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Chapter 11: Population Growth and Regulation

• Human Population

Age structure of two human populations in 2008

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Age structure of human population in Germany.

Population Dynamics

• Geometric and Exponential growth (Density independence)independence)– Discrete time, λ

– Continuous time, r

– Age‐structured birth and death• Life table

• Population projection matrix

• Logistic population growth                 (Density dependence)

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Geometric growth

Time = 0

Geometric growth (multiplicative in discrete time)

Time = 1

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Geometric growth

Time =2

Time =3

Geometric growth

Time =4

Time =5

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Geometric growth

me =6

N(t+1) = 2 × N(t)

N(1) = 2 × N(0)

N(4) = 2 × N(3)

N(2) = 2 × N(1)

N(3) = 2 × N(2)

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N(t+1) = 2 × N(t)

N(2) 2 × N(1) 2 × 2 × N(0)N(1) = 2 × N(0)

N(4) = 2 × N(3) = 2 × 2 × 2 × 2 × N(0)

N(2) = 2 × N(1) = 2 × 2 × N(0) N(3) = 2 × N(2) = 2 × 2 × 2 × N(0)

N(t) = 2 t × N(0)N(t) = 2 t × N(0)In this example, the per capita geometric growth factorfor discrete time, λ, is equal to “2”, for each individualin the population at time t, there will be 2 at time t+1.

Geometric (discrete time) growthExponential (continuous time) growth

N(t+1) = 2 × N(t)

N(t)N(t) = 2 t × N(0)

N(t)

t

N(t) = e rt N(0)( for example, if e r = 2, then r = 0.693)

dN/dt = rN(dN/dt) / N

The slope of the curve increases with NThe per capita population growth rate

N(t) = 2t N(0)

(dN/dt) / N = r The per capita population growth rate for continuous time is r.

r = b - d “little r”The intrinsic instantaneous per capita rate of population growth equals the per capita instantaneous birth rateminus the per capita instantaneous death rate.

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r ,  er =     ,  ln (     ) = r• parameters of ecological and evolutionary 

i ifi

significance

• the average realized fitness in a given environment 

• any character that affects it is subject to selectionselection

Exponential

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Figure 14.1

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Age structured birth and death

• Individuals in the population differ in birth d d th t di tand death rates according to some 

characteristic (age, size …)

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Two basic approaches

• Follow a cohort throughout its life

• Follow individuals of different ages for some subset of the life

1990

1995

2000

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2005

2010

2015

Cohort survivorship curves

vors

)(N

umbe

r of

sur

viv

Age, x

Log

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dual

s

Lx is the approximated area of each rectangle,height is (lx+ (lx+1))/2. It has dimension of individuals-time

Tx is the sum of all the rectangles to the right of age x.

opor

tion

of in

divi

Age

l x, p

ro

etc..

ex is Tx/lx(dimension individuals-time/individuals = time).

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Life Tables summarize demographic data by age

– age (x)

– number alive

– survivorship (lx): lx = s0s1s2s3 ... sx‐1– mortality rate (mx)

– probability of survival between x and x+1 (s )probability of survival between x and x+1 (sx)

– fecundity (bx)

Cohort life table parameters

x

x

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r can be calculated from the life table

Population growth rate:from life tables

r can be calculated from the life table because of Lotka’s equation.

The area under a curve given by the following equation(from age 0 to the last age) has the value 1:

1= lxbx e -rx dxe -rx = 1/e rx , where e rx is the amount the population has grown in x years, or since individuals x-yrs old were born.

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How to find out:How fast is a population growing?  

• Iteratively from Lotka’s equation

– make a guess of little r, see if it gives you an area under the curve of 1, <1 or >1

– adjust your guess

• Use net reproductive rate and generation time

• Project the population and keep tabs on its sizeProject the population and keep tabs on its size 

• Find the dominant eigenvalue of a matrix

lxbx = net reproductive rate = R0,the increase per capita per generation

T is the generation timeT = xlxbx / R0

erT = R0

rT = ln (R0)r = ln (R0)/ T

r = ln (R0)/ ( xlxbx / R0 )

r = R0 ln (R0)/ ( xlxbx)

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• Net reproductive rate, R0:

the expected total number of offspring of an individual over the course of her life span.

– R0 = 1 represents the replacement rate

– R0 < 1 represents a declining population

– R0 > 1 represents an increasing population

• Generation time

T l b / l bT = xlxbx / lxbx

r = logeR0/T

• The intrinsic rate of natural increase depends on both the net reproductive rate and the generation time

• Large values of R0 and small values of T lead to the most rapid population growth

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Table 11.7. Estimation of exponential rate of increase forhypothetical population.

Two basic approaches

• Follow a cohort throughout its life

• Follow individuals of different ages for some subset of the life

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1990

1995

follow the fates of individuals of all ages

Calculation of “r”…(cont’d)

• Structured populations have size and shape

• Population projectionPopulation projection

• Find the dominant eigenvalue of a matrix (discrete time) = λ

• Matrix models can be readily used for non‐cohort data, in discrete time 

follow the fates of individuals of all ages over a– follow the fates of individuals of all ages over a given time interval

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Age‐structured survival and fecundity per age class

Age Survival FecundityAge Survival Fecundity

0 s0 b0

1 s1 b1

2 s2 b22 s2 b2

3 s3 b3

Age‐structured survival and fecundity per age class

Age Survival Fecundity Numbe

0 0.5 0 20

1 0.8 1 10

2 0 5 3 40

= 0.5

= 2 4

realizedfecundity

2 0.5 3 40

3 0.0 2 30

= 2.4

= 1.0

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Table 11.2

Projection:Population into the future

• Stable age distribution

– if the demographic character of the environment remains constant, 

the “age‐structure” will converge to a particular “shape” (after a while)

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Table 11.3

Matrix analysis

• Computers make it easy

• Yields estimates for parameters of interestYields estimates for parameters of interest   – λ (population growth rate) (eigenvalue)

– stable age distribution (“shape”) ( right eigenvector)

– reproductive value (left eigenvector)

– sensitivity

• Flexible for populations structured by stages, sizes, p p y g , ,habitats, etc…– Not restricted to age‐structured populations

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Population projection matrix

Stage at Stage at time tgtime t+1

g

seed seedling juvenile reproductive

seed 0.1 0 0 12

seedling 0.2 0.1 0 0

juvenile 0 0.3 0.1 0

reproductive

0 0.1 0.2 0.4

Projection

n(t+1) = A n(t)

n(t+1) = n(t)n(t+1) = n(t)

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Sensitivity questions: what influences population growth the most?

• Evolutionary questions

• Ecological questions

• Conservation questions

Loggerhead Turtle, Caretta caretta

Crowder, L. , D. Crouse, S. Heppell & T. Martin, 1994

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Photos and drawings from: Carmichael, P. and W. Williamson, W. 1999. Florida’s Fabulous Reptiles and Amphibians. World Publications. Tampa, FL, p.115-116.

Summary 1• Population growth parameters 

– exponential parameter, r, for continuous time or– geometric parameter, λ, for discrete time.– Equations for density‐independent growth

• Age‐specific survival and fecundity.• Life tables as well as matrices summarize 

demographic data.• Demographic analysis permit determination of 

population growth rates life expectancy generationpopulation growth rates, life expectancy, generation time, stable age distributions, and sensitivity analyses.

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Density dependent population growth

• IF, at high densities, there is 

– less food for individuals and their offspring

– social strife

– disease spreading readily

d t tt t d– predators attracted

• Population growth may slow down and stop

Figure 11.17

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Density dependent population growth

• Population growth slower at high density• Population growth slower at high density

• Any component  may be density‐dependent

– birth rates

– death rates

– growth ratesg

Exponential vs. logistic growth

N(t)

K

t

N(t)

N(t) = K / (1+ be-rt )

N(t)

t

N(t) = ert N(0)

dN/dt = rN(dN/dt) / N = r

dN/dt = rN (1 - N/K)(dN/dt) / N = r ( 1 - N/K)r = growth rate at lowest density

K = carrying capacity: maximium sustainable population size

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The Logistic Equation

• In 1910, Raymond Pearl and L.J. Reed attempted to project the human population’s future growthto project the human population’s future growth.

• Census data suggested that r should decrease as a function of increasing N.

• Predicted stabilization at around       197 million

• Currrently, we number                       > 6 billion y,(6,000 million!)

• The concept is useful...

The Proposal of Pearl and Reed

• Pearl and Reed proposed that the relationship of r to N should take the form:N should take the form:

r = r0(1 ‐ N/K)

• the logistic equation:

dN/dt = r0N(1 ‐ N/K)

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the Logistic Equation

• The logistic equation describes a population that stabilizes at carrying capacity, K:carrying capacity, K:– below K : grows

– above K :  decreases

– at K  :      constant

• Small population exhibits sigmoid growth.

• Inflection point  (at K/2) separates the accelerating and decelerating phasesdecelerating phases 

Figure 11.15

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Figure 11.21 : white tailed deer

Density Dependence: Plants • increased mortality 

• reduced fecundityreduced fecundity

• slower growth, small individuals

• BUT: at very high densities, 

– mortality results in declining density

– Growth rates of survivors exceed the rate of fdecline of the population 

– so total weight of the planting increases

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Figure 11.24

as planting density of flax seeds is increased, the average size achieved by individual plants declinesand the distribution of sizes is altered

Figure 11.25

in horseweed, a thousand-fold increase in average plant weightoffsets a hundred-fold decrease in density

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Self‐Thinning Curvefor plants

• log (average weight) versus log (density) 

• points fall on a line with slope of Approximately ‐3/2

• this relationship is known as                   the ‐3/2 p /power law

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Density‐dependence and discrete time

Analytically, 

• Discrete time may lead to complications

– periodicity

– chaos

– independence from initial conditions

1500Trajectories of Discrete-Time Logistic Model

500

1000

Pop

ulat

ion

Siz

e

0 5 10 15 20 25 300

Time

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100

120

140Bifurcation Diagram for Discrete-Time Logistic

40

60

80

Pop

ulat

ion

Siz

e

1.8 2 2.2 2.4 2.6 2.8 30

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Intrinsic Rate of Increase, r

Summary 2

• Populations  often regulated by scarcity of d th d it d d t f tresources and other density‐dependent factors. 

• Density‐dependent population growth is described by the logistic equation.

• Both laboratory and field studies have shown how population regulation may be brought about by d i d ddensity‐dependent processes.