I'm a population geneticist by profession, currently working on a couple of population eradication projects - notably to do with leishmaniasis, sleeping sickness and malaria.
When you reduce an eradication problem down to a set of algorithms, there's several essential components -
Nc - census population size: This is the absolute number of individuals in a population and ultimately what you want to reduce
Ne - effective population size: This the number of breeding individuals actively contributing to the gene pool of a population. Reducing this reduces the genetic viability of a population
Pgr - population growth rate: The number of individuals coming into a population either through breeding or migration
Pdr - population decline rate: The number of individuals leaving a population either through death or emigration
K - carrying capacity: The absolute number of individuals which can persist in a given spatial area.
There are a series of equations and summary statistics you can generate from genetic sampling of a population to generate predictive models, to examine how a particular mechanism will impact each of these parameters. While it may be counter-intuitive at first glance, if you don't significantly increase Pdr relative to Pgr, the Nc remains at K (exactly what you want if you're say, managing a fishery). Furthermore, if you aren't removing individuals who are members of the effective population, reductions in Nc are temporary with no overall effect on the genetic density of the population. Therefore for recreational hunting to be considered an effective control measure, it needs to increase Pdr significantly in relation to Pgr, and it needs to be removing members of the effective population.
There have been a number of studies done on deer, pigs, goats, etc. which demonstrate that the impact of recreational shooting on Pdr in these populations is generally insufficient to significantly reduce populations below K, as the Pgr exceeds changes to Pdr. Furthermore, in many cases hunters prefer to remove large males rather than breeding females, thus not affecting Ne and deliberate releases to sustain populations for hunting have been observed.
E.g.
http://www.jstor.org/stable/2403610
http://www.publish.csiro.au/paper/WR02100
http://143.188.17.20/data/warehouse/mvpfgr9abr_001/mvpfgr9abr_0010111a/ggchap1_6.pdf
However in some cases, maintenance of Nc below K through recreational hunting have been observed (e.g. in prairie dogs
http://www.jstor.org/stable/3802300) but eradication was shown to be unachievable.
So, in
most cases, recreational hunting can't be relied upon as an effective population control measure for pest eradication - despite it often being a politically attractive option. The number of animals killed by hunters would need to drastically increase in many cases, and be maintained at these much higher levels for any real impact to be achieved.
The remaining arguments for hunting usually center around the right to recreation, and the economic impacts of hunting.