This is not true. The demand for research animals is huge, and growing. Due to extremely restrictive laws and procedures governing the use of animals in research in the US and the EU (source: my own experience), virtually all interesting research (=breakthrough/publishable) becomes impractical.
What has already happened is that all interesting research moved to countries with less restrictive laws, namely China (source: direct exposure to my Chinese colleagues in neuroscience). With their vast and unrestrictive funding (CAS etc.), it is the Chinese who are now coming out with some fascinating results. More than that, some of my EU/US colleagues collaborate directly with them, on the exact same research, just on the Chinese soil (read: inconvenience of relocation, but that's about it).