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Basics in population ecology

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1 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.

2 Our program Simple growth processes Outbreaks Age structured populations Harvesting and viability analysis Competition , predation and parasitism Populations in space: Metapopulation and spatial dynamics

3 Literature

4 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. 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. Carabus coriaceus in a forest Carabidae in a forest 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)

5 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)

6 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.

7 Spatially separated individuals do not form true populations
Raven (Corvus corax) A species occurring on four islands that are isolated is divided into four independently evolving populations. Ravens in different continents do not form a single population. There is no (or only limited) gene flow. Due to limited gene flow populations on two islands might be considerd as foring a single genet ically structured populations

8 Temporary separated individuals do not form populations
Omphale lugens Mikiola fagi Macrotera arcuata N Number of bees hatching from eggs N Spring Summer Spring Summer 1 2 3 Summer Eggs Hatching year Spring and summer generations have only limited overlap and thus form partly separated populations. Overlaying is connected with host change. M. fagi is univoltine. 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.

9 North atlantic salmon is semelparous
Life cycles North atlantic salmon is semelparous Man is iteroparous 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? Iteroparous populations are of age structured with each age cohorte having a different reproductive output.

10 Differences in life history
Egg Semelparous species reproduce only once and can be described by simple growth models Egg Larva 1 Juvenile Iteroparous species reproduce at least two times and might form age structured populations Larva n Adults 1 Fertility = number of eggs per female Adult Adult n Fertility = number of eggs per female Some species have age cohorts after the reproductive phase Senex Why grandparents?

11 Some basic definitions
Females only 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. Males and females 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.

12 Fertility versus population growth rate
Bacterial growth Animal growth 𝑁 𝑡+1 =2 𝑁 𝑡 𝑁 𝑡+1 =𝑅 𝑁 𝑡 Males Females 𝑁 𝑡+1 =𝑅 𝑁 𝑡 𝐹 𝑡+1 =𝑅 𝐹 𝑡 R describes the population growth rate R describes the net reproduction rate R is the average number of daughters of each female in the population In demographic analysis only females are counted. The number of females in reproductive age is called the effective population size. Net refers to the number of daughters, which reach reproductive age.

13 Discrete population growth
Birth and death dynamics Discrete population growth Natality A population growth process considers four basic variables (BIDE model) B: number of births D: number of deaths I: number of immigrations E: number of emigration N Immigration Emigration 𝑁 𝑡+1 = 𝑁 𝑡 + 𝐵 𝑡 − 𝐷 𝑡 = 𝑁 𝑡 + 𝑏 𝑡 𝑁 𝑡 − 𝑑 𝑡 𝑁 𝑡 Mortality 𝑏 𝑡 = 𝐵 𝑡 𝑁 𝑡 𝑑 𝑡 = 𝐷 𝑡 𝑁 𝑡 I, E = 0 𝑁 𝑡+1 𝑁 𝑡 = 𝑅 𝑡 𝑁 𝑡+1 = 𝑁 𝑡 (1+ 𝑏 𝑡 − 𝑑 𝑡 ) = 𝑅𝑁 𝑡 𝑁 𝑡+1 = 𝑅 𝑡 𝑁 𝑡 = 𝑏 𝑡 − 𝑑 𝑡 𝑁 𝑡 + 𝑁 𝑡 𝑅 𝑡 = 1+𝑏 𝑡 - 𝑑 𝑡 The population increases if Rt > 1. R: fundamental net population growth rate The population decreases if Rt < 1. 𝑟 𝑡 = ⁡(𝑏 𝑡 - 𝑑 𝑡 ) The population increases if rt > 0. The population decreases if rt < 0. r: intrinsic rate of population change

14 Simple population growth processes
𝑁 𝑡+1 = 𝑅𝑁 𝑡 = 𝑏 𝑡 − 𝑑 𝑡 𝑁 𝑡 + 𝑁 𝑡 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. 𝑁 𝑡 =𝑓( 𝑁 𝑡−1 ) Change equation ∆𝑁=𝑁 𝑡 − 𝑁 𝑡−1 =𝑓( 𝑁 𝑡−1 ) Difference equation Recurrence functions 𝑁 𝑡 𝑁 𝑡−1 =𝑓 (𝑁 𝑡−1 ) Ratio equation

15 Recurrence functions 𝑓 𝑥 =𝑓(𝑥−𝑛) 𝑓 𝑥 =𝑓 𝑥−1 +𝑓 𝑥−2 Leonardo Pisano (Fibonacci; ) developed this model to describe the growth of rabbit populations. Fibonacci series 1=1+0 2=1+1 3=2+1 5=3+2 8=5+3 13=8+5  This is the first model in population ecology. Assume a couple of immortal rabbits that five birth to a second couple every month. Start 1 1. month 1 2. month 2 3. month 3 4. month 5

16 The discrete form of the exponential growth model
𝑁 𝑡 = 𝑅𝑁 𝑡−1 = 𝑅 2 𝑁 𝑡−2 … =𝑅 𝑡 𝑁 𝑜 The discrete form of the exponential growth model Exponental growth is a very fast increase in population size. 𝑁 𝑡 =𝑅 𝑡 𝑁 𝑜 = 𝑁 𝑜 𝑒 𝑙𝑛𝑅×𝑡 R: fundamental net population growth rate 𝑅 0 =𝑅 𝑡 Basic reproductive rate N0 𝑟=𝑙𝑛𝑅= 𝑏−𝑑 = 𝑙𝑛 𝑅 0 𝑡 Intrinsic rate of increase per unit of time t Whooping crane (Grus americana) population in North America after protection in 1940 Scots pine (Pinus sylvestris) population in Great Britain after introduction (7500 BC)

17 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

18 Continuous population growth Exponential growth model
𝑁 𝑡 =𝑅 𝑡 𝑁 𝑜 = 𝑁 𝑜 𝑒 𝑟𝑡 𝑑𝑁 𝑑𝑡 = 𝑟𝑁 0 𝑒 𝑟𝑡 =𝑟𝑁 𝑁 𝑡+1 = 𝑅𝑁 𝑡 = 𝑏 𝑡 − 𝑑 𝑡 𝑁 𝑡 + 𝑁 𝑡 𝑁 𝑡+1 − 𝑁 𝑡 =∆ 𝑁 𝑡 = 𝑏 𝑡 − 𝑑 𝑡 𝑁 𝑡 Exponential growth model If r > 0: population increases If r < 0: population decreases 𝑟= 𝑏 𝑡 − 𝑑 𝑡 =𝑙𝑛𝑅 Intrinsic rate of increase In the lack of resource limitation a population will exponentially grow. In this case population grows is density independent. N ln N ln 𝑁 𝑡 =𝑙𝑛 𝑁 0 +𝑟𝑡 ln N0 a tan a = (r-1) a tan a = (r-1)t N0 t0 t t

19 Discrete logistic growth
N 𝑁 𝑡 = 𝑁 𝑡−1 + 𝑟𝑁 𝑡−1 𝐾− 𝑁 𝑡−1 𝐾 K The Pearl – Verhulst model of logistic population growth K/2 t0 t1/2 t Continuous logistic growth 𝑁 𝑡 = 𝐾 1+ 𝑒 −𝑟(𝑡− 𝑡 0 ) = 𝐾 1− 𝐾 𝑁 0 −1 𝑒 −𝑟𝑡 𝑑𝑁 𝑑𝑡 =𝑟𝑁 𝐾−𝑁 𝐾 Solution to this differential equation

20 𝑑𝑁 𝑑𝑡 =𝑟𝑁 𝐾−𝑁 𝐾 The logistic growth model has only two free parameters: r: net reproductive rate K: 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.

21 The logistic growth function is a discrete recursive model
The discrete version of logistic growth 𝑁 𝑡 = 𝑁 𝑡−1 + 𝑟𝑁 𝑡−1 𝐾− 𝑁 𝑡−1 𝐾 The logistic growth function is a discrete recursive model r = -0.05 K = 500 r = 0.1 K = 500

22 𝑁 𝑡 = 𝑁 𝑡−1 + 𝑟𝑁 𝑡−1 𝐾− 𝑁 𝑡−1 𝐾 r = 1 K = 500 r = 2.099 K = 500 Density dependent population regulation Stable cycling

23 r = 1.95 K = 500 r = 2.70 K = 500 Pseudochaos r = 2.85 K = 500 r = 2.87 K = 500

24 High reproductive rates imply: high population fluctuations
pseudochatotic population size no density dependent population regulation r = 3.01 K = 500 Local extinction Pseudochaos does not mean that population size is unpredictable. Very simple determinstic processes might cause pseudochaos. r-strategists often have pseudochaotic population fluctuations. A random walk is a pure stochastic process that causes unpredictable population sizes. 𝑁 𝑡+1 = 𝑁 𝑡 +𝑟𝑎𝑛(−𝑥,𝑥)

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