Volterra equations


Competitive interactions in wild nature are well described by quantitative cybernetics. Vito Volterra published the following system of differential equations which represent relations between predators and their prey living in some restricted area.


dx/dt = alpha * x - beta * x * y

dy/dt = delta * x * y - gamma * y


Here x is population of prey, y - of predators.

Their solutions have cyclic nature and were confirmed by real data.

Here x is green and y is red.

This dynamics hints an opportunity to apply them to human economy, but the task requires understanding. History knows several types of socio-economic models: slavery, feudal system, capitalism, planned socialism. The first one is the most close by the character of relations. Predator-prey obviously correspond to master-slave, but there is a substantial difference. Slave owners don't eat their servants and probably population of both is not the best value to represent the current state of the social system. Also, human life is much more complicated than life of wild animals so any model will be schematic and we need to extract the most essential variables which reflect the core of the living system.

For the beginning we will take a famous system of Anunnaki and Igigi from the Mesopotamian legend. At the dawn of human civilization, 2 peoples were tied by master-slave relations. No international trade. Nowhere to go. They were restricted in this area, dug channels, and grew grain for food. In such conditions, population of both nations is still quite representative. Let's try and imagine how it was regulated. The main difference from wild nature is production. Volterra suggested that prey animals have unlimited food and no problems to find it. Here, the grain should be produced yet and this production requires substantial efforts and spending resources such as grain from the previous harvest. The trick of slavery is that a man spends, but gives away the product. The value of compensation is decided by his master. We may suggest that the rate of reproduction for slaves depends upon their living conditions. The first member of the first equation may be left without changes. It represents some normal standard. The second member will represent variations. It is hard to imagine that a slave owner will increase compensation, but he can decrease it in such cases as poor harvest or a war. In real life, these problems emerge permanently so slaves will always receive less and the decrease will be proportional to the number of masters. As to the second equation, its second member represents natural mortality which is affected by income. When workforce is in abundance, it is low. If the number if slaves drop substantially, slave owners launch a war and actively reduce their own population. The first term here is similar to the first equation.


dx/dt = alpha * x - beta * y

dy/dt = delta * y - gamma * y / x

We will put aside the application of this theory to socialism and feudal system, but capitalism is of special interest. This type of economy is based on competitive self-regulation so is exactly what's needed. In fact, free market reproduces relations of wild nature inside artificial civilization. So what variables should we choose here? First of all, developed market relations offer yet another quantitative measure - money. Meanwhile population remains an option as well.

Again, we need 2 groups. Moving from slavery, these will be employers and workers. Interesting that on some Caribbean islands known for slavery, sugar cane remains a monoculture and the main drive of the economy, but now it runs under the capitalist model. Usually the business is run by a foreign company so the condition of restriction is broken. Capitalism is globalized. Nevertheless, we can imagine a local employer and compute the model on such presuppositions. The reality is such that the workers are paid small wages and they have no leverage to demand increase. The only difference probably will be minor changes such as greater alpha and less beta. Also, capitalism allows free migrations and people can move from one group to another. Remaining in the linear paradigm, we could add a member like + beta1 * y to the first equation. This would account for bankrupt entrepreneurs, but it just changes the value of beta. On the other hand, some workers can get rich and open own business thus adding yet another term to the second equation:


dx/dt = alpha * x - beta * y

dy/dt = delta * x * y - gamma * y + epsilon * x


This model can also render a cyclic output, but it is rather difficult to find a proper combination of parameters.

Change just one of them and it becomes unstable.



Separately, we can develop similar formulas in terms of money, that is, the total wealth of the ruling class and workers. If we combine approaches, it is possible to calculate average wellbeing for individuals from both groups.



Copyright (c) I. Volkov, March 16, 2020