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2007 Math Biology Seminar

2007 Math Biology Seminar. ODE Population Models. 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

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2007 Math Biology Seminar

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  1. 2007 Math Biology Seminar ODE Population Models

  2. 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. A: x(t) itself! more bunnies  more baby bunnies • 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?

  4. 1st Model—exponential, MalthusianSolution: 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. 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 - 2004/5 REUs • Adding a scavenger 2005/6 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<eimplies 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

  34. Malorie Winters, and in New Orleans, LA

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

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

  37. 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

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