Leslie matrices

[TN96]

There is a relatively unknown world of vampires and werewolves among us. What you may not know is that werewolves can be transformed into vampires and vampires into werewolves. In fact,each year 30% of vampires become werewolves and10% of werewolves become vampires. Vampires cannot be killed, but werewolves can die and, in fact, lose 10% of their number each year in this way. The only way into this unknown world is to be bitten by a vampire, thus becoming a vampire.Only one in five vampires manage to successfully create a new vampire in a given year.

Describe how the population of vampires and werewolves will evolve over time. In particular, determine whether their numbers will stabilize, decrease, or increase and at what rate, and what percentages of this unknown world will be comprised by vampires and by werewolves.

I have no idea how to set this out with markov chains or leslie matrices.

What I've done so far:
So 0.7 vampires stay vampires
and 0.3 become werewolves.
But vampires reproduce at a rate of 0.2

So vampire population from one year to the next is 0.7+0.2 or 0.9


From vampire to werewolves is 0.3
And werewolves to vampire is 0.1.
So vampire population is 1 which they don't die.


But werewolves do die and they lose 0.1 to vampires
and 0.1 because they die each year.
So 0.8 live.

 

[HallsofIvy]

I am afraid you have completely misunderstood the problem and are just putting numbers together at random.

Suppose, at some year, there are x vampires and y werewolves (this is the part you missed!). 70% of the vampires remain vampires and 10% of the werewolves become vampires: the next year there will be a total of .7x+ .1y vampires. 30% of the vampires become werewolves and 90% of the werewolves remain werewolves: the next year there will be a total of .3x+ .9y werewolves. Write that as a matrix product.