Background knowledge expected Population growth models

Background knowledge expected

Population growth models/equations

exponential and geometric

logistic

Refer to

204 or 304 notes

Molles Ecology Ch’s 10 and 11

Krebs Ecology Ch 11

Gotelli - Primer of Ecology (on reserve)

Habitat loss Pollution Overexploitation Exotic spp

Small fragmented isolated popn’s

Inbreeding Genetic Variation Reduced N Demographic

stochasticity

Env variation

Catastrophes Genetic processes

Stochastic processes

The ecology of small populations

How do ecological processes impact small populations?

Stochasticity and population growth

Allee effects and population growth

Outline for this weeks lectures

Immigration + Emigration -

Birth (Natality) +

Death (Mortality) -

Nt+1 = Nt +B-D+I-E

Population Nt

Demography has four components

Exponential population growth

(population well below carrying capacity, continuous reproduction

closed pop’n)

Change in population at any time

dN = (b-d) N = r N where r =instantaneous rate of increase dt

!t

!N

Cumulative change in population Nt = N0e

rt

N0 initial popn size,

Nt pop’n size at time t

e is a constant, base of natural logs

Trajectories of exponential population growth

r > 0 r = 0 r < 0

N

t

Trend

Geometric population growth (population well below carrying capacity, seasonal reproduction)

Nt+1 = Nt +B-D+I-E

!N = Nt+1 - Nt

= Nt +B-D+I-E - Nt

= B-D+I-E

Simplify - assume population is closed; I and E = 0

!N = B-D

If B and D constant, pop’n changes by rd = discrete growth factor

Nt+1 = Nt +rd Nt

= Nt (1+ rd) Let 1+ rd = !, the finite rate of increase

Nt+1 = ! Nt

Nt = !t N0

DISCRETE vs CONTINUOUS POP’N GROWTH

Reduce the time interval between the teeth and the

Discrete model converges on continuous model

! = er or Ln (!) = r

Following are equivalent r > 0 ! > 1

r = 0 ! = 1

r< 0 ! < 1

Trend

Geometric population growth (population well below carrying capacity, seasonal reproduction)

Nt+1 = (1+rdt) Nt = (1+rdt) (1+rdt-1) Nt-1

= (1+rdt) (1+rdt-1) (1+rdt-2) Nt-2

= (1+rdt) (1+rdt-1) (1+rdt-2) (1+rdt-3) Nt-3

Add data Nt-3= 10

rdt = 0.02 rdt-1 = - 0.02 rdt-2 = 0.01 rdt-3 = - 0.01

What is the average growth rate?

Geometric population growth (population well below carrying capacity, seasonal reproduction)

What is average growth rate?

= (1+0.02) + (1-0.02) + (1+0.01) + (1-0.01) = 1 4

Arithmetic mean

Predict Nt+1 given Nt-3 was 10

Geometric population growth (population well below carrying capacity, seasonal reproduction)

What is average growth rate?

Geometric mean = [(1+0.02) (1-0.02) (1+0.01) (1-0.01)]1/4 = 0.999875

KEYPOINT Long term growth is determined by the geometric not the arithmetic mean Geometric mean is always less than the arithmetic mean

Calculate Nt+1 using geometric mean

Nt+1 = !4 x 10

(0.999875)4 x10 = 9.95

Nt+1 = (1+0.02) (1-0.02) (1+0.01) (1-0.01) 10 = 9.95

DETERMINISTIC POPULATION GROWTH

For a given No, r or rd and t The outcome is determined

Eastern North Pacific Gray whales Annual mortality rates est’d at 0.089 Annual birth rates est’d at 0.13

rd=0.13-0.89 = 0.041 ! = 1.04

1967 shore surveys N = 10,000

Estimated numbers in 1968 N1= ! N0 = ?

Estimated numbers in 1990 N23= !23 N0 = (1.04)23. 10,000 = 24,462

DETERMINISTIC POPULATION GROWTH

For a given No, r or rd and t The outcome is determined

Population growth in eastern Pacific Gray Whales

-! fitted a geometric growth curve between 1967-1980

- shore based surveys showed increases till mid 90’s

In US Pacific Gray Whales were delisted in 1994

Mean r

\

SO what about variability in r due to good and bad years? ENVIRONMENTAL STOCHASTICITY

leads to uncertainty in r acts on all individuals in same way

b-d Bad 0 Good

Variance in r = "2e = "r2 -

("r)2

N N

Population growth + environmental stochasticity

Ln N

t

Deterministic 1+r= 1.06, "2

e = 0

1+r= 1.06, "2e = 0.05

Expected

Expected rate of increase is r- "2e/2

Predicting the effects of greater environmental stochasticity

Onager (200kg)

Israel - extirpated early 1900’s

- reintroduced 1982-7

- currently N > 100

RS varies with Annual rainfall

Survival lower in droughts

Global climate change (GCC) is expected to

----> changes in mean environmental conditions

----> increases in variance (ie env. stochasticity)

mean drought < 41 mm

Pre-GCC Post-GCC Mean rainfall is the same BUT

Variance and drought frequency is greater in “post GCC”

Data from Negev

Simulating impact on populations via rainfall impact on RS

Variance in rainfall

Low High

Number of quasi-extinctions

= times pop’n falls below 40

Simulating impact on populations adding impact on survival

CONC’n Environmental stochasticity can influence extinction risk

But what about variability due to chance events that act on individuals

Chance events can impact the breeding performance offspring sex ratio and death of individuals

---> so population sizes can not be predicted precisely

Demographic stochasticity

Demographic stochasticity

Dusky seaside sparrow subspecies non-migratory salt marshes of southern Florida

decline DDT flooding habitat for mosquito control Habitat loss - highway construction

1975! six left

All male

Dec 1990 declared extinct

Extinction rates of birds as a function of population size over an 80-year period!

0

30

60

1 10 100 1000 10,000

* * *

* *

* * *

10 breeding pairs – 39% went extinct

10-100 pairs – 10% went extinct

1000>pairs – none went extinct

*

Population Size (no. pairs)

%

Extinction

Jones and Diamond. 1976. Condor 78:526-549!

random variation in the fitness of individuals ("2d)

produces random fluctuations in population growth rate that are inversely proportional to N

demographic stochasticity = "2d/N

expected rate of increase is r - "2d/2N

Demographic stochasticity is density dependant

How does population size influence stochastic processes?

Demographic stochasticity varies with N

Environmental stochasticity is typically independent of N

Long term data from Great tits in Whytham Wood, UK

Partitioning variance

Species "2d "

2e

Swallow 0.18 0.024 Dipper 0.27 0.21 Great tit 0.57 0.079 Brown bear 0.16 0.003

in large populations N >> "2d / "

2e

Environmental stochasticity is more important Demographic stochasticity can be ignored

Ncrit = 10 * "2d / "

2e (approx Ncrit = 100)

Stochasticity and population growth

N0= 50 ! = 1.03

Simulations - ! = 1.03, "2e = 0.04, "

2d = 1.0

N* = "2d /4

r - ("2e /2)

N* Unstable eqm below which pop’n moves to

extinction

Environmental stochasticity -!fluctuations in repro rate and probability of mortality imposed by good and bad years -!act on all individuals in similar way -!Strong affect on ! in all populations

Demographic stochasticity -!chance events in reproduction (sex ratio,rs) or survival acting on individuals -! strong affect on ! in small populations

Catastrophes -!unpredictable events that have large effects on population size (eg drought, flood, hurricanes) -!extreme form of environmental stochasticity

SUMMARY so far

Stochasticity can lead to extinctions even when the mean population growth rate is positive

Key points

Population growth is not deterministic

Stochasticity adds uncertainty

Stochasticity is expected to reduce population growth

Demographic stochasticity is density dependant and less important when N is large

Stochasticity can lead to extinctions even when

growth rates are, on average, positive