Preliminary notes on the equilibrium vampire population

There is no question in my mind that there are vampires.  The question really should be how many vampires are there, and, as a corollary, what is the optimal strategy for a single vampire?  Essential considerations:

• I believe the basic dynamics can be adequately modeled using a slightly modified Lotka-Volterra equilibrium model.
• However, vampires can theoretically live forever.  This raises the stakes considerably for them.  Even apparently minor threats should be eliminated, as the present value of eternal life is quite high. 
• Threat #1:  Starvation.  If the Lotka-Volterra dynamics get out of hand, we will be up to our asses in vampires, and they will kill and eat every human being and then all starve to death.  So, to prevent this eventuality, each individual vampire has a strong incentive to kill any other vampire it comes across.
• Threat #2:  Discovery.  While weak, humans are somewhat intelligent and can take countermeasures if a vampire is discovered.  It is therefore paramount that vampires be discrete, operating in areas where communications and mass media are undeveloped and the victims ill-informed about vampires.  It also provides a strong incentive to kill any vampires that appear not to be sufficiently discrete.  After all, the best defense is to be so obscure that no one believes you even exist.
• Threat #3:  Other vampires.  Since Threats #1 and #2 both give vampires huge incentives to kill other vampires, it follows that the most serious threat to vampires is other vampires.  Since vampires are the greatest threat to vampires, this gives any one vampire significant additional incentive to kill other vampires.

Therefore, the vampire population is likely to be significantly lower than the Lotka-Volterra prediction.  This, in turn, leads to an overabundance of humans, and the problems resulting from that are evident.