380 likes | 556 Views
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
E N D
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 w/ initial population population function
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?
1st Model—exponential, MalthusianSolution: x(t)=x(0)exp(at)
Critique • Unbounded growth • Non integer number of rabbits • Unbounded growth even w/ 1 rabbit! Let’s fix the unbounded growth issue dx/dt = ????
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)
Solutions: • Critique: • Still non-integer rabbits • Still get rabbits with x(0)=.02
Suppose we have 2 species; one predator y(t) (e.g. wolf) and one its prey x(t) (e.g. hare)
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)
Called Lotka-Volterra Equation, Lotka & Volterra independently studied this post WW I. • Fixed points: (0,0), (c/d,a/b) (in example (1,1)).
Phase portrait y (1,1) x
A typical portrait: a ln y – b y + c lnx – dx=C
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
Generalizations of L.V. • 3-species chains - 2000 REU • 4-species chains - 2004/5 REUs • Adding a scavenger 2005/6 REUs • (other interactions possible!)
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)
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!!!!
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)
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
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.
Simple Scavenger Model lynx beetle hare
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
Scavenger Model w/ scavenger prey crowding owl opossum hare
Analysis (Malorie Winters) • Regions of periodic behavior and Hopf bifurcations and stable coexistence. Regions with multi stability and dependence on initial conditions
Lots more to do!! • Competing species • Different crowding • Previte’s doughnut
How do I learn the necessary tools? • Advanced ODE techniques/modeling course • Work independently with someone • Graduate school • REU?
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