Final answer:
The likely mating type for female spotted sandpipers that fight over territories is polyandrous mating, where one female mates with many males. This is evidenced by females controlling resource-rich territories and potentially selecting from multiple males for healthier offspring, a scenario seen in seahorses and pipefish.
Step-by-step explanation:
The mating type most likely operating in female spotted sandpipers, which fight for resource-rich territories on their beach breeding grounds, is polyandrous mating. This system involves one female mating with multiple males, which is supported by the behavior of females establishing and defending their own territories. In contrast, polygynous mating would involve one male mating with multiple females, typically in situations where the male controls a resource-rich territory that attracts several females, rather contrary to what is observed in spotted sandpipers.
Females in polyandrous systems can attract multiple males to the territories they control, thus potentially allowing them to select from a range of suitors which would ensure greater genetic variability and healthier offspring. An example of polyandrous mating can be seen in seahorses and pipefish, where after mating, males carry and care for the offspring. This leaves females free to mate with other males, theoretically resulting in many males being attracted to a single female with a desirable territory, as is the case with the spotted sandpipers.