Normally penetration of the egg by the sperm cell signals the start of the developmental process. What can replace this signal when no sperm are involved?

Answer: parthenogenesis (apomixis and automixis)

Explanation: parthenogenesis can be simply defined as a form of asexual reproduction where the developmental process begins i. e egg develop into an embryo without the prior fertilization from sperm cells.

Egg cells or sex cell (oocyte) are reproduced by replications which produces two diploid cells having full complement of chromosomes that is required for the developmental process to begin.

This form of reproduction occurs in plants as well as some lower invertebrates such as anthropods (insects) and some species of aves, amphibians and reptilians.

It should be noted that parthenogenesis doesn't occur in mammals.