Lotka Volterra Mechanism of an Oscillatory Reaction

Lotka Volterra Mechanism of an Oscillatory Reaction

Long-before the Belousov-Zhabotinsky reaction (B-Z reaction) wsa discovered, the so called Lotka-Volterra Mechanism was suggested to understand the nature of oscillatory ractions in biochemical systems.
In oscillatory reactions, the concentration of intermediate will increase and decrease alternately and periodically due to autocatalytic effect of one of the intermediates.

Lotka-Volterra Mechanism

Lotka-Volterra Mechanism involves the following steps-

This mechanism follows kinetics predicted by the predator-prey relationship. In this case, X represents the 'prey' and Y represents the 'predator'. The population of the predator can not build up unless there is a significant population of prey on which the predators can feed. Likewise, the population of predators decreases when the population of the prey falls. And finally, there is a lag, as the rise and fall of the prey population controls the rise and fall of the predator population.
The equations have been studied extensively and have applications not just in chemical kinetics, but in biology, economics, and elsewhere.
And rate of these steps can be written as-

Step-1 and step-2 are autocatalytic. The concentration of M is held constant by supplying it to the reaction. This leaves [X] and [Y] the concentrations of the intermediates as variables.
In this case we can solve the rate equations exactly for the variable concentrations of X and Y, but concentration of M is held at a constant value since it is being supplied to the reaction. After solving the equations numerically, the plot of [X] and [Y] against time is looks like as shown below-