You have recently been hired by Rigorous Mathematical Contractors (RiMaC), Inc., a mathematical contracting firm based in Lonlinc, Skanebra which is entering its second year of operation. In light of the success attained in its first year, RiMaC has started hiring and is considering a public stock offering. To boost this, the latest contract to coming pouring in is from EcoSystems, Inc., a Leseatt, Tonwashing based fish-farming company---and has been assigned to you.
31 January 1997
Rigorous Mathematical Contractors (RiMaC), Inc.
Suite 3, Strawmarket Business Plaza
Lonlinc, SK 04685
As you may know, EcoSystems, Inc., has in the past year expanded its immensely successful fish farms along the scenic Tonwathing coast near Leseatt to include a large lake near Lonlinc---therefore allowing us to also supply Lonlinc and surrounds with a needed and ecologically sound food source supplied fresh daily by our exclusively electric fleet of bright blue delivery vans. The success of this endeavor we credit primarily to a careful mathematical analysis carried out by your company last year.
In an effort to keep up with the demand that we have discovered in the Lonlinc area we are beginning a second farm near Haoma. As is our practice, we first seeking a theoretical basis for the initial stocking of the lake. During this phase of our operations we are stocking the lake with fish at regular intervals. We have found in the past that because of the disorientation of the fish being introduced to a new environment and the nature of the process of their introduction, the population of fish in the lake is changed little by breeding and must therefore be built up by sequentially adding more fish. Once a sufficient fish population exists in the lake the farm will become self sustaining.
We will be able to introduce fish in the new lake at a rate of 300 fish/2 months. During the time during which they are being introduced, however, the population decreases (due to the imposition of predators) at a rate that is for small populations proportional to the number of fish present (with an experimentally determined constant of proportionality k1 = 4x10-4 fish/day/fish) and for larger ones proportional to the square of the fish population (with constant of proportionality k2 = 1x10-5 fish/day/fish2). Initially, of course, there are no fish in the lake.
Given this information, we need from your company an analysis of how the fish population in the lake will increase over time, and the length of time it will take for the population to reach between 500 and 600 fish, at which point the farm should be self-sustaining.
To assist you in this project, we have arranged for you to be able to contact a local expert whom we met here in Leseatt last summer, Dr. Gavin LaRose, with any technical questions you might have. You must in any event contact him with a report of at least preliminary work by the 6th of February and again by or on the 13th of February. Your final 3--5 page typewritten report1 is due by or on the 28th of February, as we need it in time to begin stocking the lake before the Spring.
"Chuck" R.D. Arwin
President, EcoSystems, Inc.
last modified on 20 Jan 1997Gavin's Differential Equations Project 1, Spring 1997