Lotka-Volterra equations describe the evolution of a population of predators and preys.
The following equations dictate the variation of the number of predators (x) and preys (y) as a function of the number of predators and preys.
As an example, lets see how a population of 10 predators and 5 preys evolves over time.
In this example we are using:
- α = 1.5
- β = 1.2
- γ = 2.1
- δ = 1.8
At the starting point, the number of predators is 10 and the number of preys is 5.
It is really interesting to look at these results. As the equation dictates, the number of preys has positive influence in the number of predators, and the number of predators has a negative influence in the number of preys.
The fact that the solution does not diverge is what make co-habitation possible for all predator-prey systems in our ecosystem.
It is also interesting to look at the relationship between the number of predators and preys, by plotting a graph where the x axis represents the number of preys and the y axis represents the number of predators. This is called the phase space.
In each cycle, there is a moment when the number of preys is extremely low and the number of predators very high. As Lotka-Volterra equations assume no statistical fluctuations in population numbers, the solution is relatively stable.
In a real-life situation, the number of predators and preys may have statistical fluctuations, possibly causing extintion of the prey during this part of the cycle, and therefore, extinction of the predator.