# The Lotka-Volterra Model of Oscillating Chemical Reactions

##### This is the earliest proposed explanation for why a reaction may oscillate. In 1920 Lotka proposed the following reaction mechanism (with corresponding rate equations). Each reaction step refers to the MOLECULAR mechanism by which the reactant molecules combine to produce intermediates or products. For example, in step 1 a molecule of species A combines with a molecule of species X to yield two molecules of species X. This step depletes molecules A (and adds molecules X) at a rate proportional to the product of the concentrations of A and X.

reaction step |
molecular reaction |
step contributions to differential rate laws |

1 |
A + X 2X |
, |

2 |
X + Y 2Y |
, |

3 |
Y B |

##### The overall chemical reaction is merely A B with two transient intermediate compounds X and Y:

step 1: A + X 2X

step 2: X + Y 2Y

step 3: Y B

sum of steps: AB

##### The effective rate laws for the reactant A, the product B, and the
intermediates X and Y are found by summing the contributions from each
step:

Step 1 is called autocatalytic because X accelerates its own production.
Likewise step 2 is autocatalytic.

__Problem__: Given the mechanism it is required to solve for [A],
[X], [Y], and [B] as functions of time. Lotka obtained oscillating concentrations
for both intemediates X and Y when the concentration of reactant [A] is
constant (as, for example, A is continuously replaced from an external
source as it is consumed in the reaction). An interactive solution is provided
in the form of a mathcad file ( lotka.mcd)
(you need mathcad to view this file). You can also read a summary of the
solution (view with your browser).

##### Lotka's mechanism can be re-interpreted as a model for oscillating
populations of predators and preys as was done by Volterra. In this, A
represents the ecosystem in which prey X and predator Y live. Step 1 represents
pre procreation: prey population doubles at rate k_{1}[A] (typical
exponential growth). Then Y is the population of predators that consume
the prey in order to sustain (and expand) their population. Step 2 represents
this inclination of predators to reproduce in proportion to the availability
of prey. Finally (step 3), predators die at a certain natural rate (also
exponential) so that they are removed from the ecosystem.

Return to oscillating reactions CONTENTS , Brusselator , Oregonator

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