24 Age-structured populations (MATH5566M extra topic)

Question 1

consider the pop growth equation, what are the limitations?
e.g.: the Malthusian population growth equation:
N˙(t) = βN(t) − µN(t)
(constant birth and death rate)

Answer 1

But:
Mortality is often NOT constant
Individuals go through juvenile / fertile / sterile periods- we all age
Ageing is not a random walk!

Question 2

consider structure of a population

age structured

Answer 2

What is a ‘structure’? An individual property that uniquely characterises the vital properties of an individual.

E.g. age, size of individuals, maturation of a cell. . .

If we assume a population is age-structured, we prescribe: µ(a)
(mortality), β(a) (fertility), . . . all dependent on age

drawback of this means that at every time t we need to keep track of how the pop is distrubted over the ages e.g bell curve

If we want to know #individuals we can integrate over the denisty

Question 3

density over the continuous ages

Answer 3

p(a, t) = density of individuals of age a ∈ [0, ∞] at time t

Question 4

density over the continuous ages
finding the number number of individuals with age in the interval

Answer 4

p(a, t)Δa = number of individuals with age in the interval
[a, a + Δa] at time t

Question 5

density over the continuous ages
#individuals with age in the interval
[a1, a2] at time t

Answer 5

integral_[a_1,a_2] of
p(a, t)da = number of individuals with age in the interval
[a1, a2] at time t

Question 6

considering ages intro to distribution

Answer 6

time against age of pop.
At time t we know # individuals with age a

consider a small interval Δt
these individuals will have age a +Δt

Question 7

How can I find probabilities?
p(t+Δt,a) for age structured populations

Answer 7

p(t+Δt,a)
= p(t,a-Δt) -μp(t,a)Δt

(ageing)
(#individuals who died in this time)

[p(t+Δt,a)-p(t,a)+ p(t,a)- p(t,a-Δt)]/Δt
=-[μp(t,a)Δt]/Δt

gives limit as Δt tends to 0:
∂p(t,a)/∂t +∂p(t,a)/∂a =-μp(t,a)

holds for all a >0
differs from the diffusion equation and makes it a transport equation instead

as age is not random, you get older as time goes on older unless you die

what happens for a=0 newborns

Question 8

How can I find probabilities?
p(t+Δt,a) for age structured populations

what happens for a=0 newborns

density?

Answer 8

count how many newborns:
#in the time interval:

p(t,0)Δt =
Δt∫_[-∞,∞]

β(a)p(t,a).da

each individual of age a produces β(a) newborns
p(t,a) look at distribution of all ages

Question 9

The McKendrick–von Foerster partial differential equation
intro

Answer 9

This approach has been generalized to a population
consisting of individuals that can give birth and others that cannot pop of just females

Question 10

McKendrick–von Foerster PDE:

well defined?

Answer 10

(∂/∂t)p(a, t) =
−(∂/∂a)p(a, t) − µ(a)p(a, t).

to be well defined we need BC is at boundary of age =0

and initial condition at t=0

Question 11

McKendrick–von Foerster PDE:

p(a,0)=

Answer 11

p(a,0)=p₀(a)
means

given initial distribution)

Question 12

McKendrick–von Foerster PDE:

p(0, t)

Answer 12

p(0, t) = B(t) = ∫_[0,∞] β(a)p(a, t)da

(population birth rate)

the pop density determines the BC which makes this more complictes

Question 13

McKendrick–von Foerster PDE:
summary

Answer 13

first-order linear PDE of transport (hyperbolic) type

ageing happens with velocity 1 with respect to time

could have a maximal age amax if e.g. µ → ∞ as a → amax could be that every individual dies at some point or can be infinity

difficulty in that the BC depends on the variable p itself (pop density determines boundary condition which makes it complicated)

(not symmetric r deterministic like the diffusion eq)

Question 14

transport PDE

Answer 14

McKendrick–von Foerster PDE

The transport equation can be seen as the generalization of the conservation equation:

Convection of a scalar along a flow field moving cyclicly, from left to right.

whereas Diffusion of a scalar, becoming less concentrated at the original injection place, diffusing into the surrounding fluid.

Question 15

Integration on characteristics

Answer 15

dP(t(η),a(η))/dη

method will be used to solve McKendrick–von Foerster PDE (implicit solution):

(∂/∂t)p(a, t) =
−(∂/∂a)p(a, t) − µ(a)p(a, t).
one For now we “forget” the BC is given by how many in the population and just assume the pop birth rate is given and we know it for now

p(0, t) = B(t) = ∫_[0,∞] β(a)p(a, t)da
assumed
p(0,t)=B(t) assumed given
two the derivative is a directional derivative in the direction 1,1
time against age. So we are taking the derivative on a line of slope 1
so if individuals age 1 year as 1 year goes on the slope is 1. So actually we are just checking if these individuals who are ageing are dying or not dying
boundary of age =0 line (time axis) on this boundary we have new-borns which we have assumed as B(t)
The initial condition at time t=0 might be we have certain individuals distributed over the ages, could be a function p₀(x)

three we could simplify by thinking as if we know your age at this time we have the year you were born. This would allow us to assume we know the birth rate at negative times. We project on the diagram for these parallel lines of slope 1. Each line can be described as lines t=a+c for each c we would have the point at which the time axis was intercepted

rewrite
t=t(η)=η+c
a=a(η)=η
I have one single particle η that describes the position of this line
p(a(η),t(η))rewritten and
dp/dη written :
dp(a(η),t(η))/dη
=∂P/∂t dt/dη + ∂P/∂a da/dη

by lines dt/dη =1 and da/dη = 1
so

dp(a(η),t(η))/dη
=∂P/∂t + ∂P/∂a

− µ(a(η)) p(a(η),t(η))

gives a new ODE for the dummy variable η (family )
which is actually the LHS of our original PDE so equated this

four
to be clear call this new var p~ and we know that dp~/dη =-µ~(η) p~(η)
which gives an exponential solution

p~(η)=p~(0) exp( - ∫_[0,η] µ(σ ) dσ )

(given initial condition p(0)

If we replace p~ with the original we would have:
p(a,t) = p~
η=a =age thus we know the exponential

now what is p~(0)?
p~(0) = p(a(0), t(0))
=p(0, 0+c) = p(0,c)=p(0/t-a)
=B(t-a)
where c is the intercept of a specific linec=t-a

so this is the birth rate at time t-a as it is the pop at this time, of newborns at this time

five putting this all together

p(a,t) = B(t-a) exp(- ∫_[0,a] µ(σ ) dσ)

B(t-a)- birth rate at time t-a based on # newborns

exp term gives the probability that these newborns survive until age a ie that they are still alive now.

So this method assumes we know the birth rate at all years

we might instead say we have the initial condition for age which would give another solution

Question 16

method will be used to solve McKendrick–von Foerster PDE (implicit solution):

summary of solutions

Answer 16

The (implicit) solution reads

p(a, t) =
{p₀(a − t) exp(− ∫_[a−t,a] µ(s)ds
if a > t

{B(t − a) exp(−∫_[0,a] µ(s)ds) if a ≤ t

Can be found via integration
along characteristics

Has biological interpretation on
its own (check how many born, check how many suvrvive a years later, exp of cumulative mortality rate descibes the prob of survival up to age a)

S(a) = exp(−∫_[0,a] µ(s)ds)
is the probability to survive up to age

we look at cases η>0, η=0, η<0 etc

Question 17

does our solution
p(0,t-a)
give a pdf?

Answer 17

B(t − a) is a pop birth rate birth rate for a small interval

so if we multiply in a small time interval this rate by a small time interval it will give #newborns in this small time interval,

looking at the sol
we have a rate (#/T) from the exp integral
and then a density over time

Question 18

method will be used to solve McKendrick–von Foerster PDE (implicit solution):
we look at cases η>0, η=0, η<0 etc

Answer 18

η>0
lines of slope 1
which are above the diagonal 1,1 they intercept the birth rates B(t) on the time axis (y-axis) (positive) and we can describe as derived

η<0 those lines below this, we assume the initial condition for age is given instead then derive other part of sol

Question 19

method will be used to solve McKendrick–von Foerster PDE (implicit solution):
why implicit?

Answer 19

not explicit as i have assumed i know the birth rate B

but actually this depends on the ODE

Question 20

method will be used to solve McKendrick–von Foerster PDE (implicit solution):

looking at B(t)

Answer 20

looking at B(t)
=rate of newborns
=∫_[0,∞ ]β(a) p(a,t ) .da

integral of the (per capita fertility rate x pop density)

= ∫[0,∞ ]β(a) **B(t − a) exp(−∫[0,a] µ(σ)dσ) . da

now we have found an expression for the pop density using the survival probability

integral equation
The pop birth rate now dep on the pop birth rate in the past

fully equivalent alternative solution

Question 21

McKendrick–von Foerster PDE:

Asymptotic behaviour at t → ∞

Answer 21

cant really solve but check what happens in the long term, pop explode, extinct?

We try a solution of the form:
p(a, t) = q(a)e^(λt)
We find that λ must satisfy the Lotka–Euler equation

this sol is seperable in age and time with an exponential in time
(exp in time is typical as the model is linear)

∂p/∂t= λq(a)e^(λt)
∂p/∂a= q’(a)e^(λt)

plug into PDE

λq(a)e^(λt) + q’(a)e^(λt)
= − µ(a)q(a)e^(λt)

simplifies to
λq(a) + q’(a)
= − µ(a)q(a) (q’ satisfies an ODE)

q’(a)= − (λq(a)+µ(a)q(a))
gives sol
q(a) = q(0)exp(-∫_[0,a] (λ+µ( σ)).d σ)

and BC at age 0 : β x pop
q(0)e^(λt) =∫_[0,∞] β(a)q(a)e^(λt)

simplifies to
q(0) =∫_[0,∞] β(a)q(a).da
to be used

q(0)=∫_[0,∞] β(a)q(0) e^(-λa) exp(-∫_[0,a] µ( σ)).d σ).da

simplifies q(0) assumed not 0
1=∫[0,∞] β(a)e^(-λa) exp(-∫[0,a] µ( σ)).d σ).da

We find that λ must satisfy the Lotka–Euler equation

Question 22

Lotka–Euler equation
what is it

Answer 22

1
=∫_[0,∞] β(a)S(a) e^(−λa) da
=: F(λ)

for S(a) =
exp(− ∫_[0,a] µ(s)ds)
survival probability

important in being able to find if pops go extinct or explode

Question 23

Lotka–Euler equation
properties

1
=∫_[0,∞] β(a)S(a) e^(−λa) da
=: F(λ)

for S(a) =
exp(− ∫_[0,a] µ(s)ds)
survival probability

1=F(lambda)
if we fix, beta, survival prob then as all coeffs are positive
F is monotonically decreasing with lambda (lambda is real)

Answer 23

thinking about e^(λt):
If Re(λ)>0 grow
Re(λ)<0 extinction
as t goes to infinity

*has unique real root λ∗ which has the largest real part

**F(λ) is decreasing in λ and crosses the vertical axis at
F(0) = ∫_[0,aₘₐₓ]β(a)S(a) da =: R₀
(net reproductive rate or basic reproduction
number)
*the sign of λ∗ determines the asymptotic behaviour:

if R₀ > 1 then λ∗ > 0 and the population grows as t → ∞

if R₀ < 1 then λ∗ < 0 and the population decays as t → ∞

if R₀ = 1 then λ∗ = 0 and the population is stationary

*q(a) gives the stationary distribution of the population
over the ages (snapshot)

Question 24

The Lotka–Euler equation
graphs

Answer 24

F against lambda
decrease exponentially?
when lambda* >0 crosses yaxis at R_0 above 1

F is monotonically decreasing then as lambda tends to infinity F tends to 0
we will always have F(lambda)=1
lamdba will be positive if the intersection at y-axis F(0) is above 1 growth

if lambda* <0 then R_0 below 1 and curve shifted left, F(0) will be below 1 and this lambda* is negative
F(lambda*)=1 here too
extinction

Question 25

F(0)?

Answer 25

*F(λ) is decreasing in λ and crosses the vertical axis at
F(0) = ∫_[0,aₘₐₓ]β(a)S(a) da =: R₀
(net reproductive rate or basic reproduction
number)

F(0) is always the net reproductive rate/ basic reproductive number of the population

for one single individual how many individuals they will produce in their lifetime

and integrating over the whole lifetime of the individual counts will give you how many offspring if they survive- average number of offspring

so if on avg produce more than one offspring pop grows, on avrg less than they pop decay

Question 26

please check example sheet 5

Answer 26

a few qs sheet 5
past exams

Jan 2022
Jan 2020
on age structured populations

check if everything is clear!