Transcript Population Review - University of North Carolina at Chapel

Population Review
Exponential growth
• Nt+1 = Nt + B – D + I – E
• ΔN = B – D + I – E
For a closed population
• ΔN = 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 = rN
• Nt = Noert
1.1
1.2
Influence of r on population growth
Doubling time
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Nt = 2 No
2No = Noer(td)
2 = er(td)
ln(2) = r(td)
• td = ln(2) / r
(td = doubling time)
1.3
Assumptions
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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
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
r vs λ
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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
Environmental stochasticity
• Nt = Noert
σr2 > 2r
; where Nt and r are means
leads to extinction
Demographic stochasticity
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P(birth) = b / (b+d)
P(death) = d / (b+d)
Nt = Noert (where N and r are averages)
P(extinction) = (d/b)^No
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.
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.)
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)
Logistic growth
• dN/dT = rN (1-N/K) or rN / ((K-N) / K)
• Nt = K/ (1+((K-No)/No)e-rt)
Data ??
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.
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
Examples of small population problems
Whales,
Heath hens,
Bachmann's warbler
dN/dt = rN[(K-N)/K][(N-m)/N]
Instantaneous response is not realistic
Need to introduce time lags into the system
dN/dt = rNt[(K-Nt-T)/K]
Three time lag types
Monotonic increase of decrease: 0 < rT < e-1
Oscillations damped: e-1 < rT <  /2
Limit cycle: rT > /2
Discrete growth
with lags
1. Nt+1 = Ntexp[r(1-Nt/K)]
2. Nt+1 = Nt[1+r(1-Nt/K)]
May, 1974. Science
Logistic growth with
difference equations,
showing behavior ranging
from single stable point to
chaos
(1) Nt+1 = Ntexp[r(1-Nt/K)]
(2) Nt+1 = Nt[1+r(1-Nt/K)]
Added Assumptions
• Constant carrying capacity
• Linear density dependence