Systems Of Differential Equations

The slideshow presentation for my senior project to complete my the requirements for my degree in mathematics.

1. Systems of Differential Equations Presentation by: Joshua Dagenais

2. Systems of Differential Equations System involving several dependent variables an independent variable ( t), and rates of change of the dependent variables

3. Solutions to the System Solve for a 1 x 1 system Solve systems second order or higher

4. Solutions to the System Substitute back into original equation Theorem: Let A be an n x n matrix of constant real numbers and let X be an n-dimensional column vector. The system of equations has nontrivial solutions, that is, , if and only if the determinant of A is zero.

5. Solutions to the System Solving the determinant gives the characteristic eqn. The roots , r , are the Eigenvalues . Eigenvalues are used to solve for the associated Eigenvectors, , and the specific solutions Specific solutions as a general solution

6. Eigenvalues (Real & Distinct) Eigenvalues are of opposite signs, the origin is a saddle point, and trajectories are asymptotic to the Eigenvectors Eigenvalues are of the same signs, the the origin is a node, and trajectories converge to origin if Eigenvalues are negative and vice versa if positive

7. Eigenvalues (Complex) Eigenvalues are complex with a nonzero real point ( a + bi ) Use one of Euler’s Formulas to find real-valued solutions The origin is called a spiral point and trajectories converge to origin if Eigenvalues are negative and vice versa if positive

8. Eigenvalues (Repeated) Eigenvalues are real and repeated with multiplicity Use to solve for the specific solution of second repeated Eigenvalue The origin is called a improper node and trajectories converge to origin if Eigenvalues are negative and vice versa if positive

9. Application Predator-Prey Model also known as the Lokta-Volterra Model Part of mathematical ecology that studies populations that interact, thereby affecting each other's growth rates Model represents the "natural" growth rate and the "carrying capacity" of the environment (predators & prey)

10. Predator-Prey Model Few interactions have been recorded in nature One such set of data was taken between the Snowshoe Hare and the Canadian Lynx for almost 100 years The dominating feature is the oscillation behavior of both populations

11. Predator-Prey Assumptions x(t) will represent the number of prey at a time given by t and y(t) will represent the number of predators at a time also given by t . In the absence of the predator, the prey grows at a rate proportional to the current population; thus In the absence of the prey, the predator dies out; thus

12. Predator-Prey Assumptions The number of encounters between predator and prey is proportional to the product of their populations The growth rate of the predator is increased by a term of the form bxy , while the growth rate of the prey is decreased by a term –pxy Critical points (when ) are

13. Predator-Prey Example

14. Predator-Prey Example For The predator (green) population lags behind the prey (blue) Population for both populations are periodic (in this case about a periodicity of t=11 )

15. Predator-Prey Example Population of predators vs. prey as for Prey first increase because of small population Predators increase because of abundance of food Heavier predation causes prey to decrease Predators decrease because of diminished food supply Cycle repeats itself

16. Predator-Prey Example Analysis of the Nonzero Critical Point

17. Predator-Prey Example Analysis of the Nonzero Critical Point

18. Model with Hunters Introduced h is the effect of hunting and killing a constant amount of predators every cycle Extinction eventually occurs