Problem Set II

1. Suppose you are a predatory spider that sits in its hole waiting for prey to pass by and that there are four types of potential prey in the environment.

Prey Item

Size (milligrams)

Time to Eat (min)

Encounter Rate (items per min)

Gnats

10

1

1.0

Flies

60

3

5.0

Beetle Larvae

100

2

0.5

Sweat bees

20

4

3.0



a) Rank from high to low the set of prey items by profitability (E/h)

b) Should you ever eat flies if they are encountered? Show your calculations.

c) Should you ever eat sweat bees if they are encountered? Show your calculations.


2. Consider the following situation: Lions are cooperative hunters, and the rate of food intake for an individual lion increases up to a point as hunting group size increases. After more than three lions are in a group, the rate of food intake for an individual lion (often referred to as the per capita rate) begins to decrease.

a. Use a graph to show how you could predict the optimal hunting group size for lions. Explain all parts of the graph.
b. The average hunting group size in lions is six. Give two explanations for why lion group size may be higher than predicted by optimality models.


3. Consider a game called Dove-Bully in which bullies go around behaving like a Hawk, i.e. escalating, until somebody hits back. Then the Bully immediately runs away and never pays any cost of fighting. Doves, on the other hand, only display and retreat when confronted with an apparently attacking opponent. V is the payoff to the winner.


a. Construct the payoff matrix for the Dove-Bully game. Briefly explain any assumptions you needed to make to fill out the matrix.
b. What is the ESS if V=2?
c. Could a third strategy, Hawk - in which Hawks always escalate and pay a cost, C, half the time they confront another Hawk, invade the population found for 2b if C=4? Why?
d. Construct the payoff matrix for all three strategies - Hawk, Dove and Bully - and describe what you think the ESS will be for this game and why.


4. The red-faced twit forms winter flocks of two individuals. Individual twits can adopt one of two strategies: either watch for predators or not watch. If at least one of the pair watches, both members survive the winter. If a twit is a member of a flock in which neither watches, there is a 30 percent chance that it is killed. Non-watchers get more to eat, so if they do survive, they raise five offspring the following summer, whereas watchers raise only four offspring.

a) What is the payoff matrix for the watcher vs nonwatcher twit game?
b) What is the ESS and why?

Hint: The payoff matrix in this game can be expressed as the expected number of offspring produced in a season for each pairwise combination of strategies.