Basics in population ecology

It is not the strongest of the species that survives, nor the most intelligent, but the one most responsive to change.Charles Darwin.

Our program

1. Simple growth processes

2. Outbreaks

3. Age structured populations

4. Harvesting and viability analysis

5. Competition , predation and parasitism

6. Populations in space: Metapopulation and spatial dynamics

7. Populations in space: Metapopulation and spatial dynamics

Literature

What is a population?

A population is a group of potentially interbreeding individuals of the same species living in the same area at the same time and sharing a common gene pool.

Carabus coriaceus in a forest Carabidae in a forest

Population ecology is a sub-field of ecology that deals with the dynamics of species populations and how these populations interact with the environment.

It is the study of how the population sizes of species living together in groups change over time and space.

Basic characteristics of populations:Absolute density (individuals per unit area)Relative density (Proportion of individuals with respect to some standard)Abundance (size; total number of individuals)Age structure (triggered by natality and age dependent mortality)Dispersal (spatial dynamics)

Main axiom of population ecology:

Organisms in a population are ecologically equivalent.

Ecological equivalency means:

Organisms undergo the same life-cycle

Organisms in a particular stage of the life-cycle are involved in the same set of ecological processes

The rates of these processes (or the probabilities of ecological events) are basically the same if organisms are put into the same environment (however some individual variation may be allowed)

Sometimes species of different species interbred. These do not form a population per definition

In Sulawesi seven species of macaques (Macaca spp.) interbreed where their home ranges overlap.

Interbreedin is the cause of endangerment of Macaca nigra.

Adapted from Riley (2010) The endemic seven: four decades of research onth Sulawesi Macaques. Evol. Anthr. 19: 22.

Spatially separated individuals do not form true populations

A species occurring on four islands that are isolated is divided into four

independently evolving populations.

Due to limited gene flow populations on two islands might be considerd as foring a

single genet ically structured populations

Raven (Corvus corax)

Ravens in different continents do not form a single population. There is no (or only limited) gene flow.

Temporary separated individuals do not form populations

0 1 2 3

Number of bees hatching from eggs

Hatching year

N

Macrotera arcuata

Overlaying is a strategy to reduce risk due to unfavourable conditions.

If overlaying is genetically fixed the genotypes of the three hatching cohorts never meet.

Omphale lugens Mikiola fagi

Spring Summer Summer

N

EggsSummerSpring

Spring and summer generations have only limited overlap and thus form partly

separated populations.Overlaying is connected with host change.

M. fagi is univoltine.

North atlantic salmon is semelparous

Man is iteroparous

Iteroparous populations are of age structured with each age cohorte

having a different reproductive output.

Important questions:• What is the population rate of growth or

decline?• To what factor is the population growth• rate most responsive?• Will the population eventually go extinct?• What happened to the population in the• past?

Life cycles

Egg

Larva 1

Larva n

Adult

Egg

Juvenile

Adults 1

Adult n

Senex

Iteroparous species reproduce at least two times

and might form age structured populations

Semelparous species reproduce only once and can be described

by simple growth models

Some species have age cohorts after the

reproductive phase

Fertility = number of eggs

per female

Fertility = number of eggs per female

Differences in life history

Why grandparents?

Total fertility rate (TFR) is the total number of children a female would bear during her lifetime.

Gross Reproduction Rate (GRR) is the potential average number of female offspring per female.

Net Reproduction Rate (NRR) is the observed average number of female offspring per female. NRR is always lower than GRR. When NRR is less than one, each generation is smaller than the previous one. When NRR is greater than 1 each generation is larger than the one before.

In semelparous species age specific fertility (ASF) is the average number of offspring per female of a certain age class.

Some basic definitions

Population growth is the change in population size over time. Growth can be negative.Population growth rate is the multiplication factor that describes the magnitude of population growth. Growth rate is always positive.

Females only

Males and females

Fertility versus population growth rate

𝑁𝑡+1=2𝑁𝑡

𝑁𝑡+1=𝑅𝑁𝑡

Bacterial growth

R describes the population growth rate

𝑁𝑡+1=𝑅𝑁𝑡

𝐹 𝑡+1=𝑅𝐹 𝑡

Animal growth

R describes the net reproduction rate

MalesFemales

R is the average number of daughters of each female in the population

Net refers to the number of daughters, which reach reproductive age.

In demographic analysis only females are counted. The number of females in reproductive age is called the effective population size.

Birth and death dynamics

A population growth process considers four basic variables (BIDE model)

B: number of births D: number of deathsI: number of immigrations E: number of emigration

-𝑁𝑡+1=𝑅𝑡 𝑁𝑡=(𝑏𝑡−𝑑𝑡 )𝑁𝑡+𝑁𝑡

𝑏𝑡=𝐵𝑡

𝑁 𝑡

𝑑𝑡=𝐷𝑡

𝑁𝑡

I, E = 0𝑁𝑡+1

𝑁 𝑡

=𝑅𝑡

Discrete population growth

N

Natality

EmigrationImmigration

Mortality

R: fundamental net population growth rate

𝑁𝑡+1=𝑁𝑡+𝐵𝑡−𝐷𝑡=𝑁 𝑡+𝑏𝑡𝑁𝑡−𝑑𝑡𝑁𝑡

)

The population increases if Rt > 1.The population decreases if Rt < 1.

-

r: intrinsic rate of population change

The population increases if rt > 0.The population decreases if rt < 0.

Simple population growth processes

𝑁𝑡= 𝑓 (𝑁𝑡 −1)

∆𝑁=𝑁𝑡−𝑁 𝑡− 1= 𝑓 (𝑁 𝑡−1)

𝑁 𝑡

𝑁𝑡 −1

= 𝑓 (𝑁 ¿¿𝑡−1)¿

Change equation

Difference equation

Ratio equation

Recurrence functions

𝑁𝑡+1=𝑅𝑁𝑡= (𝑏𝑡−𝑑𝑡 )𝑁 𝑡+𝑁𝑡

Discrete growth model

The growth model has only one free parameter: R: fundamental net growth rate

• The model is simple.• The model parameter has a clear and logical ecological interpretation.• The parameter r can be estimated from field data.

Recurrence functions

𝑓 (𝑥 )= 𝑓 (𝑥−𝑛)𝑓 (𝑥 )= 𝑓 (𝑥−1 )+ 𝑓 (𝑥−2 )

Fibonacci series

12

35

8

13

1=1+02=1+13=2+15=3+28=5+3

13=8+5

Leonardo Pisano (Fibonacci; 1170-1250) developed this model to describe the

growth of rabbit populations.

Start

1. month

2. month

3. month

4. month

This is the first model in population ecology.

Assume a couple of immortal rabbits that five birth to a second couple every month.

1

1

2

3

5

𝑁𝑡=𝑅𝑁𝑡− 1=𝑅2𝑁 𝑡− 2…¿𝑅𝑡𝑁𝑜The discrete form of

the exponential growth model

N

t

N0

Exponental growth is a very fast increase in population size.

Scots pine (Pinus sylvestris) population in Great Britain after introduction (7500 BC)

Whooping crane (Grus americana) population in North America after protection in 1940

www.whoopingcrane.com

𝑁𝑡 ¿𝑅𝑡𝑁 𝑜=𝑁 𝑜𝑒

𝑙𝑛𝑅×𝑡

𝑅0¿𝑅𝑡 Basic reproductive rate

𝑟=𝑙𝑛𝑅=(𝑏−𝑑 )=𝑙𝑛𝑅0

𝑡Intrinsic rate of increase per unit of time

R: fundamental net population growth rate

The Human population growth

Human growth was hyperexponential until about 1970.

Net growth rate was not constant but increase until about 1970

Since 1970 net growth rate declined

Continuous population growth

𝑑𝑁𝑑𝑡

=𝑟𝑁0𝑒𝑟𝑡=𝑟𝑁

If r > 0: population increasesIf r < 0: population decreases

ln𝑁 𝑡=𝑙𝑛𝑁0+𝑟 𝑡

In the lack of resource limitation a population will exponentially grow. In this case population grows is density independent.

N

t

N0

tan a = (r-1)ta

ln N

t

ln N0

a tan a = (r-1)

t0

𝑁𝑡 ¿𝑅𝑡𝑁 𝑜=𝑁 𝑜𝑒

𝑟 𝑡

Exponential growth model

𝑁𝑡+1=𝑅𝑁𝑡= (𝑏𝑡−𝑑𝑡 )𝑁 𝑡+𝑁𝑡

𝑁𝑡+1−𝑁 𝑡=∆𝑁 𝑡=(𝑏𝑡−𝑑𝑡 )𝑁𝑡

𝑟=(𝑏𝑡−𝑑𝑡 )=𝑙𝑛𝑅 Intrinsic rate of increase

𝑁𝑡=𝑁𝑡− 1+𝑟𝑁 𝑡− 1

𝐾−𝑁 𝑡−1

𝐾

The Pearl – Verhulst model of logistic population growth

K

K/2

t1/2

N

tt0

Logistic growth

𝑑𝑁𝑑𝑡

=𝑟 𝑁𝐾−𝑁𝐾

Discrete logistic growth

Continuous logistic growth

𝑁 (𝑡 )= 𝐾1+𝑒−𝑟 (𝑡− 𝑡0 )

=𝐾

1−( 𝐾𝑁0

−1)𝑒−𝑟𝑡Solution to this differential equation

The logistic growth model has only two free parameters: r: net reproductive rateK: the carrying capacity.

• The model is simple.• The model parameters have a clear and logical ecological interpretations.• The parameters can be estimated from field data.

• The model does not refer to a specific group of species, but applies to all populations from Bacteria to vertebrates amd plants.

• The model is based on realistic assumptions about population growth.• The model is sufficiently precise.

Constraints:• The model refers to homogeneous environments.• Reproductive rates are supposed to be constant.• Carrying capacity is supposed to be constant.• Generations do not overtlap.

Limitation:The model is symmetrical around the point of inflection.

𝑑𝑁𝑑𝑡

=𝑟 𝑁𝐾−𝑁𝐾

The logistic growth function is a discrete recursive model

r = 0.1K = 500

r = -0.05K = 500

𝑁𝑡=𝑁𝑡− 1+𝑟𝑁 𝑡− 1

𝐾−𝑁 𝑡−1

𝐾

The discrete version of logistic growth

r = 1K = 500

r = 2.099K = 500

Density dependent population regulationStable cycling

𝑁𝑡=𝑁𝑡− 1+𝑟𝑁 𝑡− 1

𝐾−𝑁 𝑡−1

𝐾

r = 1.95K = 500 r = 2.70

K = 500

r = 2.85K = 500

r = 2.87K = 500

Pseudochaos

Local extinction

r = 3.01K = 500

High reproductive rates imply:

• high population fluctuations

• pseudochatotic population size

• no density dependent population regulation

r-strategists often have pseudochaotic population fluctuations.

Pseudochaos does not mean that population size is unpredictable.Very simple determinstic processes might cause pseudochaos.

A random walk is a pure stochastic process that causes unpredictable population sizes.

𝑁𝑡+1=𝑁𝑡+𝑟𝑎𝑛(−𝑥 , 𝑥)