It is often challenging enough to decipher the mating and reproductive habits of extant animals (humans are no exception!). To understand the same behavior in extinct species is obviously far more difficult. However, this has not prevented people from trying to understand propogation in dinosaurs. The main approach is to infer behaviors based on characteristics of crown groups, extant descendants of the extinct animals in question. For dinosaurs, two divergent crown species are birds and crocodiles, which bracket dinosaurs. If both birds and crocodiles share a particular trait, then the extinct dinosaurs could have it as well. These first order inferences are relatively the most reliable. Second order inferences, which depend on characteristics in only one crown group, or third order inferences which are not based on crown groups at all, are often less reliable. If the crown group has a recent common ancestor with the extinct group, then second order inferences can be more reliable. Hypotheses developed using crown group inferences can then be compared with those derived from geological and morphological observations in the fossil record. That being said, there are a number of interesting ideas on how dinosaurs may have propogated.

Courtship
There is fossil evidence for sexual dimorphism (females and males have distictive body shapes), as determined by a bimodal distribution of body morphologies, suggesting that this could contribute to mating behavior. If the bimodal distribution of body types is very uneven, then it might represent two different species, one being more abundant than the other.

Many dinosaurs with elaborate headgear (e.g. horns, crests, shields, spikes, etc ...) such as hadrosaurs, marginocephalians and ceratosaurs, may have used the implements in mate competition fights. Visual impressiveness is often important in mate competition ... and larger, more ornate headgear might intimidate other suitors.

Eggs and Nests
Because of the extensive fossil record of extinct dinosaur eggs and eggshells, it is well established that dinsaurs layed eggs. Additionally, this is a first order inference based on extant crown groups. However, only nine species of dinosaur have been directly linked to egg laying by fossilized ova still in the egg. The eggshells of dinosaurs also have a microarchitecture that is significantly different from that of either crocodiles or birds. Features such as size, microstructure, pore distribution, surface patterns and such help scientists understand the embryonic stages of dinosaur development, based on inference.

In terms of nesting, dinosaur eggs are either found isolated, in disorganized groups, or in well organized clutches. Some dinosaur eggs have been found in rows of various lengths ... suggesting a linear deposition during laying. This suggests that nesting is species specific, with some dinosaurs dropping single eggs that ovulated and shelled quickly, and some which were layed en masse, as reptiles do. There is geological evidence that egg aggregations are often found in shallow pits, which could be evidence of nest building.

Incubation
Crocodiles incubate eggs passively by storing them in covered mounds. Birds have specfic brooding behaviors for active incubation. Fossil evidence exists of adult theropods straddling egg clutches, although this has been interpreted as either incubating, or nest plundering.

Geographic Extent
Some dinosaur eggs are found over huge geographic areas, but there are some examples of "site fidelity", where a specific local has a large accumulation of eggs, suggesting it may have been used for periods of thousands of years.

Embryos
Very few embryonic or neonatal dinosaur skeletons have been found. Even for those found, its often difficult to assign the adult species as morphologies can change drastically during growth. Juveniles across species often look very similar. As a result, no positive identifications of species at this level have been successfully made.

Parental Care
If dinosaurs grew at rates equivalent to their extant ancestors, it is believed that they wouldn't reach maturity for several decades, although other studies have suggested that dinosaurs grew more quickly than modern relatives. Thus, it is unclear from growth rates alone whether extended parenting was necessary.

Evidence of nestlings, young dinosaurs whose bodies had not developed enough to allow significant locomotion, have been discovered. Although it is hard to imagine, given the huge discrepancy between adult and child body sizes, that a parent could be able to attend the needs of tiny offspring without crushing them.

In comparison, crocodile hatchlings do receive protective care by adults. Probably, parental care among dinosaurs was again a varied phenomenon, depending on the specific taxa.

For a good review of this topic upon which this node is largely based, read:
Dinosaur Reproduction and Parenting by John R Horner, in Annual Reviews of Earth and Planetary Sciences, (2000) v28, pp 19-45

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