1. 2007 Math Biology Seminar
ODE Population Models

2. Differential Equations!
Intro
Often know how populations change over time (e.g. birth rates, predation, etc.), as opposed to knowing a ‘population function’
Differential Equations!
Knowing how population evolves over time
w/ initial population  population function

3. Example – Hypothetical rabbit colony lives in a field, no predators.
Let x(t) be population at time t;
Want to write equation for dx/dt
Q: What is the biggest factor that affects
dx/dt?
A: x(t) itself!
more bunnies  more baby bunnies

4. 1st Model—exponential, Malthusian Solution:
x(t)=x(0)exp(at)

5. Critique Unbounded growth Non integer number of rabbits
Unbounded growth even w/ 1 rabbit!
Let’s fix the unbounded growth issue
dx/dt = ????

6. Logistic Model dx/dt = ax(1-x/K) K-carrying capacity
we can change variables (time) to get
dx/dt = x(1-x/K)
Can actually solve this DE
Example:
dx/dt = x(1-x/7)

7. Solutions: Critique: Still non-integer rabbits
Still get rabbits with x(0)=.02

8. Suppose we have 2 species; one predator y(t) (e. g
Suppose we have 2 species; one predator y(t) (e.g. wolf) and one its prey x(t) (e.g. hare)

9. Actual Data

10. Model Want a DE to describe this situation
dx/dt= ax-bxy = x(a-by) dy/dt=-cy+dxy = y(-c+dx)
Let’s look at: dx/dt= x(1-y) dy/dt=y(-1+x)

11. Called Lotka-Volterra Equation, Lotka & Volterra independently studied this post WW I.
Fixed points: (0,0), (c/d,a/b) (in example (1,1)).

12. Phase portrait y
(1,1) x

13. A typical portrait:
a ln y – b y + c lnx – dx=C

14. Solution vs time

15. Critiques Nicely captures periodic nature of data
Orbits are all bounded, so we do not need a logistic term to bound x.
Periodic cycles not seen in nature

16. Generalizations of L.V. 3-species chains - 2000 REU
4-species chains /5 REUs
Adding a scavenger /6 REUs
(other interactions possible!)

17. 3-species model 3 species food chain!
x = worms; y= robins; z= eagles
dx/dt = ax-bxy =x(a-by) dy/dt= -cy+dxy-eyz =y(-c+dx-ez) dz/dt= -fz+gyz =z(-f+gy)

18. Critical analysis of 3-species chain
ag > bf → unbounded orbits
ag < bf → species z goes extinct
ag = bf → periodicity
Highly unrealistic model!! (vs. 2-species)
Adding a top predator causes possible unbounded behavior!!!!

19. ag ≠ bf ag=bf

20. 2000 REU and paper

21. 4-species model
dw/dt = aw-bxw =w(a-bx) dx/dt= -cx+dwx-exy =x(-c+dw-ey) dy/dt= -fy+gxy - hyz =y(-f+gx-hz)
dz/dt= -iz+jyz =z(-i+jy)

22. 2004 REU did analysis Orbits bounded again as in n=2
Quasi periodicity (next slide)
ag<bf gives death to top 2
ag=bf gives death to top species
ag>bf gives quasi-periodicity

23. Even vs odd disparity
Hairston Smith Slobodkin in 1960 (biologists) hypothesize that
(HSS-conjecture)
Even level food chains (world is brown)
(top- down)
Odd level food chains (world is green)
(bottom –up)
Taught in ecology courses.

24. Quasi-periodicity

25. Previte’s doughnut conjecture (ag>bf)

26. Simple Scavenger Model
lynx
beetle
hare

27. Semi-Simple scavenger– Ben Nolting 2005
Know (x,y) -> (c, 1-bc) use this to see
fc+gc+h=e every solution is periodic
fc+gc+h<e implies z goes extinct
fc+gc+h>e implies z to a periodic on the cylinder

28. Dynamics trapped on cylinders

29. Several trajectories

30. Ben Nolting and his poster in San Antonio, TX

31. Scavenger Model with feedback (Malorie Winters 2006/7)

32. Scavenger Model w/ scavenger prey crowding
owl
opossum
hare

33. Analysis (Malorie Winters)
Regions of periodic behavior and
Hopf bifurcations and stable coexistence.
Regions with multi stability and dependence on initial conditions

35. Malorie Winters, and in New Orleans, LA

36. Lots more to do!! Competing species Different crowding
Previte’s doughnut

37. How do I learn the necessary tools?
Advanced ODE techniques/modeling course
Work independently with someone
Graduate school
REU?

38. R.E.U.? Research Experience for Undergraduates Usually a summer
100’s of them in science (ours is in math biology)
All expenses paid plus stipend $$$!
Competitive
Good for resume
Experience doing research