BIOL 4120: Principles of EcologyBIOL 4120: Principles of Ecology

Lecture 10: Population Lecture 10: Population GrowthGrowth

Dafeng HuiDafeng Hui

Office: Harned Hall 320Office: Harned Hall 320

Phone: 963-5777Phone: 963-5777

Email: dhui@tnstate.eduEmail: dhui@tnstate.edu

World populationWorld population

Population growthPopulation growth

Definition: how the number of individuals Definition: how the number of individuals in a population increases or decreases in a population increases or decreases with timewith time

Growth is controlled by rates of birth, Growth is controlled by rates of birth, immigration and death and emigration.immigration and death and emigration.

Open or closed population: no immigration Open or closed population: no immigration and emigration, or immig.=emig.and emigration, or immig.=emig.

In closed population, growth is determined In closed population, growth is determined by birth rate and death rate. by birth rate and death rate.

10.1 Population growth reflects the 10.1 Population growth reflects the difference between rates of birth and deathdifference between rates of birth and death

Model developmentModel development A population of freshwater hydra growing in an aquarium in the A population of freshwater hydra growing in an aquarium in the

laboratory. laboratory.

Population size N(t) when time is t.Population size N(t) when time is t.

This is a closed population. This is a closed population.

Population size change is related to birth rate (b) and death rate (d)Population size change is related to birth rate (b) and death rate (d)

dN/dt=(b-d)N=rN dN/dt=(b-d)N=rN

The difference between birth rate and death rate is the intrinsic The difference between birth rate and death rate is the intrinsic growth rate (r) (instantaneous per capita rate of growth).growth rate (r) (instantaneous per capita rate of growth).

r=b-dr=b-d

In a closed population, population size In a closed population, population size change is related to birth rate (b) and change is related to birth rate (b) and death rate (d)death rate (d)

The difference between birth rate and The difference between birth rate and death rate is the intrinsic growth rate (r) death rate is the intrinsic growth rate (r) (instantaneous per capita rate of growth).(instantaneous per capita rate of growth).

r=b-dr=b-d Population growth is related to this Population growth is related to this

intrinsic growth rate (r).intrinsic growth rate (r).

Exponential population growthExponential population growth

Equations:Equations:• 1) dN/dt=rN (differential equation form)1) dN/dt=rN (differential equation form)

• 2) N2) N(t)(t)=N=N(0) (0) exp(rt) (exponential growth exp(rt) (exponential growth model)model)

Conditions:Conditions:• Initial population is smallInitial population is small• No food or resource limitationNo food or resource limitation

An exampleAn example

Reindeer, St. Paul, Alaska.

Started in 1910 with only 4 males and 22 females

In 1940, there were nearly 2000

Whooping crane, an Whooping crane, an endangered speciesendangered species

recovered from near recovered from near extinction in 1941extinction in 1941

How to calculate r?

Software, Excel (trendline)

Aransas National Wildlife Refuge

Properties of exponential growthProperties of exponential growth

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.

Properties of exponential growthProperties of exponential growth

Widely used in biology

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, ?

Prediction of population growthPrediction of population growth

44524

10.2 Life table10.2 Life table

Life table is an age-Life table is an age-specific account of specific account of mortality.mortality.

Purpose of life Purpose of life table: to provide a table: to provide a clear and clear and systematic picture systematic picture of mortality and of mortality and survival within a survival within a populationpopulation

How to construct a life table?How to construct a life table?

1. start with a 1. start with a cohort: a group of cohort: a group of individuals born in individuals born in the same period of the same period of time;time;

2. Add a column of 2. Add a column of llxx as the probability as the probability at birth of surviving at birth of surviving to any given age;to any given age;

How to construct a life table How to construct a life table (cont.)?(cont.)?

3. calculate d3. calculate dxx, a , a measure of age-measure of age-specific mortalityspecific mortality

4. Calculate age-4. Calculate age-specific mortality specific mortality rate, qrate, qxx

10.3 Different types of life table10.3 Different types of life table

Two typesTwo types• Cohort or dynamic life tableCohort or dynamic life table

as the above gray squirrelas the above gray squirrel

• Time-specific life tableTime-specific life table

Elf opine

10.4 Life tables provide data for 10.4 Life tables provide data for mortality and survivorship curvesmortality and survivorship curves

Table is better than words, but a Table is better than words, but a graph is worth one thousand words.graph is worth one thousand words.

Mortality curve and survivorship Mortality curve and survivorship curve.curve.

Mortality curvesMortality curves

Survivorship curvesSurvivorship curves

Log scale for Y axis

Three basic types of survivorshipThree basic types of survivorshipType 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)

10.5 Birthrate is age-specific10.5 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.

10.6 Birth rate and survivorship 10.6 Birth rate and survivorship determine net reproductive ratedetermine net reproductive rate

Fecundity table: Fecundity table: take survivorship take survivorship column, lcolumn, lxx, from life , from life table and add age-table and add age-specific birthrate, bspecific birthrate, bxx..

Net reproduction Net reproduction rate, Rrate, R00: : number of female number of female

offspring a female at birth can offspring a female at birth can produce (or average # of produce (or average # of females that will be produced females that will be produced (left) during a lifetime by a (left) during a lifetime by a newborn females.)newborn females.)

RR00: depends on : depends on survivorship and survivorship and fecundityfecundity

R0=1, >1 or <1R0=1, >1 or <1

10.7 Project population growth10.7 Project population growth

Given a population with age structure and Given a population with age structure and some other information (age-specific some other information (age-specific mortality rates and birthrates), we can mortality rates and birthrates), we can project future changes of the population project future changes of the population size.size.

For example, a population of squirrel with For example, a population of squirrel with 10 adults (1-yr) and 20 juveniles females, 10 adults (1-yr) and 20 juveniles females, what would happen in the next 10 years? what would happen in the next 10 years?

What do we need to project future What do we need to project future population size change?population size change?

Calculate age-specific survivor rate: sx=1-qxbx is age-specific birthrate

How to construct a population How to construct a population projection table?projection table?

How to construct a population How to construct a population projection table?projection table?

Population size (N) increases every year.

Lambda (finite multiplication rate): =N(t+1)/N(t).

Age distributionAge distribution

Stable age distribution: by year 7, the proportion of each age group remain the same year after year.

Population is still growing.

Geometric growth vs exponential Geometric growth vs exponential growthgrowth

N(t)=N(0) N(t)=N(0) tt

N(t)=N(0)exp(rt)N(t)=N(0)exp(rt)

=exp(r) or r=ln(=exp(r) or r=ln())

These models are used to describe These models are used to describe dynamics of populations. Geometric growth dynamics of populations. Geometric growth is used for population generations not is used for population generations not overlap (discrete time interval), exponential overlap (discrete time interval), exponential growth model is for continuous population.growth model is for continuous population.

Fig. 11.3Fig. 11.3

10.8 Stochastic processes can 10.8 Stochastic processes can influence population dynamicsinfluence population dynamics

What’s stochastic process?What’s stochastic process?

Deterministic process: Given a set of initial Deterministic process: Given a set of initial conditions (N(0), r), the exponential growth conditions (N(0), r), the exponential growth will predict only one exact outcome. will predict only one exact outcome.

But the age-specific mortality rates, birth But the age-specific mortality rates, birth rates represent probability and averages rates represent probability and averages derived from the cohort or population under derived from the cohort or population under study (bx=2? 0,1,2,3).study (bx=2? 0,1,2,3).

StochasticityStochasticity

Demographic stochasticity: stochastic (or Demographic stochasticity: stochastic (or random) variations in birth and death rates random) variations in birth and death rates that occur in populations from year to year. that occur in populations from year to year. (Cause change in r).(Cause change in r).

Environmental stochasticity: Random Environmental stochasticity: Random variation in the environment, such as annul variation in the environment, such as annul variation in climate and natural disasters variation in climate and natural disasters can have a direct influence on average can have a direct influence on average birth and death rates within the population. birth and death rates within the population.

10.9 Population extinction10.9 Population extinction

If r becomes negative If r becomes negative (birth rate < death rate), (birth rate < death rate), population declines and population declines and will go extinction. will go extinction.

Factors: Extreme Factors: Extreme environmental events environmental events (droughts, floods, cold or (droughts, floods, cold or heat etc), heat etc), loss of loss of habitat habitat (human). (human).

Small populations are Small populations are susceptible to extinctionsusceptible to extinction Allee effect, genetic drift,

inbreeding (mating between relatives)

Overgraze, only 8 in 1950

Hackney and McGraw (West Virginia Hackney and McGraw (West Virginia University) examined the reproductive University) examined the reproductive limitations by small population size on limitations by small population size on American ginseng (American ginseng (Panax Panax quinquefoliusquinquefolius))• Fruit production per plant declined with Fruit production per plant declined with

decreasing population size due to decreasing population size due to reduced visitation by pollinationreduced visitation by pollination

Small population size may result in Small population size may result in the breakdown of social structures the breakdown of social structures that are integral to successful that are integral to successful cooperative behaviors (mating, cooperative behaviors (mating, foraging, defense)foraging, defense)

The The Allee effectAllee effect is the decline in is the decline in reproduction or survival under reproduction or survival under conditions of low population densityconditions of low population density

There is less genetic variation in a There is less genetic variation in a small population and this may affect small population and this may affect the population’s ability to adapt to the population’s ability to adapt to environmental changeenvironmental change

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Geometric GrowthGeometric Growth When generations do not overlap, growth When generations do not overlap, growth

can be modeled geometrically.can be modeled geometrically.

NNt t = N= Noott

• NNtt = Number of individuals at time t. = Number of individuals at time t.

• NNoo = Initial number of individuals. = Initial number of individuals. = Geometric rate of increase.= Geometric rate of increase.• t = Number of time intervals or generations.t = Number of time intervals or generations.

Exponential GrowthExponential Growth

Continuous population growth in an Continuous population growth in an unlimited environment can be modeled unlimited environment can be modeled exponentially.exponentially.

Appropriate for populations with Appropriate for populations with overlapping generations.overlapping generations.• As population size increases, rate of As population size increases, rate of

population increase gets larger.population increase gets larger.