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MTH 256 Supplement to Judson's “The Ordinary Differential Equations Project”

David Froemke, Heiko Spoddeck

Section 1.3 Geometric and Quantitative Analysis

Exercises Exercises

Supplemental HW Problems.

1.
My koi pond contains a population of fish that grows over time. I find that the growth rate of my fish is proportional to the product of how many fish are currently in my pond, and the difference between the number of fish and \(200\text{.}\) If the growth constant of my koi is \(0.005\text{,}\) that gives the following differential equation for the fish growth:
\begin{equation*} P’ = 0.005P(200-P) \end{equation*}
Where \(P(t)\) is the population of fish in my pond during month \(t\text{.}\)
  1. If I start with a population of \(40\) fish, do I expect my population of fish to grow, shrink, or be stable?
  2. Answer the same question above for the following initial amounts of fish: \(10, 100, 180, 200, 250, 1000\text{?}\)
  3. Over time, what is the long-term amount of fish I expect to have in my pond?
2.
I begin to sell my koi once they’re big enough, harvesting \(20\) fish per month. Answer the same questions as above. Note that part (c) might depend on the initial number of fish!
3.
My fishpond workers mis-read my instructions and harvest \(60\) fish per month by accident. Repeat the analysis from question 1 under these conditions.
4.
I get that sorted out, but my next worker also mis-reads my instructions and harvests \(20\%\) of the total population of fish each month. Repeat the analysis from question 1 under these conditions.
5.
My business is so good that now I actually am finding former customers with too many fish on their hands, so I am adding an additional \(50\) fish per month. Because of this, I up my harvest to \(50\%\) of the total number of fish I have in the pond. Perform the same analysis as in question 1 for these conditions.
6.
I change my fish food formulation, and it changes the growth rate of my fish. My new differential equation that models my fish population is:
\begin{equation*} P’ = 0.008P^2(P-200)(100-P) \end{equation*}
Are there any initial conditions under this new model that lead to extinction (\(0\) fish, over time)? That lead to doomsday (infinite fish, over time)? That lead to stability? Identify those conditions, and what the stability solutions are (the amount of fish the pond tends to over time) if any of those exist.
7.
Finally, I get a new set of pumps and filters for my pond. Now, my model is:
\begin{equation*} P’ = 0.005P(P-200)(P-100) \end{equation*}
Given this model, perform the same analysis as in the exercise above.
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