Exploring a Predator-Prey Model | Visualization of the Lotka-Volterra Equations.
In these clips, colored particles show the evolution of the system state of a predator-prey model described by the Lotka-Voltera equations. This simple basis model aims to model the most basic interactions of a predator-prey relationship.
200 x 200 points are uniformly distributed to visualize the trajectory of the system in state space, also called phase space. Contour plots show lines where the magnitude of the rate of change is equal and a vector field showing the gradient, the direction of the rate of change, at regular intervals.
The initial conditions are colored as a function of the distance to the equilibrium point to illustrate the different trajectories.
Numerical integration of the differential equations shown in the video was performed using the Euler scheme with a time step of 0.005 for 10,000 iterations. Assuming the time unit was years then the time frame of the simualtion would cover 50 years.
By integrating with the Euler method, numerical artifacts are created and the popualtions do not oscillate constantly but diverge which gives a cool viseulen effect
00:00 Growth and Death Rate = 1
00:58 Predator Death Rate = 2
01:39 Pred. Death Rate = 0.5, Prey. Growth Rate = 2
02:00 Pred. Death/Growth Rate = 0.5, Prey. Growth/Growth Rate = 2
This code was written in Matlab