# Lotka-Volterra system

The Lotka-Volterra system was derived by Volterra in 1926 to describe the relationship between a predator and a prey, and independently by Lotka in 1920 to describe a chemical reaction.

Suppose that $N(t)$ is the prey population at time $t$, and $P(t)$ is the predator population. Then the system is

$\frac{dN}{dt}$ | $=$ | $N(a-bP)$ | ||

$\frac{dP}{dt}$ | $=$ | $P(cN-d)$ |

where $a$, $b$, $c$ and $d$ are positive constants. The term $aN$ is the birth of preys, $-bNP$ represents the diminution of preys due to predation, which is converted into new predators with a rate $cNP$. Finally, predators die at the natural death rate $d$.

Local analysis of this system is not very complicated (see, e.g., [1]). It is easily shown that it admits the zero equilibrium (unstable) as well as a positive equilibrium, which is neutrally stable. Hence, in the neighborhood^{} of this equilibrium exist periodic solutions (with period $T=2\pi {(ad)}^{-1/2}$).

This system is , and has obvious limitations, one of the most important being that in the absence of predator, the prey population grows unbounded^{}. But many improvements and generalizations^{} have been proposed, making the Lotka-Volterra system one of the most studied systems in mathematical biology.

## References

- 1 J.D. Murray (2002). Mathematical Biology. I. An Introduction. Springer.
- 2 Lotka, A.J. (1925). Elements of physical biology. Baltimore: Williams & Wilkins Co.
- 3 Volterra, V. (1926). Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei. Ser. VI, vol. 2.

Title | Lotka-Volterra system |
---|---|

Canonical name | LotkaVolterraSystem |

Date of creation | 2013-03-22 13:22:25 |

Last modified on | 2013-03-22 13:22:25 |

Owner | jarino (552) |

Last modified by | jarino (552) |

Numerical id | 9 |

Author | jarino (552) |

Entry type | Definition |

Classification | msc 92D40 |

Classification | msc 92D25 |

Classification | msc 92B05 |