I am not into all the tech details of communication via RF with implant chips (tracking chips) and so would like to ask how the known atmospheric pertubations of shortwave radio affect the distances over which one can closely and continuously track an animal with an implanted chip (e.g. a chip transmitting physiological information).
I think you are confusing a few different technologies.
GPS only receives signals from the GPS satellites, for this it just needs an external antennae. It isn't going to work with an implanted device in a dog.
Radio directional finding (RDF) transmitters used to track migrating animals broadcast a continuous signal which is followed by simply looking for the direction that the signal is strongest and assuming the animal is in that direction. Again you need an external antenna and more power than GPS (modern tag systems actually find the animals position by GPS and broadcast it by cell phone)
RFID comes in two types, powered and unpowered. If the tag has power it can broadcast a stronger signal or use a smaller antennae and work over 10s of meters. An unpowered RFID tag - like the tags used to 'chip' pets uses power broadcast from the reader and only work over a few cm.
There needs to be more information to answer the question as stated, and it seems to have some misconceptions embedded.
It is highly unlikely that a device IN the dog is receiving GPS signals. "Shortwave radio" is typically though of as running from about 0.5 MHz to 30 MHz, and GPS operates at 1200 and 1500 MHz, so shortwave propagation issues don't generally apply to GPS. GPS receivers need power from somewhere, and implanting batteries in dogs would be unusual. It takes a fair bit of compute power and signal processing to access the GPS signal.
The thing in the dog is likely an RFID implant which knows nothing about where the dog is, but it will send a response when powered by an interrogator. These interactions take place in the near field, so "shortwave propagation" issues don't really apply. Finding the dog becomes an exercise like playing "Hot and Cold" where all you really know is the signal is stronger (closer) or weaker (farther away). This may answer your question.
Now if we are talking about a thing ON a dog, then we have enough power and space for GPS reception, and then we need to know something about the transmitter frequency and modulation (method of encoding the data). The transmission range could be yards, miles, or global depending on method.