LS 30B Homework 5 Solutions
Exercise 6.3.1 Come up with a Leslie matrix model for a fictional species with two life
stages and describe the meaning of its entries, as above.
For example, for every year:
• 65% of the babies will stay as babies
• Every adult on average will give birth to 0.2 babies
• 15% of the babies will become adults
• 80% of the adults will remain adults
The matrix will be:
0.65 0.2
M =
0.15 0.80
Exercise 6.3.2 Explain why the entries in each column of a transition matrix such as
equation (6.1) must add up to one. (Hint: Label the rows and columns, writing “from” and
“to” where appropriate.)
The columns in the matrix represent the contribution from a specific variable/species/life stage to
the population present in the next state. For example, the first column is the contribution from
SN to the values of SN +1 and IN +1 .
On the diagram, we see that for the population S and I, they only have two choices: (1) they either
go to the other variable, or (2) they stay put at where they are. Thus, a fraction of them will go to
other state (e.g. S → I or I → S), and a fraction will stay the same (e.g. S stays as S, and I stays
as I). But all of the fractions come from the same pool (e.g. S or I), so the fractions will need to
add up to a whole (i.e. 1). Just like if you cut a cake, you can have two fractions (e.g. a · S and
b · S), but they add up together into one pie (e.g. (e.g. a · S + b · S = 1 · S).
The values in the matrix represent the fractions of the population that make the decision to stay
or to leave. However, since the values in a column belong to the same original variable, they have
to add up to 1 (“whole”).
1
,LS 30B Homework 5 Solutions
Exercise 6.3.3 Starting with 20 susceptible and 40 infected individuals, iterate MSI 15
times in SageMath. What steady state does the system reach? Do the same for 50 susceptible
and 60 infected individuals. How do your results compare to the simulations in Figure 6.7?
S = 20 and I = 40:
The system seems to reach the steady state at S = 40 and I = 20.
S = 50 and I = 50:
The system seems to reach the steady state at S = 73.3 and I = 36.7.
The first initial condition seems to follow the long-term behavior seen on the left graph, when the
second initial condition seems to follow the long-term behavior seen on the right graph.
2
, LS 30B Homework 5 Solutions
Exercise 6.3.4 What is the behavior of the total population (S + I) over time?
The total population (S + I) seems to stay constant over time:
• Start at S + I = 60 (S = 20 and I = 40) → End at S + I = 60 (S = 40 and I = 20)
• Start at S + I = 110 (S = 50 and I = 60) → End at S + I = 110 (S = 73.3 and I = 36.7)
Exercise 6.3.7 If a species is going extinct, what equilibrium is the population size
approaching? Is this equilibrium stable or unstable?
If a species is going extinct, the Leslie model will have a stable equilibrium point at the origin, and
the population size will approach 0.
Exercise 6.3.FE 1 Giant pandas are a vulnerable species famous for their consumption of
large amounts of bamboo. Write a discrete-time matrix model of a giant panda population
using the following assumptions. We are modeling only the female population.
• Pandas have three life stages: cubs, subadults, and reproductively mature adults.
• Cubs remain cubs for only one year. They have a mortality rate of 17%.
• Pandas remain subadults for three years. Thus, about 33% of subadults mature into
adults each year.
• 28% of subadults die each year.
• On average, adults give birth to 0.5 female cubs each year.
• 97.7% of adults survive from one year to the next.
When working with discrete time models, we always have to look at how the population from this
state will contribute to the population in the next state. Let’s define the following three variables:
• C ≡ cubs
• S ≡ subadults
• A ≡ adults
3
Exercise 6.3.1 Come up with a Leslie matrix model for a fictional species with two life
stages and describe the meaning of its entries, as above.
For example, for every year:
• 65% of the babies will stay as babies
• Every adult on average will give birth to 0.2 babies
• 15% of the babies will become adults
• 80% of the adults will remain adults
The matrix will be:
0.65 0.2
M =
0.15 0.80
Exercise 6.3.2 Explain why the entries in each column of a transition matrix such as
equation (6.1) must add up to one. (Hint: Label the rows and columns, writing “from” and
“to” where appropriate.)
The columns in the matrix represent the contribution from a specific variable/species/life stage to
the population present in the next state. For example, the first column is the contribution from
SN to the values of SN +1 and IN +1 .
On the diagram, we see that for the population S and I, they only have two choices: (1) they either
go to the other variable, or (2) they stay put at where they are. Thus, a fraction of them will go to
other state (e.g. S → I or I → S), and a fraction will stay the same (e.g. S stays as S, and I stays
as I). But all of the fractions come from the same pool (e.g. S or I), so the fractions will need to
add up to a whole (i.e. 1). Just like if you cut a cake, you can have two fractions (e.g. a · S and
b · S), but they add up together into one pie (e.g. (e.g. a · S + b · S = 1 · S).
The values in the matrix represent the fractions of the population that make the decision to stay
or to leave. However, since the values in a column belong to the same original variable, they have
to add up to 1 (“whole”).
1
,LS 30B Homework 5 Solutions
Exercise 6.3.3 Starting with 20 susceptible and 40 infected individuals, iterate MSI 15
times in SageMath. What steady state does the system reach? Do the same for 50 susceptible
and 60 infected individuals. How do your results compare to the simulations in Figure 6.7?
S = 20 and I = 40:
The system seems to reach the steady state at S = 40 and I = 20.
S = 50 and I = 50:
The system seems to reach the steady state at S = 73.3 and I = 36.7.
The first initial condition seems to follow the long-term behavior seen on the left graph, when the
second initial condition seems to follow the long-term behavior seen on the right graph.
2
, LS 30B Homework 5 Solutions
Exercise 6.3.4 What is the behavior of the total population (S + I) over time?
The total population (S + I) seems to stay constant over time:
• Start at S + I = 60 (S = 20 and I = 40) → End at S + I = 60 (S = 40 and I = 20)
• Start at S + I = 110 (S = 50 and I = 60) → End at S + I = 110 (S = 73.3 and I = 36.7)
Exercise 6.3.7 If a species is going extinct, what equilibrium is the population size
approaching? Is this equilibrium stable or unstable?
If a species is going extinct, the Leslie model will have a stable equilibrium point at the origin, and
the population size will approach 0.
Exercise 6.3.FE 1 Giant pandas are a vulnerable species famous for their consumption of
large amounts of bamboo. Write a discrete-time matrix model of a giant panda population
using the following assumptions. We are modeling only the female population.
• Pandas have three life stages: cubs, subadults, and reproductively mature adults.
• Cubs remain cubs for only one year. They have a mortality rate of 17%.
• Pandas remain subadults for three years. Thus, about 33% of subadults mature into
adults each year.
• 28% of subadults die each year.
• On average, adults give birth to 0.5 female cubs each year.
• 97.7% of adults survive from one year to the next.
When working with discrete time models, we always have to look at how the population from this
state will contribute to the population in the next state. Let’s define the following three variables:
• C ≡ cubs
• S ≡ subadults
• A ≡ adults
3
