how mathematics help us understand the world around uswtang/files/basistalk.pdfhow mathematics help...

How mathematics help us understand the

world around us

Wenbo Tang

School of Mathematical & Statistical Sciences,Arizona State University

2009-10 speaker series, Basis Scottsdale High School, Sept. 29th, 2009

Wenbo Tang (ASU) Math Modeling of Dynamics 09/29/09 1 / 14

The World As a Dynamical System

The world is in constant motion

Want to know the solution of a given variable at a given time

Solution often complex, yet dynamics may be governed by simple rules

E.g. Newton’s second law of motion

F = ma = md2x

Tricky part — F always non-constant!

F = ma = md2x

Mathematical Preparations Towards Dynamics

Motions change continuously (Calculus)

For physical problems, we need 4 dimensions (Multivariable Calculus)

Most important information is not on the axis (Linear Algebra)

1 0.5 0 0.5 11

Complex numbers denoting oscillation (Complex Variables)

dt2+ ω2x = 0; x = A cos(ωt) + B sin(ωt) = A1e

iωt + B1e−iωt

Ordinary and partial differential equations

1 0.5 0 0.5 11

iωt + B1e−iωt

1 0.5 0 0.5 11

iωt + B1e−iωt

1 0.5 0 0.5 11

iωt + B1e−iωt

1 0.5 0 0.5 11

iωt + B1e−iωt

How Is Mathematics Involved

Physicists, chemists and biologists find general rules

I Newton’s laws of motionI Laws of thermodynamics (zero-mean random walk)

These laws are translated into mathematical language

Differential equations are constructed to model processes

Equations usually appear in the form

dt= F − D

Differential equations are solved using analytical or numericalprocedures

Simplifications make problems easier to solve

Solutions are interpreted in physical language

Physicists, chemists and biologists find general rulesI Newton’s laws of motion

I Laws of thermodynamics (zero-mean random walk)

dt= F − D

Physicists, chemists and biologists find general rulesI Newton’s laws of motionI Laws of thermodynamics (zero-mean random walk)

dt= F − D

Example I: Modeling Population DynamicsThe logistic model has two important ingredients:

dt= RN(1− N

I Exponential growth when population is smallI Growth rate decreases with size

R,K determined from experiments/observationsThis is overly simple, but captures the essential dynamicsDecrease can be in many ways, dynamical behaviors are similar

G.Rate G.Rate

dt= RN(1− N

R,K determined from experiments/observations

This is overly simple, but captures the essential dynamicsDecrease can be in many ways, dynamical behaviors are similar

G.Rate G.Rate

dt= RN(1− N

R,K determined from experiments/observationsThis is overly simple, but captures the essential dynamics

Decrease can be in many ways, dynamical behaviors are similar

G.Rate G.Rate

dt= RN(1− N

R,K determined from experiments/observationsThis is overly simple, but captures the essential dynamicsDecrease can be in many ways, dynamical behaviors are similar

G.Rate G.Rate

Let’s Add Some Predation

In a modified model we considerloss of biomass by predation

dt= RN(1−N

K)− BN2

A2 + N2

The functional response modelscertain predation behavior

This model supports two stablestates in some parameter range

Also for some parameters a largepopulation is the onlyequilibrium

The states are essentiallydetermined by the intersectionof two functions

Predation function and flow

P dNdt

Bifurcation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

BNA2+N 2

R(1 ! NK )

dt= RN(1−N

K)− BN2

A2 + N2

P dNdt

Bifurcation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

BNA2+N 2

R(1 ! NK )

dt= RN(1−N

K)− BN2

A2 + N2

P dNdt

Bifurcation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

BNA2+N 2

R(1 ! NK )

dt= RN(1−N

K)− BN2

A2 + N2

P dNdt

Bifurcation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

BNA2+N 2

R(1 ! NK )

dt= RN(1−N

K)− BN2

A2 + N2

P dNdt

Bifurcation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

BNA2+N 2

R(1 ! NK )

Example II: A Mathematically Twisted Love Story

True dynamics of love is of course COMPLEX

But, let’s approach love mathematically with a weird example

Verbal statement: Romeo loves Juliet when she loves him, he backsoff when she doesn’t love him; Juliet is protective when Romeo lovesher, but will approach him when he walks away.

Mathematical model:

dt= aJ,

dt= −bR, a, b > 0

Differentiate left eq. w.r.t. t and use the right eq. we get

dt2+ abR = 0

The solution is an oscillation, R, J never equal to zero together,unless they start together, and if Romeo tries to love Juliet more,they will break up!

Mathematical model:

dt= aJ,

dt= −bR, a, b > 0

dt2+ abR = 0

Mathematical model:

dt= aJ,

dt= −bR, a, b > 0

dt2+ abR = 0

Mathematical model:

dt= aJ,

dt= −bR, a, b > 0

dt2+ abR = 0

Mathematical model:

dt= aJ,

dt= −bR, a, b > 0

dt2+ abR = 0

Mathematical model:

dt= aJ,

dt= −bR, a, b > 0

dt2+ abR = 0

Example III: Motion in the environment

Internal wave motions are important in several ways

1015 1016 1017 1018 1019 1020 10214000

Observations tell us that seawater or air density vary with height

We idealize a seawater drop as ping-pong ball, and neglect damping

The motion is governed by the buoyancy force, and for simplicity, weassume linear density stratification

ρ(z) = ρ0 − kz , k > 0

minus since seawater denser at depth, ρ0 is a reference density,chosen to be the density of the ping-pong ball

z = 0 at equilibrium density

1015 1016 1017 1018 1019 1020 10214000

ρ(z) = ρ0 − kz , k > 0

1015 1016 1017 1018 1019 1020 10214000

ρ(z) = ρ0 − kz , k > 0

1015 1016 1017 1018 1019 1020 10214000

ρ(z) = ρ0 − kz , k > 0

1015 1016 1017 1018 1019 1020 10214000

ρ(z) = ρ0 − kz , k > 0

Constructing Differential Equation

Motion is induced by buoyancy force

FB = ma = ρ0vd2z

dt2= [ρ(z)− ρ0]gv

→ d2z

ρ0z = 0

It’s an oscillation with frequency√

A lot more to learn before we can discuss internal gravity waves

Important to remember, we have idealized the problem a lot!

FB = ma = ρ0vd2z

→ d2z

ρ0z = 0

FB = ma = ρ0vd2z

→ d2z

ρ0z = 0

FB = ma = ρ0vd2z

→ d2z

ρ0z = 0

Example IV: Synchronization

Synchronization in fireflies, clapping hands

We model cyclic behavior through angular speed, for Tom and TJ

dt= Ω + K1 sin(θ −Θ)

dt= ω + K2 sin(Θ− θ)

To begin, we subtract the first equation from the second by writingφ = Θ− θ:

dt= Ω− ω − (K1 + K2) sinφ

Looking for dφ/dt = 0, we find that if |(Ω− ω)/(K1 + K2)| ≤ 1

φ? = arcsin(Ω− ω

K1 + K2)

Two equilibrium points, one stable, corresponding to phase lock

K1 + K2)

Reality Check

‘Realistic’ models are built on these simple models

Although independence is important, collaboration is crucial

Problems are interesting, walking through takes patience(constructing models, coding and debugging)

Important to keep track of things, as your memory power worksagainst you when you get older

You need to stay at the forefront, but being the lead expert on yourown field feels great!

Reality Check

Roles And Challenges

Identify new dynamical processes and translate into mathematicallanguage

For complex problems, deal with one aspect, study certain range ofbehaviors

Many fronts in the application of mathematicse.g. Math Awareness Month themes: Mathematics and theEnvironment, Mathematics and Manufacturing, Mathematics andMedicine, Mathematics and Decision Making, Mathematics and theOcean, Mathematics and the Genome, Mathematics and Climate,Mathematics and Sports, ...

Holistic approach to modeling the environment, involvingatmosphere-ocean dynamics, natural resources, urban developmentand decision making, etc.

Uncertainties arise in nonlinear dynamical systems for the environment

Learn Good Math and Solve Important Problems!

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