If an apocalypse were to occur, and the only survivors were a father, mother, son, and daughter, they would indeed face significant challenges in repopulating the Earth. Repopulating the human species would require a genetically diverse and sustainable population, and in this scenario, with only four survivors, they would have a very limited gene pool.
The challenges they would face include:
Limited genetic diversity: With only four individuals, there would be a severe lack of genetic diversity. This increases the risk of genetic disorders and reduces the overall fitness of the population. Over time, inbreeding could lead to harmful genetic mutations and reduced fertility.
Reproductive limitations: The mother and father may be of reproductive age, but the son and daughter might not be old enough to have children yet. Waiting for the son and daughter to reach reproductive age could delay the population growth.
Small population size: A small population is more vulnerable to diseases, environmental changes, and other hazards. A single catastrophic event could easily wipe out the entire population.
Lack of resources: After an apocalypse, resources like food, clean water, and medical supplies may be scarce. Meeting the basic needs for survival would be a significant challenge.
Psychological and emotional challenges: The trauma of surviving an apocalypse and losing the rest of humanity could take a severe toll on the mental health of the survivors.
Social and educational limitations: With only four people, it would be challenging to pass on a diverse set of skills, knowledge, and culture to future generations.
Considering these challenges, it would be extremely difficult for just four individuals to repopulate the Earth and build a sustainable human civilization. To achieve a viable population, a minimum number of genetically diverse individuals is necessary. Estimates vary, but it is generally believed that a few hundred individuals would be required to have a reasonable chance of long-term survival and avoid the pitfalls of inbreeding depression.
To ensure the survival of the human species, a larger group of diverse survivors, with skills and knowledge to sustain and rebuild society, would be needed. Cooperation with other surviving groups, if any, would also be crucial for the long-term success of repopulation efforts.