A mathematical explanation of macro-evolution?

If I understand macro-evolution rightly, it means that some divine force switch from one species to another. Recalling that the development from fertilized egg (a unicellular organism) to adult individual (many-cellular) may be seen as a stepwise modified recapitulation of the evolution of the individual, there is perhaps no need for any macro-evolution, at least not for our own species. Because there are no big jumps in the recapitulation.
Nevertheless, we may speculate in some possible illusion of macro-evolution, due to the properties of Gaussian daptation.If I understand macro-evolution rightly, it means that some divine force switch from one species to another. Recalling that the development from fertilized egg (a unicellular organism) to adult individual (many-cellular) may be seen as a stepwise modified recapitulation of the evolution of the individual, there is perhaps no need for any macro-evolution, at least not for our own species. Because there are no big jumps in the recapitulation. Nevertheless, we may speculate in some possible illusion of macro-evolution due to the properties of Gaussian adaptation.

Nevertheless, we may speculate in some possible illusion of macro-evolution.The Darwinian evolution has been seen as a random process climbing a phenotypic value landscape with very many peaks and hollows. That such a landscape exists may be understood from the fact that certain DNA-messages more probably multiply and survive due to a higher fitness or adaptation of the corresponding individual. This ability may in the metaphor represent a higher altitude, which should not be confused with the equal dignity of all human beings.

If the gene pool of a large population is situated on a slope of the landscape, then more offspring will be generated on the higher side, which – because the variability and number of individuals must always be limited – causes the gene pool to move upwards towards some peak(s) in the landscape. As a result the mean fitness (i. e. the mean value of individual fitness taken over the whole population) tends to increase in the population. This, in turn, gives room for more phenotypic disorder (diversity), which may increase if the mutation rate is sufficiently high. Thus, we may have a simultaneous maximization of mean fitness and diversity.

Experiments with macro-evolution are hardly possible, but may perhaps be simulated on computers. Mean fitness may be calculated provided that the distribution of parameters and the structure of the landscape is known. The real landscape is not known, but the image below shows a fictitious profile (blue) of a landscape along a line (x) in a room spanned by such parameters. The red curve is the mean (lowered to improve readability) based on the red bell curve at the bottom of image. It is obtained by letting the bell curve slide along the x-axis, calculating the mean at every location.

As can be seen, small peaks and pits are smoothed out. Thus, if evolution is started at A with a relatively small variance (the red bell curve), then climbing will (at least theoretically) take place on the red curve. The process may get stuck for millions of years at B or C, as long as the hollows to the right of these points remain, and the mutation rate is too small.

If the mutation rate is sufficiently high, the disorder or variance may increase and the parameter(s) may become distributed like the green bell curve. Then the climbing will take place on the green curve, which is even more smoothed out. Because the hollows to the right of B and C have now disappeared, the process may continue up to the peaks at D. But of course the landscape puts a limit on the disorder or variability. Besides – dependent on the landscape – the process may become very jerky, and if the ratio between the time spent by the process at a local peak and the time of transition to the next peak is very high, it may as well look like a punctuated equilibrium as suggested by Gould (see Ridley, 1996) or a macro-evolution.

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