ecology reflection
Alyssa Johnson
F18Lecture11PopulationGrowth.pptxPopulation 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
represent life history tradeoffs?
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
