lecture3

lecture3 - From Gehard Ertl’s Chemistry Nobel Prize...

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Unformatted text preview: From Gehard Ertl’s Chemistry Nobel Prize Lecture (2007) In catalysis experiment, CO = hares, O2 = lynxes Code for Euler approach… Declarations ! Euler method (forward-centered) for Lotka-Volterra IMPLICIT NONE INTEGER, PARAMETER :: Prec14=SELECTED_REAL_KIND(14) REAL(Kind=Prec14), PARAMETER :: dt=0.01d0 ! timestep REAL(Kind=Prec14), PARAMETER :: time=200.0d0 ! total simulation time REAL(Kind=Prec14), PARAMETER :: x0=0.1d0 ! initial x value REAL(Kind=Prec14), PARAMETER :: y0=0.2d0 ! initial y value REAL(Kind=Prec14), PARAMETER :: alpha1=1.0d0,alpha2=1.0d0,beta1=1.0d0,be REAL(Kind=Prec14) :: x,xn ! current population x, next population xn REAL(Kind=Prec14) :: y,yn ! current population y, next population yn REAL(Kind=Prec14) :: t ! current time INTEGER :: nsteps,n Execution part… nsteps=anint(time/dt) ! number of integration steps, integer value x=x0 ; y=x0 ! set initial populations x,y t=0.0d0 do n=1,nsteps t=t+dt ! step forward time t ! Integrate step using Euler xn=x+(alpha1*x-alpha2*x*y)*dt ! step forward x population yn=y+(beta1*x*y-beta2*y)*dt ! step forward y population x=xn y=yn ! Output time and x,y populations write(6,100) t,x,y enddo 100 format(f8.4,3f12.6) ! Format statement, f for floats, gives total number of columns and those after decimal point stop end Unstable results! Growth with Euler method inherently unstable ! Red=hares, Green=lynxes Midpoint method (2nd order Runge-Kutta) ! Integrate step using 2nd order Runge-Kutta xp=x+0.5d0*(alpha1*x-alpha2*x*y)*dt ! step forward x population yp=y+0.5d0*(beta1*x*y-beta2*y)*dt ! step forward y population xn=x+(alpha1*xp-alpha2*xp*yp)*dt ! step forward x population yn=y+(beta1*xp*yp-beta2*yp)*dt ! step forward y population Results appear to be stable, populations not growing without bound Red=hares, Green=lynxes ...
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