Lotka Volterra Models: A mental model

In 1910, Alfred J Lotka discovered something interesting. And he found it in a corner which was so unlikely that it almost lay in plain sight.

But before that, a bit of background. In 1828,a man named Fechner described an electrochemical cell which produced oscillating current.  At that time it was the first report on a chemical oscillation.

Then a few years later reports of oscillatory behaviour of chromium in acid dissolution came up. In other words, the amount of chromium which dissolved in acid was increasing and decreasing in a predictable manner.

To say that it baffled scientists would be an understatement.

 

However Lotka, was an eclectic character. He was a trained biostatistician, mathematical biologist and then turned into a patent examiner,editor and later worked in an insurance company for the major part of his life. And this insurance fellow discovered something which will seem as fresh as today.

He found that in chemical reactions where

1. One reactant is in abundance [A], if reacts with another reactant X to yield another molecule of X itself (which implies post reaction there are two molecules of X), and

2. X reacts with Y to yield another molecule of Y itself (which again implies that post reaction there will be two molecules of Y). and

3 Lastly, Y decays and transforms into P (dropping out of the system)

Then, in that case X and Y will show an oscillatory behaviour. And that is the reason why reactions like the one above display such kind of oscillations.

However, this phenomenon is extremely fertile. In 1928, Vito Volterra discovered that such kind of phenomenon is not only limited to chemical reactions but also in nature. Specifically the population of salmons in Adriatic Sea.  And this phenomenon was called thereafter Lotka-Volterra Model.

Its easy to draw analogies between nature and this chemical reaction.

Consider an ecosystem with plentiful of grass [A], cows [X] and some wolves[Y]. The initial condition is that grasses have to be abundant in the system and the wolves have to be miniscule in number to cows.

So what happens is initially, because there is a lot of food(grass) and very less natural predator(wolves), cows thrive ( remember “A reacting with X to yield another molecule of X itself”). Thus their population keeps on increasing.

But increasing population of cows also prove to be beneficial for wolves. Because for wolves the food is aplenty. As a result the population of wolves grow.

However as the number of wolves grow, the cow population undergoes a fall (because they are getting eaten a lot). As a result a time comes when cow population falls below a certain number, triggering in a slow fall in wolf population.

As the wolf population undergoes a fall, the cow population slowly starts heading up. This continues till the whole system repeats itself.

So if we trace their population with respect to time, the following behaviour will emerge:

The Lotka Volterra Model displays a oscillatory behaviour between predator and prey

The Lotka Volterra Model displays a oscillatory behaviour between predator and prey

Does it hold in real life?

You bet!  Huffaker showed the efficacy of this model in real life using two species of mites and found this :

Huffaker's work in 1958

Huffaker’s work in 1958

But what has this to do with micro economics!

A lot actually. Consider two sectors which compete with each other, e.g. ebook publishers and physical publishers.  Then it can be shown that under certain circumstances- the number of ebook publishers and physical publishers can

  • enter into a symbiotic competition (co-existence) or can lead to
  • extinction of one of the industries.

Think of  the system as this:

The rate of change of ebook publishers depends upon natural growth rate of ebook users minus self influencing effect of ebook users plus the proportion of users switching from physical books to ebooks.

Similarly,

The rate of change of physical book publishers depend upon natural growth rate of physical book users minus self influencing effect of physical book users plus the proportion of users switching from ebooks to physical books.

So, the self influencing effects can be called as “lock in effect” or how strong it is. It is possible that there is no lock in effect in a particular area. Then that parameter will be zero. Now lets think about it- if we have an ebook are we locked out of using or enjoying physical books? I doubt so.

However, if I am a physical book user, doing vertical reading on computer monitors is difficult. Hence there is a definite switching benefit and hence a non zero parameter.

Similarly a network effect can also be thought about ebook industry. Being surrounded by more ebook users makes my preference for ebooks more convenient.

Here Physical book industry is the cow(predator), ebook industry is the wolf (predator) and say some part of ebook market undergoes a natural decay (and not growth) e.g. most of ebook readers are young adults and as readers grow up, they stop reading books(assume).

It can be shown:

1. If the (predation rate/decay rate) of predator industry  is less than or equal to (the network effects/growth rate) of prey industry, then predators will go extinct, regardless of initial condition.

2. In all other circumstances, there will be a symbiotic competition.

The logic of such a condition makes inherent sense. What does \frac{predation rate}{decay rate} \leq \frac{network  effect}{growth rate} mean?

It just means, either I am a very bad predator(i.e. I just can’t hunt), or that I die away very fast i.e my life expectancy is low- so I might be a good hunter, but I just dont live beyond a certain maximum which makes it incredibly difficult for me to survive,let alone make a reasonable dent in the prey population.

Or that, the preys develop certain advantages as they grow in scale(network effect) which leads to the fact that preys get hunted less. Or the growth rate of preys are so low, that they cannot support any population of predators over them.

So drawing business analogies:

Predator firm Characteristics:

1. Predation Rate: Switching rate of customers from original firm to the new firm.

2. Decay Rate: The internal growth rate of predator firm is too low, e.g. Return on Capital for predator firm to attract customers from incumbent firm will be negative, i.e. a presence of very strong and robust moat.

Prey firm characteristics

3. Network Effect:  The benefit received to customers by being part of the large customer base served by the firm. e.g. an exchange business. There is a strong “winner takes all” syndrome seen, because there is a phenomenon of increasing returns at play.

4. Growth Rate: That is the business as such is too unattractive to really poach on. For example at the beginning of 19th century, any industry trying to “live off” a capital goods industry will automatically die off, because by 1800s Industrial Revolution didnt even start!

It can also be remembered that if a prey firm has very less margins, then also it stimulates this effect. As they say there is “safety in low margins”.

In this case of ebook industry vs publishing book industry, I am inclined to think that the market of ebook users is growing, predation rate of physical book publishers are high.

Also, there is very low network effect of physical book industry.

As a result, I believe that there will be a symbiotic competition in ebook industry.

However the model parameters will keep changing, till the point where there will be a huge critical mass of ebook publishers and ebook users. This will generate its own network effects thus making physical books the predator and ebook industry the prey.

And in that system, the physical book industry will die out.

Already we are seeing the rise of network effects with the coming of Kindle and other such readers.

The paper connecting Lotka-Volterra Mental Model with MicroEconomics can be found here. Do give it a read.

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