Euler's Method for a system of differential equations 

In this video, I show how to use Euler's Method to approximate solutions to a system of ordinary differential equations, particularly the Lotka-Volterra equations modeling the interaction between two species in an ecosystem (predators and prey). (Review Euler's Method for a single first-order equation here: • Euler's method for fir... )
Using the example of rabbits (R) as prey and foxes (F) as predators, I present this pair of first-order differential equations modeling population dynamics and explain the parameters (a, b, c, d) involved. Then we work through Euler's method: this algorithm relies on estimating incremental changes in population at discretized moments in time (with time step Delta t) from a starting pair of initial populations.
We implement the method, starting with 40 rabbits and 8 foxes, approximating their populations daily over 60 weeks. (Error in my computations: For R3, I wrote 39.03 instead of 39.30 -- another reason why I would rather use MATLAB!) Then with MATLAB, I compare the results of Euler's Method with the ode45 solver, a more accurate numerical technique. Overall Euler's Method captures some qualitative aspects of the species behavior but misses some of the true dynamics.
#mathematics #differentialequations #numericalmethods #numericalanalysis #EulersMethod #LotkaVolterra #populationdynamics #predatorprey #matlabsimulation #mathtutorial