Suppose the resource is rice.
If competition for rice increases, there are several possibilities.
First, the supply for rice would decrease, enough that there is a possibility that the supply would not be enough for the demand. If this continues, high demand and decreasing supply, there will be a shortage, meaning the rice will not be enough for everyone who wants it.
Second, To try and slow down the decrease of the supply, those who sell rice will increase the price, making those who want to buy it, or who are competing for it, a bit more skeptical in buying it. But if they really really need the rice...
Third, those who sell the rice will try and look for ways to get more rice to meet the demand for rice. Either they will try and buy rice from far places, or they will develop ways to produce more rice in a shorter amount of time.
This is the same with other resources.