Is there any formal definition of a relationship between resource availability and demand? I do not have any background on economics. So my question might sound too simple and I hope it to be clear enough. I am curious about the following: 

Is there any formal definition of a relationship between resource availability and demand? Intuitively I would expect that resource shortage, demand would increase. But this is not expected by the law of demand, assuming that the price of the resource would increase. Am I confusing something here?

Thank you,
best

EDIT 1: @Brythan Thank you for your answer. @EnergyNumbers and @erik thank you for your contributions. They were very useful to understand these concepts deeper. I was reading a little more about this and I wonder if the concept of scarcity is instead better applied here. I reformulate my question: 

(1) does supply-induced scarcity implies that demanded is greater than supply?

(2) what is the correlation between supply-induced scarcity and quantity demanded?

EDIT 2: To be more clear, I understand that scarcity does not necessarly refer "to whether a good is in shortage or surplus, nor to whether there are only a few or whether there are many". That is to say that scarcity is not the same as availability. Instead, scarcity "refers to the possible existence of conflict over the possession of a finite thing" (https://mises.org/library/goods-scarce-and-nonscarce). The scarcity of a good (or a resource) happens when supply is not sufficient to meet the demand. "Supply-induced scarcity occurs when environmental degradation, pollution, natural variation or a breakdown in the delivery infrastructure constrains or reduces the total supply or local availability of a specific resource. As the supply of natural resources is reduced, options for pursuing productive livelihood strategies are undermined, potentially creating competition between livelihood groups." (http://www.un.org/en/events/environmentconflictday/pdf/GN_Renewable_Consultation.pdf). That is, supply-induced scarcity is correlated with the resource availability.

Now concerning the questions (1) and (2), it seems reasonable to say that in circumstances of supply-induced scarcity, when a decrease on the availability of a resource happens, because supply is not sufficient to meet the demand, the quantity demanded is higher than in circumstances closer to equilibrium. Does this make sense to you?

Now returning to my original question:

Is there any formal definition of a relationship between resource availability and demand?


I do not know how to answer it. But taking into account the conclusions above it seems plausible to say that there is a relationship between resource availability and quantity of demand (instead of demand): in circumstances of supply-induced scarcity the quantity of demand may increase.

What do you think?
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.