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 � dt = r B N − p ( N ) K B Eddie Koch Spruce Budworm

Predation Ludwig’s Suggested Form for p ( N ) BN 2 p ( N ) = A 2 + N 2 Eddie Koch Spruce Budworm

Predation Graph of BN 2 / ( A 2 + N 2 ) p ( N ) B N Figure 1: Behavior of predation as budworm population increases Eddie Koch Spruce Budworm

Budworm Population Budworm Population is Goverened by BN 2 � � dN 1 − N dt = r B N − A 2 + N 2 K B Eddie Koch Spruce Budworm

Saturation Differentiate p ( N ) to see where function is increasing BN 2 p ( N ) = A 2 + N 2 p ′ ( N ) = ( A 2 + N 2 )(2 BN ) − ( BN 2 )(2 N ) ( A 2 + N 2 ) 2 2 A 2 BN p ′ ( N ) = ( A 2 + N 2 ) 2 Eddie Koch Spruce Budworm

Saturation Differentiate p ′ ( N ) to check for concavity 2 A 2 BN p ′ ( N ) = ( A 2 + N 2 ) 2 p ′′ ( N ) = ( A 2 + N 2 ) 2 (2 A 2 B ) − (2 A 2 BN )[2( A 2 + N 2 )2 N ] ( A 2 + N 2 ) 4 p ′′ ( N ) = 2 A 2 B ( A 2 − 3 N 2 ) ( A 2 + N 2 ) 3 Eddie Koch Spruce Budworm

Roots of Equation A 2 − 3 N 2 = 0 � 1 N = ± 3 A 2 � A N c = 3 Eddie Koch Spruce Budworm

Critical value Threshold value is at N c . p ( N ) B N N c Figure 2: The population value N c is an approximate threshold value. For N < N c predation is small, while for N > N c it is switched on. Eddie Koch Spruce Budworm

Scaling Convert to nondimensional terms. BN 2 dN � 1 − N � dt = r B N − A 2 + N 2 K B Eddie Koch Spruce Budworm

Scaling Introduction of nondimensional terms. u = N A , r = Ar B B , q = K B A , τ = Bt A With these substitutions, BN 2 dN � 1 − N � dt = r B N − A 2 + N 2 K B u 2 � � du 1 − u d τ = ru 1 + u 2 . − q Eddie Koch Spruce Budworm

Steady States Finding equilibrium points � � 1 − u � u � 0 = u r − q 1 + u 2 Either u = 0 or � 1 − u � u = 1 + u 2 . r q Eddie Koch Spruce Budworm

Graph of r and q 0.6 0.5 0.4 0.3 0.2 0.1 0 u 1 0 2 4 6 8 10 12 u−axis Figure 3: There is an asymptotically stable equilibrium point at u 1 . Eddie Koch Spruce Budworm

Graph of r and q 0.6 0.5 0.4 0.3 0.2 0.1 0 u 1 u 2 0 2 4 6 8 10 12 u−axis Figure 4: There is a additional semi-stable equilibrium point at u 2 . Eddie Koch Spruce Budworm

Graph of r and q 0.6 0.5 0.4 0.3 0.2 0.1 u 2 u 3 0 u 1 0 2 4 6 8 10 12 u−axis Figure 5: Three equilibrium points. Eddie Koch Spruce Budworm

Graph of r and q 0.6 0.5 0.4 0.3 0.2 0.1 u 2 u 3 0 u 1 0 2 4 6 8 10 12 u−axis Figure 6: As r increases u 1 and u 2 move closer together. Eddie Koch Spruce Budworm

Graph of r and q 0.6 0.5 0.4 0.3 0.2 0.1 u 1 u 3 0 0 2 4 6 8 10 12 u−axis Figure 7: Increasing r u 1 and u 2 coalesce into one semi stable equilibrium point. Eddie Koch Spruce Budworm

Graph of r and q 0.6 0.5 0.4 0.3 0.2 0.1 u 3 0 0 2 4 6 8 10 12 u−axis Figure 8: Increasing r we are back to one stable equilibrium point at u 3 . Eddie Koch Spruce Budworm

Hysteresis 0.8 D C 0.6 r−axis 0.4 B 0.2 A 0 0 10 20 30 40 q−axis Figure 9: Path of r Along ABCD Eddie Koch Spruce Budworm

Hysteresis D 8 C 6 u−axis B 4 2 B C A 0 0 0.5 1 r−axis Figure 10: Path of r Along ABCD Eddie Koch Spruce Budworm