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Begin with predator prey model between trees (T) and people 
where
represents the rate of tree growth, represents the number of trees harvested per person and represents the contribution each harvested tree has towards population growth, and is the death rate per person.Note this is similar to the classic predator prey model, if
which says that the number of trees chopped per person per day is directly proportional to the total number of trees (or equivalently their density)Now we can add the health of the environment as a lower rung on the food chain that affects the growth rate of trees and of humans
where
represents the Environments ability to regenerateand
represents the harm to the environment per consumption of treeand
represents an additional death rate per trees based on the state of the environmentand
represents an additional death rate per person based on the state of the environmentAnd then we can add oil as an alternative fuel source (oil)
However, at this point we have assumed that people will harvest both trees and oil independent of each other.
In reality, however, the rate at which trees are harvested may depend on the availability of oil and vice versa…
So we can generalize our
and functions to be functions of both the amount of trees and oil
Beta functions…
One possibility - and perhaps the simplest - is that only trees or oil are harvested exclusively based on whichever can yield more reproductive energy
This assumes that the rate of tree harvest per person is just
and the same for oil.
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