1. ODE and Population Models
2008 REU
ODE and 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. Fixed Points (equilibria)
In Previous example:
x=0 and x=7 are fixed points
Fixed Point: dx/dt = 0 (so it’s fixed!)
Stability: stable – near solutions tend to fixed point
unstable = not stable

9. Stability
Note: near x=7
d/dx ( du/dt) <0 (stable)

10. Stability
Note: near x=0
d/dx ( du/dt) > 0 (unstable)

11. Taylor series at x* dx/dt=f(x) (no dependence on t)
dx/dt = f(x)= c0+c1(x-x*)+c2(x-x*)^2+ ….
(c0 = 0)
If c1≠0, we can tell stability.

12. Moral: If dx/dt = f(x) and f(x*)=0
1) d/dx( f(x)) <0 at x* then x* is stable.
2) d/dx( f(x) ) >0 at x* then x* is unstable.

13. x’ versus x
For first order autonomous equations, plotting x’ versus x encapsulates all this info
x’ positive (unstable)
x’ negative (stable)

14. Reality check Find and classify all equilibria of dx/dt = sin (x(t))
Firefly example (tomorrow)

15. Rabbit vs. Deer

16. compete for the same food source.
Let x(t) rabbits
and y(t) deer
compete for the same food source.
dx/dt =
dy/dt =
Ax(1-x/K)
-Cxy
By(1-y/W)
-Dxy
Or…. (after changes of coordinates…)
dx/dt = x(1-x-ay)
dy/dt = y(b-by-cx)

17. Analysis of one case dx/dt = x(1-x-2y) dy/dt = y(2-2y-5x)
Equlibria/Fixed Points: (0,0) , (0,1), (1,0), (1/4,3/8)
Q: How do we know if these are stable or unstable?
A: Linear approximation (derivative)

18. Linear Systems
dx/dt= Ax (given by matrix mult)
Fixed Point(s)?

19. What’s an eigenvalue again?
Ax = λx
(λ,x) are eigenvalue eigenvector pair
Who cares?
Think about:
x(t) = exp (λt)x
(Handout/Maple)

20. Other Tools Trapping regions Poincare Bendixson Nullclines
Series solutions ,etc.
Invariant Sets
Bifurcations

21. 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)

22. Actual Data

23. 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)

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

25. Phase portrait y
(1,1) x

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

27. Solution vs time

28. 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

29. Previte’s Population Projects
3-species chains REU
3 Competing Species 2002/3 REU
4-species chains /5 REUs
Adding a scavenger /7 REUs
(other interactions possible!)

30. 3-species model (REU 2000) 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)

31. Critical analysis of 3-species chain
ag > bf → unbounded orbits
ag < bf → species z goes extinct
ag = bf → periodicity

32. ag ≠ bf ag=bf

33. 2000 REU and paper

34. Tools used in analysis Linearization Trapping regions Invariant sets
Liapunov functions (“energy” functions)

35. One open conjecture ag>bf y tends to a limit as time increases
all numerical evidence shows this, but
no analytic proof.

36. 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)

37. 2004/5 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

38. Quasi-periodicity

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

40. This is wide open Project never finished
Proof seems too hard, may involve deep topics such as
KAM theory, Hamiltonian systems

41. Simple Scavenger Model
lynx
beetle
hare

42. 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

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

44. Scavenger Model with feedback (Malorie Winters & James Greene 2006/7)

45. Biological Example (crowding prey)
crayfish
Predator of mayfly nymph
Scavenger of trout carcasses
Rainbow Trout (predator)
Mayfly nymph (Prey)
Crayfish are scavenger & predator

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

48. Malorie Winters, and in New Orleans, LA

49. REU 2007
James Greene finds a model that exhibits chaos

50. 2007 scavenger system dx/dt=x(1-bx-y-z) b, c, e, f, g, β > 0
dy/dt=y(-c+x)
dz/dt=z(-e+fx+gy-βz)

51. Period Doubling cascade and attractor

52. TO DO Finish up the analysis from 2007
Including Hopf Bifurcation analysis, boundedness of orbits, and compare onset of chaos with other models
Crowd the predator