#!/usr/bin/env python
# coding: utf-8

# # Predator-Prey System
# 
# Copyright (C) 2020 Andreas Kloeckner
# 
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# <summary>MIT License</summary>
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# In[11]:


import numpy as np
import matplotlib.pyplot as plt


# This is the "right-hand side" of the [Lotka-Volterra predator-prey system](https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations)
# $$\begin{align*}
# y_1' &= y_1 (\alpha _1 - \beta _1 y_2)\\
# y_2' &= y_2 (- \alpha _2 + \beta _2 y_1)
# \end{align*}$$
# written in vector form, with somewhat arbitrarily chosen coefficients:

# In[7]:


alpha1 = 0.5
beta1 = 0.5
alpha2 = 0.5
beta2 = 0.5

def predator_prey_rhs(t, y):
    y1, y2 = y
    return np.array([
        y1*(alpha1-beta1*y2),
        y2*(-alpha2+beta2*y1)
    ])


# We will integrate this with the help of the fourth-order Runge-Kutta method, which we will meet in more detail later in the chapter:

# In[2]:


def rk4_step(y, t, h, f):
    k1 = f(t, y)
    k2 = f(t+h/2, y + h/2*k1)
    k3 = f(t+h/2, y + h/2*k2)
    k4 = f(t+h, y + h*k3)
    return y + h/6*(k1 + 2*k2 + 2*k3 + k4)


# Next, integrate the IVP for some time:

# In[13]:


times = [0]
ys = [np.array([0.1, 0.9])]

dt = 0.1

while times[-1] < 100:
    y = ys[-1]
    ynext = rk4_step(y, times[0], dt, predator_prey_rhs)
    ys.append(ynext)
    times.append(times[-1] + dt)

ys = np.array(ys)
times = np.array(times)


# Lastly, plot the result:

# In[14]:


plt.plot(times, ys[:, 0], label="Prey")
plt.plot(times, ys[:, 1], label="Predator")


# In[ ]: