Numerical Methods Contents - SAM

Model:
autonomous Lotka-Volterra ODE:
˙u = (α − βv)u
˙v = (δu − γ)v
population sizes:
u(t) → no. of prey at time t,
v(t) → no. of predators at time t
↔ ẏ = f(y) with y =
( u
v)
, f(y) =
( )
(α − βv)u
. (11.1.3)
(δu − γ)v
State of heart described by quantities:
with parameters:
Phenomenological model:
l = l(t) ˆ= length of muscle fiber
p = p(t) ˆ= electro-chemical potential
˙l = −(l 3 − αl + p) ,
ṗ = βl ,
α ˆ= pre-tension of muscle fiber
β ˆ= (phenomenological) feedback parameter
(11.1.4)
This is the so-called Zeeman model: it is a phenomenological model entirely based on macroscopic
observations without relying on knowledge about the underlying molecular mechanisms.
v
vector field f for Lotka-Volterra ODE
✄
α/β
Vector fields and solutions for different choices of parameters:
Solution curves are trajectories of particles carried
along by velocity field f.
γ/δ
u
Fig. 123
Parameter values for Fig. 123: α = 2,β = 1, δ = 1,γ = 1
Ôº¾ ½½º½
Ôº¾ ½½º½
6
5
u = y 1
6
5
2.5
2
1.5
Phase flow for Zeeman model (α = 3,β=1.000000e−01)
3
2
Heartbeat according to Zeeman model (α = 3,β=1.000000e−01)
l(t)
p(t)
4
4
1
1
0.5
y
3
v = y 2
3
p
0
l/p
0
2
v = y 2
Fig. 124
2
−0.5
−1
−1
1
1
−1.5
−2
−2
0
1 2 3 4 5 6 7 8 9 10
t
Solution ( u(t)
v(t)
)
for y0 := ( u(0) ) (
v(0) = 42 )
0
0 0.5 1 1.5 2 2.5 3 3.5 4
u = y 1
Fig. 125
Solution curves for (11.1.3)
−2.5
0
l
2 2.5
Fig. 126
−2.5 −2 −1.5 −1 −0.5 0.5 1 1.5
−3
0 10 20 30 40 50 60 70 80 90 100
time t
Fig. 127
Parameter values for Figs. 125, 124: α = 1, β = 1, δ = 1,γ = 2
Example 11.1.3 (Heartbeat model). → [10, p. 655]
stationary point
✸
Ôº¾ ½½º½
Ôº¾ ½½º½