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IDEA: Internet Differential Equations Activities

The Lotka-Volterra Model of Oscillating Chemical Reactions

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


A + X




X + Y





The overall chemical reaction is merely A B with two transient intermediate compounds X and Y:
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 ( (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 k1[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.

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