Task 3

Differing Seed Sizes

 

 On an island in the Galapagos, a group of finches with beaks about 1.5 cm long eat primarily seeds that are 3‐7 mm long. However, the plant that makes seeds of this size will only reproduce in wet years (> 30 cm rain). After a series of unusually dry years, there are no seeds 3‐7 mm long anymore. However, plants that produce smaller seeds (thistles, seeds <2 mm) and longer (cactus, seeds >10 mm) seeds are still able to reproduce during the dry years.

Researchers have found that the survival of these finches depends on their ability to eat the larger cactus seeds or smaller thistle seeds, and finches with different beak lengths will eat seeds of corresponding size. In finches, fitness (survival and reproduction) is determined by beak length. They measured a relative fitness term (F) that is dependent on the difference (D, in mm) in the beak length from the average (1.5 cm). Their data is described by the equation:

1. D is the difference in beak length from the average of 1.5 cm. Finches were only found to vary in beak length between 1.0 and 2.0 cm. Using this information, determine the domain of F and find all critical points of the function F(D). 

F’(x)= -0.4D3+0.2D2+3D=0

D=0

D=-2.5

D=3

2. What are the increasing and decreasing intervals of F(D)? 

Increasing intervals: (0,3)

Decreasing intervals: (3, ∞)

3. Use the first or second derivative test to determine the relative minimum or relative maximum.

F’(x)= -0.4D3+0.2D2+3D=0

D=0

D=-2.5

D=3

Relative maximum: (3,-7.2)

4. What is the absolute minimum and absolute maximum of F(D) within this domain? How did you know?

absolute maximum: (3,-7.2)

5.F(D) represents fitness. Natural selection for a given trait, such as beak length, tends to maximize fitness for that trait. Who are the most fit among the finches, and what are their beak sizes?

F(3)=(-0.1)(3)4+(2/30)(3)3+(1.5)(3)2

F(3)=-7.2