2007 Math Biology Seminar ODE Population Models

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

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

1st Model—exponential, Malthusian Solution: 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 Suppose we have 2 species; one predator y(t) (e.g. wolf) and one its prey x(t) (e.g. hare)

Actual Data

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

Solution vs time

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!!!!

ag ≠ bf ag=bf

2000 REU and paper

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.

Quasi-periodicity

Previte’s doughnut conjecture (ag>bf)

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<e implies z goes extinct fc+gc+h>e implies z to a periodic on the cylinder

Dynamics trapped on cylinders

Several trajectories

Ben Nolting and his poster in San Antonio, TX

Scavenger Model with feedback (Malorie Winters 2006/7)

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

Malorie Winters, and in New Orleans, LA

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