This case can be used as a fine example of factual research, economic theory and applied econometrics.
Since the authors examine a specific real-world case, the actual facts can help to establish a plausible causal sequence.
For example: was there a rush in the numbers of elephants killed, in response to which authorities announced and implemented a more aggressive and effective seizure policy?
Or, although the numbers of elephants killed remained relatively stable per period, but, the authorities, eventually yielded to political pressure and implemented this more aggressive and effective seizure policy?
Presumably, there are data and information to conclude on the above. Assume that it is the second situation. Then...
Argue that "Economic Theory tells us that..." and lay out the theoretical argument that the OP described in his question. Then
Estimate the relationship between the magnitudes of interest, in order to check whether the theoretical argument is consistent with the available data. Here, one should control for other factors, for example as @denesp writes, was there a rush of unemployment in other sectors of the economy that could have contributed to labor turning for work to the black market for ivory?
Illegal goods are very interesting, because supply deviates from production by the amount of goods confiscated prior to reaching the market. As long as the demand is there, increased confiscations will tend to increase production also, which tends in turn to offset the intended result of the seizure policy.
Assume now that the record shows that first we had an increase in the numbers of elephants killed, and then the authorities responded with increased seizures. Then one should establish first the likely reasons for the initial increase in production/supply. Are there indications that demand shifted? Are there indications that supply increased because of unemployment in other sectors? Assume that it was unemployment that pushed people to this activity, and that there is no indications that demand shifted. Then economic theory would predict that price should tend to initially fall. Then the authorities stepped in harder, and mitigated the increased supply, pushing the price back up.
But if it was demand that originally shifted, then we should observe an initial rise in price, then an increase in production and a tendency for the price to fall, and then the effect from the harder seizure policy.
Hopefully, since things in the real world do not happen all at once, there could be indications in the data on these price movements.