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Basics Numerics Oscillators! Chemistry! Rabbits vs. Foxes! Zombies! Pursuit! Chaos! Rabbits vs. Sheep!

2015 Holl Mini-Conference @ ISU MSMAmes, Iowa

A Few Good ODEs: An Introduction toModeling and Computation

James A. Rossmanith

Department of MathematicsIowa State University

June 20th, 2015

J.A. Rossmanith | ISU 1/22


Outline

1 Ordinary differential equations: some basics

2 Ordinary differential equations: numerical solutions

3 Harmonic and Van der Pol oscillators

4 Chemical reaction

5 Population dynamics: Rabbits vs. Foxes

6 Spreading disease: Human-Zombie-Removed

7 Non-trivial pursuit: 1 Fox chasing 1 Rabbit

8 Lorenz equations: Chaotic water wheel

9 Phase diagrams: Rabbits vs. Sheep



Outline




4 Chemical reaction








Ordinary differential equationsScalar initial value problem

Scalar ordinary differential equation (ODE):

A scalar ODE is an equation of the form:

dudt

= f (u), u(0) = u0

The solution is a function of a single-variable t (≡ time)

u0 is the initial condition

f (u) is a known function of u: (e.g., f (u) = u, f (u) = u2, . . . , f (u) = eu)

In principle (i.e., if you can evaluate the integrals and solve for u), solution is

duf (u)

= dt =⇒∫

1f (u)

du =∫

dt = t + c

For example f (u) = u:∫1u

du = t + c =⇒ ln(u) = t + c =⇒ u = ec et =⇒ u(t) = u0et .



Ordinary differential equationsSystem of initial value problems

System of ordinary differential equations:

A system of ODEs is an equation of the form:dudt

= f(u), u(0) = u0

The solution is a vector-valued function of a single-variable t (≡ time)

u0 is the initial condition (and it is a vector)

A solution is a trajectory in d-dimensional space (where u0 ∈ Rd ):

Trajectory: u(t) =(x(t),y(t),z(t)

)



Outline




4 Chemical reaction








Numerical approximation of ODEsEuler’s method

Recall: Taylor series for comparing the solution at t and t + k

u(t + k) = u(t)+ ku′(t)+12

k2u′′(t)+16

k3u′′′(t)+124

k4u(4)(t)+ . . .

Euler’s method

Let’s say that we have an approximation at time t = tn: Un ≈ u(tn)

Now, we want to get Un+1 ≈ u(tn + k):

u(tn +k)≈ u(tn)+k u′(tn) = u(tn)+k f (u(tn)) =⇒ Un+1 = Un +k f (Un)

Algorithm and example with f (u) = u:

1: U0 = u0, t0 = 02: for n = 0,1,2, . . . do3: tn+1 = tn + k4: Un+1 = Un + k f (Un)5: end for6: plot(t ,U)



Numerical approximation of ODEsRunge-Kutta method

Four-stage Runge-Kutta method:

Y1 = f (Un) , Y2 = f

(Un +

k2

Y1

), Y3 = f

(Un +

k2

Y2

), Y4 = f (Un + kY3) ,

Un+1 = Un +k6(Y1 +2Y2 +2Y3 +Y4)

Applied to f (u) = u:

Y1 = Un, Y2 =

(1+

k2

)Un, Y3 =

(1+

k2+

k2

4

)Un, Y4 =

(1+ k +

k2

2+

k3

4

)Un,

Un+1 =

(1+ k +

k2

2+

k3

6+

k4

24

)Un

=⇒ matches first five terms in Taylor series of u(t + k) = u(t)+ . . .

Extension to systems:

Y1 = f(Un) , Y2 = f(

Un +k2

Y1

), Y3 = f

(Un +

k2

Y2

), Y4 = f(Un + kY3) ,

Un+1 = Un +k6(Y1 +2Y2 +2Y3 +Y4)



Outline




4 Chemical reaction








Harmonic and Van der Pol OscillatorsExample from physics

Generic model (Newton’s second law F = Ma):

Mx ′′(t) =−RM x ′(t)− k x(t), u(t) =[

x(t)x ′(t)

],

ddt

[u1u2

]=

[u2

−RMM u2− k

M u1

]Example 1: Harmonic oscillator (no friction) (M = 1, k = 1):

RM ≡ 0, u(0) = (1,5) =⇒ harmonic_ex.m, harmonic.m

Example 2: Van der Pol oscillator (nonlinear friction) (M = 1, k = 1):

RM ≡ 10(u21 −1), u(0) = (1,5) =⇒ vanderpol_ex.m, vanderpol.m



Outline




4 Chemical reaction








Chemical reactionsExample from chemistry

Consider concentrations of the following chemicals:

S := substrate, E := enzyme, C := complex, P := product

Here is an example of a chemical reaction with these chemicals:

S+Ek1−→ C

Ck2−→ S+E

Ck3−→ P +E

Assuming everything is homogeneous in space this can be modeled as ODEs:

S′(t) =−k1 S(t)E(t)+ k2 C(t)

E ′(t) =−k1 S(t)E(t)+ k2 C(t)+ k3 C(t)

C′(t) = k1S(t)E(t)− k2 C(t)− k3 C(t)

P′(t) = k3C(t)

Example:(k1 = 106, k2 = 10−4, k3 = 0.1

)u(0) =

(5×10−7,2×10−7,0,0

)=⇒ chem_ex.m, chem.m



Outline




4 Chemical reaction








Population dynamicsRabbits vs. Foxes

Consider the following prey and predator species:

R := population of rabbits, F := population of foxes

Model assumptions:

(time rate of change

of population

)=

(birthrate

)−(

deathrate

)In the absence of foxes, rabbit population will grow exponentially:(

birthrate

)= k1 R(t) and

(deathrate

)= k2 R(t)F(t)

Death rate increases with larger rabbit pop. and larger fox pop.

In the absence of rabbits, fox population will decay exponentially:(deathrate

)= k4 F(t) and

(birthrate

)= k3 R(t)F(t)

Birth rate increases with larger rabbit pop. and larger fox pop.

The resulting model:

R′ = k1 R− k2 R F and F ′ = k3 R F − k4 F

Example:(k1 = 3, k2 = 2×10−3, k3 = 6×10−4, k4 = 0.5

)u(0) = (1000,500) =⇒ predprey_ex.m, predprey.m



Outline




4 Chemical reaction








Infectious disease modelingHuman-Zombie-Removed

Consider three populations:

H := humans, Z := zombies, R := temporarily removed zombies

Three conversion rate constants:

k1 := rate of conversion from (human) to (zombie)

k2 := rate of conversion from (temporarily removed zombie) to (zombie)

k3 := rate of conversion from (zombie) to (temporarily removed zombie)

Here is a model (again, assuming everything is space homogeneous):

H ′ =−k1 H Z

Z ′ = k1 H Z + k2 R− k3 H Z

R′ = k3 H Z − k2 R

Example: (k1 = 0.01, k2 = 0.02, k3 = 0.005)

(H(0), Z(0), R(0)) = (500, 10, 0) =⇒ zombie_ex.m, zombie.m



Outline




4 Chemical reaction








Pursuit problems1 Fox chasing 1 Rabbit

Rabbit moves along a prescribed path: (r(t), s(t))

Fox chases rabbit so that

1 speed of fox is always k times speed of rabbit

2 tangent vector to fox position points at rabbit

Tangent: x ′ = α(r − x) y ′ = α(s− y)

Speed: (x ′)2 +(y ′)2 = α2[(r − x)2 +(s− y)2

]= k

[(s′)2 +(t ′)2

]Full equations:

ddt

[xy

]= k

√(r ′)2 +(s′)2

(x− r)2 +(y− s)2

[r − xs− y

]

Example: k = 1.1, r(t) = (1+ t)12 cos(t), s(t) = (1+ t)

12 sin(t)

(x(0), y(0)) = (3, 0) =⇒ pursuit_ex.m, pursuit.m



Outline




4 Chemical reaction








Chaotic water wheelLorenz equation

x ′ = σ(y− x)

y ′ = r x− y− x z

z′ = x y−b z

Lorenz equation: simplified model of convection in the troposphere

Lorenz equation: simplified model of a leaky water wheel

Example: (σ = 10, b = 8/3, r = 28)

lorenz_ex.m, lorenz.m

See video of an experiment here: Link


https://www.youtube.com/watch?v=zhOBibeW5J0


Outline




4 Chemical reaction








Resource competition modelRabbits vs. Sheep

Consider two populations competing for the same food source:

x := rabbits, y := sheep

Each species follows logistic growth plus a competition term

x ′ = x(3− x−2y)

y ′ = y(2− x− y)

There are four fixed points in this system:

x(3− x−2y) = 0 and y(2− x− y) = 0

(x ,y) = (0,0)−unstable node

(x ,y) = (0,2)− stable node

(x ,y) = (3,0)− stable node

(x ,y) = (1,1)−unstable saddle

Create phase diagram:

pplane8.m, dfield8.m


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