4b Spruce budworm model L5 ADDED

Question 1

SPRUCE BUDWORM MODEL
insect pest
first approximation of the model

meaning of the terms

Answer 1

B(t) is the Budworm density
we only consider this

dB(t)/dt =
r_B* B(t)*
[1 - (B(t)/Κ_B] - β[B²(t)/{α+B²(t)}]
positive constants α,r_B ,Κ_B and β
Meaning:
* Similar to logistic growth, we have a growth rate r_B.
* At the beginning B(t)≈0 which implies dB/dt ≈ r_B B(t) and we would have exponential growth at the beginning
*In the long term we would have it converging to carrying capacity Κ_B
*However we have a loss term which is stress/negative pressure acting on population growth, may affect whether we reach a carrying capacity of Κ

Question 2

SPRUCE BUDWORM MODEL dimensions

Answer 2

AS B(t) is density IT IS NOT DIMENSIONLESS

[α]=[B]
[r_B] = 1/T
[Κ_B]=[B]
[β]= [B]/T

Question 3

SPRUCE BUDWORM
SCALING change of vars
scale the variables B(t) and t so that the initial model becomes
db(𝜏)/d𝜏 =
b(𝜏) x [R(1-[b(𝜏)/Q]) - [b(𝜏)/(1+b²(𝜏))] ]

where R=(r_B)(α)/β and Q= K_B/α

Answer 3

Choose B(t)=Ab(t)
t=C𝜏

in this case A= 1/α and C=α/β

Question 4

SPRUCE BUDWORM
dynamics of scaled system
Find the steady states
db(𝜏)/d𝜏 =
b(𝜏) x [R(1-[b(𝜏)/Q]) - [b(𝜏)/(1+b²(𝜏))] ]

Answer 4

set db(𝜏)/d𝜏 =0
we have
b*=0
but the hint says define two functions f_1(b) and f_2(b) and plot them.
f_1(b): linear (0,R) to (Q,0)
f_2(b): not linear (0,0) max at (1, 1/2) decreases and tends to 0 after long time
#SS dep on if these intersect:
5 different cases!
If
differentiate to find sign or note for values before and after the intersections
SS STABLE:
If pop is increasing to steady state
before steady state f_1-f_2 >0
and after SS f_1-f_2 <0
SS UNSTABLE
Before SS
f_1-f_2 <0 and after SS f_1-f_2>0

Note: one example is semistable «

Question 5

SPRUCE BUD
WORM HABITAT MODEL

Consider the slow variable S, surface habitat for the larvae and food energy

assumption?

Answer 5

S- surface habitat, branch surface area
E-food energy reserves available to the budworm
We assume that B is near a steady and therefore constant.

dS(t)/dt =
r_S.S(t)[1 - (S(t))/(κ_E(E(t)/κ_E))]
dE(t)/dt =
r_E.E(t)[1 - (E(t))/(κ_E)] - p (B/S(t))

positive real constants r_S,r_E,κ_S,κ_E and p

Question 6

SPRUCE BUDworm
WORM HABITAT MODEL
1)Explain in your own words the meaning of all the terms

dS(t)/dt =
r_S.S(t)[1 - (S(t))/(κ_E(E(t)/κ_E))]
dE(t)/dt =
r_E.E(t)[1 - (E(t))/(κ_E)] - p (B/S(t))

Answer 6

*Surface habitat models follows logistic growth for dS/dt
*dS/dt seems to have a carrying capacity which depends on variable E (comparing to logistic κ_E(E(t)/κ_E)

*food energy reserves ODE seems to follow logistic growth carrying capacity might be κ_E but there some negative values acting against it depending on the term pB/S ie budworms eating/causing stress on the rate E

Question 7

SPRUCE BUDworm
WORM HABITAT MODEL
2) find the dimensions

Answer 7

[r_S] =1/T
[r_E]=1/T
[κ_S]=[S]
[κ_E] =[E]
[p]=[E][S]/([B]T)

Question 8

SPRUCE BUDWORM
WORM HABITAT MODEL
3) find the SS

Answer 8

Look for the nullclines

s-nullclines:
ds/dt=0
s=0 and
s=(κ_s/κ_E)E
(linear)

E-nullcine: dE/dt = 0
S = pB/(r_E E (1-(E/κ_E)))
(this will have a minimum at E=κ_E/2)

We sketch the nullclines
Could find the regions pos and neg for derivatives:
Linear S nullcline s=(κ_s/κ_E)E
dS/dt negative above and positive below this line
E nullcline has dE/dt positive inside the U shape and negative below the U shape

Note that S(E) tends to infinity as E tends to 0 and as E tends to K_E
phase portrait 2 scenarios:
Line and E nullcline don’t intersect
no SS-E becomes negative?
occurs if 4pB/r_Ek_E > K_s/2

Line and U shape intersect
if 4pB/r_Ek_E<K_S/2

2 SS
Consider trajectories, arrows
SS1 unstable
SS2 stable