$C_{E}PT$ Symmetry of the Simple Ecological Dynamical Equations

Abstract

It is shown that all simple ecological, i.e. population dynamical equations (unlimited exponential population growth (or decrease) dynamics, logistic or Verhulst equation, usual and generalized Lotka-Volterra equations) hold a symmetry, called $C_{E}PT$ symmetry. Namely, all simple ecological dynamical equations are invariant (symmetric) in respect to successive application of the time reversal transformation - $T$, space coordinates reversal or parity transformation - $P$, and predator-prey reversal transformation - $C_{E}$ that changes preys in the predators or pure (healthy) in the impure (fatal) environment, and vice versa. It is deeply conceptually analogous to remarkable $CPT$ symmetry of the fundamental physical dynamical equations. Further, it is shown that by more accurate, "microscopic" analysis, given $C_{E}PT$ symmetry becomes explicitly broken.