Is there any formal definition of a relationship between resource availability and demand?
I'm going to take a stab at providing the information that I think that you want, but I won't be answering the question that you asked. If this doesn't help you, consider changing the question as I don't think that we are necessarily understanding it.  

TL;DR:  Resource availability and demand are mostly unrelated.  If anything, low resource availability will reduce demand (in the long term, not the short term).  

Resource availability has no direct effect on the demand curve.  The demand curve reflects how people value the resource.  

Availability determines the supply curve, which measures the cost of production.  I.e. at a specific quantity supplied, the supply curve will give a specific price.  That is often a marginal price.  That is to say, it is the price at which the last unit can be produced.  

In general, in a market economy, the quantity supplied and the quantity demanded are the same.  And that value occurs when the demand and supply curves intersect.  

I've been using a term, quantity demanded, with which you may not be familiar.  It may be that what I call quantity demanded is what you are thinking about when you say demand.  That happens sometimes.  Demand in economics is generally a graph of how the quantity demanded will change based on price.  

In general, if a resource is constrained, this moves the supply curve (and may change its shape).  A constraint will move the supply curve to the left.  Since supply curves generally slope up and demand curves slope down (the higher the price, the more people will find a way to supply; the higher the price, less people will demand at that price), moving supply to the left will reduce the quantity supplied/demanded.  

Meanwhile, there will be a bunch of people who would like to buy the resource but not at that price.  These also may be the people you think of as "increased demand."  From an economic perspective, this unmet demand is not measured beyond that.  It is the portion of the demand curve to the right of the quantity demanded.  That area will increase if the demand curve stays the same while the quantity demanded shifts left.  Most economists would not refer to this as increased demand, but you may be.  

Now all that's the short term reaction.  In the long term, lack of a resource may cause demand for the resource to drop.  

Consider gasoline.  Let's assume that it becomes constrained.  Initially, in the short term, my only real options are to pay more or drive less.  But in the long term, I can get a different car.  My new car may run on natural gas or electricity.  But if I do that, then the gasoline demand curve changes.  Not by a lot, as I'm only one person.  But if many other people do the same thing, collectively we shift the demand curve.  It would be reasonable then to say that lack of availability reduced demand in the long term.