The Lokta Volterra Model describes competition & growth of different species in nature. It’s created by connecting species-specific logarithmic growth model(in the first equation below) with that of another species.
Imagine a natural habitat with only one species, it’s easy to imagine how the population of that species will grow exponentially in the beginning, and start to saturate when certain numbers are reached. This is expressed in a logarithmic equation:
And this is the curve for N with different initial N0, where the constant K=10.
When there’s more than one species in the habitat, and they like the same type of food, there will be competition. Two scientists Lokta and Volterra developed such a model independently, and it goes:
Clearly in these two equations, the number of both species affected the numerator on the right hand side, while keeping other parts of the equation untouched. Now the growth rates of two species are coupled. α and β are the relative competitiveness of species.
The equilibrium of the competion could be easily found by setting dN1/dt = 0 and dN2/dt = 0, resulting in:
Simutaneously solve these two simple equations for N1 and N2 will predict the equilibrium:
If one of N1 and N2 is negative, this means one of the species will totally loose in the competition and go extinct; if both N1 and N2 stay positive, this means the two species are at a balance.
VARIANTS OF LOKTA VOLTERRA MODEL
There’re different variants of the original problem, for example, the Lokta Volterra Fishery problem, and the Preditor and Pray problem, details could be found in these following posts: