1 Exponential growth

Question 1

IMMIGRATION MODEL

Answer 1

Single population
dN(t)/dt = -(d-b)N(t) +γ
γ is the immigrant arrival rate

solution found by sep of vars, diff again or integrating factor

Question 2

Dimensionality analysis

Answer 2

[dN/dt]
= dimensions of dN/dt
= [N]T^−1

arguments of functions are dimensionless
scaling vars affects domain

the dimensions of the left-hand and right-hand side are equal,

all the quantities in a sum have the same dimensions,

and the dimensions of a product is the product of the dimensions of each factor.

A number is considered to be “dimensionless”, which is written as:
[N] = 1
If N(t) is a density or weight of some quantity, it will not be dimensionless, but that will not
change (

Question 3

EXPONENTIAL GROWTH MODEL

Answer 3

Single population
dN/dt = bN(t)-dN(t) = (b-d)N(t)
=rN(t) where r = b -d
In this model the rate doesn’t depend on t

solution by separation of variables
N(t) = N(0)exp((b-d)t)

Pop grows or decreases exponentially if b>d or d>b (shape N(t))

Question 3

dimensionality: number of cells in an experiment

Answer 3

number of cells in an experiment = rate of cell division × average lifetime of a cell

1 = T^−1 × T. This does not prove that the equation is a correct description of reality,
but at least it makes sense

Question 4

Half life and doubling time

Answer 4

half life:if d>b
τ>0 s.t N(t+τ) = 0.5N(t)

Doubling time:if b<d>0 s.t N(t+τ) = 2N(t)
in exp model: they dont dep on t
τ_0.5 = log(0.5)/(b-d) = -log2/(b-d) = log2/(d-b)</d>

τ_2=log2/(b-d) but dep on r = b-d

Question 5

Integrating factor method
dy/dx + Py = Q
P and Q are functs of c only

Answer 5

multiply both sides by IF
I = exp( integral P(x).dx)
notice dI/dx= PI

I(dy/dx) + IPy = IQ
using product rule

∫(Idy/dx + IPy) dx = ∫IQ dx

Iy = ∫IQ dx

Question 6

Integrating factor method example:
dy/dx + (3y/x) = (exp(x))/x^3
solve

Answer 6

I=exp(3lnx) = exp(ln(x^3))

y= (e^x + c)/(x^3)

Question 7

IMMIGRATION MODEL
solution

Answer 7

dN(t)/dt = -(d-b)N(t) +γ
γ is the immigrant arrival rate

N(t) =
[γ/(b-d)](1- exp(-(d-b)t) + N(0)exp(-(d-b)t)
*found by differentiating N’ again then using sep of vars N’‘/N’ on LHS to integrate.

As t → ∞, N → γ/(d−b)
So, assuming that d − b and γ are constant, we say that the population
approaches a “steady state”.

Question 8

Immigration model ss

dN(t)/dt = -(d-b)N(t) +γ
γ is the immigrant arrival rate

Answer 8

N* =γ/(d−b) (doesnt dep on t)

If N(0)=N* then pop stays there
decrease from N(0) towards ss on axis then below

Question 9

If N(0)=N* then

Answer 9

If N(0)=N* then pop stays there