1. FW364 Ecological Problem Solving Class 5: Population Growth
Sept. 16, 2013

2. Outline for Today Goal for Today: Objective for Today:
Introduction to population growth – Discrete Growth
Objective for Today:
Derive a simple model of discrete population growth between consecutive time periods (Nt and Nt+1)
Objective for Next Class:
Derive an equation to forecast population growth (still discrete growth)
Objective for Class after Next:
Derive continuous population growth equation
Text (optional reading):
Next three classes correspond to Chapter 1

3. Muskox Case Study
s: Excessive hunting eliminated muskox from most of their natural North American range
: Last individuals killed in Alaska
1930: Territory of Alaska Legislature authorized re-introduction from Greenland to Nunivak Island
Ovibos moschatus
1936: 31 muskox introduced to Nunivak Island
1965: Population size of 514 muskox!
: 48 animals removed for re-introductions elsewhere in Alaska
Great opportunity for population analysis because:
Closed population
Regular census from
Nunivak Island

4. Populations What is a population?
Individuals of the same species in a defined area
A group of interbreeding organisms
Why is it important to define “the population” for modeling and management?
Definition of the population in part determines whether immigration & emigration need to be accounted for
 The larger the area over which the population is defined, the less important immigration and emigration become

5. Today’s Focus: Populations
What is a population?
Individuals of the same species in a defined area
A group of interbreeding organisms
Why is it important to define “the population” for modeling and management?
Definition of the population in part determines whether immigration & emigration need to be accounted for
 The larger the area over which the population is defined, the less important immigration and emigration become
Today’s Focus:
Modeling (i.e., building an equation) to describe
how populations grow (or shrink)

6. Basic Population Growth
General form:
Density at time in future = function of density at present time
Nt+1 = f(Nt)
Nt = density of the population at time t (e.g., year 3 or day 12)
Nt+1 = density of the population at time t+1 (e.g., year 3+1 or day 12+1)
f(Nt) means a function “dependent on Nt”
Note: It is common to refer to Nt and Nt+1 as “densities”
because we are working with abundance in a pre-determined area
[remember the definition of a population]
However, the spatial aspect is implicit in the model
… We’ll treat Nt and Nt+1 as abundances (i.e., numbers of individuals)

7. Basic Population Growth
General form:
Density at time in future = function of density at present time
Nt+1 = f(Nt)
Nt = density of the population at time t (e.g., year 3 or day 12)
Nt+1 = density of the population at time t+1 (e.g., year 3+1 or day 12+1)
f(Nt) means a function “dependent on Nt”
What are general factors that might affect abundance at t+1
if you know abundance at time t?
Inputs to population Outputs from population
Births (B) Deaths (natural, harvesting, other) (D)
Immigration (I) Emigration (E)
Nt+1 = Nt + B – D + I – E
f(Nt)

8. Basic Population Growth
Nt+1 = Nt + B – D + I – E
Remember from last week:
When trying to understand an ecological system, should start simple
For now, will assume we are working with closed population:
No immigration or emigration; only births and deaths
Nt+1 = Nt + B – D
Let’s start by building an equation for discrete growth
Discrete models – recap:
Useful for predicting quantities over fixed intervals
Time is modeled in discrete steps; Intervening time is not modeled
Good for populations that reproduce seasonally, like moose, salmon…
… and muskox

9. Basic Population Growth
: Removals
Nt+1 = Nt + B – D
Population closed to immigration / emigration
So we can use
to model muskox population size at discrete intervals
e.g., one year to the next
1936: First introduction
Fig 1.3 in text
Plot curves upward:
Suggestive of “multiplicative growth”, but not diagnostic (we’ll check later)
Multiplicative (geometric) growth: population size increases (or decreases) by a constant fraction per year (rather than adding, e.g. 50 individuals, per year)

10. Birth Rate Nt+1 = Nt + B – D Let’s look closer at B, the birth term
What two most general factors determine B, the number of births?
 The number of individuals (or just number of females), N
Whether all individuals or just females are considered varies (we’ll do all individuals)
(males are often ignored in models of population growth... sorry guys)
 The number of offspring per individual in one time step, b’
Per capita birth rate (i.e., per individual)
Prime distinguishes discrete rate from instantaneous (continuous) rate (used later)
Birth rate is a population parameter: an average rate for the population
(all individuals will not produce the same number of offspring in any time step)
b’ = (total number of births/time step) / total number of individuals
For a yearly time step:
b’ = (total number of births/year) / total number of individuals
Nt+1 = Nt + B – D
Per capita is a Latin prepositional phrase: per (preposition, taking the accusative case, meaning "by, by means of") and capita (accusative plural of the noun caput, "head"). The phrase thus means "by heads" or "for each head", i.e. per individual or per person.

11. We can now create an equation for B:
Birth Rate
Let’s look closer at B, the birth term
What two most general factors determine B, the number of births?
 The number of individuals or just number of females, N
 The number of offspring per individual in one time step, b’
Nt+1 = Nt + B – D
We can now create an equation for B:
B = b’ Nt
where:
B is the number of births
Nt is population size at time, t
b’ is per capita birth rate
In other words, the number of births is the product of the number of individuals and the average number of offspring each individual has

12. Concept check: What are some of the assumptions we have made for B?
Birth Rate
B = b’ Nt
Concept check: What are some of the assumptions we have made for B? b
(a)-(e) are possible b’ values
many e c
Which of these options could fit our equation for B? B a d
Which of these options is/are impossible to ever have?
many Nt

13. Concept check: What are some of the assumptions we have made for B?
Birth Rate
B = b’ Nt
Concept check: What are some of the assumptions we have made for B? b
(a)-(e) are possible b’ values
many e c
Which of these options could fit our equation for B? B
Either (b) or (c): linear function a d
Which of these options is/are impossible to ever have?
(a): cannot have births at Nt = 0
many Nt

14. Concept check: What are some of the assumptions we have made for B?
Birth Rate
B = b’ Nt
Concept check: What are some of the assumptions we have made for B? b
(a)-(e) are possible b’ values
many e c
What is the difference between (b) and (c)? B a d
Which option(s) make the most sense ecologically?
many Nt

15. Concept check: What are some of the assumptions we have made for B?
Birth Rate
B = b’ Nt
Concept check: What are some of the assumptions we have made for B? b
(a)-(e) are possible b’ values
many e c
What is the difference between (b) and (c)? B
b’ in (b) is larger than (c) a d
Which option(s) make the most sense ecologically?
(e): # births decreases as a function of density (carrying capacity)
(d): # births increases as a function of density (finding mates)…
(d) realistic a low pop size, not high
many Nt

16. Death Rate Nt+1 = Nt + B – D Let’s look closer at D, the death term
What two most general factors determine D, the number of deaths?
 The number of individuals, N
The average death rate, d’
Per capita death rate; Probability that one individual will die during the time step
Proportion of all individuals from population dying
d’ = (total number of deaths/time step) / total number of individuals
Are there bounds on d’?
Can d’ be negative?
Can d’ be > 1?
Nt+1 = Nt + B – D

17. Death Rate Nt+1 = Nt + B – D Let’s look closer at D, the death term
What two most general factors determine D, the number of deaths?
 The number of individuals, N
The average death rate, d’
Per capita death rate; Probability that one individual will die during the time step
Proportion of all individuals from population dying
d’ = (total number of deaths/time step) / total number of individuals
Are there bounds on d’?
Can d’ be negative?  No (that would be inverse death)
Can d’ be > 1?
Nt+1 = Nt + B – D
 No, d’ must be ≤ 1 (cannot have >100% of population dying)

18. Death Rate Nt+1 = Nt + B – D Let’s look closer at D, the death term
What two most general factors determine D, the number of deaths?
 The number of individuals, N
The average death rate, d’
Per capita death rate; Probability that one individual will die during the time step
Proportion of all individuals from population dying
d’ = (total number of deaths/time step) / total number of individuals
Are there bounds on d’?
Can d’ be negative?  No (that would be inverse death)
Can d’ be > 1?
Are there bounds on b’?
Can b’ be negative?
Can b’ be > 1?
Nt+1 = Nt + B – D
 No, d’ must be ≤ 1 (cannot have >100% of population dying)

19. Death Rate Nt+1 = Nt + B – D Let’s look closer at D, the death term
What two most general factors determine D, the number of deaths?
 The number of individuals, N
The average death rate, d’
Per capita death rate; Probability that one individual will die during the time step
Proportion of all individuals from population dying
d’ = (total number of deaths/time step) / total number of individuals
Are there bounds on d’?
Can d’ be negative?  No (that would be inverse death)
Can d’ be > 1?
Are there bounds on b’?
Can b’ be negative?  No (cannot have negative births)
Can b’ be > 1?
Nt+1 = Nt + B – D
 No, d’ must be ≤ 1 (cannot have >100% of population dying)
 Absolutely! Individuals can have > 1 offspring

20. Like for B, we can create an equation for D:
Death Rate
Let’s look closer at D, the death term
What two most general factors determine D, the number of deaths?
 The number of individuals, N
The average death rate, d’
Per capita death rate; Probability that one individual will die during the time step
Proportion of all individuals from population dying
d’ = (total number of deaths/time step) / total number of individuals
Nt+1 = Nt + B – D
Like for B, we can create an equation for D:
D = d’ Nt
where:
D is total number of deaths
Nt is population size at time, t
d’ is per capita death rate

21. Putting Everything Together
Let’s plug in our new equations!
Nt+1 = Nt + B – D
Nt+1 = Nt + b’Nt – d’Nt
B = b’ Nt
D = d’ Nt
Rearrange to get:
Nt+1 = Nt (1 + b’ – d’)
(note: birth and death rates are “additive”)
Now, we can define a new parameter, r’
r’ = b’ – d’
(r’ is net population change)
and plug r’ into equation:
Nt+1 = Nt (1 + r’)
We can define another new parameter, λ (lambda)
λ = 1 + r’
and plug λ into equation:
Nt+1 = Nt λ
Note: Text refers to λ as R, then later calls it λ. We’ll use λ always.

22. Putting Everything Together
Nt+1 = Nt λ
We have created a useful model of population growth!
Let’s take a moment to think about what we have just done…
…We derived an important equation using “first principles”
…We started by thinking about simple additions and losses to populations
…and have arrived at a powerful (though still simple) equation

23. Population Growth Break!
Let’s think about birth and death rates in an example population…humans!

24. r’ = b’ – d’ Pick a Country Card
Each card has a country with its birth and death rate.
Using the birth and death rate, calculate net population change and be ready to comment and share:
r’ = b’ – d’

25. Simple model of multiplicative population growth (discrete type)
More on λ
Nt+1 = Nt λ
Simple model of multiplicative population growth (discrete type)
Multiplicative means the population increases in proportion to its size
Equation allows us to predict this year from last year, or next year from this year
Let’s talk about λ (lambda)
λ is the finite growth rate of population
λ is the factor by which the population grows (or shrinks) each year
Like r’, λ is a net rate: the net of inputs (b’) and outputs (d’)
λ is also the ratio of population sizes for consecutive time steps:
λ = Nt+1 / Nt

26. What is happening to a population with a growth rate, λ, of:
More on λ
Nt+1 = Nt λ
λ = Nt+1 / Nt
What is happening to a population with a growth rate, λ, of:
λ = 1 ?
λ > 1 ?
0 < λ < 1 ?
λ < 0 ?

27. What is happening to a population with a growth rate, λ, of:
More on λ
Nt+1 = Nt λ
λ = Nt+1 / Nt
What is happening to a population with a growth rate, λ, of:
λ = 1 ? Population is stable (not changing, Nt+1 = Nt )
λ > 1 ? Population is increasing (growing)
0 < λ < 1 ? Population is decreasing (shrinking)
λ < 0 ? Not possible; cannot have negative population size

28. More on λ Nt+1 = Nt λ λ = Nt+1 / Nt λ = 1.01 ? λ = 1.23 ? λ = 0.95 ?
By what percentage is a population changing size with a growth rate, λ, of:
λ = 1.01 ?
λ = 1.23 ?
λ = 0.95 ?
(Post – Pre)/Pre

29. More on λ Nt+1 = Nt λ λ = Nt+1 / Nt
By what percentage is a population changing size with a growth rate, λ, of:
λ = 1.01 ? Population increasing by 1% per time step
λ = 1.23 ? Population increasing by 23% per time step
λ = 0.95 ? Population decreasing by 5% per time step

30. More on λ Nt+1 = Nt λ λ = Nt+1 / Nt
By what percentage is a population changing size with a growth rate, λ, of:
λ = 1.01 ? Population increasing by 1% per time step
λ = 1.23 ? Population increasing by 23% per time step
λ = 0.95 ? Population decreasing by 5% per time step
Recall that: λ = 1 + r’
so r’ is the proportion by which
the population changes (grows
or shrinks) each time step
From above:
when λ = 1.01 , r’ = 0.01 or 1% per time step
when λ = 1.23 , r’ = 0.23 or 23% per time step
when λ = 0.95 , r’ = or -5% per time step

31. Correspondence with Book
Note: The book uses different symbols!
The book uses this equation for “long-lived” populations:
Nt+1 = Nt(s + f)
where: s is survival rate (proportion surviving)
f is the birth rate (fecundity rate)
f is equivalent to our b’ (we use b’ because birth begins with “b”)
s is equivalent to 1 – d’ (proportion surviving is 1 - proportion dying)
So our equation:
Nt+1 = Nt(1 + b’ – d’)
is equivalent to the book's equation:
Nt+1 = Nt(s + f), when substituting b’ = f and s = 1 – d’

32. Looking Ahead Next Class: Tomorrow: Lab 3
Derive an equation to forecast population growth (still discrete growth)
so we can predict population growth between non-consecutive time steps (e.g., 10 years in the future, not just last year to this year, or this year to next year)
Tomorrow: Lab 3
Bring exercise, pencil, paper, and calculator