MATH3202/7232 Operations Research & Mathematical Planning 2021
Week 11 – Web Building and a Family Drive
Web Building
Female orb-weaving spiders need to gain energy to grow in weight from 35 mg, at their sexual maturity, to 80 mg, where they can lay a first batch of eggs. These spiders are sit- and-wait predators, building a web to catch prey and gain energy. They replace their web every day and so can change the size of web they use. Larger webs are more likely to catch prey but require more energy to build and more time in the open, making them more vulnerable to their own predators.
Suppose the spiders have a choice of building webs where the total length of the sticky spiral is 4 m, 8 m or 12 m. Building a web of size 𝑤 requires an energy cost of
𝛼!(𝑠) = −0.125 𝑤 + 0.005 𝑤 𝑠,
where 𝑠 is the current weight of the spider in mg. Each day a spider also has a basal
metabolic expenditure, 𝛼”, of 0.4 mg.
The table below shows the daily probability, 𝜆! , of catching a prey and the daily
probability, 𝛽! , that the spider will be eaten for the different web sizes.
Web size, 𝑤 4 8 12 𝜆! 0.66 0.77 0.82 𝛽! 0.01 0.02 0.03
The energy value of a prey item is 6 mg. We assume that at most one prey can be caught each day. If the weight of a spider drops below 25 mg then it dies from starvation.
At the end of each day let 𝑠 be the weight of the spider in mg with 𝑝 = 𝑠 − ⌊𝑠⌋. Then the weight of the spider at the start of the next day will be ⌊𝑠⌋ + 1 with probability 𝑝 and ⌊𝑠⌋ with probability 1 − 𝑝. (This keeps the possible states more manageable.)
Suppose a spider starts a day with weight s. What is the maximum probability that she will reach the egg-laying weight of 80 mg within the next 10 days? What strategy should she pursue to achieve this?
Reference
This model is a finite-time version of one presented in Venner, S., Chadès, I., Bel-Venner, M., & Pasquet, A. (2006). Dynamic optimization over infinite-time horizon: Web-building strategy in an orb-weaving spider as a case study. Journal of Theoretical Biology doi:10.1016/j.jtbi.2006.01.008.
OOppttimimisisininggththeeFfamillyDdrive
A family is setting out to drive between two cities – X and Y. These cities are joined by two
A family is setting out to drive between two cities, X and Y. These cities are joined by two
main highways, with some options to cross over between the highways. The highways can
main highways, with some options to cross over between the highways. The highways can
be drawn as follows, with distances marked in kilometres.
be drawn as follows, with distances marked in kilometres.
AAlolonnggttheway the familywililllnneeeddtotostsotpopfofrofrufeulealnadnfdoofodo. dT.hTehceocstossotsf othfetshesvearvyaarsy as follows:
follows:
amilCy imtyeal
CFousetlocfofsatmily meal ($)
Fuel cost (cents/L)
$
A
cents per litre
80
152
80 43
B
152 152
43
152
40
C
186
40
186
40
D
153
40
153
74
E
123
74
123
78
F
186
72
143
45
G
124
78
186
73
H
191
45
124
55
y traffic, the
I
126
car travels at an average s
73
191
peed of 60km/hr. The family
J
tting off and
55
can go at most 4 hours be
126
tween meals – end of one m
Cost of f City
A
B
C
D
E
F 72 143 G
H I J
Due to holida has just
fed before se eal to start
of the next. They will eat with friends on arrival at Y. The car uses 10 litres per 100
kilometres and can safely go at most 400km on a tank. Whenever the family stops to refuel
Due to holiday traffic, the car travels at an average speed of 60 km/hr. The family has just
fed before setting off and can go at most 4 hours between meals (end of one meal to start odf ethsteinaetxiotn).iTt hweilyl nweielldetaotbweirtehffureiellendsatoancaorsrtivoafl$a1t.2Y0. Tpherelcitarre.uses 10 litres per 100 km and can safely go at most 400 km on a tank. Whenever the family stops to refuel it
it completely fills the tank. The tank is full on setting off. When the car arrives at the
completely fills the tank. The tank is full on setting off. When the car arrives at the dYe.stSionlavteiotnhisitpwroilbllnemee.d to be refuelled at a cost of $1.20 per litre.
Write down a general purpose formulation for calculating the cheapest way to get from X to
a) Write down a general-purpose formulation for calculating the cheapest way to get
Perhaps the family wants to have the fewest stops possible (where a stop for both food and
from X to Y.
petrol counts as only one stop). Modify your formulation to calculate the cheapest way to travel, amongst journeys that also have the smallest number of stops. Solve this problem.
b) Perhaps the family wants to have the fewest stops possible (where a stop for both food and petrol counts as only one stop). Modify your formulation to calculate the cheapest way to travel, amongst journeys that also have the smallest number of stops.