Invariance of Population Survival Functions

The present study investigates an invariant property of the class of population survival functions that characterize the dynamics of population systems. Building on our previous group-theoretical analysis, we establish that the set of all monotonic survival functions is closed under the operations of translation (shift), rotation by an angle \(\pi\), and the transformation \(1 – \alpha(z)\), thereby forming the group \(K \times G\). It is shown that this group admits a representation in the form of the group \(SM(3, R)\), which provides a constructive description of the corresponding transformations. Within this framework, it is proven that an arbitrary monotonic survival function can be represented as α(z) = a₂₃ + a₂₂ exp(−|a₁₁z(t) − a₁₃|), taking into account the elements of the transformation group \(SM(3, R)\). The obtained results suggest the existence of a universal invariant property of the class of population survival functions, which is preserved irrespective of species identity.