Question

Consider our model ds/dt=fS=kS(1-S/n)(S/m-1) of a fox squirrel population. Suppose that the parameters M and k remain relatively constant over the long term but as more people move into the area, the parameter N (the carrying capacity) decreases.

(a) Assuming that M ≤ N, sketch the graph of the function f(S) for fixed values of k and M and several values of N.

Answer 1

The function f(S) = kS(1 - S/n)(S/m - 1) for a fox squirrel population, with fixed values of k and M and varying values of N (the carrying capacity), results in a graph exhibiting a sigmoidal shape. As N decreases while maintaining M ≤ N, the curve shifts towards the left along the S-axis, showcasing a steeper increase in population size for lower values of N.

Step-by-step explanation:

The function f(S) = kS(1 - S/n)(S/m - 1) represents the rate of change of the fox squirrel population S over time t. Considering fixed values of parameters k and M, the graph illustrates the behavior of f(S) for various values of N (the carrying capacity) while ensuring M ≤ N.

As N decreases, the carrying capacity for the squirrel population diminishes, resulting in a more pronounced impact on population growth. When N decreases, the function f(S) exhibits a leftward shift along the S-axis. This shift implies that for smaller values of N, the squirrel population reaches its maximum size (carrying capacity) at lower population counts, thus displaying a steeper increase in population size before leveling off.

The shape of the graph for f(S) remains sigmoidal, depicting a characteristic S-shaped curve often observed in population growth models. This sigmoidal shape indicates a gradual increase in population size initially, followed by a rapid growth phase before reaching a plateau due to environmental constraints. The varying values of N influence the steepness and position of the curve, showcasing the sensitivity of the fox squirrel population growth to changes in carrying