TD7

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**e. In the model of the Easter Island seen in class, there is a threshold amount of resource such that population is growing above that threshold, and decreasing below that threshold.**

*Answer***True:**You can easily see it from the phase diagram. We have a line along which is fixed, above that line it increases and below it decreases. The expression of that line is , therefore if is greater than the population is growing, otherwise it is decreasing.

**f. In the model of the Easter Island seen in class, the net growth of the resource (i.e. net of harvesting by humans) is decreasing in the numbers of humans present in the ecosystem, for a given value of the amount of the resource.**

*Answer***True:**The net growth is given by the replenish of the resource minus the harvest, which is the part affected by the number of humans (population).We can clearly see that an increase in

decreases the net growth. If you want to be more precise its enough to take the derivative with respect to :What you get is how much growth is smaller after an infinitesimal change in

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In this exercise, we just have to reason with the phase diagram to get the answers. In the rest of the solution, I indicate with dashed lines the old equations, while continuous lines represents the new ones after the variable changes. We assume we are in the nontrivial internal steady state.

**a. What happens if **

Let's start by breaking down all the components of the relevant equations.

We can see that

Figure 1: Phase diagram with new

What we see is that in the new steady state

**b. What happens if **

The rate

Figure 2: Phase diagram with new

Figure 3: Phase diagram with new

In the new steady state we have unambiguously lower

**c. What happens if **

The capacity