overview of population growth: discretecontinuous density independent density dependent geometric...
TRANSCRIPT
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Overview ofpopulation growth:
discrete continuous
densityindependent
densitydependent
Geometric Exponential
DiscreteLogistic
LogisticNew Concepts:
- Stability- DI (non-regulating)
vs. DD (regulating) growth
- equilibrium
Variability in growth
(1) Individual variation in births and deaths(2) Environmental (extrinsic variability)(3) Intrinsic variability
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How do populations grow – a derivation of geometric growth
N1 = N0 + rN0
Growth rate (r) = birth rate – death rate
N0 = initial population density (time = 0)
N1 = population density 1 year later (time =1)
(express as per individual)
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How do populations grow?
N1 = N0 + rN0 = N0 (1 + r)
Growth rate (r) = birth rate – death rate
![Page 4: Overview of population growth: discretecontinuous density independent density dependent Geometric Exponential Discrete Logistic New Concepts: - Stability](https://reader035.vdocuments.mx/reader035/viewer/2022062309/56649eb35503460f94bba9b6/html5/thumbnails/4.jpg)
How do populations grow?
N1 = N0 + rN0 = N0 (1 + r)
N2 = N1 + rN1 = N1 (1 + r)
Growth rate (r) = birth rate – death rate
![Page 5: Overview of population growth: discretecontinuous density independent density dependent Geometric Exponential Discrete Logistic New Concepts: - Stability](https://reader035.vdocuments.mx/reader035/viewer/2022062309/56649eb35503460f94bba9b6/html5/thumbnails/5.jpg)
How do populations grow?
N1 = N0 + rN0 = N0 (1 + r)
N2 = N1 + rN1 = N1 (1 + r)
Can we rewrite N2 in terms of N0 ???
Growth rate (r) = birth rate – death rate
![Page 6: Overview of population growth: discretecontinuous density independent density dependent Geometric Exponential Discrete Logistic New Concepts: - Stability](https://reader035.vdocuments.mx/reader035/viewer/2022062309/56649eb35503460f94bba9b6/html5/thumbnails/6.jpg)
How do populations grow?
N1 = N0 + rN0 = N0 (1 + r)
N2 = N1 + rN1 = N1 (1 + r)
Growth rate (r) = birth rate – death rate
substitute
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How do populations grow?
N1 = N0 + rN0 = N0 (1 + r)
N2 = N1 + rN1 = N1 (1 + r)
Growth rate (r) = birth rate – death rate
N2 = N0 (1 + r)(1 + r) = N0 (1 + r)2
substitute
rewrite:
![Page 8: Overview of population growth: discretecontinuous density independent density dependent Geometric Exponential Discrete Logistic New Concepts: - Stability](https://reader035.vdocuments.mx/reader035/viewer/2022062309/56649eb35503460f94bba9b6/html5/thumbnails/8.jpg)
How do populations grow?
N1 = N0 + rN0 = N0 (1 + r)
N2 = N1 + rN1 = N1 (1 + r)
Growth rate (r) = birth rate – death rate
N2 = N0 (1 + r)(1 + r) = N0 (1 + r)2
substitute
or
Nt = N0 (1 + r)t}
= , finite rate of increase
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Discrete (geometric) growth
N
time
Nt = N0t
12
3
4
5
= finite rate of increase
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Continuous (exponential) growth
N
time
12
3
4
5
Nt = N0ert
r = intrinsic growth rate
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Continuous (exponential) growth
N
time
12
3
4
5
1 dNN dt
= rdN dt
= rN;
populationgrowth rate
per capitagrowth rate
N Per capita growth is constant and independent of N
dN dt
Read as change in N (density) over change in time.
1 dNN dt
1 dNN dt
= r
Y = b + mX
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Discrete Continuous
Nt = N0t Nt = N0ert
> 1 r > 0 < 1 r < 0
Increasing:Decreasing:
Where: = er r = ln
Every time-step (e.g., generation)Time lag:
None Compounded instantaneously
Applications: Populations w/ discrete breeding season
No breeding season - at any time there are individuals in all stages
of reproduction
Examples: Most temperate vertebrates and plants Humans, bacteria, protozoa
Mathematics: Often intractable;simulations Mathematically convenient
Comparison
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Geometric (or close to it)growth in wildebeest populationof the Serengeti following Rinderpest inoculation
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Exponential growth in the total human population
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Simplest expression of population growth: 1 parameter, e.g., r = intrinsic growth rate
Population grows geometrically/exponentially, but the Per capita growth rate is constant
First Law of Ecology: All populations possessthe capacity to grow exponentially
The Take Home Message:
Exponential/geometric growth is a model to which we build on
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Overview ofpopulation growth:
discrete continuous
densityindependent
densitydependent
Geometric Exponential
DiscreteLogistic
LogisticNew Concepts:
- Stability- DI (non-regulating)
vs. DD (regulating) growth
- equilibrium
Variability in growth
(1) Individual variation in births and deaths(2) Environmental (extrinsic variability)(3) Intrinsic variability
XX
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Variability in space In time
No
mig
rati
onm
igra
tion
![Page 18: Overview of population growth: discretecontinuous density independent density dependent Geometric Exponential Discrete Logistic New Concepts: - Stability](https://reader035.vdocuments.mx/reader035/viewer/2022062309/56649eb35503460f94bba9b6/html5/thumbnails/18.jpg)
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Variability in space In time
No
mig
rati
onm
igra
tion
Source-sink structure
![Page 20: Overview of population growth: discretecontinuous density independent density dependent Geometric Exponential Discrete Logistic New Concepts: - Stability](https://reader035.vdocuments.mx/reader035/viewer/2022062309/56649eb35503460f94bba9b6/html5/thumbnails/20.jpg)
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Variability in space In time
No
mig
rati
onm
igra
tion (arithmetic)
Source-sink structurewith the rescue effect
Source-sink structure
![Page 22: Overview of population growth: discretecontinuous density independent density dependent Geometric Exponential Discrete Logistic New Concepts: - Stability](https://reader035.vdocuments.mx/reader035/viewer/2022062309/56649eb35503460f94bba9b6/html5/thumbnails/22.jpg)
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Variability in space In time
No
mig
rati
onm
igra
tion
Source-sink structure
(geometric)
G < A G declines with increasing variance
(arithmetic)
Source-sink structurewith the rescue effect
![Page 24: Overview of population growth: discretecontinuous density independent density dependent Geometric Exponential Discrete Logistic New Concepts: - Stability](https://reader035.vdocuments.mx/reader035/viewer/2022062309/56649eb35503460f94bba9b6/html5/thumbnails/24.jpg)
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Variability in space In time
No
mig
rati
onm
igra
tion (arithmetic)
Source-sink structurewith the rescue effect
(geometric)
G < A G declines with increasing variance
Temporal variability reduces population growth rates
Cure – populations decoupled with respect to variability, but coupled with respect to sharing individuals
Source-sink structure
(arith & geom)Increase the number of subpopulations increases the growth rate (to a point),and slows the time to extinction