The three most basic and important things that control evolution in nature are : Selection, Replication and Mutation. They have been proved applicable to any biological organization in this universe or anywhere else. They frame the fundamental blocks to any growth of an organism, residing in any ecosystem. But these, and especially one of the three, also form the groundwork for the famous Darwin theory “ Survival of the Fittest “. I will, kind of, explain to you the maths behind it. But before we plunge into the proof I will give you some rudimentary concepts and ideas that would make your understanding easy.

The growth law for a sample expansion can be written as a recursive equation. This depicts a case where a mother cell gives birth to two daughter cells.

x(t+1) = 2x(t)

This is a discrete function where we consider that the population will always be double after the time gap. We may also take an exponential function or a differential equation which will be more accurate to produce the exact numbers and trend of growth.

Differential Equation depicting an ideal population growth :

x’ = f*x

Here x’ depicts the rate of change of population, x being the population. Here f is a factor that we will talk of next. This factor takes into consideration that there are deaths as there are births.

But for the sake of proof and to make it easy to look and understand I will be considering the differential equation written above. For your interest, you may try and prove the same with the help of discrete function or exponential function and it would give the same result.

Next thing that I would like to explain before I use is a factor called fitness factor. Mathematically, it is the ratio of births to deaths in a given span of time. Ideally, a fractional fitness factor would lead to extinction of the race ( as there are more deaths than births ). A fitness factor more than one will lead to growth of the race. A stable state is when the factor is 1, where the race will have a fixed number of organisms alive at any point of time. Any deviation ( be it more death or more birth, that is deviation from the ideal ) would result in either exponential growth or extinction of the race. We need to keep this in mind as this factor plays a huge role in depicting the rate of growth of the population.

Now we are ready to proceed further into the details of the proof. Keep the two things, I mentioned above, in mind and let's do it !!

Let us take two subpopulations of organisms , A and B. Their respective fitness factors be a and b. Without loss of generosity, let us take a > b ( which implies A reproduces faster than B ). x denotes the population of A at any point of time. y implies the population of B.

From the equations discussed above,

(x)’ = a*x

(y)’ = b*y

These equations depict the growth of subpopulations A and B.

Now we define another factor D, which is the population density of A with respect to B. Mathematically,

D = (x)/(y)

Now

D’ = {(x)’(y) - (x)(y)’}/ (y*y) = (a-b)*D

This equation derived above helps us compare the relative growth of populations A and B. If we closely observe the above equation, we may dawn upon these three results :

● When a>b, A grows substantially high and deletes the existence of subpopulation B.

● When a<b, B grows exponentially and A nears extinction.

● Lastly, if a=b, their relative populations remains constant ( not necessarily the exact population of A and B)

( Note: We consider a finite earth, i.e., the space containing the subpopulations is not infinite and has a limit as to how much population it can hold )

Among all these, the unstable equilibrium state, i.e., case 3, has the least chances of happening and continuing in an ecosystem ( because in the real world, both face challenges to survival and it has the least probability that both survive the challenges equally ) . So, we happily rule out that case. For each of the other two cases one of the two populations exists while the other depletes.

Till this stage, we were considering only 2 subpopulations. But in reality, there are thousands more and for proving the statement in all possible cases we take n subpopulations.

For every subpopulation, i = 1,2,3………,n, we have their respective fitness factors. Now taking a combination of any two of these n subpopulation, we can safely say one of the two exists, while the other depletes. We take the winner from this combination, and compare it with another existing subpopulation ( we consider the loser from the combinations to be non-existent as it will eventually go extinct ). We get the winner here and keep doing these steps till we have only one existing subpopulation. We have the winner which can rightfully claim all the space to itself, while all others become vulnerable to extinction.

What we proved above is what we know as the “ Survival of the fittest “. That is, in a finite ecosystem containing n populations, the master, which has the highest fitness factor and hence the highest ability to safely reproduce, survives the Endgame.

Bibliography :

Evolutionary Dynamics, Harvard University Press