Lotka–Volterra equations

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Lotka Volterra dynamics

Lotka-Volterra model (1.1, 0.4, 0.4, 0.1)

Predator prey dynamics

Lotka-Volterra orbits 01

Lotka–Volterra equations, also known as the predator-prey equations, are a pair of first-order, non-linear, 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 independently proposed by Alfred J. Lotka in 1925 and Vito Volterra in 1926. The equations have since been applied to various disciplines, including biology, ecology, economics, and chemistry, illustrating how complex interactions can be understood through relatively simple mathematical models.

Formulation of the Equations

The Lotka–Volterra system of equations consists of two equations that describe the population dynamics of two species in a predator-prey model. Let \(x\) represent the number of prey (for example, rabbits), and let \(y\) represent the number of predators (for example, foxes). The equations are:

\[ \frac{dx}{dt} = \alpha x - \beta xy \]

\[ \frac{dy}{dt} = \delta xy - \gamma y \]

where:

• \(\alpha\) is the growth rate of the prey population in the absence of predators.
• \(\beta\) is the rate at which predators destroy prey.
• \(\gamma\) is the mortality rate of the predators in the absence of prey.
• \(\delta\) is the rate at which predators increase by consuming prey.

Assumptions

The Lotka–Volterra model makes several simplifying assumptions:

• The prey population will grow exponentially in the absence of predators.
• The predator population will starve in the absence of prey but will increase proportionally to the amount of prey available.
• The rate of change of population is proportional to its size.
• The environment does not change in favor of one species, and genetic adaptation is negligible.

Applications and Extensions

The original Lotka–Volterra model has been extended in various ways to include additional biological factors such as carrying capacity, disease, and genetic variation. These extensions make the model more realistic but also more complex. The equations are used in many fields to model different scenarios where two entities interact in a predator-prey-like relationship, even outside of biology, such as in economics to model competition between companies or in cybersecurity to model interactions between attackers and defenders.

Limitations

While the Lotka–Volterra equations provide valuable insights into predator-prey dynamics, they have limitations. The model's assumptions often do not hold in real-world ecosystems, where numerous factors can affect population sizes. Additionally, the model does not account for environmental variability or more complex interactions between species, such as competition, mutualism, or the presence of more than two species.

Conclusion

The Lotka–Volterra equations have played a significant role in the development of mathematical biology and continue to be a fundamental tool in theoretical ecology. Despite their simplicity and limitations, these equations have paved the way for more sophisticated models and have enhanced our understanding of dynamic systems.