POPULATION DYNAMICS

POPULATION DYNAMICS 

Today we focus on population 

censuses and models that count all 

individuals equally (using the 

variable N 

only, i.e. without age or 

sex) and that do not measure 

resource availability. 

Exam 3 lecture #4 UIC BioS 101 Nyberg 1

Reading 

Chapter 52.2. 

Box 52.2 (p1043-44) -read carefully 

Box 52.3 (p1048) on Mark-Recapture 

Review of the x1 07 lecture may be 

useful to understand population growth. 

Chapter 52.1 introduces demographic 

models using age of individuals. 

Exam 3 #4 UIC BioS 101 Nyberg 2

Changes in population size 

In a sustainable world, we expect population 

sizes of animals and plant species to stay 

about the same, N constant thru time. 

Reproduction gives organisms the potential to 

grow exponentially. 

Population growth eventually exhausts the 

resources and maintenance of population is 

dependent on renewal of resources. 


Determining population size 

Census = Count all the individuals 

Generally tough to do 

Sampling 

Create subpopulations (often based on 

area), count individuals in subpopulations, 

extrapolate to entire population/area 

Mark-Recapture Studies 

Sample, mark, release, resample 


Mark-recapture basics 

Box 52.3 (p1048) 

Perhaps simplest to understand in small lake 

Capture individuals, mark (tag) them, release n 1 

marked individuals, now n 1 / N is fraction marked. 

Assume the marked animals disperse and 

mix with unmarked animals in lake 

Capture n 2 individuals, if m 2 is # marked, the fraction 

marked is m 2 / n 2 & should be equal to n 1 / N . 

then N = total # in population = n 1 •n 2 / m 2 


Mark-recapture example 

You catch 14 butterflies and mark the 

thorax with white paint and then release 

the butterflies. 

Two days later you go out and net 18 

butterflies and find 4 of them marked. 

The Estimate of the total population = 

14 x 18/4 = 63 butterflies. 


Census Histories 

The Census history is the record of the 

numbers of individuals through time 

Some populations show a pattern of 

constant doubling for a period of time. 

Many populations are stable in size. 

Some species have a pattern of steady 

decrease. 

Many insects have population “outbreaks” 

with large fluctuations from year to year 


Census history examples 


Minimum Doubling Time 

The time it takes for a species to double 

the number of individuals even when 

resources are abundant is called the 

doubling time. 

Both geometric and exponential growth 

imply a constant doubling time, doubling 

time simply related to r, growth rate, is a 

parameter of the species. 


Population Size 

at a particular time 

N is the symbol for the variable population 

size 

N t = means the population size at time t, as a 

subscript it implies the geometric or discrete 

time model. 

N t + 1 = population size one generation after t 

The exponential or continuous time model 

would be written as N(t), verbally ‘N as a 

function of time’. 


Population Change 

using birth & deaths and migration 

Plus Births, B is number of births 

Minus Deaths, D is number that died 

Plus Immigrants, I = # that moved in 

Minus Emigrants, E = # that left 

N t+1 

= N t + B – D + I - E 

If population is closed, N t+1 = N t + B – D 

ΔN = N t+1 -N t = B – D = change in size 


The Whooping Crane 

The species is ENDANGERED 

according to US law. 

The population was once at least 

10,000 birds (always rare). 

The population was reduced to only 20 

birds in the 1940s. 

The population has been growing 

exponentially for about 60 years. 


Census history of Whooping Crane 


Geometric Growth Model 

N t+1 = λ• N t 

λ, lambda,is the multiplier from 

one generation to the next. 

= 1, the population size 

stays the same = is constant. 

If λ 


Exponential Growth Model 

N(t) = N 0 •e r•t 

The parameter that measures growth is r, 

which measure the instantaneous per capita 

growth rate per unit time. 

If r = 0.04 yr -1 the population grows 4% per 

year, as e 0.04 = 1.04 approximately. 

If r = 0, the population size does not change 


Exponential Growth Model 

Can also be written in “differential” form: 

dN/dt = r•N where dN/dt is the change 

in abundance per unit time change and 

r is the per capita growth rate. 

Note that if r =0 the population is not 

changing in size, i.e. dN/dt =0, and if r is 

negative the population is decreasing. 


Relationships of λ 

and r 

λ = e rt or e r if time is one unit long. 

If λ < 1, then r will be negative, i.e. the 

population is declining (exponential decay). 

If λ > 1, then r will be greater than zero and 

the population will increase geometrically. 


Modifications of growth model 

Populations don’t grow indefinitely, but 

rather reach a maximum density. 

The logistic model is a simple 

modification of exponential growth that 

leads to curve (sometimes referred to a 

‘s’ shaped) that conforms to 

observations of batch cultures. 


The “logistic” 

model of growth 

We take our exponential model (per 

capita growth constant) and add a new 

parameter, a, that reduces the growth 

rate in proportion to the population size 

dN/dt = r•N – a•N2 per capita growth rate dN/N•dt = r – a•N 

dN/ 

N•dt 

N 


Paramecium 


N 

The Logistic Equation 

Time 

See Fig. 52.7a 


Metapopulations 

Species are usually made up of patches 

of populations with few individuals found 

in the area between the patches. 

The population is said to have a 

metapopulation structure if population 

extinction and colonization of empty 

suitable patches is frequent. 


Meta- 

popula 

tions 


Population Abundance Cycles 

Some populations show regular 

fluctuations of population size. 

Evenly repeated highs and lows are 

known as population cycles. 

The Lynx (a cat that eats hares) and 

hare (rabbit) populations cycle with an 

11 year period. The Lynx hi and low 

trails the hi and low of the hare. 


Hare & Lynx populations 


Outbreaks 

It is not unusual for populations of 

insects to vary greatly from year to year 

Very great increases in abundance are 

called “outbreaks” 

Examples in 2003 include the Painted 

Lady butterfly & the Asian Ladybug 


Census History with Outbreak 

N 

Time 


Vocabulary 

Constant Doubling time 

Geometric growth 

Exponential growth 

metapopulations 

parameter 

Emigrants, immigrants 

Census history 

Outbreak 

N 0 •e r•t 

Logistic growth 

Cycle 

λ, lambda

POPULATION DYNAMICS

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