Population Review

38
Population Review

description

Population Review. Exponential growth. N t+1 = N t + B – D + I – E Δ N = B – D + I – E For a closed population Δ N = B – D. dN/dt = B – D B = bN ; D = dN (b and d are instanteous birth and death rates) dN/dT = (b-d)N dN/dt = rN1.1 N t = N o e rt 1.2. - PowerPoint PPT Presentation

Transcript of Population Review

Page 1: Population Review

Population Review

Page 2: Population Review

Exponential growth

• Nt+1 = Nt + B – D + I – E

• ΔN = B – D + I – E

For a closed population

• ΔN = B – D

Page 3: Population Review

• dN/dt = B – D

• B = bN ; D = dN (b and d are instanteous birth and death rates)

• dN/dT = (b-d)N

• dN/dt = rN 1.1

• Nt = Noert 1.2

Page 4: Population Review

Influence of r on population growth

Page 5: Population Review

Doubling time

• Nt = 2 No

• 2No = Noer(td) (td = doubling time)

• 2 = er(td)

• ln(2) = r(td)

• td = ln(2) / r 1.3

Page 6: Population Review
Page 7: Population Review

Assumptions

• No I or E

• Constant b and d (no variance)

• No genetic structure (all are equal)

• No age or size structure (all are equal)

• Continuous growth with no time lags

Page 8: Population Review

Discrete growth

• Nt+1 = Nt + rdNt (rd = discrete growth factor)

• Nt+1 = Nt(1+rd)

• Nt+1 = λ Nt

• N2 = λ N1 = λ (λ No) = λ2No

• Nt = λtNo 1.4

Page 9: Population Review

r vs λ

• er = λ if one lets the time step approach 0• r = ln(λ)• r > 0 ↔ λ > 1• r = 0 ↔ λ = 1• r < 0 ↔ 0 < λ < 1

Page 10: Population Review

Environmental stochasticity• Nt = Noert ; where Nt and r are means

σr2 > 2r leads to extinction

Page 11: Population Review

Demographic stochasticity

• P(birth) = b / (b+d)

• P(death) = d / (b+d)

• Nt = Noert (where N and r are averages)

• P(extinction) = (d/b)^No

Page 12: Population Review
Page 13: Population Review

Elementary Postulates 

• Every living organism has arisen from at least one parent of the same kind.

• In a finite space there is an upper limit to the number of finite beings that can occupy or utilize that space.

Page 14: Population Review

Think about a complex model approximated by many terms in a potentially infinite series. Then consider how many of these terms are needed for the simplest acceptable model.

dN/dt = a + bN + cN2 + dN3 + ....

 

From parenthood postulate, N = 0 ==> dN/dt = 0, therefore a = 0.

Simplest model ===> dN/dt = bN, (or rN, where r is the intrinsic rate of increase.)

Page 15: Population Review

Logistic Growth

There has to be a limit. Postulate 2.

Therefore add a second parameter to equation.

  dN/dt = rN + cN2

define c = -r/K

  dN/dt = rN ((K-N)/K)

Page 16: Population Review

Logistic growth

• dN/dT = rN (1-N/K) or rN / ((K-N) / K)

• Nt = K/ (1+((K-No)/No)e-rt)

Page 17: Population Review
Page 18: Population Review

Data ??

Page 19: Population Review
Page 20: Population Review
Page 21: Population Review
Page 22: Population Review
Page 23: Population Review
Page 24: Population Review
Page 25: Population Review
Page 26: Population Review
Page 27: Population Review
Page 28: Population Review

Further Refinements of the theory

 Third term to equation?

More realism? Symmetry?

No reason why the curve has to be a symmetric curve with maximal growth at N = K/2.

Page 29: Population Review

What if the population is too small? Is r still high under these conditions?

• Need to find each other to mate

• Need to keep up genetic diversity

• Need for various social systems to work

Page 30: Population Review

Examples of small population problems

Whales, Heath hens, Bachmann's warbler

dN/dt = rN[(K-N)/K][(N-m)/N]

Page 31: Population Review

Instantaneous response is not realistic

 Need to introduce time lags into the system

dN/dt = rNt[(K-Nt-T)/K]

Page 32: Population Review
Page 33: Population Review

Three time lag types

Monotonic increase of decrease: 0 < rT < e-1

Oscillations damped: e-1 < rT < /2

Limit cycle: rT > /2

Page 34: Population Review
Page 35: Population Review

Discrete growth with lags

Page 36: Population Review

May, 1974. Science

1. Nt+1 = Ntexp[r(1-Nt/K)]

2. Nt+1 = Nt[1+r(1-Nt/K)]

Page 37: Population Review

(1) Nt+1 = Ntexp[r(1-Nt/K)]

(2) Nt+1 = Nt[1+r(1-Nt/K)]

Logistic growth with difference equations, showing behavior ranging from single stable point to chaos

Page 38: Population Review

Added Assumptions

• Constant carrying capacity

• Linear density dependence