Population GrowthPopulation Growth and Regulation and Regulation

BIOL400BIOL400

31 August 201531 August 2015

PopulationPopulation

Individuals of a single species sharing time Individuals of a single species sharing time and spaceand space

Ecologists must define limits of Ecologists must define limits of populations they studypopulations they study Almost no population is closed to immigration Almost no population is closed to immigration

and emigration and emigration

Exponential Population Growth Exponential Population Growth

Equations:Equations: NNtt = N = N00eertrt

ddN/N/dtdt = = rrNN Model terms:Model terms:

rr = per-individual rate of change (= = per-individual rate of change (= bb – – dd))

= intrinsic capacity for increase, given the= intrinsic capacity for increase, given the

environmental conditionsenvironmental conditions N = population size, at time N = population size, at time tt ee = 2.718 = 2.718

Fig. 8.13 p. 131Fig. 8.13 p. 131

Exponential Population Growth Exponential Population Growth

Assumptions of the Model:Assumptions of the Model: Constant per-capita rate of increase, Constant per-capita rate of increase,

regardless of how high N getsregardless of how high N gets Continuous breedingContinuous breeding

Geometric Population GrowthGeometric Population Growth

Model modified for discrete annual breedingModel modified for discrete annual breeding NNtt = N = N00tt

= = eerr

is the annual rate of increase in Nis the annual rate of increase in N

Example: NExample: N00 = 1000, = 1000, = 1.10 = 1.10• NN11 = 1100 = 1100 NN22 = 1210 = 1210

• NN33 = 1331 = 1331 NN44 = 1464 = 1464

• NN55 = 1611 = 1611 NN1010 = 2594 = 2594

• NN2525 = 10,834 = 10,834 NN100100 = 13,780,612 = 13,780,612Q: Can you spot the oversimplification of nature here?

Fig. 8.10 p. 129Fig. 8.10 p. 129

Fig. 9.1 p. 144Fig. 9.1 p. 144

R0 = per-generation multiplicative rate of increase

Logistic Population GrowthLogistic Population Growth

Equations:Equations: NNtt = K/(1 + = K/(1 + eea-rta-rt))

• K = K = karryingkarrying kapacitykapacity of the environment of the environment• aa positions curve relative to origin positions curve relative to origin

ddN/N/dtdt = N = Nrr[(K-N)/K][(K-N)/K] Assumption: Growth rate will slow as N Assumption: Growth rate will slow as N

approaches Kapproaches K

Fig. 9.4 p. 146Fig. 9.4 p. 146

Table p. 148Table p. 148

As N increases, As N increases, per-capitaper-capita rate of increase declines, but the rate of increase declines, but the absoluteabsolute rate of increase always peaks at ½ K rate of increase always peaks at ½ K

Data from Populations Data from Populations in the Fieldin the Field

Fig. 9.8 p. 150Fig. 9.8 p. 150

Cormorants in Lake Cormorants in Lake HuronHuron

Low numbers due to Low numbers due to toxinstoxins

Increase is not Increase is not strongly sigmoidstrongly sigmoid

Fig. 9.9 p. 150Fig. 9.9 p. 150

Ibex in SwitzerlandIbex in Switzerland Reintroduced after Reintroduced after

elimination via elimination via huntinghunting

Roughly sigmoid Roughly sigmoid (=logistic) but with big (=logistic) but with big decline in 1960s decline in 1960s

Fig. 9.10 p. 151Fig. 9.10 p. 151

Whooping cranes of Whooping cranes of single remaining wild single remaining wild populationpopulation 15 in 1941, now over 15 in 1941, now over

200200

rr increased in 1950s increased in 1950s Every mid-decade, Every mid-decade,

there is a mini-crash there is a mini-crash Apparently related to Apparently related to

predation cycles predation cycles

Fig. 9.15 p. 154Fig. 9.15 p. 154

CladoceransCladocerans Predominant lake Predominant lake

zooplanktonzooplankton

No constant K; big No constant K; big swings seasonally swings seasonally

Can We Improve Our Models?Can We Improve Our Models?

1) Theta logistic model1) Theta logistic model 2) Time-lag logistic model2) Time-lag logistic model 3) Stochastic models3) Stochastic models 4) Population projection matrices4) Population projection matrices

Theta Logistic ModelTheta Logistic Model

New term, New term, , defines , defines curve relating growth curve relating growth rate to Nrate to N

ddN/N/dtdt = N = Nrr[(K-N)/K][(K-N)/K]

Fig. 9.12 p. 152

Fig. 9.13 p. 152Fig. 9.13 p. 152

Time-Lag ModelsTime-Lag Models

Logistic model in which population growth Logistic model in which population growth rate depends not on present N, but on N rate depends not on present N, but on N one (or more) time periods priorone (or more) time periods prior

Assumes population’s demographic Assumes population’s demographic response to density may be delayedresponse to density may be delayed

Fig. 9.14 p. 153Fig. 9.14 p. 153

With time lag, stable With time lag, stable ups and downs may ups and downs may occuroccur

Fig. 11.14 p. 170Fig. 11.14 p. 170

Water fleas show Water fleas show stable approach to K stable approach to K at 18at 18CC

Time lag effect occurs Time lag effect occurs at 25at 25CC DaphniaDaphnia store energy store energy

to use when food to use when food resources collapse resources collapse

Stochastic ModelsStochastic Models

Predict a range of Predict a range of possible population possible population projections, with projections, with calculation of the calculation of the probability of each probability of each

Fig. 9.17 p. 156

Population Projection MatricesPopulation Projection Matrices

Use matrix algebra to project population Use matrix algebra to project population growth, based on fecundity and age-growth, based on fecundity and age-specific survivorshipspecific survivorship

• Fig. 9.18A p. 157

Application: Determining whether Application: Determining whether changes in one aspect or another of the changes in one aspect or another of the life history of an organism have the greater life history of an organism have the greater impact on impact on r r (calculate “elasticity” of each (calculate “elasticity” of each life-history parameter)life-history parameter)

Fig. 9.19 p. 159Fig. 9.19 p. 159

HANDOUT—Biek et al. 2002HANDOUT—Biek et al. 2002

Survivorship in a PopulationSurvivorship in a Population

Three types of Three types of curves are curves are recognized recognized following Pearl following Pearl (1928)(1928)

Examination of Examination of the survivorship the survivorship of various of various species shows species shows that most have a that most have a mixed patternmixed pattern

Fig. 8.6 p. 124

Fig. 8.8 p. 126Fig. 8.8 p. 126

Life TableLife Table

Used to project population growthUsed to project population growth Can be used to determine RCan be used to determine R00, from which , from which rr or or

can be calculatedcan be calculated 1) Vertical (=Static):1) Vertical (=Static): useful if there is long-term useful if there is long-term

stability in age-specific mortality and fecunditystability in age-specific mortality and fecundity 2) Cohort:2) Cohort: data taken from a population data taken from a population

followed over time (ideally, a cohort followed followed over time (ideally, a cohort followed until all have died)until all have died) Observing year-year survivorship, orObserving year-year survivorship, or Collecting data on age at deathCollecting data on age at death

Table 8.5 p. 128Table 8.5 p. 128

Table 8.3 p. 122Table 8.3 p. 122