Simulation of Competitive Lotka-Volterra Model¶

Author: Huarui Zhou

Description of the model¶

Suppose there are two species that live in the same area and will compete for food, space and other resources. Let $N_1$ and $N_2$ be their population size, $r_1$ and $r_2$ be their ideal growth rate, $K_1$ and $K_2$ be their maximal population size. We can establish the following differential equations.

$$ \left\{ \begin{split} \frac{\mathrm{d} N_1}{\mathrm{d} t} & =r_1N_1(1-\frac{N_1+\alpha N_2}{K_1}) \\ \frac{\mathrm{d} N_2}{\mathrm{d} t} & = r_2N_2(1-\frac{N_2+\beta N_1}{K_2}) \end{split}\right. $$

Simulation¶

In [4]:
from scipy import spatial as sp
import numpy as np
from scipy import integrate
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import math
In [5]:
def LV_deriv(x, t0, alpha=1, beta=1, K1 = 100, K2=100,r1=1, r2=1):
    return (r1*x[0]*(1-(x[0]+alpha*x[1])/K1),
            r2*x[1]*(1-(x[1]+beta*x[0])/K2))

def LV_plot(alpha,beta,K1,K2,r1,r2, max_time=20,start_x = 10, start_y=10,run=True) :
   
    
    if not run:
        return "not run"
    
    x1_0 = [start_x,start_y]
    # starting vector
    
    
    t = np.linspace(0, max_time, max_time*1000) # time step is fixed
    x1_t = integrate.odeint(LV_deriv, x1_0, t, args=(alpha,beta, K1,K2,r1,r2))
    
    x, y = np.meshgrid(np.linspace(0, K1,2500), np.linspace(0, K2, 2500))
    u = r1*x*(1-(x+alpha*y)/K1)
    v = r2*y*(1-(y+beta*x)/K2)
    
    fig =plt.figure(figsize=(12,5))
    
    ax1= fig.add_subplot(1,2,1)
    ax1.plot(x1_t[:,0], x1_t[:,1])
    plt.streamplot(x, y, u, v)
    plt.xlim([0, K1])
    plt.ylim([0, K2])
    ax1.set_xlabel("$N_1$")
    ax1.set_ylabel("$N_2$")
    
    plt.title('$\\alpha =' + str(alpha) + '$, $\\beta =' + str(beta)
              + '$, $K_1=' + str(K1) + '$, $K_2=' + str(K2) + '$')
    
    
    ax2 = fig.add_subplot(1,2,2)
    ax2.plot(t,x1_t[:,0], label = 'Species 1')
    ax2.plot(t,x1_t[:,1], label = 'Species 2')
    ax2.legend()
    ax2.set_xlabel("$t$")
    ax2.set_ylabel("Population size")
    
    plt.show()   
 
    
from ipywidgets import interact, interactive, Checkbox
from IPython.display import clear_output, display, HTML
interact(LV_plot, alpha=(0.0,5.0),beta=(0.0, 5.0),
         r1=(0.0, 10.0),r2=(0.0, 10.0),K1=(0.0,500.0),K2=(0.0,500.0),
         max_time=(10,100,1), start_x = (0.0,500.0), start_y = (0.0,500.0), run=Checkbox(value=False))

interactive(children=(FloatSlider(value=2.5, description='alpha', max=5.0), FloatSlider(value=2.5, description…
Out[5]:
<function __main__.LV_plot(alpha, beta, K1, K2, r1, r2, max_time=20, start_x=10, start_y=10, run=True)>