Growth & Decay of Fish in Lake Ontario
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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 8
Citations ………………………..…….………………… Page 9
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 . What we will prove is that this systemis unstable.
The values of and individually do not necessarily matter when considering the stability, and behavior of this system. However, the product of multiplied by are extremely crucial toward molding the behavior and stability of this system. Since the birth rates of each population of this system are, as we will prove, always going to be positive, the only types of behavior warranted are source, spiral source, and saddle; all of which are inherently unstable. The real choice to be made regarding the values of and for this system are whether we wish for the population of the fish to either continuously grow, decrease until zero, or a varying of the two that will eventually lead to continuous growth.
We could form a stable system, where each population hovers around a specified value, if we could control more than just the constants and. If we were able to somehow manipulate the birth rates of population and population then we would be able to form a stable environment. However, with the given conditions we will show the possible outcomes depending on the different values of and as well as the product of .
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. If we assume and are both positive values then and . If both values are assumed to be negative then and. 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. If we assume that both and are positive values then: and. If is negative and is positive, then and. If is assumed negative and is positive, then and . However if and are both then their values are and .If 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. If we assume that is the negative value then and. If we assume the opposite is true then and.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. These eigenvalues are found by subtracting an identity matrix multiplied by the eigenvalues from the given matrix, and then finding the determinant of the newly made matrix. Once, we have found the determinant, we set it equal to zero and find the corresponding eigenvalues. We then used those eigenvalues to find eigenvectors, by inserting them into the matrix, and reducing the matrix to reduced row echelon form. For the eigenvalue, we found the eigenvector to equal:. For the second eigenvalue, , we found the corresponding eigenvector to equal: .
These eigenvalues and eigenvectors caused the general solution to this system to be,
(2)
where is the exponential function and and are undetermined constants.
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 between and . 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:
No matter the values we manipulate and to equal, the behavior of the solution to this system is always going to be unstable. In order to sustain any stability to this system, our model would have to be less simplistic. Realistically the only two factors affecting the population of these two species of fish are not just the birth rate and the interactions between the two species represented by constants and . Their populations are affected by other predator and prey species within Lake Ontario, overpopulation from their own species, as well as by human harvesting, such as fishing. Through our manipulations, we would prefer that the populations of each fish increased to a certain value and then just hovered around that value. However because of our model, we can only vary and so our solution will either cause continual growth, or a decrease in population until it is zero. Both of these behaviors are extremely unlikely, and very nearly impossible to occur in nature. Our model, and our solution fail because no matter how we manipulate the system using constants and , the system will always be unstable.
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.