Solutions to the Lotka-Volterra equations

The Lotka-Volterra equations are a pair of first-order, non-linear differential equations that are used to describe the dynamics of any biological system where there are interactions between predator and prey populations. They constitute an example of the more-generic Kolmogorov model.

[caption id="attachment_21073" align="aligncenter" width="301" caption="Variation of population of predator w.r.t. change in population of prey: 'x' is the number of predators and 'y' is the number of their prey."][/caption]

[caption id="attachment_21084" align="aligncenter" width="600" caption="As the number of prey (y) decreases, the number of predators (x) increases marginally, but after a point, a self-regulating mechanism kicks in, and the number of hunters grows tremendously with smaller increases in the number of prey. This is because the quantity of food available per hunter goes up, lifting with it the population of hunters."][/caption]

[caption id="attachment_21075" align="aligncenter" width="285" caption="Variation of number of prey w.r.t. number of predators"][/caption]

[caption id="attachment_21076" align="aligncenter" width="600" caption="As the number of prey (y) increases, so does the number of predators (x), but after a point, a self-regulating mechanism kicks in, and the number of hunters comes down as well even though there is an increase in the amount of game available."][/caption]

Note that, if superimposed, the blue curve will lead the red curve, implying that the predator population changes faster than the prey population.

Solved using the Wolfram Mathematica.