1. NONLINEAR SYSTEMS IN THREE DIMENSIONS
Courtney Hutton, Evan Lubin, Jake Chambers

2. Lotka-Volterra system in three dimensions
OVERVIEW
Lotka-Volterra system in three dimensions
Solutions to the general model
Nonlinear food chain model
Solutions to the nonlinear model
Conclusion

3. Lotka-Volterra in three dimensions Solutions to the general model
OVERVIEW
Lotka-Volterra in three dimensions
Solutions to the general model
Nonlinear food chain model
Solutions to nonlinear model
Conclusion
Ultimately, we will show that the behavior of the nonlinear system was effectively predicted by the generalized linear system, with one exception. To reach equilibrium the linear system must reduce to a two-dimensional system of equations, however, the nonlinear example illustrates that the species can coexist at equilibrium in three dimensions without reduction to the xy-plane.

4. General Equations in Three Dimensions1
LOTKA-VOLTERRA MODEL
General Equations in Three Dimensions

5. LOTKA-VOLTERRA MODEL GENERAL LINEAR EQUATIONS
x is the lowest level prey, y is a mid level predator, z is the apex predator

6. for the linear equation
GENERAL SOLUTION
for the linear equation

7. LOTKA-VOLTERRA MODEL EQUILIBRIUM SOLUTIONS
Everything is dead.
Flattens to a two dimensional system
Three cases associated with this solution, we will examine the first two
RIP EVERYTHING

8. LOTKA-VOLTERRA MODEL THE JACOBIAN MATRIX
x is the lowest level prey, y is a mid level predator, z is the apex predator

9. LOTKA-VOLTERRA MODEL THE TRIVIAL SOLUTION
x is the lowest level prey, y is a mid level predator, z is the apex predator

10. LOTKA-VOLTERRA MODEL THE INTERESTING SOLUTION
x is the lowest level prey, y is a mid level predator, z is the apex predator

11. LOTKA-VOLTERRA MODEL IGNORING THE IMAGINARY ANIMALS...
Three possible cases:
Stable curve
Unstable curve
Outside our scope

12. LOTKA-VOLTERRA MODEL FIRST CASE Solution for initial condition
The growth rate of the prey population exceeds the effect of predation
Basically, the top-level predator cannot support itself
Solution for initial condition

13. LOTKA-VOLTERRA MODEL FIRST CASE Solution for initial condition
All solutions spiral down towards the xy-plane
Solution for initial condition

14. LOTKA-VOLTERRA MODEL SECOND CASE Solution for initial condition
The predator population and the prey population tend towards infinity
The population of the mid-level species fluctuates over time with increasing amplitude
Solution for initial condition
in both cases, what happens to the predator is determined by the parameters relating to species x and z (species y is irrelevant)

15. Nonlinear Equations in Three Dimensions2
THE FOOD CHAIN MODEL
Nonlinear Equations in Three Dimensions

16. FOOD CHAIN MODEL

17. modeling a three species food chain
NONLINEAR SYSTEM
modeling a three species food chain

18. The system used to map this food chain model is:
...but how does this relate to the general system?

19. FOOD CHAIN MODEL THIS YIELDS THE NONLINEAR SYSTEM
Recall the generalized system:
Set the parameters as follows:
THIS YIELDS THE NONLINEAR SYSTEM

20. FOOD CHAIN MODEL EQUILIBRIUM SOLUTIONS
There are six total solutions:
Two dimensional solutions:
in the form of: (Trees, Moose, Wolves)
Any scalar multiple of the following:
(0,0,0) - All are dead (trivial solution) (1,0,0) - Only trees exist (0,1,0) - Only moose exist (0,0,1) - Only wolves exist (1,0,1) - Only trees and wolves exist
Solutions where all species coexist:
in the form of: (Trees, Moose, Wolves)
Any scalar multiple of:
(2 trees, 1 moose, 4 wolves)

21. FOOD CHAIN MODEL THE JACOBIAN MATRIX

22. FOOD CHAIN MODEL AT THE POINT OF INTEREST...

23. FOOD CHAIN MODEL SOLVING THE CHARACTERISTIC POLYNOMIAL
One negative eigenvalue and two imaginary eigenvalues with imaginary real parts yields:
A Stable Sink
The intuition of this is such: so long as all three species are present and non-extinct, they will tend towards a harmonious coexistant equilibrium in the ratio of 2 trees to 1 moose to 4 wolves.

24. FOOD CHAIN MODEL GRAPHS OF OUR SOLUTION OF INTEREST
As long as the initial point is positive for all X,Y, and Z, the populations will converge to (⅔, ⅓, 4/3) …. thus the graph backs up the algebra… it is a sink...

25. 3
CONCLUSION

26. There are two solutions to the general equation:
CONCLUSION
There are two solutions to the general equation:
There are three cases for the second solution. In all cases, the population of the apex predator is unrelated to species y
The nonlinear food chain model has six solutions, but only one allows all species to coexist
Graphically, this solution is a stable sink

28. 3
REFERENCES

29. REFERENCES
Ultimately, we will show that the behavior of the nonlinear system was effectively predicted by the generalized linear system, with one exception. To reach equilibrium the linear system must reduce to a two-dimensional system of equations, however, the nonlinear example illustrates that the species can coexist at equilibrium in three dimensions without reduction to the xy-plane.