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Lotka-Volterra Model


The Lotka-Volterra Model 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 the mathematicians Alfred Lotka and Vito Volterra.

The equations are as follows:

  1. For the prey (population \(x\)):
  2. $$\frac{dx}{dt} = \left( \alpha x - \beta xy \right) = \left( ax - bxy \right)$$

    • \(\alpha\), \(a\) 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).
  3. For the predator (population \(y\)):
  4. $$\frac{dy}{dt} = \left( \delta xy - \gamma y \right) = \left( dxy - cy \right)$$

    • \(\gamma\), \(c\) 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

Simulation

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:


See also

Alfred James Lotka

Differential Calculus

Integral Calculus

Logistic map

Kuramoto Model

Vito Volterra