Lotka-Volterra Equations

The Lotka-Volterra equations are a set of differential equations that model the dynamics of biological systems in which two species interact, typically a predator and a prey. They were independently developed by mathematician Alfred Lotka in 1910 and biologist Vito Volterra in 1926.

The equations are as follows:

1. For the prey (population \(x\)):

$$\frac{dx}{dt} = \left( \alpha x - \beta xy \right) = \left( ax - bxy \right)$$

• \(\alpha\), \(a\) is the growth rate of the prey in the absence of predators.
• \(\beta\), \(b\) is the predation rate (the frequency with which predators hunt the prey).
2. For the predator (population \(y\)):

$$\frac{dy}{dt} = \left( \delta xy - \gamma y \right) = \left( dxy - cy \right)$$

• \(\gamma\), \(c\) is the death rate of predators in the absence of prey.
• \(\delta\), \(d\) is the growth rate of the predators based on the number of prey hunted.

Interpretation

• The prey population \(x\) grows exponentially if there are no predators \((\alpha x)\), but decreases when there is interaction with predators \((-\beta xy)\).
• The predator population \(y\) grows proportionally to the number of prey available \( (\delta xy) \), but decreases naturally in the absence of prey \((- \gamma y)\).

Instructions

Enter the initial conditions for the predator quantity \( y_{0} \) and the prey quantity \( x_{0} \) the prey quantity, then press the preset button to reset the values. To control the animation, use the 'skip backward' button \(\left(\vert \!\!\! \blacktriangleleft \! \blacktriangleleft \right) \) button to restart and stop the animation, the 'play' button \( \left( \blacktriangleright \right) \) to start, and the ⏸ symbol button to pause. Adjust the sliders to modify the system's behavior: