Chaos theory is considered by many modern mathematicians and scientists as one of the four pillars of modern science. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect. Chaos theory has applications in multiple scientific disciplines including: geology, mathematics, microbiology, biology, computer science, economics, engineering, finance, meteorology, philosophy, physics, politics, population dynamics,psychology, and robotics.
This article is an examination of the significance of chaos theory in the evolution of sexes in sexually reproducing species including humans.
Chaos theory in simple terms states that dynamical system which are deterministic (having finite numeber of states and depends only on the initial condition of the system) are unpredictable by nature. In real world this has big consequences because many systems in nature are extremely sensitive to initial condition (such as weather,stock markets,chemical reactions etc.) and which makes it very difficult to predict and control.
Bifurcation and strange attractors
One interesting property came from the study of chaotic systems is that there is a phenomenon called strange attractors. Some systems tends to be chaotic with most of input space(called phase space) except certain region of the phase space. The diagram below shows the two attractors of the system represented by the logistic equation x = 4 x (1 – x)
[image courtesy wikimedia.org]
In many of the systems this process tends to be centered around two attractors. And this is called bifurcation process. This phenomenon is of interest and significance in this article.
[image courtesy wikimedia.org]
Evolution of sexes
Evolution of sexual reproduction and sexes are of great interest among modern biologists and is one of the mysterious area in biological evolution. There are many competing theories trying to explain this process. Here the attempt is to give a mathematical dimension to the search.
As we now, through evolution, living organisms takes all possible forms and tactics to survive and propagate. In most of the species, it is striking that the number of sexes are limited to two(male and female). This is either by chance or limited by certain constraints exerted by nature(like mitochondrial efficiency in the nucleus). But the number two gives rise to the question of its relationship with mathematics.
Here are some patterns that connect the chaos theory with evolution
- The biological system(morphology) of any organism is dynamical system.
- Evolution process is mostly in the change in DNA.
- Sensitivity of the morphogenesis of organisms to initial conditions(the DNA).
- Sensitivity of the sexual dimorphism of organisms to initial conditions(X,Y chromosome region of the DNA)
- Large number of iterations over time to support bifurcation(the evolutionary timeline is of the order of billions of years).
- The evolutionary process tries to takes all forms(phase space of the life form) based on the change in DNA(initial condition)
- The number of sexes tends to attract to two(strange attractors,bifurcation?).
The figure below show the hypothetical bifurcation of the evolution of x and y chromosomes. Start from a common ancestor of chromosome pair x’x’ the ancestors of x and y sex chromosomes. Assuming there were no sexual dimorphism at this phase and the evolutionary change can be in any direction. But the number of sexes converges to 2 rather than n types able to reproduce with each other leading to xx(female) and xy(male).
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The chaos theory of evolution: An interesting article published in in New Scientist discusses similarity of evolution of life and non-linear systems, fractal nature of life and chaos.
My own conversation in TED.com ideas :
- Evolutionary origins of Cooperators and Defectors
- Cooperation and the evolution of anisogamy
- A chapter from the book The Score by Faye Flam