Eli HolmesNational Marine Fisheries Service

Population modeling of marine mammal populations

Lecture 1: simple models of population counts

nmfs.noaa.gov

• Density-independent growth– Exponential growth

• Density-dependent growth– Hockey Stick– Logistic Growth– Optimal sustainable population

• Depensation (Allee effects)

Today’s lecture topics:

Some real population counts for long-lived species

0 10 20 30 40 500

5

10

15x 10

4 Northern fur seals

0 5 10 15 20 251

2

3

4x 10

4 California sea lions

0 5 10 15 20 25 30100

150

200

250Northern Resident Killer Whales up to 2000

0 20 40 60 800

100

200

300African Elephants

0 10 20 30 40 500

50

100

150Whooping Cranes

0 5 10 15 20 25 30500

1000

1500

Wandering Albatross

Real data

Exponential growth

λ > 1

λ < 1

Classical models for population counts

Numbers this year = Numbers last year + births - deaths

Models differ in how these depend on

population densityDENSITY-DEPENDENCE

Classical models for population counts

Births = b x Numbers last yearDeaths = d x Numbers last year

For example, b=0.1 (10%) and d=0.2 (20%)

Year 2010 2011 2012 2013 2014 ….Count 100 90 81 72 65 …. to 0

Exponential growth (or decline)Rate of population growth is not influenced by the number or density of individuals in the population. Density-independent

Numbers this year = Numbers last year + births - deaths

exponentialdecline

exponentialincrease

Classical models for population counts

Nt+1 = Nt + (b-d)Nt = 1+(b-d)Nt= Nt + rNt

r is the “intrinsic rate of increase”r>0 the population is increasingr<0 the population is decreasing

(1+r) = λλ >0 the population is increasingλ <0 the population is decreasing

Exponential growth (or decline)Rate of population growth is not influenced by the number or density of individuals in the population. Density-independent

Numbers this year = Numbers last year + births - deaths

exponentialdecline

exponentialincrease

Classical models for population counts“Hockey Stick”Rate of population growth is not influenced by the density of individuals in the population until it hits some threshold. Density-dependent

exponential

linear

Nt+1 = Nt + rNt exponentialNt+1 = Nt + r linear

Until N reaches K then

Nt+1 = K

K

carryingcapacity

Classical models for population countsLogistic Growth ModelRate of population growth is influenced by the density of individuals in the population at all densities. Density-dependent

exponential

sigmoidal

Nt+1= Nt + rNt(1-[Nt/K])

Compare to exponential

Nt+1 = Nt + rNt

K

carryingcapacity

Exponential versus logistic growth

Time

Num

bers

S. Holt

Effects of growth rate (“r”) on the shape of the logistic curve

6

Small r

big r

Comparing exponential and logistic curves

5

The change in N is highest hereThis is where the “net productivity” is highest

The per capita rate of increase is highest here

Number

The per capita rate of increase is lower here

Optimum sustainable population (OSP)

OSP: Key managementelement of the MarineMammal Protection Act of1972:

“the number of animals which will result in the maximum productivity of the population or the species, keeping in mind the carrying capacity of the habitat and the health of the ecosystem”

Optimum sustainable population*

Working definition developed by the NationalMarine Fisheries Service:

Population size between MNPL and K, where MNPL is the population size that produces the maximum net productivity level

K

MNPL

OSP

*This is a concept from the field of Fisheries. In non-Fisheries fields (Conservation Biology, Ecology), this concept is not taught nor used. A K/2 decline is reason to consider a species for the IUCN Red List!

Nt+1= Nt + Ntr(1-[Nt/K]z)

z<1, r big = “high growth rate at low population size, but higher density-dependence at high population size”

z>1, r small = “lower growth rate at low population size, but less density-dependence at high population size”

Different shapes of density-dependence lead to significant changes OSP!

OSP

z=0.5r=0.32

z=1r=.2

z=2r=0.15

10

Optimum sustainable population (OSP) depends on the “shape” of density-dependence

OSP

Time (yrs)

Num

bers

Nt+1= Nt + Ntr(1-[Nt/K]z)

z<1, r big = “high growth rate at low population size, but higher density-dependence at high population size”

z>1, r small = “lower growth rate at low population size, but less density-dependence at high population size”

Marine mammals density-dependence tends to look like the red curve.

OSP

z=0.5r=0.32

z=1r=.2

z=2r=0.15

10

Optimum sustainable population (OSP) depends on the “shape” of density-dependence

OSP

Time (yrs)

Num

bers

Depensation (Allee effects) = reduced per capita production at small population size

Per capita production(Nt+1‐Nt)/N

11

Num

bers

threshold

Pop. Growth

rate

logistic

exponential

Depensation (Allee effects) = reduced per capita production at small population size

Potential causes include

• inbreeding depression• difficulty finding a mate• intensive predation (same #

of predators but fewer prey)• lack of social cohesion (esp.

for social hunters)• or just chance due to small

numbers (demographic stochasticity)

11

Num

bers

threshold

Severe reduction in population size can lead to the “extinction vortex”

Where low densities produce a significant risk of extinction if the population becomes depleted

There is much current research trying to determine how to define these “extinction vortex”thresholds.

J.D. Flores

N

13

Per c

apita

gro

wth

Extinction vortexthreshold

• Age-structures models of marine mammals

• How we figure out what’s going wrong with a population and how best to reverse declines

Monday’s lecture topic:

nmfs.noaa.gov