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

1. Presentation by: Joshua Dagenais

2. Systems of Differential Equations
 System involving several dependent variables ( x1, x2 ,..., xn ),
an independent variable (t), and rates of change of the
dependent variables ( x1, x2 ,..., xn )
x1 p11 (t ) x1 ... p1n (t ) xn g1 (t )
 x P(t ) x g (t )
xn pn1 (t ) x1 ... pnn (t ) xn g n (t )
x1 p11 (t )  p1n (t ) x1 g1 (t )
x  P (t )   x  g (t ) 
xn pn1 (t )  pnn (t ) xn g n (t )

3. Solutions to the System
 Solve for a 1 x 1 system
dx pt
px x ce
dt
 Solve systems second order or higher
x1 p11 (t ) x1 ... p1n (t ) xn
1 r1t n rn t
 x1 e ,..., xn e
xn pn1 (t ) x1 ... pnn (t ) xn

4. Solutions to the System
rt
 Substitute x e back into original equation
r ert P(t ) ert ( P(t ) rI ) 0
 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, X 0 , if and only if the determinant of A is zero.
p11 (t ) r  p1n (t )
   0
pn1 (t )  pnn (t ) r

5. Solutions to the System
 Solving the determinant gives the characteristic eqn.
a0r n a1r n 1 ... an 1r an 0
 The roots , r, are the Eigenvalues r1 ,..., rn .
 Eigenvalues are used to solve for the associated
Eigenvectors, 1 ,..., n , and the specific solutions
n rnt
x(1) (t ) 1er1t ,..., x( n) (t ) e
 Specific solutions as a general solution
(1) (k ) (1) ( n)
x c1x (t )  ck x (t ) if W[ x ,..., x ] 0

6. Eigenvalues (Real & Distinct)
 Eigenvalues are of opposite signs,
the origin is a saddle point, and
trajectories are asymptotic to the
Eigenvectors
1 t 2
x(t ) c1 e c2 e 2t
2 1
 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 1 4t 3 1 2t
x(t ) e e
2 1 2 3

7. Eigenvalues (Complex)
 Eigenvalues are complex with a
nonzero real point (a+bi)
 Use one of Euler’s Formulas
e( i )t e t cos( t ) i e t sin( t )
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
cos(2t ) sin(2t )
x(t ) c1et c2et
sin(2t ) cos(2t ) sin(2t ) cos(2t )

8. Eigenvalues (Repeated)
 Eigenvalues are real and
repeated with multiplicity 2
 Use x(t ) tert ert 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 x(t ) c1
1
2
e6t c2
1
2
t
0
1
e6t
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
dx
ax, a 0, when y 0
dt
 In the absence of the prey, the predator dies out; thus
dy
cy, c 0, when x 0
dt

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
dx dy
ax bxy x(a by ) and cy pxy y( c px)
dt dt
 Critical points (when x(a by) 0 and y( c px) 0)
are (0,0) and (c / p, a / b)

13. Predator-Prey Example
dx
x 0.03 xy
dt
dy
0.4 y 0.01xy
dt
a 1, b 0.03
c 0.4, p 0.01
cp (40,100 3) 1 ln y 0.03 y 0.4 ln x 0.01 x C

14. Predator-Prey Example
 For
x(0) 15 and y(0) 15
 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 t for
x(0) 15 and y(0) 15
 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 c p (40,100 3)

17. Predator-Prey Example
Analysis of the Nonzero Critical Point c p (40,100 3)

18. Model with Hunters Introduced
dx
ax bxy
dt
dy
cy pxy h
dt
 h is the effect of
hunting and killing a
constant amount of
predators every cycle
 Extinction eventually
occurs