# Lotka–Volterra equation, Paradox of enrichment and Communism

“Under capitalism, man exploits man. Under communism, it’s just the opposite.”

– John Kenneth Galbraith, American Economist.

Two mathematicians Alfred Lotka and Vito Volterra applied mathematics to biology esp. study of population giving rise to what is known as Lotka-Volterra equation.

This equation is very useful to study population of prey and predator ex. if density of prey ex. hare is x and density of predator ex. lynx is y, then equation giving their relationship is

Growth of prey over period time t is dx/dt= ax+bxy, while that of predator will be dy/dt= cxy-dy.

We will not go into details (plenty of information is available on web); application of these equations helps us to understand what is known as paradox of enrichment.

Suppose population of hare goes up, this will result in plenty of food for lynx, as a result it will increase the number of lynxes, but this situation i.e. plenty of hare and lynxes does not last for long. Hunting by lynxes brings down population of hare, resulting in shortage of food and ultimately resulting in deaths of lynxes due to starvation.  Reduction of number of lynxes again increases population of hare, resulting in increase in population of lynxes and so on. This cycle has been going on for centuries i.e. decline and growth of prey and predators is cyclic and correlated.

One more application of this is paradox of insecticide. To kill pest a farmer sprays insecticide, the insecticide due to rains reaches soil and water table killing natural predators of these pests. Over a period of times pest develop resistance to pesticide and by then all its natural predators are eliminated, so population of pests grows more than before.

In economy you have cycles of boom and recession. Communists believed that they could do away with such cycles through planned economy; unfortunately it was a very costly experiment which ultimately resulted in downfall of most of the communist regimes in the world.