# 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 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

##### 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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This project is supported, in part, by the National Science Foundation. Opinions expressed are those of the authors, and not necessarily those of the Foundation.