Natural tendency that adults have to take care of their small ones

Solution 1:

Maternal bonding

The maternal bond (or motherly bond) is typically the relationship between a mother and her child.

While it typically occurs due to pregnancy and childbirth, it may also occur between a woman and an unrelated child, such as in adoption.

The above term is included in the larger sense of bonding in general

In 1935, British developmental psychologist John Bowlby published the ground-breaking paper "the Nature of the Child's Tie to his Mother," in which the precursory concepts of "attachment theory" were developed. This included the development of the concept of the affectional bond, [or] emotional bond [...]. Attachment theory has some of its origins in the observation of and experiments with animals, but is also based on observations of children who had missed typical experiences of adult care

A more scientific term which describes the phenomena of parents investing time, energy, special attention and care on their off-spring is

Parental investment

Parental investment theory accounts for many of the differences between males and females: [...] Human males spend more time caring for their offspring than other male mammals. This higher parental investment is the result of extended childhood of human offspring. [...] However, this requires parental investment in the form of parents ‘leading the way’- teaching and protecting children. Abandoned children may be left to die, though in some cases societies have developed various means of caring for them. Males do spend time caring for their children but to a much smaller degree than mothers. This translates into a general observation that females’ parental investment is much greater than that of males, both before and after childbirth.

Solution 2:

Maternal (or paternal, or parental) instinct seems to work fine for me.

Solution 3:

Tending or tending instinct are used to reference mature creatures who care for young creatures.

http://newsroom.ucla.edu/releases/What-Motivates-Human-Behavior-3328

Solution 4:

Nurturing instinct and (depending on the context) altruism.