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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
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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 1.1
• Nt = Noert 1.2
Influence of r on population growth
Doubling time
• Nt = 2 No
• 2No = Noer(td) (td = doubling time)
• 2 = er(td)
• ln(2) = r(td)
• td = ln(2) / r 1.3
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
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
• 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 ; where Nt and r are means
σr2 > 2r leads to extinction
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
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
May, 1974. Science
1. Nt+1 = Ntexp[r(1-Nt/K)]
2. Nt+1 = Nt[1+r(1-Nt/K)]
(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
Added Assumptions
• Constant carrying capacity
• Linear density dependence
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