1. BIOL 4120: Principles of Ecology Lecture 10: Population Growth
Dafeng Hui
Office: Harned Hall 320
Phone:
Abundance, distribution & density
Space and time
In one place, population size varies. How to describe the population size change mathmatically (model), and to predict population size in the future.

2. World population

4. Population growth
Definition: how the number of individuals in a population increases or decreases with time
Growth is controlled by rates of birth, immigration and death and emigration.
Open or closed population: no immigration and emigration, or immig.=emig.
In closed population, growth is determined by birth rate and death rate.

5. 10.1 Population growth reflects the difference between rates of birth and death
Model development
A population of freshwater hydra growing in an aquarium in the laboratory.
Population size N(t) when time is t.
This is a closed population.
Population size change is related to birth rate (b) and death rate (d)
dN/dt=(b-d)N=rN
The difference between birth rate and death rate is the intrinsic growth rate (r) (instantaneous per capita rate of growth).
r=b-d
N=1000, in one month, 50 death, 100 birth, b=10/100=0.10, death rate=5/100=0.05, r=0.05, population growth rate=50 per 1000 per month.
t, t+1
N(t)=100,
N(t+1)=105
N(t+1)=N(t) +B(t)- D(t)
B(t)=bN(t)*delta(t)
D(t)=dN(t)*delta(t)

6. Population growth is related to this intrinsic growth rate (r).
In a closed population, population size change is related to birth rate (b) and death rate (d)
The difference between birth rate and death rate is the intrinsic growth rate (r) (instantaneous per capita rate of growth).
r=b-d
Population growth is related to this intrinsic growth rate (r).
N0=100, in one year, 5 death, 10 birth, b=10/100=0.10, death rate=5/100=0.05, r=0.05, population growth rate=5 per year.

7. Exponential population growth
Equations:
1) dN/dt=rN (differential equation form)
2) N(t)=N(0) exp(rt) (exponential growth model)
Conditions:
Initial population is small
No food or resource limitation
First one: predict the rate of population size change with time
Second one: predict the population size change with time
Derive the exponential growth model

8. An example Started in 1910 with only 4 males and 22 females
In 1940, there were nearly 2000
island
Reindeer, St. Paul, Alaska.

9. Whooping crane, an endangered species recovered from near extinction in 1941
How to calculate r?
Software, Excel (trendline)
Aransas National Wildlife Refuge

10. Properties of exponential growth
4/21/2017
Properties of exponential growth
R is the parameter that determining the shape of the growth form

11. Properties of exponential growth
4/21/2017
Properties of exponential growth
R is the parameter that determining the shape of the growth form
Beer-larmert law, metabolic rate and body mass, growth,
(decomposition)
Widely used in biology
r determines the shape of the growth.
r=0, no change in population size
r<0, decrease in population size; r>0, increase in population size.

12. Prediction of population growth
4/21/2017
Prediction of population growth
N(t)=N(0)Exp(rt)
Give a time t, we can predict the population size.
An Example:
Deer population: N(0)=300, r=0.5, after 5 years, what’s the population size?
N(5)=N(0)Exp(rt)=300*exp(0.5*5)=3655
(495, 815, 1344, 2216, 3655) t=10, ?
R is the parameter that determining the shape of the growth form
44524 12

13. 10.2 Life table Life table is an age-specific account of mortality.
Purpose of life table: to provide a clear and systematic picture of mortality and survival within a population
Population growth model provides a general picture of population size change without considering age structure and difference in birth and death rate at different ages. To obtain a better picture of mortality (survival) within a population, another approach is commonly used, life table.
Life table provide a summary of how survial and reproductive rates vary with the age of the organisms.
A life table is an age-specific account of mortality
It includes several columns. We will explain what these mean and how to construct a life table.
A cohort is a group of individuals born in the same period of time
x = age classes
nx = the number of individuals from the original cohorts that are alive at the specified age (x)
lx = the probability at birth of surviving to any given age (x)
dx = age-specific mortality = the difference between the number of individuals alive for any age class (nx) and the next older age class (nx + 1)
qx = age-specific mortality rate = the number of individuals that died in a given time interval (dx) divided by the number alive at the beginning of that interval (nx)

14. How to construct a life table?
4/21/2017
How to construct a life table?
1. start with a cohort: a group of individuals born in the same period of time;
2. Add a column of lx as the probability at birth of surviving to any given age;
Cohort
X represent age in unit of year
nx represent the # of individuals from the original cohort that are alive at the specific age (x).
Track all individuals

15. How to construct a life table (cont.)?
3. calculate dx, a measure of age-specific mortality
4. Calculate age-specific mortality rate, qx

17. 10.3 Different types of life table
4/21/2017
10.3 Different types of life table
Two types
Cohort or dynamic life table
as the above gray squirrel
Time-specific life table
Pupae instars elf orpine
Time specific life table??? is a snap-shot of a long-lived species. It usually starts out as a survivorship curve and they usually have a normalization to 1,000 individuals. Time-specific life tables work best for long-lived species, and they assume that there are equal birth rates among all age classes.
A cohort or dynamic life table is used to track the fate of a group of individuals born at a given time
These individuals are followed from birth to death
Modified version: A dynamic composite life table constructs a cohort from individuals born over several time periods
A time-specific life table is a distribution of age classes during a single time period
Several assumptions are made in this approach
Each age class was sampled in proportion to its numbers in the population
Age-specific mortality rates (and birthrates) are constant over time
Many animals (e.g., insects) live only one breeding season. Because generations do not overlap, all individuals belong to the same age class
nx is measured by estimating the population size several times over its annual season
The life table is useful for studying several areas of plant demography
Seedling mortality and survival
Population dynamics of perennial plants marked as seedlings
Life cycles of annual plants
Elf opine

18. 10.4 Life tables provide data for mortality and survivorship curves
Table is better than words, but a graph is worth one thousand words.
Mortality curve and survivorship curve.
Life table data are generally presented as:
A mortality curve that plots the qx column against age (x)
A survivorship curve that plots the lx column against age (x)
Life tables and curves are based on data from one population at a specific time and under certain environmental conditions

19. 4/21/2017
Mortality curves
Rosette
Sedum: Crassullceae

20. Survivorship curves Log scale for Y axis Good for comparison
Example of survivorship curves
Next slide for:
I. deer, sheep, human, convex
II: squire and adult birds, linear, not change with age
III. Plants, fish, young bird, concave
Log scale for Y axis

21. Three basic types of survivorship
4/21/2017
Three basic types of survivorship
Type I (convex)
Humans and other mammals and some plants (k-selection)
Type II (survival rates do not vary with age)
Adult birds, rodent, and reptiles, perennial plants
Type III. Concave
Mortality rate high in the beginning (r-selection)
Oysters, Fish, many plant species (most trees)
There are three idealized types of survivorship curves
Type I: typical of populations in which individuals have long life spans, survival rate is high throughout the life span with heavy mortality at the end
Humans, other mammals, some plants
Type II: survival rates do not vary with age
Adult birds, rodents, reptiles, perennial plants
Type III: mortality rates are extremely high in early life
Fish, many invertebrates, and plants
Some show different types at different stages, like bird in the previous slide

22. 10.5 Birthrate is age-specific
Crude birthrate (demographers): # of birth over a period of time divided by population size at the beginning of the period*1000
Age-specific birthrates, bx
Mean # of females birth to a female in each age group.
(Only females give birth; birth rates vary with ages)
Gross reproduction rate: sum of the bx values across all age classes, provides an estimate of average offspring born to a female over her lifetime.
Life table can also be used to estimate the popultaiton size change
Next is the reproduction
The crude birthrate is expressed as births per 1000 population per unit time
Only females give birth
Birthrate of females generally varies with age
Birthrate is better expressed as the number of births per female of age x
bx = mean number of females born to a female in each age group
Continuing with the gray squirrel example
 = gross reproductive rate = the average number of female offspring born to a female over her lifetime
Assume that the female will survive and live a full life

23. 10.6 Birth rate and survivorship determine net reproductive rate
4/21/2017
10.6 Birth rate and survivorship determine net reproductive rate
Fecundity table: take survivorship column, lx, from life table and add age-specific birthrate, bx.
Net reproduction rate, R0: number of female offspring a female at birth can produce (or average # of females that will be produced (left) during a lifetime by a newborn females.)
Total number of offsprings produced not only depends on the age-specific birthrate, but also the survivorship.
Lxbx: mean number of females born in each age group adjusted for survisorship
R0=1, replace itself
A fecundity table combines the survivorship (lx) with the age-specific birthrates (bx)
lxbx = mean number of females born in each age group, adjusted for survivorship
R0 = net reproductive rate = the average number of females that will be produced during a lifetime by a newborn female
R0 = 1; on average, females will replace themselves in the population
R0 < 1; females are not replacing themselves in the population
R0 > 1; females are more than replacing themselves in the population
R0: depends on survivorship and fecundity
R0=1, >1 or <1

24. 10.7 Project population growth
Given a population with age structure and some other information (age-specific mortality rates and birthrates), we can project future changes of the population size.
For example, a population of squirrel with 10 adults (1-yr) and 20 juveniles females, what would happen in the next 10 years?

25. What do we need to project future population size change?
For simplicity, age-specific mortality (qx) is converted to age-specific survival (sx)
sx = 1 – qx
A population projection table can be constructed using sx and bx
Calculate age-specific survivor rate: sx=1-qx
bx is age-specific birthrate

26. How to construct a population projection table?
Only females here
We will use the life and fecundity table values of the gray squirrel to illustrate a hypothetical population of squirrels introduced into an unoccupied forest
Females are only used to construct the population projection table

27. How to construct a population projection table?
Population size (N) increases every year.
Lambda (finite multiplication rate): =N(t+1)/N(t).

28. Age distribution
An age distribution for each successive year can be calculated from a population projection table
Age distribution is the proportion of individuals in the various age classes for any one year
A stable age distribution is attained when the proportions of each age group in the population remain the same year after year (even though N(t) increases)
Stable age distribution: by year 7, the proportion of each age group remain the same year after year.
Population is still growing.

29. Geometric growth vs exponential growth
N(t)=N(0) t
N(t)=N(0)exp(rt)
=exp(r) or r=ln()
These models are used to describe dynamics of populations. Geometric growth is used for population generations not overlap (discrete time interval), exponential growth model is for continuous population.
An estimate of population growth can be derived from a population projection table
 = finite multiplication rate = N(t +1)/N(t)
Once the population reaches a stable age distribution, the value of  remains constant
 > 1.0 indicates a population that is growing
 < 1.0 indicates a population in decline
= 1.0 indicates a stable population size through time
The population projection table demonstrates two concepts of population growth
 (estimated population growth rate) is a function of sx and bx
The constant rate of population increase from year to year and the stable age distribution are results of sx and bx that are constant through time
If  does not vary (under conditions of stable age distribution), population size in the future can be projected
N(t) = N(0) t
Describes a pattern of population growth similar to the exponential growth model
Geometric population growth = N(t) = N(0) t
Finite
Exponential population growth = N(t) = N(0)ert
Continuous
 = er or r = ln

30. Fig. 11.3
Plant population size change of phlox

31. 10.8 Stochastic processes can influence population dynamics
What’s stochastic process?
Deterministic process: Given a set of initial conditions (N(0), r), the exponential growth will predict only one exact outcome.
But the age-specific mortality rates, birth rates represent probability and averages derived from the cohort or population under study (bx=2? 0,1,2,3).
Population dynamics represent the combined outcome of many individual probabilities
Age-specific survival rates (sx) represent the probability that a female of that age will survive to the next age class
This reality has led ecologists to develop probabilistic or stochastic models of population growth to account for these variations
Demographic stochasticity is the random (stochastic) variations in birth and death rates from year to year
The variations in d and b cause populations to deviate from the predictions based on deterministic models
Environmental stochasticity is the random variations in the environment or the occurrence of natural disasters
These events directly influence d and b

32. Stochasticity
Demographic stochasticity: stochastic (or random) variations in birth and death rates that occur in populations from year to year. (Cause change in r).
Environmental stochasticity: Random variation in the environment, such as annul variation in climate and natural disasters can have a direct influence on average birth and death rates within the population.
Individual survival and reproduction
Average birth and death rates caused by environmental factors
Under the following conditions, a population can become so small that it declines toward extinction:
When deaths exceed births, populations decline
R0 becomes less than 1.0
r becomes negative
Extreme environmental events
Severe shortage of resources
Introduction of a novel predator, competitor, or parasite (disease)
Habitat loss (due to human activities)
Small population size

33. 10.9 Population extinction
If r becomes negative (birth rate < death rate), population declines and will go extinction.
Factors: Extreme environmental events (droughts, floods, cold or heat etc), loss of habitat (human).
Small populations are susceptible to extinction
Allee effect: decline of either reproduction or survive under conditions of small population. Reasons: difficult to find a mate for animal, produce less seeds for plants, breakdown social structure for animals (hard to defend predators etc).
Small populations are more susceptible to both demographic and environmental stochasticity
When only a few individuals make up a population, the fate of each individual can be crucial to population survival
Over large territories, it can be impossible to find a mate (large cats)
Chemical signals will not be intercepted (insects)
Pollination is unlikely (plants)
Inbreeding cause some rare, recessive, deleterious genes become expressed. decease in fertility, slow growth, even deaths.
Overgraze, only 8 in 1950
Allee effect, genetic drift, inbreeding (mating between relatives)

34. Hackney and McGraw (West Virginia University) examined the reproductive limitations by small population size on American ginseng (Panax quinquefolius)
Fruit production per plant declined with decreasing population size due to reduced visitation by pollination

35. Figure 10.13

36. Small population size may result in the breakdown of social structures that are integral to successful cooperative behaviors (mating, foraging, defense)
The Allee effect is the decline in reproduction or survival under conditions of low population density
There is less genetic variation in a small population and this may affect the population’s ability to adapt to environmental change

37. END

38. Geometric Growth
When generations do not overlap, growth can be modeled geometrically.
Nt = Not
Nt = Number of individuals at time t.
No = Initial number of individuals.
 = Geometric rate of increase.
t = Number of time intervals or generations.

39. Exponential Growth
Continuous population growth in an unlimited environment can be modeled exponentially.
Appropriate for populations with overlapping generations.
As population size increases, rate of population increase gets larger.