Issuu on Google+

Test file for mfpic4ode package Robert Maˇr´ık April 15, 2009 See the source file demo.tex for comments in the TEX code.

1

Logistic equation

Here we draw a simple picture which describes stability of stationary points of teh equation and then draw phase portrait of the equation.

 x x x0 = r · 1 − K

(1)

Stability and sign of the right–hand side. f (x)

K

x

Phase portrait x

K

t

1


2

Logistic equation with harvesting

Similar to the previous picture, but both pictures are drawn together to see the relations between them.

 x x−p x0 = r · 1 − K

x

(2) x

K

K

f (x) t

3

Three numerical methods for ODEs

Here we draw solution of ODE using all three available numerical methods. We use big step to see the difference between Euler, Runge–Kutta and fourth order Runge–Kutta method.

xn+1 = xn + h

y0 = x + y3

yn+1 = yn + kh

y(0) = 1

h = 0.2

2


2.4

Exact solution

2.2 RK4 RK

2 1.8

Euler

1.6 1.4 1.2

k1 for second step

1 k 1 0.8

4

0

k2 0.2

0.4

Autonomous systems

We draw the phase portrait of autonomous system, nulclines, invariant set between nulclines, trajectories. We draw arrows in regular grid and add few more arrows on nulclines and outside the regular grid.

3


y

Competing species

a c

α β

α γ

y

x

a b

Pedator prey system with HollingII response function

x

4


demo