Invariance of Population Survival Functions

This study investigates a fundamental property of population systems—namely, the invariance of the survival function. 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 $mathrm{G}$. It is shown that this group admits a representation in the form of the group $mathrm{G}_s$, 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 $alpha(z) = exp(-k|z-z_0|)$, taking into account the elements of the transformation group $mathrm{G}_s$.The obtained results are generalized in the form of an invariance principle of the survival function, which holds independently of the type of environmental factors and the species identity of the population. This supports its interpretation as a fundamental biological law.