We now turn to an interesting coupled system of ODE, namely, the Volterra-Lotka model of prey-predator interactions. This is modelled with the pair of coupled ODE:
\begin{align} \frac{dx}{dt} &= \alpha x - \beta x y, \\ \frac{dy}{dt} &= \delta x y - \gamma y, \end{align}
The functions are implemented as:
function f1(t,y1, y2) α*y1-β*y1*y2 end function f2(t,y1, y2) δ*y1*y2-γ*y2 end
Some common initialization for all methods, in particular, we now need two arrays of values to store the evolution of $x$ (in y1) and $y$ (in y2):
tmax=100 h=.001 npts=round(Int,tmax/h) y1=0.0*collect(1:npts); y2=0.0*collect(1:npts); y1[1]=1; y2[1]=.1; α=2/3; β=4/3; γ=1; δ=1;
Euler's method in this case reads:
$$\begin{align} y_{1,n+1}&=y_{1,n}+hf_1(t_n,y_{1,n},y_{2,n})\\ y_{2,n+1}&=y_{2,n}+hf_2(t_n,y_{1,n},y_{2,n}) \end{align}$$
In code:
@time for i=1:npts-1 y1[i+1]=y1[i]+h*f1((i-1)*h,y1[i],y2[i]) y2[i+1]=y2[i]+h*f2((i-1)*h,y1[i],y2[i]) end
Heun's version reads:
$$\begin{align} y_{1,n+1} &= y_{1,n} + {h\over2} [f(t_n, y_{1,n}, y_{2,n}) + f(t_n + h, y_{1,n} + h f(t_n, y_{1,n}, y_{2,n}), y_{2,n})]\\ y_{2,n+1} &= y_{2,n} + {h\over2} [f(t_n, y_{1,n}, y_{2,n}) + f(t_n + h, y_{1,n}, y_{2,n} + h f(t_n, y_{1,n}, y_{2,n}))] \end{align}$$
In code:
@time for n=1:npts-1 y1[n+1]=y1[n]+(h/2)*(f1((n-1)*h,y1[n],y2[n])+f1(n*h,y1[n]+h*f1((n-1)*h,y1[n],y2[n]),y2[n])) y2[n+1]=y2[n]+(h/2)*(f2((n-1)*h,y1[n],y2[n])+f2(n*h,y1[n],y2[n]+h*f2((n-1)*h,y1[n],y2[n]))) end
RK4's version reads:
Results can be shown in two ways:
plot([[y1[i] for i=1:npts], [y2[i] for i=1:npts]])
plot([(y1[i], y2[i]) for i=1:npts])
http://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx
http://calculuslab.deltacollege.edu/ODE/7-C-3/7-C-3-h.html
Backward Euler method, or Implicit Euler method,