SLIDE 1
Spruce Budworm
Eddie Koch May 14th, 2008
Eddie Koch Spruce Budworm
Logistic Equation
Logistic Equation dN dt = rBN
- 1 − N
KB
- Eddie Koch
Spruce Budworm
Spruce Budworm Eddie Koch May 14th, 2008 Eddie Koch Spruce - - PowerPoint PPT Presentation
Spruce Budworm Eddie Koch May 14th, 2008 Eddie Koch Spruce - - PowerPoint PPT Presentation
Spruce Budworm Eddie Koch May 14th, 2008 Eddie Koch Spruce Budworm Logistic Equation Logistic Equation dN 1 N dt = r B N K B Eddie Koch Spruce Budworm Adding Predation Logistic Equation with Predation dN 1 N
SLIDE 2
SLIDE 3
Adding Predation
Logistic Equation with Predation dN dt = rBN
- 1 − N
KB
- − p(N)
Eddie Koch Spruce Budworm
SLIDE 4
Predation
Ludwig’s Suggested Form for p(N) p(N) = BN2 A2 + N2
Eddie Koch Spruce Budworm
SLIDE 5
Predation
Graph of BN2/(A2 + N2)
N p(N) B Figure 1: Behavior of predation as budworm population increases
Eddie Koch Spruce Budworm
SLIDE 6
Budworm Population
Budworm Population is Goverened by dN dt = rBN
- 1 − N
KB
- −
BN2 A2 + N2
Eddie Koch Spruce Budworm
SLIDE 7
Saturation
Differentiate p(N) to see where function is increasing p(N) = BN2 A2 + N2 p′(N) = (A2 + N2)(2BN) − (BN2)(2N) (A2 + N2)2 p′(N) = 2A2BN (A2 + N2)2
Eddie Koch Spruce Budworm
SLIDE 8
Saturation
Differentiate p′(N) to check for concavity p′(N) = 2A2BN (A2 + N2)2 p′′(N) = (A2 + N2)2(2A2B) − (2A2BN)[2(A2 + N2)2N] (A2 + N2)4 p′′(N) = 2A2B(A2 − 3N2) (A2 + N2)3
Eddie Koch Spruce Budworm
SLIDE 9
Roots of Equation
A2 − 3N2 = 0 N = ±
- 1
3A2 Nc =
- A
3
Eddie Koch Spruce Budworm
SLIDE 10
Critical value
Threshold value is at Nc.
N p(N) B Nc Figure 2: The population value Nc is an approximate threshold value. For N < Nc predation is small, while for N > Nc it is switched on.
Eddie Koch Spruce Budworm
SLIDE 11
Scaling
Convert to nondimensional terms. dN dt = rBN
- 1 − N
KB
- −
BN2 A2 + N2
Eddie Koch Spruce Budworm
SLIDE 12
Scaling
Introduction of nondimensional terms. u = N A , r = ArB B , q = KB A , τ = Bt A With these substitutions, dN dt = rBN
- 1 − N
KB
- −
BN2 A2 + N2 du dτ = ru
- 1 − u
q
- −
u2 1 + u2 .
Eddie Koch Spruce Budworm
SLIDE 13
Steady States
Finding equilibrium points 0 = u
- r
- 1 − u
q
- −
u 1 + u2
- Either u = 0 or
r
- 1 − u
q
- =
u 1 + u2 .
Eddie Koch Spruce Budworm
SLIDE 14
Graph of r and q
2 4 6 8 10 12 0.1 0.2 0.3 0.4 0.5 0.6 u1 u−axis Figure 3: There is an asymptotically stable equilibrium point at u1.
Eddie Koch Spruce Budworm
SLIDE 15
Graph of r and q
2 4 6 8 10 12 0.1 0.2 0.3 0.4 0.5 0.6 u1 u2 u−axis Figure 4: There is a additional semi-stable equilibrium point at u2.
Eddie Koch Spruce Budworm
SLIDE 16
Graph of r and q
2 4 6 8 10 12 0.1 0.2 0.3 0.4 0.5 0.6 u1 u2 u3 u−axis Figure 5: Three equilibrium points.
Eddie Koch Spruce Budworm
SLIDE 17
Graph of r and q
2 4 6 8 10 12 0.1 0.2 0.3 0.4 0.5 0.6 u1 u2 u3 u−axis Figure 6: As r increases u1 and u2 move closer together.
Eddie Koch Spruce Budworm
SLIDE 18
Graph of r and q
2 4 6 8 10 12 0.1 0.2 0.3 0.4 0.5 0.6 u1 u3 u−axis Figure 7: Increasing r u1 and u2 coalesce into one semi stable equilibrium point.
Eddie Koch Spruce Budworm
SLIDE 19
Graph of r and q
2 4 6 8 10 12 0.1 0.2 0.3 0.4 0.5 0.6 u3 u−axis Figure 8: Increasing r we are back to one stable equilibrium point at u3.
Eddie Koch Spruce Budworm
SLIDE 20
Hysteresis
10 20 30 40 0.2 0.4 0.6 0.8 A B C D q−axis r−axis Figure 9: Path of r Along ABCD
Eddie Koch Spruce Budworm
SLIDE 21
Hysteresis
0.5 1 2 4 6 8 A C C B B D r−axis u−axis Figure 10: Path of r Along ABCD
Eddie Koch Spruce Budworm