Review and Threshold Population

In Lab $5$ we learned about exponential growth. That is when the rate of change of a population is directly proportional to

the current size of the population $($whether that be of humans, algae, or even pickles$)$:

dP

dt $$ P $($t$)$

We found that this resulted in the following differential equation:

dP

dt $=$ rP $($t$)$

In Lab $5$ and Assignment $5$ we used this relationship to solve problems using this equation. In Lab $7$ we expanded on this

idea to include a carrying capacity K$,$ some value which represented the maximum population that can be sustained based

on environmental limitations$.$ This makes sense because there isn$$t anything that can grow exponentially forever because we

have a limited amount of space and resources in all natural systems. This gave us the following differential equation:

dP

dt $=$ rP $($t$)$

$($

$1$ P $($t$)$

K

$)$

Here we will introduce another concept known as a threshold population T $,$ that is the minimum population necessary for a

species to survive. Once again this makes sense. For sexually reproducing species to survive we need at absolute minimum

of two individuals $($you know where I am going here$)$ but even with only two individuals the lack of genetic diversity and

inbreeding will quickly result in the species dying off. Thus, we introduce a threshold population and arrive at the following

differential equation:

dP

dt $=$rP $($t$)$

$($

$1$ P $($t$)$

K

$)($

$1$ P $($t$)$

T

$)$

The Pickle Farmers of Ontario

You recently acquired a cucumber farm under totally legitimate circumstances and you are excited because pickles are your

favourite whether they be bread and butter pickles, baby dills, or even the ones with extra garlic. You relish this new

opportunity but also find yourself missing your differential equations class. Though you have an opportunity to merge your

interests. A survey of the farm determines that the maximum number of cucumber plants that you can sustain is $50$ and

that you need a minimum of $5$ cucumber plants to sustainably maintain a population of cucumbers long term.

First, draw a direction field of the possible solutions to this differential equation, carefully showing where solutions are in$-$

creasing and decreasing and calculating curvature. Assume that r $>0.$

Second, solve the differential equation providing a general solution implicitly defining the relationship between P and t$.$

Hint: for the first part follow along with the direction field examples from class. After taking the derivative df

dP use the

quadratic formula to find the roots, then determine the curvature. For the second part follow along with the notes from lab

on solving the differential equation that just includes carrying capacity. Go slowly with the algebra. Take your time and

think through the problem. You got this!