what is a population
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What is a Population A collection of potentially interacting organisms of one species within a defined geographic area. Figure 14.1. Estimates of Population Size Recall the Lincoln index from recitation: M/N = m/n M is the total number marked; m is the number marked in the sample; - PowerPoint PPT PresentationTRANSCRIPT
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What is a Population
A collection of potentially interacting organisms of one species within a defined geographic area.
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Figure 14.1
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Estimates of Population Size
Recall the Lincoln index from recitation:
M/N = m/n
M is the total number marked;
m is the number marked in the sample;
n is the sample size.
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Elementary Postulates
1. Every living organism has arisen from at least one parent of the same kind.
2. In a finite space there is an upper limit to the number of finite beings that can occupy or utilize that space.
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Populations grow by multiplication.A population increases in proportion to its size, in a manner analogous to a savings account earning interest on principal:
at a 10% annual rate of increase:a population of 100 adds 10 individuals in 1 year
a population of 1000 adds 100 individuals in 1 year
allowed to grow unchecked, a population growing at a constant rate would rapidly climb toward infinity
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Geometric population growth
Usually think of animals increasing by distinct generations.
N(1) = N(0) + B – D + I – E
N(1) = N(0)R
N(2) = N(0)RR
N(t) = N(0)Rt
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Critical parameter is = net replacement rate (Ro).
N(t+1) = N(t)
where:N(t + 1) = number of individuals after 1 time unit
N(t) = initial population size
= ratio of population at any time to that 1 time unit earlier, such that λ = N(t + 1)/N(t)
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To calculate the growth of a population over many time intervals, we multiply the original population size by the geometric growth rate for the appropriate number of intervals t:
N(t) = N(0) t
For a population growing at a geometric rate of 50% per year ( = 1.50), an initial population of N(0) = 100 would grow to N(10) = N(0) 10 = 5,767 in 10 years.
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Given No = 5000; N1 = 6000;
What is N2 ?
N2 = (6/5)2 x 5000 = 7200
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Exponential Growth
Generations overlap, usually not discrete generations.
For convenience, most of our models are continuous.
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Think about a complex model approximated by may term 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.)
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Solve equation: N = No e rt
Alternative form: dN/dt = bN - dN = (b-d)N
Rarely do b and d remain constant,
but if well below what environment can support, then OK assumption.
Each species has optimum environment with r = max
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Figure 14.3
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Figure 14.6
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Human lice; r=.111/day
How fast will a population that starts at 100 lice increase?
(i.e., what is rate of increase of 100 lice?)
dN/dt = rN = .111 x 100 = 11.1 lice/day
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Human population in 1993 = 5,600,000,000
b = 26/1000, d = 9/1000
How fast was the population growing?
dN/dt = rN = (.017)(5,600,000,000) = 95,200,000
(i.e., in excess of 1/3 US population per year)
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Humans currently have b and d of 26 and 9 per 1000. How many years to double the population?
N = No e rt = Nox2
2 = ert
ln2/.017 = 40.77 yrs
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1700-1800 Human population from 600,000,000 ==> 900,000,000.
Calculate r.
r = ln (N/No) / t
= ln(9/6)/100
= .0040547
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Logistic Growth
There has to be a limit. Postulate 2.
Therefore add a second parameter to equation.
dN/dt = rN + cN2
call c = -r/K
dN/dt = rN ((K-N)/K)
Nt = K/[1+((K-No)/No)e-rt]
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Figure 14.18
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Figure 14.16
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Optimal yield problem.
dN/dt = rN - rN2/K
d2N/dt2 = r - 2rN/K
set = 0 N = K/2
If want maximum yield, should exercise continual cropping around N = K/2
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Figure 14.17
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Data ??
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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.
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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
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Examples of small population problems
Whales, Heath hens, Bachmann's warbler
dN/dt = rN[(K-N)/K][(N-m)/N]
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Instantaneous response is not realistic.
Need to introduce time lags into the system
dN/dt = rNt[(K-Nt-T)/K]
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Three time lag types
Monotonic increase of decrease: 0 < rT < e-1
Oscillations damped: e-1 < rT < /2
Limit cycle: rT > /2
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Finite difference equations and Chaos
Nt+1 = aNt(1-Nt)
Models populations with discrete, nonoverlapping generations, like many temperate zone insects.
if 1<a<3, population settles to a steady state.if 3<a<3.57.., population settles into a stable cycle.if 3.57..<a<4, population apparently random or chaotic.if 4< a, N runs away to minus infinity.
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This weird range of behaviors is generic to most difference equations that describe a population with a propensity to increase at low values and to decrease at high values. Similar behavior arises if there are many discrete but overlapping generations.