Ecology 8310Population (and Community) Ecology

• Introductions (Craig, students)• Course website, content, goals, and logistics• Today's lecture

• Dynamics (basic terminology)• Population growth (review)

• Exponential• Logistic• Discrete (e.g., logistic map)• Equilibrium and stability• Limiting factors vs. regulation

Dynamics:

N

• N is a state variable (i.e., it describes the state of a dynamical system)

• Dynamics is concerned with how N changes through time, t, and across space, x.

• Thus, N expressed as a function of time (and space): N(t) or N(x,t)

Dynamics:

N(t)

• N can be affected (positively or negatively) by environmental parameters (or other state variables)

• We can write out a model describing how N(t) changes through time (in relation to these parameters and variables).

+ −

Dynamics:

N(t)

• Continuous vs. Discrete time models• Deterministic vs. Stochastic models• Solutions can be found analytically or numerically• Let's remind ourselves of a derivative…

+ −

Dt

DN

Time, t

Ab

un

dan

ce,

Nd

N/d

t

0

Slope = dN/dt

Derivative?

What does dN/dt do through time?

Geometric growth:

Nt

• Discrete time, and deterministic.• Assume N changes by a constant multiplier each time

step, l. • Then: Nt+1=lNt

• Or more generally, Nt=ltN0

• So, a simple parameter, l, determines how population grows (in addition to the starting conditions)

l=1.17

0 5 10 15 20 250

1000

2000

3000

4000

5000

6000

Time (years)

Ab

un

da

nc

e

Geometric growth:

N0=100

l=1.04

l=1.02

l=1.00

l=0.90

Geometric growth:

0 3 6 9 12 15 18 21 24 27 30 33 36 39 420

100

200

300

400

500

600

Time, t

Den

sity

, N

l>1 (increase); l=1 (no change); l<1 (decrease)

N0=100

l=1.04

l=1.02

l=1.00

l=0.90

Geometric growth:

0 3 6 9 12 15 18 21 24 27 30 33 36 39 421

10

100

1000

Time, t

Den

sity

, N

What is the slope of each line? Log(l)

Moving to continuous time:

1) To start: Nt+1 = lNt = (1+(l-1))Nt

2) Now, break up a time step into 2 smaller steps

3) And divide up the 'growth' (l-1) by the number of steps

4) Then: Nt+0.5=(1+((l-1)/2))Nt and Nt+1 = (1+((l-1)/2)Nt+0.5

5) Thus, Nt+1 = (1+((l-1)/2)2Nt

Moving to continuous time:

6) Let the number of steps increase…

7) Take the limit of this process as the number of steps increase to infinite:

An example:

8) Let l=2 (e.g., population doubles every year)

9) Let's slowly increase n from 1 to infinite…

Number of substeps, n Nt+1

1 2

2 1.5x1.5=2.25

3 1.3333=2.37

4 1.254=2.44

5 1.205=2.49

6 1.176=2.52

7 1.147=2.55

An example:

10) Keep going…

Number of substeps, n Nt+1

10 1.110=2.5937

20 1.0520=2.6533

30 1.033330=2.6743

100 1.01100=2.7048

200 1.005200=2.7115

500 1.002500=2.7156

1000 1.0011000=2.7169

∞ 2.7182818284590… aka e

An example:

11) In this example with l=2, the continuous time model becomes: Nt+1/Nt = e

12) Notice that the annual increase for the continuous model (2.72) exceeds the rate for the annual model (2.00).

13) Why?

N0=100

r=0.0392 l=1.04

r=0.0198 l=1.02

r=0.0000 l=1.00

r=-0.1054 l=0.90

Exponential growth:

0 4 8 12 16 20 24 28 32 36 400

100

200

300

400

500

600

Time, t

Den

sity

, N

r>0 (increase); r=0 (no change); r<0 (decrease)

N0=100

r=0.0392 l=1.04

r=0.0198 l=1.02

r=0.0000 l=1.00

r=-0.1054 l=0.90

Exponential growth:

0 5 10 15 20 25 30 35 401

10

100

1000

Time, t

Den

sity

, N

What is the slope of each line?

r = [ ln(Nt+Dt)-(ln(Nx)]/Dt = ln(Nt+Dt/Nt)/Dt = ln(Nt/N0)/t

Exponential growth:

r = ln(Nt/N0)/t

Nt=N0ert

Exponential growth:

If Nt=N0ert, then, what is dN/dt?

= d(N0ert)/dt= N0d(ert)/dt= N0[d(rt)/dt]ert

= N0rert

= rN0ert

= rNt

i.e., dN/dt = rN or dN/Ndt=r

Geometric vs. Exponential Growth:

Discrete (Geometric)

Continuous (Exponential)

Nt = ? N0lt N0ert

Pop. gro. rate DN/Dt = N(l-1) dN/dt = rN

Per capita g.r. l-1 r

No growth l = 1 r = 0

Increases l > 1 r > 0

Decreases l < 1 r < 0

Range 0 < l < +∞ -∞ < r < +∞

Units None time-1

Doubling Time:

Nt/N0 = 2 = ert

t = ln(2)/r = 0.69/r

r Doubling time

0.01 yr-1 69 yrs

0.05 yr-1 14 yrs

0.10 yr-1 6.9 yrs

1.0 yr-1 .69 yrs

NOT!

100 yrs

20 yrs

10 yrs

1 yr

Doubling time: r vs. l

r ~ l – 1So, which grows faster:

1) l = 22) r = 1

Why?

Compound Interest

Does growth rate remain constant?

Darwin:

…Linnaeus has calculated that if an annual plant produced only two seeds - and there is no plant so unproductive as this - and their seedlings next year produced two, and so on, then in twenty years there would be a million plants. The elephant is reckoned to be the slowest breeder of all known animals… assume that it breeds when thirty years old, and goes on breeding till ninety years old, bringing forth three pairs of young in this interval; if this be so, at the end of the fifth century there would be alive fifteen million elephants, descended from the first pair.

0

Density (N)

dN

/Nd

t

Density-independent (exponential)

Density-dependent (negative)

Per capita rate of increase as function of density:

Density-dependent (positiv

e)

Implications?Causes?

1. Density independence: dN/Ndt (or its components) are independent of N

2. Density dependence: dN/Ndt (or its components) vary with N

3. Via effects on birth, death, emigration, immigration

Definitions:

1. Competition (reduce birth, incr. death…)

2. Allee effect (inc. birth, dec. death at low N)

3. Predator attraction or disease (increase death; decrease birth?)

4. Predator satiation (decrease death, increase birth?)

5. Biotic vs. Abiotic

Causes of density-dependence:

• Stability• Population regulation: N kept within bounds (tends to increase from low N; decrease at high N)

Implications:

Population growth:

Population growth:

Equilibrium and Stability:

1. Equilibrium: N at which dN/Ndt = 0

2. No net growth (dynamic equilibrium)

3. Stability: does the system return to equilibrium following a small perturbation?

Unstable equil. Stable equil. Neutrally stab. eq.

Local vs. Global vs. Regional Stability:

1. Local: returns following small perturbation

2. Global: returns following any perturbation

3. Regional: unstable, but remains in "region" (will discuss again for >1 spp)

Locally stable Globally Stable Regionally stable

0

Density (N)

dN

/Nd

tPositive Density Dependence

Equilibrium?

Stable?

Example:

Unstable equilibrium

0

Density (N)

dN

/Nd

t

Example:

Negative Density Dependence

Equilibrium?

Stable?

Stable equilibrium

0

Density (N)

dN

/Nd

t

Logistic model:

r

Linear decrease; each individual reduces per capita growth by ‘a’

units

Equilibrium

K (or r/a)

Slope = -a

dN/Ndt = r – aN

dN/Ndt = r(K-N)/K

where a=r/K

Vary K:

Vary r:

00

Density (N)d

N/d

t

K1/2K

00

Density (N)

r

dN

/Nd

t

Exponential

Logistic

K

Exponential vs. Logistic:

More generally:

0

Density (N)

0

dN

/Nd

t

Causes?

Equilibria?

Stability?

Outcome?

Invasive species?

30 250130

00

Density (N)

( co

mponents

of

dN

/Ndt)

Per

cap

ita r

ate

s

Death

Birth

Decompose growth:

N* N*

Natural history?

1. 3 treatments (reduced, increased, ambient N)

2. Monitored population size

Will they observe convergence?Why?

Stimson and Black (1975):

Treatment N before N start N end Change in N

Decreased

142228

00

228300

228300

Control 154145

154145

322345

168200

Increased 129122

239207

268257

2950

Results:

1. Is this what we expected?

2. Why?

3. It's not because of competition…

Discuss:

00

Density (N)

(Com

ponen

ts o

f d

N/d

t)

Rate

Settlement

N*

Death

Regulation in "Open Systems":

00

Density (N)

(Com

ponen

ts o

f d

N/N

dt)

Per

Cap

ita R

ate

N*

Death

Settlement

Regulation in "Open Systems":

But what about discrete time models with density-dependence?

0

Nt

Nt+

1

Density-dependence in Discrete Time:

What would a map with density-independence look like?

l constant and >1?l constant and <1?l constant and =1?

What if there was density-dependence?

What would the dynamics look like (go to board)?

Nt

Nt+

1

Density-dependence, Discrete Time: Stability

dF/dNt|N* > 1 Unstable

(geometric growth)

dF/dNt|N* < -1 Unstable

(divergent oscillations)

-1 < dF/dNt|N* < 0 Stable

(damped oscillations)

0 < dF/dNt|N* < 1Stable

(geometric decay)

Stability requires:-1 < dF/dNt|N* < +1

Oscillations if dF/dNt|N* negative

Stability analysis: How did we get this?

Nt

Nt+

1

At equilibrium…Apply a small perturbation:

nt=Nt-N*How will system respond?

Nt+1 = F(N*+nt)Rewrite as (substract N*)nt+1 = Nt+1-N*=F(N*+nt)-N*

But what is F(N*-nt)?

An aside…

Taylor expansion

Nt

Nt+

1

At equilibrium…Apply a small perturbation:

nt=Nt-N*How will system respond?

Nt+1 = F(N*+nt)Rewrite as (substract N*)nt+1 = Nt+1-N*=F(N*+nt)-N*

But what is F(N*-nt)?

An aside…

Taylor Series Expansion

How can we approximate our function F (at N*) using a polynomial?

Taylor series expansion (around N*):

F(N*+n) ≈ F(N*) + F’(N*)(n) + F’’(N*)n2/2! + F’’’(N*)n3/3! …

Drop higher order terms… (because n is small, n2 and n3 … are very small compared to n)

What does this mean?

Taylor Series Expansion

Taylor Series: the cartoon

We are trying to estimate F(N*+n)

We make a guess based upon F'(N*)

F(N*+n) ≈ F(N*) + nF’(N*)

F(N)

N*

F(N*)

N*+n

F(N*+n)F(N*)+nF'(N*)

Stability analysis: How did we get this?

Nt

Nt+

1

So, now we can rewrite:

nt+1 =F(N*+nt) - N*

As:

nt+1 ≈ F(N*) + ntF'(N*) - N*

But, at equilibrium, F(N*)=N*

So

nt+1 ≈ ntF'(N*)

Stability analysis: How did we get this?

nt+1 ≈ ntF'(N*)

This is just a geometric growth model with a growth parameter given by F'(N*)

So, |F'(N*)| < 1 means that the perturbation is SMALLER the next time step.

F'(N*) < 0 means that direction of the deviation switches sign (N oscillates around N*).

Hence…

Nt

Nt+

1

Density-dependence, Discrete Time: Stability

dF/dNt|N* > 1 Unstable

(geometric growth)

dF/dNt|N* < -1 Unstable

(divergent oscillations)

-1 < dF/dNt|N* < 0 Stable

(damped oscillations)

0 < dF/dNt|N* < 1Stable

(geometric decay)

Stability requires:-1 < dF/dNt|N* < +1

Oscillations if dF/dNt|N* negative

One specific example…

0

Nt

Nt+

1

Discrete Logistic (one form):

Three maps with same K, but different R's

What would the dynamics look like (go to board)?

(Nt+1 – Nt)/Dt = RNt(1-Nt/Kt)

Nt+1 = Nt + RNt(K-Nt)/K

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

0

Nt

Nt+

1

Discrete Logistic: Stability:

Three maps with same K, but different R's

Stability depends on the slope of the function at N*.

What is the slope at N*?

Slope = 1-R

R<2: locally stableR=2: limit cycle with

period 2 beginsR=2.449: limit cycle of

period 4 beginsR=2.544: 8R=2.564: 16R=2.5687: 32R>2.57: Chaos

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

0

Nt

Nt+

1

Discrete Logistic: Stability:

Three maps with same K, but different R's

Stability depends on the slope of the function at N=K.

What is the slope at N=K?

Slope = 1-RR<2: locally stableR=2: cycle with period

2 beginsR=2.449: cycle of

period 4 beginsR=2.544: 8R=2.564: 16R=2.5687: 32R>2.57: Chaos

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

Bifurcation diagram: Note: this for a slightly different logistic map (May 1976), so parameter has different meaning and transitions occur at different values.

Link here.

Chaos• Chaos is not "random"• It arises from a purely deterministic model• A point very close to another gives no information about the

future position of the adjacent point:• "When the present determines the future, but the

approximate present does not approximately determine the future." (Wikipedia)

• E.g., if you plot consecutive points in a chaotic time series….

0

Nt

Nt+

1

Homework #1

The Ricker equation is another commonly used discrete time model: Nt+1=Nter(1-(Nt/K))

a) Plot the equaition (aka return map): i.e., Nt+1 vs. Nt

b) Solve for the equilibrium.c) For what values of r is there an equilibrium?d) Is the equilibrium locally stable? Do a stability analysis

(as we did with the Taylor series expansion).e) Simulate the dynamics for r=2 and K=100.f) Do these dynamics match your expectation from e and

f?g) Email your answers to Craig by 5pm this Monday.

Discussion:

Persson et al. (1988)

• …predator and prey are regulated by different factors. Top carnivores are predicted to be resource-limited, while plants … should be controlled by grazers…

What do we mean by "limited", "controlled" and "regulated"?