world.

Finally a discussion of predator-prey models.

5.2. Pure exponential growth

Discus Malthus, what “malthusian” means.

Then we review what the exponential function looks like compared to this progression.

At the python3 interpreter type:

P0 = 10
rate = 1.2

P = P0
print('## generation population')
for generation in range(100):
    P = P + P * (rate-1)
    print(generation, '    ', P)

Now put that code in a file and run the program, saving its output so that we can plot it.

Now compare that to what would have come out of a straight exponential function \(f(x) = P0 \times e^{rt}\). Can you make the rate and initial population correspond between the difference equation?

https://en.wikipedia.org/wiki/Population_growth

\(dP/dt = rP(1 - P/K)\)

or

\(df(x)/dx = f(x) (1 - f(x))\)

Solution is:

f(x) = exp(x) / (exp(x) + C)

discuss the exponential phase initially, then the cooling down phase. talk about fidget spinner fads.

P = P0

K = 3.7 is the “carrying capacity”

https://en.wikipedia.org/wiki/Logistic_function

limit t -> infinity P(t) = K

From wikipedia: “In ecology, species are sometimes referred to as r {displaystyle r} r-strategist or K {displaystyle K} K-strategist depending upon the selective processes that have shaped their life history strategies.”

5.3. Checked growth

5.3.1. Checked by lack of resources

5.3.2. Checked by competition with other species

5.4. Simple predator-prey interactions

5.4.1. The Lotka-Volterra equations

The relationship between populations of predators and prey can be expressed by the Lotka-Volterra equations, a set of paired differential equations that approximate the interaction between predator and prey populations over many generations.

The Lotka-Volterra equations are commonly expressed as two paired differential equations, where \(h\) is the population of prey and \(p\) is the population of predators. Notice the parralels between the first and second terms of each equation.

\[ \begin{align}\begin{aligned}\frac{dh}{dt} = ah - bhp\\\frac{dp}{dt} = -cp + dhp\end{aligned}\end{align} \]

\(a\), \(b\), \(c\), and \(d\) are variables which dictate the nature of the interaction. In the case of the prey, \(a\) represents the growth of the prey population. Since it is multiplied by the current prey population \(h\), the population growth of the prey is exponential at its base, excepting the second term. Conversely, \(c\) represents the exponential decay of the predator population in the absence of any prey to eat.

The second term of each equation reflects how often predators and prey encounter each other and the effect these encounters have on their respective populations. Before we look into what \(hp\) is doing in the second term of both differentials, it’s important to understand what the general purpose of the second term is in each case.

For the prey, the exponential population growth from the first term is checked by predation. In the case of the predator population, the exponential decay they experience without external input is curbed by the same force of predation. This is the basis of why both differentials include the term \(hp\).

However, this makes more sense when we look more closely at what \(hp\) means in the context of this equation. By multiplying the populations of the predators and prey, we can find the dimension of the populations. On the surface this seems like a nonsensical calculation, but it is integral to the function of the Lotka Volterra equations.

If there are many more prey than predators (a large dimension), it is easy for predators to find something to eat. Similarly, we also notice a large dimension if there are mant more predators than prey. In this case, it is easy for prey to get caught by a predator. The dimension of the populations reflects the number of interactions we can expect between predators and prey, but it doesn’t necessarily tell us about which side benefits most.

We can only get this information by looking back at the differential equations. b and d are variables which affect how strongly the dimension of the populations effect each population.