Thus, first we need to estimate the function value at the middle point using simple Euler's method, and then we can estimate the derivative at the middle point. The most popular is the Runge-Kutta method of the fourth order. However, for our purposes it is enough to use the second order method. The only difference is in initial density of prey. This figure shows relative changes in prey predator density for both initial conditions.
Trajectories are closed lines.
The model of Lotka and Volterra is not very realistic. It does not consider any competition among prey or predators. As a result, prey population may grow infinitely without any resource limits. The rate of prey consumption is proportional to prey density. Thus, it is not surprising that model behavior is unnatural showing no asymptotic stability.