Contents

Idea

Showing an easy way of doing predator-prey modeling in Sage. Right now it is a generic version and one version which is the competitive Lotka-Volterra.

Details

Original Lotka-Volterra

This is the original Lotka-Volterra phase map for a non-dimensional form. This was posted on Marshall Hampon on the ask.sagemath.org site. Lotka-Volterra has many coarse “flaws” and has also been modified over time as we’ll see below. Dimensionless format of Lotka-Volterra. default for Sage ode_solver is to use runga-kutta-felhberg (4,5)

Plot and Code

T = ode_solver()
T.function = lambda t, y: [y[0]-y[0]*y[1], -y[1]+y[0]*y[1]]
sol_lines = Graphics()
for i in srange(0.1,1.1,.1):
    T.ode_solve(y_0=
    [i,i],t_span=[0,10],num_points=1000)
    y = T.solution
    sol_lines += line([x[1] for x in y], rgbcolor = (i,0,1-i))
show(sol_lines+point((1,1),rgbcolor=(0,0,0)), figsize = [6,6], xmax = 6, ymax = 6)

Interactive and more realistic Lotka-Volterra

Here we enable choice of the exponential growth in the original Lotka-Volterra and logistic growth. We also added a parameter g which is used in both and k which is the scaled carrying capacity.

See if you can find the fixed point for the latter (using the code below)?

It should look like this:

def lv(g,k,growth):
   Tg = ode_solver()
   if growth == "Malthusian":
       Tg.function = lambda t, y: [g*y[0]*(1-y[1]), (-1.0/g)*(1-y[0])]
   else:
       Tg.function = lambda t, y: [g*y[0]*(1-y[0]/k - y[1]), (-1.0/g)*(1-y[0])]
   sol_lines = Graphics()
   for i in srange(0.1,1.1,.2):
      Tg.ode_solve(y_0=[i,i],t_span=[0,10],num_points=1000)
      y = Tg.solution
      sol_lines += line([x[1] for x in y], rgbcolor = (i,0,1-i))
   return sol_lines

@interact
def _(g = (0.1,1.,0.1), k = (0.1,4.0,0.1), Growth=["Malthusian","Logistic"]):
    show(lv(g,k,Growth),legend_label='Lv')

Next step

We will look at cases where Lotka-Volterra might lead to Hopf bifurcation and also see if we can add the Allee effect.

References

• Sage public worksheet containing the complete code so far which you can view or download.