SLIDE 1
Exponential growth can’t last forever
10000 20000 30000 40000 50000 60000 70000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Tempo
Nt
4 , 2 = = = N N N
t t
Exponential growth cant last forever 70000 60000 = = = t - - PowerPoint PPT Presentation
Exponential growth cant last forever 70000 60000 = = = t - - PowerPoint PPT Presentation
Exponential growth cant last forever 70000 60000 = = = t N N 2 , N 4 t 0 0 50000 40000 N t 30000 20000 10000 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Tempo Regulation factors N t Negative feedback
SLIDE 2
SLIDE 3
Mechanisms that may induce density dependent regulation
Limited food resources:
– Less consumption per capita, longer time periods searching for prey, with longer exposition to predators (affects S and b)
Less space:
- Smaller average territory or greater number of individuals without
territory
Greater predator and/or parasite pressure:
- Predators “shift” to denser prey populations; greater incidence of
infectious diseases.
Greater use of marginal habitats of lesser quality
- etc. etc...
SLIDE 4
Exponential growth: constant b and d
t r t
e N N =
constants d b r − =
b0 d0 N time N0
growth l exponentia ⇒ > ⇒ > r d b
SLIDE 5
Density dependent regulation
b0 d0 ) (N f d = ) (N f b = N
increase d b ⇒ >
decrease d b ⇒ <
Equilibrium, N = K
N=K
Stable equilibrium
SLIDE 6
Carrying Capacity, K
Carrying capacity ≈ Population density which is sustained by the resources available
10 20 30
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
tempo N
K
SLIDE 7
How to model density-dependence
- 1. State the mechanisms of density dependence explicitly
Example: what are the mechanisms of intra-specific competition ?
- 2. Assume simple analytical functions for b=f(N) and d=f(N)
Etc.
SLIDE 8
Continuous breeding
b0 d0
d b r − =
r Substituting in
( ) ( ) ( ) [ ] ( ) ( ) [ ]
t t t t t t t t
N N q p d b dt dN N qN d pN b dt dN N d b dt dN + − − = + − − = − =
we get
t t t t
pN b b qN d d − = + =
N
SLIDE 9
Introducing K
N K
At K, dN/dt = 0
when does
? = dt dN
( )
[ ]
t t N
N q p r dt dN + − =
Nt = 0 Trivial equilibrium q p r Nt + = Non-trivial equilibrium K itself
SLIDE 10
The logistic equation of continuos breeders (Verhulst, 1838)
K r q p q p r K = + ∴ + =
( ) [ ]
t t N
N q p r dt dN + − =
Substituting here
− = K N rN dt dN 1
Unregulated growth Regulating factor
SLIDE 11
Growth per capita
− = K N rN dt dN 1
Contribution of 1 individual for population growth
≡
N K r r dt dN N − = 1
dt dN N 1
r
K r −
N
slope =
K Contrib. per capita
SLIDE 12
Solution of the logistic equation
− = K N rN dt dN 1
4 8 12 16
1 21 41 61 81 101 121 141
Tempo
N
K r=0.7 r=1 r=1.2
( )
t r t
e N K N KN N
−
− + =
Solution:
SLIDE 13