Peter and Anti-Peter Principle as the Discrete Logistic Equation
aa r X i v : . [ phy s i c s . g e n - ph ] J u l PETER AND ANTI-PETER PRINCIPLE AS THEDISCRETE LOGISTIC EQUATION
Vladan Pankovi´cDepartment of Physics, Faculty of Sciences, 21000 Novi Sad,Trg Dositeja Obradovi´ca 4. , Serbia, [email protected]
Abstract
In this work Peter principle (in the hierarchical structure any competent member tendsto rise to his level of incompetence) is consistently interpreted as the discrete form of thewell-known logistic (Verhulst or Maltusian) equation of the population dynamics. Accordingto such interpretation anti-Peter principle (in the hierarchical structure any incompetentmember tends to rise to his level of competence) is formulated too.
As it is well-known remarkable Peter principle [1], [2] states that in the hierarchical structureany competent member tends to rise to his level of incompetence. Even if Peter principle is seem-ingly paradoxical it is in a satisfactory agreement with situations existing in real social hierarchicalstructures. There are different attempts of the interpretation or mathematical foundation of Peterprinciple. In this work an original interpretation will be suggested. Namely, in this work Peterprinciple will be consistently interpreted as the discrete logistic (Verhulst or Maltusian) equationof the population dynamics. According to such interpretation anti-Peter principle (in the hierar-chical structure any incompetent member tends to rise to his level of competence) is formulatedtoo.Thus, as it is well-known logistic (Verhulst or Maltusian) equation in the population dynamicshas form dxdt = a x (1 − xr ) for a, r > x ≤ r (1)where t represents the time moment, x - (human or some other species) population, a - growthparameter and r carrying capacity. Simple solution of this equation, representing a sigmoid func-tion, is x = x r exp[ a t ] 1 r − x − x exp[ a t ] (2)where x represents the initial population smaller than r . Obviously, when t tends toward infinity x tends toward r and dxdt toward zero. Given logistic dynamics describes population growth limitedby negative species self-interaction.It is well known too that there is anti-logistic equation corresponding to (1) dxdt = − a x ( xr −
1) for a, r > x ≥ r (3)1here − a represents the decrease parameter. It holds simple solution x = x r x − ( x − r ) exp[ a t ] (4)where x represents the initial population smaller than r . Obviously, when t tends toward infinity x tends toward r and dxdt toward zero. Given logistic dynamics describes population decreaselimited by positive species self-interaction.Finally, it is well-known that both, logistic and anti-logistic, equations have significant ap-plication not only in population dynamics, i.e. in the biology and demography, but also in thechemistry, mathematical psychology, economics and sociology.We observe that sigmoid form of the solution of logistic equation satisfactory corresponds topredicted and observed form of the successful member competence time evolution. For this reasonwe suppose that, in the first approximation, successful member competence time evolution isdescribed by (1). But now x represents competence in the time moment t , x - initial competence, a - competence growth parameter and r - level of incompetence, all of which are characteristic forgiven member.But in distinction of a biological species where individuals number can be very large so thatpopulation can be effectively treated as a continuous variable satisfying logistic differential equation(1), number of the members in a hierarchical sociological structure (e.g. factory, university, etc.)can be relatively small. For this reason member competence can be a discrete function thatsatisfies a discrete dynamics. Nevertheless, it is not hard to see that such discrete dynamicsmust correspond to the discretized form of the logistic equation (1) (which will be not discusseddetailedly). According to such discretization any competent member of the hierarchical structurecan rise to its incompetence level in a finite time interval.Moreover, mentioned interpretation of the Peter principle (by discretized form of the logisticequation) admits formulation of the anti-Peter principle by discretized anti-logistic equation (3).But now x represents competence in the time moment t , x - initial competence, − a - competencedecrease parameter and r - level of competence, all of which are characteristic for given member.This principle states that in the hierarchical structure any incompetent member tends to rise tohis level of competence. It is, of course, in full agreement with discretized form of (4) (which willbe not discussed detailedly). Also, according to such discretization, any incompetent member ofthe hierarchical structure can rise to its competence level in a finite time interval.In conclusion, it can be shortly repeated and pointed out the following. In this work Peter prin-ciple (in the hierarchical structure any successful member tends to rise to his level of incompetence)is consistently interpreted as the discrete form of the well-known logistic (Verhulst or Maltusian)equation of the population dynamics. According to such interpretation anti-Peter principle (inthe hierarchical structure any non-successful member tends to rise to his level of competence) isformulated too. [1 ] L. J. Peter, R. Hul, The Peter Principle: why things always go wrong (William Morrowand Company, New York, 1969)[2 ] A. Pulchino, A. Rapisarda, C. Garofalo,