LotkaVolterran

Lotka-Volterra equations, also known as the predator-prey equations, are a pair of first-order nonlinear differential equations frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The equations were formulated independently by Alfred Lotka in 1925 and Vito Volterra in 1926.

The Lotka-Volterra equations are typically written as:

dx/dt = alpha * x - beta * x * y

dy/dt = delta * x * y - gamma * y

where:

- x is the population of prey,

- y is the population of predators,

- alpha, beta, gamma, and delta are positive parameters describing the interaction between the two species.

The first equation describes the growth of the prey population, which increases at a rate proportional to its current size (alpha * x) and decreases at a rate proportional to the rate of encounters between predators and prey (beta * x * y). The second equation describes the growth of the predator population, which increases at a rate proportional to the rate of encounters between predators and prey (delta * x * y) and decreases at a rate proportional to its current size (gamma * y).

The Lotka-Volterra equations can exhibit a range of behaviors, including stable points, periodic solutions, and chaotic behavior, depending on the values of the parameters. They have been widely used in ecology and theoretical biology to model the dynamics of predator-prey interactions and have also been applied in other fields, such as chemistry and economics.