Growth & Decay of Fish in Lake Ontario

This report is our work. Any help, insight, or aid that we have received is explicitly acknowledged. To do otherwise is plagiarism, and we understand that we will receive a zero on this report if it is found that we did not adequately acknowledge the contribution of anybody else.

MA232, Discussion Section 17Due: November 21, 2013

Shane Garrabrant Date: .

Zachary Rauen Date: .

Allen Perry _____ Date: .

TABLE of CONTENTS

Title ..…………………………………………..….……..Page 1

Signatures ………………….….……………….………Page 1

Table of Contents …………..……………..….……Page 2

Introduction …………………………..………………Page 3

Determining the Ranges …….…………….…….Page 4

Saddle ……………………………………………………. Page 5

Source ……………………………………………………. Page 5

Spiral Source …………………….……………………. Page 5

Solutions …………………….…….……………………. Page 6

Conclusions …………….…………………….……..… Page 7

Citations ……………………………….………………… Page 8

Introduction:

Many species of fish live in the great lakes with many species making their lives in Lake Ontario. Every species of fish interacts with one another in separate ways. These interactions between species combined with the birth rate of the fish can help depict the population of the fish. Without knowing how the fish interact we can model the system as,

(1)

where and are unknown constants representing the interactions between population and population . From the information in equation 1 we can determine ranges of values for and including what each range means for the interaction between populations and .

Determining the Ranges:


In order to determine the ranges possible for constants and it’s necessary to check the matrix’s values against the trace-determinant plane. After finding the trace and variable determinant of the matrix in equation 1 the following figure, figure 1, can be produced.


From observation of figure 1, the system has the possibility of being a ‘saddle,’ a ‘source,’ or a ‘spiral source.’ After some simple calculations, we get that the system will be a saddle when a source when and a spiral source when

Saddle:

Knowing that a saddle occurs when , we can then conclude that and must both be negative or both be positive. This occurs because 0.07 is positive and we are looking for a value that is greater than that. This gives a range of size infinite for both and where neither of them can equal zero. Interpreting this particular outcome is as easy as observing the possible values of and. For instance, if both and are positive then that means the two populations of fish are both predators of some outside fish. The magnitude of and represents the amount of growth they experience. On the other hand, if both values are negative that means they are both prey of outside fish. The magnitude in this case would represent the amount of decay they experience. In either saddle case this system is unstable.

Source:

For a source we desire to be true. Similar to saddle there is an infinite range except this time we include zero as a possible value. For a source you can have any combination of positive and negative numbers. For both negative numbers, the fish populations are both prey to outside fish and the magnitudes indicate their decay. The opposite happens for both positive numbers; the magnitude indicates growth and the species are both predators of outside fish. When is negative and is positive population is the prey of population . The magnitudes indicate decay and growth respectively. This situation can flip around causing to be the predator of population meaning is negative and is positive. This system is also unstable.

Spiral Source:

In a spiral source situation that means that . For this situation one value has to be negative. This indicates that one population is prey while the other is the predator. For a negative value the population ends up as the prey and when is negative the population is the prey. This system is quite unstable.

Solutions:

The solutions to the original equation, equation 1, are the eigenvaluesand . Where and are the original constants from the equation. Looking at the first eigenvalue we would get an imaginary number if was less than zero. Solving that would give that when is less than there is an imaginary number. When there is a complex number it means it will be a spiral solution and the only possible option is the spiral source, which is exactly what was predicted.

Testing both eigenvalues with values for we find that both eigenvalues are positive only for the range betweenand . Having both eigenvalues positive is the condition for a source solution, which is what was predicted for this exact range.

After , the first eigenvalue goes negative forever and the other goes positive forever. Having one value positive and one negative is the qualification for a saddle solution. This is the same as our predicted condition.

All of these value possibilities give unstable solutions meaning that the original system given in equation 1 is unstable.

Conclusions:

Citations

Black, Kelly, Guangming Yao. "Differential Equations." MA232. Clarkson University.Science Center, Potsdam.Class lecture.

Stewart, James.Essential calculus: early transcendentals. Belmont, CA: Thomson Higher Education, 2007. Print.