Lecture 9: Logistic Growth & Life Tables Flashcards
Define carrying capacity. What variable is used for it?
the maximum number of individuals that can be supported by a particular environment
represented by K
How does carrying capacity vary? Why?
with the quality of the environment
K will be lower if the environment is of poor quality and vice versa
If there are less resources in an environment, less individuals will be able to survive there
Describe the logistic growth model?
A method of modelling population growth that factors in the limitations on growth that carrying capacity presents
what is the equation for logistic growth? List the variables
dN/dt = rN(1-N/K)
dN/dt = population growth r = per capita rate of increase N = number of individuals in the population K = carrying capacity
Which part of the logistic growth equation is responsible for constraining population growth as pop size increases?
(1-N/K) acts as a modifier of r
When (1-N/K) is a small number, how is r effected?
Negligibly
What happens when (1-N/K) is a # approaching K?
Fewer and fewer individuals are added per unit time until N = K
When N = K, what is the per capita rate of increase?
0
ie, no individuals are being added to the population per unit time
What kind of shape does the logistic growth curve have?
An S-shape curve
What does (1-N/K) tell us about the population?
How close the population is to carrying capacity
if it’s closer to 1 = further from K
if it’s closer to 0 = closer to K
What occurs as the population size (N) approaches K?
a) per capita birth rate DECREASES
and/or
b) per capita death rate INCREASES
Which results in the per capita rate of increase (r) DECREASING
(r = b-d) so
if b
T or F: population growth speeds up as the size approaches K
FALSE
because the death rate will increase and/or the birth rate will decrease
So less individuals are being added to the population per unit time
At what population size (N) is growth the fastest? Explain
At K/2
When N is small, the population is far from K, and r will be large because individuals are able to produce many offspring but the amount of individuals reproducing is small - slow growth
When N is approaching K, r is small because the number of deaths is increasing and number of births is decreasing = growth is slow
When N is halfway to K, N and r are intermediate values so growth is the fastest
What are the 3 phases of the logistic growth model?
- establishment phase
- boom phase
- stabilization phase
Describe the establishment phase of the logistic growth model
Population size (N) is small, so growth is slow because even if everyone who can produce is reproducing, there are still only a few individuals reproducing
bottom of the curve closest to axes (left)
Describe the boom phase of the logistic growth model
Growth is fastest because N is big, but not approaching K yet
Everyone is not producing at max rate because r is being modified
halfway up the curve (when K/2)
Describe the stabilization phase of the logistic growth model
Growth is slow because N is large and approaching K
Individuals are not reproducing at max rate and many individuals will be dying
the plateau at the top of the curve
if the environment has a carrying capacity, then the population growth is dependent on _____ ____
population density
What are birth and death rates dependent on?
population density
When does negative density dependence occur?
When the growth rate DECREASES as population size increases
most common
What two kinds of factors affect negative density dependence? Explain them
- intrinsic factors: physiological responses to high density (stress)
- extrinsic factors: environmental factors impacting birth/death rates as population density increases
Give an example of how intrinsic factors affect negative density dependence
High density of white-footed mice induces hormonal changes that:
- delay sexual maturity (reduce birth rate)
- depress immune system (increase death rate)
- r will decrease
Give some examples of extrinsic factors that affect negative density dependence
- resource availability (more individuals, less food/water/shelter/breeding territory = more death/less birth)
- patterns of predation can change when population is large (more vulnerable; more death)
- disease/parasite transmission is higher (more death)
What are the 3 ways density dependence can occur?
- Death rate increases as N approaches K (death rate is density dependent)
- Birth rate decreases as N approaches K (birth rate is density dependent)
- both death rate increases and birth rate decreases as N approaches K
Describe positive (inverse) density dependence. When is the usually seen?
Density dependence that occurs when the intrinsic growth rate (r) INCREASES as population size increases
Usually occurs in very small population sizes
What is the Allee effect?
A decline in reproduction or survival that occurs when the population size is SMALL
Give an example of the Allee effect?
The positive correlation between seed production and population size in cowslip plants
When the population size is large, the number of seeds produced per individual is also large.
When the pop size is small, the number of seeds produced per individual is also small
Describe density independent factors and give some examples
Factors that are unrelated to population size that influence the births and deaths of a population
- flood
- drought
- hurricane
- tornado
- heat wave
- fire
T or F: density independent factors, like density dependent factors, have a regulatory effect on population size?
FALSE, these factors occur regardless of population size and density
What assumption does the logistic growth model have? why is this problematic?
It assumes that all members of a population have the same intrinsic growth rate (ie, every individual is contributing equally to the birth and death rate of the population)
This is problematic because survival and fecundity vary per individual with life stage and age
Define age structure
the proportion of the population that occurs in different age classes
What are the 2 types of populations in regards to generation structure?
- Non-overlapping generations
2. Overlapping generations
Describe non-overlapping generations and give an example
There is no age structure because all the individuals are in the same age class
ex. annual species of insects and plants
- a brood of an insect species will all hatch in the same short period of time and grow and develop through life stages together, reproduce and die around the same time
Describe overlapping generations and give an example
When the population has individuals of all life classes alive at the same time
ex. humans exist as babies, teenagers, adults, and elderlies all at once
In overlapping generations, what factor influences the population growth?
the proportion of each age class that is currently alive
What are the 3 age classes in overlapping generations?
- pre-reproductive
- reproductive
- post-reproductive
Which age classes are more vulnerable to death? why?
pre-reproductive and post-reproductive
young and old individuals are more vulnerable to predation and disease
Which age class is least common in the wild? Why?
Post-reproductive because most species life histories do not out live the reproduction stage
Once they have reproduced, they have little energy left to invest in maintaining their own body
Give examples of species that naturally have a post-reproductive age class
Long lived species, such as humans, elephants, and whales
In all cases, the grandparents help with raising offspring
If there is a large proportion of individuals in the pre-reproductive and reproductive age classes, what will happen to the population?
the population will INCREASE in size because there are many female individuals that will soon be of reproductive age
If there is a relatively even proportion of individuals in the pre-reproductive and reproductive age classes, what will happen to the population?
The population will STABILIZE
If there is a small proportion of individuals in the pre-reproductive and reproductive age classes, what will happen to the population?
The population will DECREASE (or grow at slower rate) because there are less female individuals who will be entering the reproductive age class
What are life tables? What biological sex are they usually based on?
Tables that summarize age-specific schedules of survival, mortality, and fecundity of a population
Usually based on females because maternity is much easier to determine in nature
What are the functions of life tables?
- model population growth and predict future numbers based on current age structure
- understand observed trends in population structure
What are the 2 types of life tables?
- Cohort life table
2. Static life table
Describe the cohort life table and give an example
Life tables based on a group of individuals (cohort) throughout their life, from birth to the death of the very last individual
ex. The cactus finch on Daphne Major study by the Grants
Why can cohort life tables be problematic?
- If the species of interest is long-lived, it can take a long time to collect the data, so it is more useful for organisms with short lifespans
- It doesn’t work well for highly mobile animals (mobility makes catching and finding challenging)
- Predictive value reduced in variable environments
Describe the static life table and give an example
Life tables based on a cross-section of a population at a specific time
ex. the Dall mountain sheep, Alaska, study that used 608 skeletons to determine the age structure of the population
Why are static life tables more convenient than cohort tables?
- Static life tables can be made for organisms that are alive or dead, whereas cohort life tables require living organisms
- cohort life tables require studying the population for their entire lifespan (this could be many years), whereas the static life tables require only data from a particular time
What are 2 problems with static life tables?
- requires a method for determining individual age (especially if studying a live population)
- Assumes survival rate does not change over time (it’s measured from only one point in time)
- this can be overcome by conducting multiple static life tables at different points in time
How can the data from a life table be used?
to plot age specific survivorship
What are the 3 types of survivorship curves that can be made based off life table data?
Type 1: high survival until old age, then high mortality (ex. wolverines)
Type 2: relatively constant mortality over lifespan (ex. small animals)
Type 3: mortality highest in juvenile stages (ex. insects, plants)
Give an example of how survivorship curves vary among populations with different environmental conditions
Survivorship in human populations vary between populations in developing countries vs. populations in developed countries
developing: curve represents type 2) because of disparities in safe water, nutritional foods, medical access
developed: curve represents type 1
What categories within life tables do we need to know for this course?
age/age stage number alive at each stage survivorship survival rate fecundity (will be given)
Describe survival rate
the proportion of individuals surviving from one age class to the next
What is the formula for survival rate?
Sx = nx+1 / nx
ie., survival rate = # of individuals alive in year 2 / # individuals alive in year 1
Describe survivorship
the probability of living from birth to another specific age class
or the proportion of original population still alive at each age class
What is the formula for survivorship?
lx = nx/n1
ex. survivorship = # individuals alive in year 4 / # individuals alive in first year
How is the net reproductive rate (R0) calculated?
The sum of the survivorship (lx) multiplied by fecundity (bx) values for the entire lifetime
sum of all lxbx values
What is the variable for survival rate?
Sx
What is the variable for survivorship?
lx
What is the variable for the life stage/age?
x
What is the variable for the number alive at age x?
nx
What is the variable for fecundity?
bx
What does fecundity tell us?
how many, on average, female offspring each female gives birth to in each age class
What is the purpose for the variable lxbx?
Since fecundity can only apply to females that survive to each age class, the birth rate needs to be corrected by multiplying survivorship by a given age
What is the net reproductive rate?
the average number of females produced per female in a life time
(R0)
What does the net reproductive rate tell us?
whether the population is increasing, decreasing, or stable based on its specific age structure because it essentially is the RATE OF FEMALE REPLACEMENT
What happens to the population if R0 = 1?
it will stabilize because each female replaces herself
What happens to the population if R0 > 1?
increase because females are reproducing more female offspring than needs to replace themselves (on average)
What happens to the population if R0 < 1?
decrease because females are not replacing themselves
