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