Simple population models Births Deaths Population increase Population increase = Births – deaths t Equilibrium N: population size b: birthrate d: deathrate The net reproduction rate R = (1+b t -d t ) If the population is age structured and contains k age classes we get The numbers of surviving individuals from class i to class j are given by

Leslie matrix Assume you have a population of organisms that is age structured. Let f X denote the fecundity (rate of reproduction) at age class x. Let s x denote the fraction of individuals that survives to the next age class x+1 (survival rates). Let n x denote the number of individuals at age class x We can denote this assumptions in a matrix model called the Leslie model. We have  -1 age classes,  is the maximum age of an individual. L is a square matrix. Numbers per age class at time t=1 are the dot product of the Leslie matrix with the abundance vector N at time t

v The sum of all fecundities gives the number of newborns v n 0 s 0 gives the number of individuals in the first age class N  -1 s  -2 gives the number of individuals in the last class v The Leslie model is a linear approach. It assumes stable fecundity and mortality rates The effect pof the initial age composition disappears over time Age composition approaches an equilibrium although the whole population might go extinct. Population growth or decline is often exponential

An example At the long run the population dies out. Reproduction rates are too low to counterbalance the high mortality rates Important properties: 1.Eventually all age classes grow or shrink at the same rate 2.Initial growth depends on the age structure 3.Early reproduction contributes more to population growth than late reproduction

Does the Leslie approach predict a stationary point where population abundances doesn’t change any more? We’re looking for a vector that doesn’t change direction when multiplied with the Leslie matrix. This vector is the eigenvector U of the matrix. Eigenvectors are only defined for square matrices. I: identity matrix

Exponential population growth r=0.1 r=-0.1 The exponential growth model predicts continuous increase or decrease in population size. What is if there is an upper boundary of population size?

Saccharomyces cerevisiae Maximum growth We assume that population growth is a simple quadratic function with a maximum growth at an intermediate level of population size The Pearl-Verhulst model of population growth The logistic growth equation Second order differential equation Logistic population increase and decrease K is the carrying capacity (maximum population size)

Time lags The time lag model assumes that population growth might dependet not on th eprevious but on some even earlier population states.

Low growth rates generates a typical logistic growth High growth rates can generate increasing population cycles Intermediate growth rates give damped oscillations High growth rates give irregular but stable oscillations Certain high growth rates produce pseudochaos Too high growth rates lead to extinction A simple deterministic model is able to produce very different time series and even pseudochaos

m = rK / 4 Critical harvesting rate Constant harvesting Constant harvesting rate m Where is the stationary point whwere fish population becomes stable -4m + rK > 0

Alfred James Lotka ( ) Vito Volterra ( ) Life tables Age Observed number of animals Number dying Mortality rate Cumula- tive mortality rate Propor- tion survi- ving Cumula- tive proportion surviving Mean number alive Cumula- tive L t Mean further life expec- tancy tNtNt DtDt mtmt MtMt ltlt stst LtLt TtTt EtEt Demographic or life history tables

Cumulative mortality rate M t l t = 1 - m t-1 is the proportion of individuals that survived to interval t Cumulative proportion surviving s t is 1 - m t Mean number of individuals alive at each interval

Mean life expectancy at age t

Net reproduction rate of a population Mean generation length is the mean period elapsing betwee the birth of prents and the birth of offspring G = 30.2 years Age Pivotal age (class mean) Observed number at pivotal age Fraction surviving No. of female offspring Female offspring per female R t  NtNt ltlt DtDt btbt ltbtltbt ltbtltbt Sum Generation time

The Weibull distribution is particularly used in the analysis of life expectancies and mortality rates

The two parametric form The characteristic life expectancy T is the age at which 63.2% of the population already died. For t = T we get B: shape paramter T: time T: characteristic life time

How to estimate the parameter  and the characteristic life expectancy T from life history tables? We obtain b from the slope of a plot of ln[ln(1-F)] against ln(t)