# Population Growth

Lecture 12 ∙ October 11, 2018

1

Announcements

Exam Tuesday

Bring scantron, #2 pencil

No other tools/notes allowed

Review session Monday 6-8pm, HH 320

Peer Reviews due Tuesday at start of class

2

Study tips

Review the objectives from each lecture, make sure you can meet each objective

Practice interpreting life tables, survivorship curves, and population growth curves

Go over the problem sets

today’s objectives

Interpret population growth rate from a life table

Infer life history traits from a population growth curve

Predict population growth patterns based on life history traits

4

Previous population size (Nt-1)

Number of births (B)

Number of deaths (D)

Number of immigrants/joiners(I)

Number that emigrate/leave (E)

What processes determine current population size (Nt)?

Population dynamics

Nt = Nt-1 + (B-D) + (I-E)

5

Previous population size (Nt-1)

Number of births (B)

Number of deaths (D)

Number of immigrants/joiners(I)

Number that emigrate/leave (E)

What processes determine current population size (Nt)?

Population dynamics

Nt = Nt-1 + (B-D) + (I-E)

What effects the rate of change?

6

The answers to this questions help us:

Protect biodiversity through conservation efforts

Mitigate harmful effects of human population growth

Understand why certain populations are declining

Understand how organisms interact with each other and their environments to predict the impact of environmental change

What processes determine future population size?

Population dynamics

7

Two kinds of life table are useful

Cohort (dynamic) life table – good for plants and other

sessile organisms

Survivorship patterns

You fill in these, calculate the rest

Survivorship from one period to the next: 0.625/0.857 = 0.729

Mortality from one period to the next: 1 – 0.857 = 0.729

mx

8

Two kinds of life table are useful

Cohort (dynamic) life table – good for plants and other

sessile organisms

Survivorship patterns

You fill in these, calculate the rest

Survivorship from one period to the next: 0.064/0.171 = 0.374

Mortality from one period to the next: 1 – 0.456 = 0.544

# alive / # started in cohort

527/843 = 0.625

mx

9

Two kinds of life table are useful

Cohort (dynamic) life table – good for plants and other

sessile organisms

Survivorship patterns

You fill in these, calculate the rest

Survivorship from one period to the next: 0.064/0.171 = 0.374

Mortality from one period to the next: 1 – 0.456 = 0.544

# alive / # started in cohort

527/843 = 0.625

This comes in when we talk about growth rate

mx

10

Two kinds of life table are useful

Cohort (dynamic) life table – good for plants and other

sessile organisms

Survivorship patterns

Based on direct observation

Fecundity schedule = age-specific birth rates over lifespan

mx

11

Net reproductive rate (R0) = average number of offspring produced by an individual organism over lifespan

Sum (∑) of the average number of offspring produced by each individual in each age class(mx), weighted by the proportion surviving in each age class (Lx)

How does net birth rate interact with survivorship to influence population growth?

Net reproductive rate

R0 = ∑ Lxmx

Fecundity schedule = age-specific birth rates over lifespan; called mx DO NOT confuse this with mortality!

12

How does net birth rate interact with survivorship to influence population growth?

Net reproductive rate

R0 = ∑ Lxmx=2.4177

13

R0 > 1  population is growing

R0 < 1  population is declining

R0 = 1  population is stable

How does net birth rate interact with survivorship to influence population growth?

Net reproductive rate

R0 = ∑ Lxmx

14

The answers to this questions help us:

Protect biodiversity through conservation efforts

Mitigate harmful effects of human population growth

Understand why certain populations are declining

Understand how organisms interact with each other and their environments to predict the impact of environmental change

What processes determine future population size?

Population dynamics

15

Geometric rate of increase () is the future population size (Nt+1) divided by the current population size (Nt)

200 / 100 = 2  the population is doubling

This equation applies to populations with non-overlapping generations

How do we calculate Nt+1?

If each individual leaves an average of R0 offspring, then Nt+1 is NtR0

What is the population growth rate based on population size (Nt) and reproductive rate (R0)?

Geometric growth rate

 = Nt+1 / Nt

How do we calculate Nt+1?

If each individual leaves an average of R0 offspring, then Nt+1 is NtR0

Nt = 124

Each individual leaves an average of 2 offspring over the course of lifespan

What is R0?

What is Nt+1?

What is ?

What is the population growth rate based on population size (Nt) and reproductive rate (R0)?

Geometric growth rate

 = Nt+1 / Nt

How do we calculate Nt+1?

If each individual leaves an average of R0 offspring, then Nt+1 is NtR0

Nt = 124

Each individual leaves an average of 2 offspring over the course of lifespan

What is R0?  2

What is Nt+1?  248

What is ?  2

What is the population growth rate based on population size (Nt) and reproductive rate (R0)?

Geometric growth rate

 = Nt+1 / Nt

How does net birth rate interact with survivorship to influence population growth?

Geometric growth rate

Nt+1 =NtR0 = 996 x 2.4177 = 2408

Geometric rate of increase () is the future population size (Nt+1) divided by the current population size(Nt)

This equation applies to populations with non-overlapping generations

How do we calculate Nt+1?

If each individual leaves an average of R0 offspring, then Nt+1 is NtR0

What is the population growth rate based on population size (Nt) and reproductive rate (R0)?

Geometric growth rate

= Nt+1 / Nt

 = 2408/ 996 = 2.41

Increased survivorship will decrease the geometric rate of increase

Increased survivorship will increase the geometric rate of increase

Increased survivorship will not affect the geometric rate of increase

How would population growth be affected by increased survivorship?

Geometric growth rate

Increased survivorship will decrease the geometric rate of increase

Increased survivorship will increase the geometric rate of increase

Increased survivorship will not affect the geometric rate of increase

How would population growth be affected by increased survivorship?

Geometric growth rate

R0 = ∑ Lxmx

Nt+1 =NtR0

 = Nt+1 / Nt

What else will increase geometric growth rate?

increase fecundity (mx)

How would population growth be affected by increased survivorship?

Geometric growth rate

R0 = ∑ Lxmx

Nt+1 =NtR0

 = Nt+1 / Nt

What else will increase geometric growth rate?

increase fecundity (mx)

Can organisms do both? How do Lx and mx

How would population growth be affected by increased survivorship?

Geometric growth rate

R0 = ∑ Lxmx

Nt+1 =NtR0

 = Nt+1 / Nt

How does reproductive rate affect population growth?

Net reproductive rate

vs.

= Nt+1 / Nt

 = 3/1

 = 9/3

= Nt+1 / Nt

= 2/1

 = 4/2

R0 = 3

R0 = 2

The rate of population growth needs to be adjusted for generation time

Generation time (T) = average time from birth to when an organism reproduces

Average age of first time mothers

Multiply fecundity schedule (Lxmx) by age class (X), sum over lifespan, and divide by the net reproductive rate (R0)

What if generations overlap?

Exponential growth rate

T = ∑ XLxmx / R0

 Age (X) Survivorship (Lx) Birth Rate (mx) Fecundity schedule (Lx mx) X Lx mx 0 1.000 0 0 0 1 0.628 3 1.88 1.88 2 0.258 11 2.84 5.68 3 0.147 14 2.06 6.18 4 0.198 7 1.39 5.56 Total R0 = 8.17 ∑ XLxmx = 19.3

What if generations overlap?

Exponential growth rate

T = ∑ XLxmx / R0

T = 19.3/8.17 = 2.36

The rate of population growth needs to be adjusted for generation time

Generation time (T) = average time from birth to when an organism reproduces

Average age of first time mothers

The per capita rate of population growth (r) is the natural log (Ln) of the net reproductive rate (R0), adjusted for generation time (T)

What if generations overlap?

Exponential growth rate

r = Ln(R0) / T

 Age (X) Survivorship (Lx) Birth Rate (mx) Fecundity schedule (Lx mx) X Lx mx 0 1.000 0 0 0 1 0.628 3 1.88 1.88 2 0.258 11 2.84 5.68 3 0.147 14 2.06 6.18 4 0.198 7 1.39 5.56 Total R0 = 8.17 ∑ XLxmx = 19.3

What if generations overlap?

Exponential growth rate

T = ∑ XLxmx / R0

T = 19.3/8.17 = 2.36

The rate of population growth needs to be adjusted for generation time

Generation time (T) = average time from birth to when an organism reproduces

Average age of first time mothers

The per capita rate of population growth (r) is the natural log (Ln) of the net reproductive rate (R0), adjusted for generation time (T)

What if generations overlap?

Exponential growth rate

r = Ln(R0) / T

r = Ln(8.17) / 2.36 = 0.89

r > 0  population is growing

r < 0  population is declining

r = 0  population is stable

How does per capita rate of increase reflect population growth?

Exponential growth rate

r = Ln(R0) / T

How does generation time affect population growth?

Exponential growth rate

20 yrs

20 yrs

20 yrs

60 yrs

How does generation time affect population growth?

Exponential growth rate

30 yrs

30 yrs

60 yrs

Same time span, same reproductive rate, longer generation time, fewer individuals

Comparing growth rate in populations with overlapping and non-overlapping generations

Exponential growth rate

 λ r Name: Finite rate of increase Per capita growth rate Used to model: Discrete generations Continuous generations Measures: Growth compounding at constant intervals Growth compounding continuously What is it? Equivalent to net birth rate per individual over discrete time period Per capita difference between birth and death rates during a fixed time period Estimated by: =Nt+1/Nt = ln λ Population does not change when: = 1 = 0 Population increases when: > 1 > 0 Population decreases when: < 1 < 0

How does generation time affect population growth?

Which type of organisms do you expect to have shorter generation times?

r selected species

K selected species

Exponential growth rate

How does generation time affect population growth?

Which type of organisms do you expect to have shorter generation times?

r selected species

K selected species

Exponential growth rate

Decrease T through rapid growth rate or earlier reproductive maturity

Body size is positively correlated with generation time

Exponential growth rate

Larger species take longer to grow to reproductive size

Unrestricted population growth is exponential

Exponential growth rate

Human population growth is exponential

Exponential growth rate

Human population growth is exponential

Exponential growth rate

But not all population growth is exponential

Logistic population growth rate

But not all population growth is exponential

Logistic population growth rate

Suggests

limit

But not all population growth is exponential

Logistic population growth rate

Environmental factors limit exponential growth

As resources are depleted, population growth rate slows and eventually stops

Density-dependent factors

Influence a population in proportion to its size

Disease, resource competition, predation, etc..

Negative feedback; “Population regulation”

Density-independent factors

Influence a population regardless of population size

Natural disasters (e.g., flood, hurricane)

Influence growth rate, but do not “regulate population”

This is logistic population growth

Sigmoid (S-shaped) population growth curve

Reflects the carrying capacity (K)

The number of individuals the environment can support in a population

But not all population growth is exponential

Logistic population growth rate

Environmental factors limit exponential growth

As resources are depleted, population growth rate slows and eventually stops

Density-dependent factors

Influence a population in proportion to its size

Disease, resource competition, predation, etc..

Negative feedback; “Population regulation”

Density-independent factors

Influence a population regardless of population size

Natural disasters (e.g., flood, hurricane)

Influence growth rate, but do not “regulate population”

This is logistic population growth

Sigmoid (S-shaped) population growth curve

Reflects the carrying capacity (K)

The number of individuals the environment can support in a population

But not all population growth is exponential

Logistic population growth rate

Environmental factors limit exponential growth

As resources are depleted, population growth rate slows and eventually stops

Density-dependent factors

Influence a population in proportion to its size

Disease, resource competition, predation, etc..

Negative feedback; “Population regulation”

Density-independent factors

Influence a population regardless of population size

Natural disasters (e.g., flood, hurricane)

Influence growth rate, but do not “regulate population”

This is logistic population growth

Sigmoid (S-shaped) population growth curve

Reflects the carrying capacity (the K in K-selected species)

The number of individuals the environment can support in a population

Which is an example of a density-dependent factor that could limit growth rate?

Seasonal drought creates arid conditions that lead to decreased transpiration efficiency

Global climate change leads to an increased amount of solar radiation in part of the population range

Increasing pollution in rivers reduces survival in fish

Nesting sites are limited to a particular species of tree

Logistic population growth rate

Which is an example of a density-dependent factor that could limit growth rate?

Seasonal drought creates arid conditions that lead to decreased transpiration efficiency

Global climate change leads to an increased amount of solar radiation in part of the population range

Increasing pollution in rivers reduces survival in fish

Nesting sites are limited to a particular species of tree

Logistic population growth rate

Logistic population growth curve

Logistic population growth rate

Logistic population growth curve

Logistic population growth rate

dN/dt = rN(1-N/K)

dN/dt represents the instantaneous growth rate at a particular time point

rN represents exponential growth

Number of individuals in the population (N) multiplied by the per capita rate of population growth (r)

1-N/K represents the limits on exponential growth made by K

When N nears K, the right side of the equation nears zero.

As population size increases, the rate of population growth (dN/dt) slows until N=K, when population growth is zero.

dN/dt is highest when N=K/2.

Logistic population growth curve

Logistic population growth rate

Population growth rate is highest here

Logistic population growth curve

Logistic population growth rate

Logistic population growth curve

Logistic population growth rate

today’s objectives

Interpret population growth rate from a life table

Infer life history traits from a population growth curve

Predict population growth patterns based on life history traits