intraspecific competition individuals in a population have same resource needs combined demand for a...
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INTRASPECIFIC COMPETITION
Individuals in a population have same resource needs
Combined demand for a resource influences its supply – leads to competition
Competition affects population processes
1 : 20
5 : 20
FITNESS
8 : 20
12 : 20
17 : 20
Characteristics of Competition
Increases in density – decrease in individual fitness (growth, survivorship or fecundity)
Resource/s in limiting supply
All individuals inherently equal
Effects of competition on an individual’s fitness density dependent
Population size / density
Nu
mb
ers
dyi
ng
Nu
mb
ers
dyi
ng
p
er i
nd
ivid
ual
Which line shows density independent mortality?
If N = 100, and number dying = 15: q = 15 / 100 = 0.15If N = 300, and number dying = 45: q = 45 / 300 = 0.15If N = 300 and number dying = 90: q = 90 / 300 = 0.30
Population size / density
Mo
rtal
ity
rate
I
II
III
Nu
mb
ers
Dyi
ng
I II
III
Population size / density
I
IIIII
I = Independent
II and III - Dependent
II = under-compensatingIII = over-compensating
Population size / density
Exactly compensating
Population size / density
Rat
eBirth
Death
K
Define K
Born
Population size / density
Nu
mb
ers
Dying
Difference = NET Recruitment
0
200
400
600
800
1000
1200
1400
1600
1800
1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106
Time
N
S-Shaped Growth CurvesCharacteristic of intra-specific competition
0
5
10
15
20
25
0 200 400 600 800 1000
Population Size
Net
Rec
ruit
men
t
N - Shaped
K
Palmblad Data – Is Competition Occurring?
Sowing Density 1 5 50 100 200Log Density 0.00 0.70 1.70 2.00 2.30% Germination 100 100 83 86 83No Germinated 1 5 41.5 86 166% Mortality 0 0 1 3 8No Mature 1 5 41 83 150% Reproducing 100 100 82 83 73No Reproducing 1 5 41 83 146% Vegetative 0 0 0 0 2No Vegetative 0 0 0 0 4Dry Weight 2.01 3.44 4.83 4.51 4.16Mean No Seeds 23741 6102 990 451 210Total No Seeds 23741 30510 40590 37433 30660Mean No Seeds 23741 6102 990 451 204
Capsella bursa-pastoris
Is there any evidence that an increase in density results in a reduction in fitness?
Is there any evidence that the reduction in fitness is density dependent?
0
20
40
60
80
100
120
0 50 100 150 200
Sowing Density
Pe
rce
nta
ge
-1
0
1
2
3
45
6
7
8
9
Pe
rce
nta
ge
Germination
Mortality
Reproducing
0
1
2
3
4
5
6
0 50 100 150 200 250
Sowing Density
Yie
ld
0
20
40
60
80
100
120
140
160
180
De
nsi
ty o
f su
rviv
ors
Competition affects QUALITY of individuals
Is there any evidence that the population reaches a carrying capacity?
Law of Constant Yield – Plants
Sowing Density 1 5 50 100 200Log Density 0.00 0.70 1.70 2.00 2.30% Germination 100 100 83 86 83No Germinated 1 5 41.5 86 166% Mortality 0 0 1 3 8No Mature 1 5 41 83 150% Reproducing 100 100 82 83 73No Reproducing 1 5 41 83 146% Vegetative 0 0 0 0 2No Vegetative 0 0 0 0 4Dry Weight 2.01 3.44 4.83 4.51 4.16Mean No Seeds 23741 6102 990 451 210Total No Seeds 23741 30510 40590 37433 30660Mean No Seeds 23741 6102 990 451 204
Capsella bursa-pastoris
If competition is occurring – is density dependence over-, under- or exactly compensating?
How do you tell?
Plot k values against (log10) sowing density – if slope of the line < unity, under-compensating; if > unity, over-compensating; if = 1, exactly compensating
What are k-values?
k killing power – reflects stage specific mortality and can be summed
Stage (x) ax lx dx qx px log10ax log10lx log10ax - log10ax+1 Fx mx lxmx
Eggs (0) 44000 1.0000 0.9201 0.9201 0.0799 4.6435 0 1.0975 0 0 0Instar I (1) 3515 0.0799 0.0224 0.2805 0.7195 3.5459 -1.098 0.1430 0 0 0Instar II (2) 2529 0.0575 0.0138 0.2400 0.7600 3.4029 -1.241 0.1192 0 0 0Instar III (3) 1922 0.0437 0.0105 0.2399 0.7601 3.2838 -1.36 0.1191 0 0 0Instar IV (4) 1461 0.0332 0.0037 0.1102 0.8898 3.1647 -1.479 0.0507 0 0 0Adults V (5) 1300 0.0295 -- -- -- 3.1139 -1.53 -- 22617 17.3977 0.51
K1 5 50 100 200
% Germination 100 100 83 86 83
% Mortality 0 0 1 3 8
% Reproducing 100 100 82 83 73
Mean No Seeds 23741 6102 990 451 210
No Germinated 1 5 41.5 86 166No Mature 1 5 41 83 150No Reproducing 1 5 41 83 146Mean No Seeds 23741 6102 990 451 210
Sowing Density 0 0.69897 1.69897 2 2.30103
No Germinated 0 0.69897 1.618048 1.934498 2.220108
No Mature 0 0.69897 1.612784 1.919078 2.176091
No Reproducing 0 0.69897 1.612784 1.919078 2.164353
Mean No Seeds 4.375499 3.785472 2.995635 2.654177 2.322219
Germination 0 0 0.080922 0.065502 0.080922Mortality 0 0 0.005264 0.01542 0.044017Vegetative 0 0 0 0 0.011738Fecundity 0 0.590027 1.379864 1.721322 2.065018Total 0 0.590027 1.46605 1.802244 2.201695
0 0.69897 1.69897 2 2.30103
Pe
rce
nta
ge
Capsella bursa-pastoris
Sowing Density
Nu
mb
ers
One to One
Lo
g10
Nu
mb
ers
K
0.00
0.50
1.00
1.50
2.00
2.50
0.00 0.50 1.00 1.50 2.00 2.50
Log Density
K
K
1:1
1 5 50 100 200
% Germination 100 100 83 86 83
% Mortality 0 0 1 3 8
% Reproducing 100 100 82 83 73
Mean No Seeds 23741 6102 990 451 210
No Germinated 1 5 41.5 86 166No Mature 1 5 41 83 150No Reproducing 1 5 41 83 146Mean No Seeds 23741 6102 990 451 210
Sowing Density 0 0.69897 1.69897 2 2.30103
No Germinated 0 0.69897 1.618048 1.934498 2.220108
No Mature 0 0.69897 1.612784 1.919078 2.176091
No Reproducing 0 0.69897 1.612784 1.919078 2.164353
Mean No Seeds 4.375499 3.785472 2.995635 2.654177 2.322219
Germination 0 0 0.080922 0.065502 0.080922Mortality 0 0 0.005264 0.01542 0.044017Vegetative 0 0 0 0 0.011738Fecundity 0 0.590027 1.379864 1.721322 2.065018Total 0 0.590027 1.46605 1.802244 2.201695
0 0.69897 1.69897 2 2.30103
Pe
rce
nta
ge
Capsella bursa-pastoris
Sowing Density
Nu
mb
ers
One to One
Lo
g10
Nu
mb
ers
K
Exactly -
Under -
Over -
70
40
0 1200
125
50
Density (no. m-2)
Bio
mas
s (g
. m
-2)
Mea
n S
hel
l L
eng
thScutellastra cochlear
Log Density
K g
amet
e o
utp
ut
1225 m2365 m2125 m2
Reproductive Asymmetry
Nt = N0.Rt
Exponential Growth
Models built to date, constant R
Not realistic, because R varies with population size due to competition
How do we build a model where R varies?
R 1.12
Time N0 251 282 313 354 395 446 497 558 629 6910 78
96 132699897 148623798 166458699 1864336
100 2088057101 2338623102 2619258103 2933569104 3285598105 3679869106 4121454107 4616028108 5169951109 5790346
0
1000000
2000000
3000000
4000000
5000000
6000000
7000000
0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105
Time
N
Populations showing discrete breeding (pulse)
When Nt =A (very small), R = R, A = 1/R
A
Nt+1 = Nt.RtNt
Nt+1
= 1/R
Nt
Nt+1
Nt
1/R
Equation for a straight line: Y = mx + c
Equation for a straight line: y = c + mx
Nt
Nt+1
= 1/R + .Nt(1 – 1/R)
K[ ]
K
1
When Nt = B, R = 1
B
Equation for a straight line: y = c + mx
Nt
Nt+1
= 1/R + .Nt(1 – 1/R)
K[ ]
Therefore:
Nt+1 = Nt / {(1/R) + [Nt(1/R)(R-1)(1/K)]}
Simplify Denominator on RHS
(1/R) + [Nt(1/R)(R-1)(1/K)] = (1/R) {1 + [Nt(R-1)/K]}
Therefore:
Nt+1 = Nt / {(1/R)[1 + (Nt.(R-1)/K)]}
Nt+1 = (Nt R) / {1 + [Nt.(R-1)/K]}
Nt+1 / Nt = R = R / {1 + [Nt.(R-1)/K]}
Simplify
[1 – (1/R)] = [(R/R) – (1/R)] = (1/R)(R-1)
Nt+1 = (Nt R) / {1 + [Nt.(R-1)/K]}
The expression [(R-1)/K] is often written as a
Nt+1 = (Nt R) / [1 + (Nt.a)]
Nt+1 / Nt = R = R / {1 + [Nt.(R-1)/K]}
Reproductive rate not constant!
Rearrange
R 1.12 K 796
Time N N New R Additions0 25 25 1.1158 31 28 28 1.1153 32 31 31 1.1148 43 35 35 1.1142 44 39 39 1.1135 45 44 43 1.1128 56 49 48 1.1120 57 55 53 1.1111 68 62 59 1.1101 79 69 66 1.1090 7
10 78 73 1.1078 8
96 1326998 796 1.0001 097 1486237 796 1.0001 098 1664586 796 1.0000 099 1864336 796 1.0000 0
100 2088057 796 1.0000 0101 2338623 796 1.0000 0102 2619258 796 1.0000 0103 2933569 796 1.0000 0104 3285598 796 1.0000 0105 3679869 796 1.0000 0106 4121454 796 1.0000 0107 4616028 796 1.0000 0108 5169951 796 1.0000 0109 5790346 796 0.0000
0
1000000
2000000
3000000
4000000
5000000
6000000
7000000
0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105
Time
N
0
100
200
300
400
500
600
700
800
900
Using Constant R
Using Variable R
0
5
10
15
20
25
0 200 400 600 800 1000
N
Rec
ruit
men
t
Stock – Recruit Curve
Shape of Growth Curve depends on R and K
0
200
400
600
800
1000
0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105
Time
N
0
5
10
15
20
25
30
R=1.12 R=1.55 Series4 R=0.95
K = 796
The higher the R, the faster the population reaches K
0
500
1000
1500
2000
0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105
Time
N
K=256 K=796 K=1870
R=1.12
The higher the K, the bigger the N for a given t AND the slower it takes to reach K for a given R
Models assume instantaneous responses of Nt+1 to Nt
Population lags
What if the amount of resources available to a population at time t (which, after all, determines the size of the population at time t+1 – through R) is determined by the size of the population at time t-1
i.e . R is dependent NOT on Nt but on Nt-1
Nt+1 = (Nt R) / [1 + (Nt-1.a)]
0
200
400
600
800
1000
1200
1400
0 5 10 15 20 25 30 35
Time
N
R = 2.8
Time lags promote fluctuations in population size
WHY?
Fluctuations common in models of DISCRETE breeding because the population still responding at the end of a time
interval to the density at its start
0
100
200
300
400
500
600
700
0 5 10 15 20 25 30 35
Time
N
R = 1.15
0
200
400
600
800
1000
1200
1400
0 5 10 15 20 25 30 35
Time
N
R = 2.8
0
100
200
300
400
500
600
700
800
900
0 5 10 15 20 25 30 35
Time
NR = 1.55
0100200300400500600700800900
1000
0 5 10 15 20 25 30 35
Time
N R = 2.0
0
100
200
300
400
500
600
700
800
900
0 5 10 15 20 25 30 35
Time
N
R = 1.25
Magnitude of fluctuations dependent on R
Model is realistic for EXACTLY compensating density-dependence
Nt+1 = (Nt R) / [1 + (Nt.a)]
Is this realistic?
Sowing Density 1 5 50 100 200Log Density 0.00 0.70 1.70 2.00 2.30% Germination 100 100 83 86 83No Germinated 1 5 41.5 86 166% Mortality 0 0 1 3 8No Mature 1 5 41 83 150% Reproducing 100 100 82 83 73No Reproducing 1 5 41 83 146% Vegetative 0 0 0 0 2No Vegetative 0 0 0 0 4Dry Weight 2.01 3.44 4.83 4.51 4.16Mean No Seeds 23741 6102 990 451 210Total No Seeds 23741 30510 40590 37433 30660Mean No Seeds 23741 6102 990 451 204Kgermination 0.00 0.00 0.08 0.07 0.08Kmortality 0.00 0.00 0.01 0.02 0.04Kvegetative 0.00 0.00 0.00 0.00 0.01Kfecundity 0.00 0.59 1.38 1.72 2.07Ktotal 0.00 0.59 1.47 1.80 2.20One to One 0.00 0.70 1.70 2.00 2.30
Capsella bursa-pastoris
0.00
0.50
1.00
1.50
2.00
2.50
0.00 0.50 1.00 1.50 2.00 2.50
Log Density
K
K 1:1
Under-compensating
Over-compensating
In the absence of competition, Potential recruitment can be calculated from Nt+1 = Nt.R
The difference between Nt+1 and Nt is due to net recruitment (+ or -)
Actual recruitment is calculated from Nt+1 = (Nt R) / [1 + (Nt.a)]
k = log10 (Produced) – log10 (Surviving)
k = log10(NtR) – log10 {(Nt R) / [1 + (Nt.a)]}
k = log10Nt + log10R – {log10 Nt +log10R – log10(1 + aNt)}
k = log10(1 + aNt) = b
The difference between Potential and Actual = k
Substituting
or
or
Substituting
Nt+1 = (Nt R) / {1 + [Nt.(R-1)/K]}b