Variability in space In time

No

mig

rati

onm

igra

tion (arithmetic)

Source-sink structurewith the rescue effect

(geometric)

G < A G declines with increasing variance

Temporal variability reduces population growth rates

Cure – populations decoupled with respect to variability, but coupled with respect to sharing individuals

Source-sink structure

(arith & geom)Increase the number of subpopulations increases the growth rate (to a point),and slows the time to extinction

Overview ofpopulation growth:

discrete continuous

densityindependent

densitydependent

Geometric Exponential

DiscreteLogistic

LogisticNew Concepts:

- Stability- DI (non-regulating)

vs. DD (regulating) growth

- equilibrium

Variability in growth

(1) Individual variation in births and deaths(2) Environmental (extrinsic variability)(3) Intrinsic variability

XX

X

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Series1

BUT, most populations appear more regulated than this…..

And

THERE ARE LIMITS TO GROWTH!!!!

e.g., Australian sheep

Limits are manifestedin (-) density dependence in population vital rates:

mortality/survivorshipreproduction

At higher densities, song sparrows:

(a) smaller % reproductive males (b) fewer young fledged/female(c) lower juvenile survivorship

Density dependence often affects more than a single component of those rates:

How do populations grow?

time

N

Logistic Growth

dNdt

(K-N)K

rN=

1 dNN dt

(K-N)K

r=

population

per capita

N

1 dNN dt 0

K

K

K = Carrying capacity: themaximum density of individuals that the environment can support

If N = 0 (K-(0))K

= r

1 dNN dt

(K-N)K

r=

KK

= r

= r

N

If N = 0 (K-(0))K

= r

1 dNN dt

(K-N)K

r=

KK

= r

= r

That’s Exponential Growth}

Exponentialgrowth-like

time

If N = K (K-(K))K

= r

1 dNN dt

(K-N)K

r=

0K

= r

= 0

N

K

If N = K (K-(K))K

= r

1 dNN dt

(K-N)K

r=

0K

= r

= 0

That’s Zero Growth}

Zerogrowth

time

N

K

1 dNN dt

(K-N)K

r=

Put the two together

LOGISTIC GROWTHtime

N

1 dNN dt 0

K

1 dNN dt

(K-N)K

r=

r

(= r K _K

NK)

= r 1 _ 1K )( N

= r _ r K N

Y = b + m X

- growth

+ growth

2nd Simplest expression of population growth: 2 parameters: r = intrinsic growth rate and K = carrying capacity

Per capita growth rate is (-) density dependent

Second Law of Ecology: There are limits to growth

N

K

time

N

time

EQ stability regulationLog.

Exp.

Rinderpestinnoculation

Severe drought

Rainfall

Total food

per capita food

So what aboutDensity-dependence?

0.0

0.2

0.4

0.6

0.8

1.0P

ropo

rtio

n of

ani

mal

s Live wildebeest

Solid

, whi

te fa

t

Opaqu

e gela

tinou

s

Trans

luce

nt g

elatin

ous

0.0

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0.8

1.0

Lion/hyena killed

Trans

luce

nt g

elatin

ous

Opaqu

e gela

tinou

s

Solid

, whi

te fa

t

http//www.cbs.umn.edu/populus/download/download.html

To download a version of Populus:

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r=0.2 r=1.0

r=1.8 r=2.0

Dampedoscillations 2-point

limit cycle

Den

sity

time

Discrete Logistic Growth

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r=2.2

r=2.8 r=4.0

r=2.5

Chaos

4-pt cycle

extinction

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r=2.8ChaosChaos – “unpredictable” populationdynamics incurred through very highgrowth rate and time lags between growth and negative feedback.

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sity

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Extrinsicvariability

time

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sity

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K=1000; r=3.0

Islands < 1.0 ha support too few shrews to persist

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sity

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K=1000; r=3.0 Population culled by 25%

time

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Population culled by 25%Extrinsicvariability

Variability comes in 2 flavors: Extrinsic and Intrinsic

Recognizing the type of variability is important because different types require different solutions.

Intrinsic – growth rate or population size

Extrinsic – migration, # populations, population size

Overview ofpopulation growth:

discrete continuous

densityindependent

densitydependent

Geometric Exponential

DiscreteLogistic

LogisticNew Concepts:

- Stability- DI (non-regulating)

vs. DD (regulating) growth

- equilibrium

Variability in growth

(1) Individual variation in births and deaths(2) Environmental (extrinsic variability)(3) Intrinsic variability

XX

XX

XX

REVIEW

- Populations consist of sources ( > 1) and sinks (<1), the latter doom to extinction……..

- Populations have good years and bad years and temporal variation is bad ……………………………

- Populations can grow chaotically by over- and under-shooting Carrying capacity………………….

- Populations with an Allee Effect can decline to extinction if N is too low………………………………..

- Cure: Dispersal from sources can Rescue sinks

- Cure: Many populations that share individuals (dispersal)

- Cull the population or otherwise reduce its growth

- Recognize and keep density above the critical density

N

time

2 Models of growth

Exponential – all populations have the capacity to growth exponentially, but

Growth has no limits and is density independent

N

1 dNN dt

1 dNN dt

= rSustained Exponential

growth is unrealist

ic

time

N

K

1 dNN dt

(K-N)K

r=

Logistic – recognizes limits to growth (Carrying capacity) and incorporates the negative effect individuals have on their growth rate

N

1 dNN dt 0

K

r

(- Density Dependence)

Stable EQ @ K

N

1 dNN dt

0

K

One other variation is the ALLEE EFFECT where individuals also have + Density Dependence at low density

+ DDe.g., social behaviorsafety in numbers

- DD Individuals inhibit

their growth

aahh

hhh…

.

Important Concepts we have touch upon under Population Growth

- Life Tables: Understanding how patterns of age-specific survivorship and maternity has consequences for population growth and can be manipulated to achieve a management goal

- Variability: In space, populations exist as sources ( > 1) and sinks ( < 1), the latter of which must receive migrants to persist (Rescue Effect)

In time, environmental variation is an anathema to population growth, but it too has a cure: increase the number of populations, migration,

- Intrinsic Variability: Appreciate the difference between external and internal variation arising from time lags and delayed density dependence. Its cure is radically different than for external variation – and requires

culling population size or otherwise reducing the growth rate.

Important Concepts we have touch upon under Population Growth

- EQ, stability, and Pop. regulation: Attainable only under (-) density dependence. Negative feedback is Universal

- Domains of Attraction: Specifically, under the Allee Effect, population extinction is an “attractant” below some critical density

The concept of the limits to growth is manifested in the Carrying Capacity

Species Social Behavior is manifested in the Allee Effect

But otherwise, we have incorporated the biology of species as phenomena and have not appreciated the actual details

------------------------------------------------------------------------

But we will……

Where’s the Biology?

Wildebeest populations growth

competition for grass occurs

Individuals are energy stressed

Lions kill off weak individuals

1 dNN dt

(K-N)K

r=Lions?Grass?

??Energy/stress??

The Phenomenological Approach

THE GOOD: Modeling the phenomena allows us to look past the details … we don’t need separate models for

every organism

THE BAD: We only get a superficial understanding …. when the details matter we’re left scratching our heads

This tradeoff between DETAIL and GENERALITY Is pervasive throughout science