2015 Holl ISU MSM Ames, Iowa. A Few Good ODEs: An Introduction to Modeling and Computation
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1 2015 Holl ISU MSM Ames, Iowa A Few Good ODEs: An Introduction to Modeling and Computation James A. Rossmanith Department of Mathematics Iowa State University June 20 th, 2015 J.A. Rossmanith ISU 1/22
2 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 J.A. Rossmanith ISU 2/22
3 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 J.A. Rossmanith ISU 3/22
4 Ordinary differential equations Scalar initial value problem Scalar ordinary differential equation (ODE): A scalar ODE is an equation of the form: du = f(u), u(0) = u 0 dt The solution is a function of a single-variable t ( time) u 0 is the initial condition f(u) is a known function of u: (e.g., f(u) = u, f(u) = u 2,..., f(u) = e u ) In principle (i.e., if you can evaluate the integrals and solve for u), solution is For example f(u) = u: 1 du f(u) = dt = 1 f(u) du = dt = t + c u du = t + c = ln(u) = t + c = u = ec e t = u(t) = u 0 e t. J.A. Rossmanith ISU 4/22
5 Ordinary differential equations System of initial value problems System of ordinary differential equations: A system of ODEs is an equation of the form: du = f(u), u(0) = u 0 dt The solution is a vector-valued function of a single-variable t ( time) u 0 is the initial condition (and it is a vector) A solution is a trajectory in d-dimensional space (where u 0 R d ): Trajectory: u(t) = ( x(t),y(t),z(t) ) J.A. Rossmanith ISU 5/22
6 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 J.A. Rossmanith ISU 6/22
7 Numerical approximation of ODEs Euler s method Recall: Taylor series for comparing the solution at t and t + k u(t + k) = u(t) + ku (t) k 2 u (t) k 3 u (t) k 4 u (4) (t) +... Euler s method Let s say that we have an approximation at time t = t n : U n u(t n ) Now, we want to get U n+1 u(t n + k): u(t n +k) u(t n )+k u (t n ) = u(t n )+k f(u(t n )) = U n+1 = U n +k f(u n ) Algorithm and example with f(u) = u: 1: U 0 = u 0, t 0 = 0 2: for n = 0,1,2,... do 3: t n+1 = t n + k 4: U n+1 = U n + k f (U n ) 5: end for 6: plot(t,u) J.A. Rossmanith ISU 7/22
8 Numerical approximation of ODEs Runge-Kutta method Four-stage Runge-Kutta method: ( Y 1 = f (U n ), Y 2 = f U n + k ) ( 2 Y 1, Y 3 = f U n + k ) 2 Y 2, Y 4 = f (U n + ky 3 ), U n+1 = U n + k 6 (Y 1 + 2Y 2 + 2Y 3 + Y 4 ) Applied to f(u) = u: ( Y 1 = U n, Y 2 = 1 + k ) U n, Y 3 = (1 + k2 2 + k 2 ) 4 U n+1 = (1 + k + k k k 4 24 U n, Y 4 = (1 + k + k 2 = matches first five terms in Taylor series of u(t + k) = u(t) +... ) U n 2 + k 3 4 Extension to systems: ( Y 1 = f(u n ), Y 2 = f U n + k ) ( 2 Y 1, Y 3 = f U n + k ) 2 Y 2, Y 4 = f(u n + ky 3 ), U n+1 = U n + k 6 (Y 1 + 2Y 2 + 2Y 3 + Y 4 ) ) U n, J.A. Rossmanith ISU 8/22
9 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 J.A. Rossmanith ISU 9/22
10 Harmonic and Van der Pol Oscillators Example from physics Generic model (Newton s second law F = Ma): [ ] Mx (t) = R M x x(t) (t) k x(t), u(t) = x, (t) [ ] d u1 dt u 2 [ = u 2 R M M u 2 k M u 1 ] Example 1: Harmonic oscillator (no friction) (M = 1, k = 1): R M 0, u(0) = (1,5) = harmonic_ex.m, harmonic.m Example 2: Van der Pol oscillator (nonlinear friction) (M = 1, k = 1): R M 10(u1 2 1), u(0) = (1,5) = vanderpol_ex.m, vanderpol.m J.A. Rossmanith ISU 10/22
11 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 J.A. Rossmanith ISU 11/22
12 Chemical reactions Example 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 + E k 1 C C k 2 S + E C k 3 P + E Assuming everything is homogeneous in space this can be modeled as ODEs: S (t) = k 1 S(t)E(t) + k 2 C(t) E (t) = k 1 S(t)E(t) + k 2 C(t) + k 3 C(t) C (t) = k 1 S(t)E(t) k 2 C(t) k 3 C(t) P (t) = k 3 C(t) ( Example: k 1 = 10 6, k 2 = 10 4, k 3 = ) 0.1 u(0) = ( ,2 10 7,0,0 ) = chem_ex.m, chem.m J.A. Rossmanith ISU 12/22
13 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 J.A. Rossmanith ISU 13/22
14 Population dynamics Rabbits vs. Foxes Consider the following prey and predator species: R := population of rabbits, F := population of foxes ( ) ( ) ( ) time rate of change birth death Model assumptions: = of population rate rate In the absence of foxes, rabbit population will grow exponentially: ( ) ( ) birth death = k rate 1 R(t) and = k rate 2 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: ( ) ( ) death birth = k rate 4 F(t) and = k rate 3 R(t)F(t) Birth rate increases with larger rabbit pop. and larger fox pop. The resulting model: R = k 1 R k 2 R F and F = k 3 R F k 4 F Example: ( k 1 = 3, k 2 = , k 3 = , k 4 = 0.5 ) u(0) = (1000, 500) = predprey_ex.m, predprey.m J.A. Rossmanith ISU 14/22
15 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 J.A. Rossmanith ISU 15/22
16 Infectious disease modeling Human-Zombie-Removed Consider three populations: H := humans, Z := zombies, R := temporarily removed zombies Three conversion rate constants: k 1 := rate of conversion from (human) to (zombie) k 2 := rate of conversion from (temporarily removed zombie) to (zombie) k 3 := rate of conversion from (zombie) to (temporarily removed zombie) Here is a model (again, assuming everything is space homogeneous): H = k 1 H Z Z = k 1 H Z + k 2 R k 3 H Z R = k 3 H Z k 2 R Example: (k 1 = 0.01, k 2 = 0.02, k 3 = 0.005) (H(0), Z(0), R(0)) = (500, 10, 0) = zombie_ex.m, zombie.m J.A. Rossmanith ISU 16/22
17 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 J.A. Rossmanith ISU 17/22
18 Pursuit problems 1 Fox chasing 1 Rabbit Rabbit moves along a prescribed path: Fox chases rabbit so that (r(t), s(t)) 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 = α 2 [ (r x) 2 + (s y) 2] = k [ (s ) 2 + (t ) 2] Full equations: d dt [ ] x (r = ) 2 + (s ) 2 k y (x r) 2 + (y s) 2 [ ] r x s y Example: k = 1.1, r(t) = (1 + t) 1 2 cos(t), s(t) = (1 + t) 1 2 sin(t) (x(0), y(0)) = (3, 0) = pursuit_ex.m, pursuit.m J.A. Rossmanith ISU 18/22
19 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 J.A. Rossmanith ISU 19/22
20 Chaotic water wheel Lorenz 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 J.A. Rossmanith ISU 20/22
21 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 J.A. Rossmanith ISU 21/22
22 Resource competition model Rabbits 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 J.A. Rossmanith ISU 22/22
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