Effect of Time Delay on a Model of Bone Homeostasis

time t

200

time delay τ

0.

This Demonstration analyzes the effect of time delay on a model of bone homeostasis. The model consists of two delay-ordinary differential equations coupled to an ordinary differential equation [1]:

dx

dt

=

c

1

m

1

+y

-

d

1

x

,

dy

dt

=

c

2

x

m

2

+

2

x

y(t-τ)z(t-τ)-

d

2

y

,

dz

dt

=

c

3+

c

4

x

m

3

+x

z(t-τ)-

d

3

y

,

with initial condition

x(0)=0.1

and initial history functions

y(t≤0)=2

and

z(t≤0)=1

.

Here

x

is the level of vitamin D above the basal level in blood,

y

is the number of active bone-resorbing osteoclasts,

z

is the number of active bone-forming osteoblasts,

t

is time, and

τ

is the time delay necessary for cells to become active in the bone resorption and formation process. All parameters are positive constants defined in the reference. The time trajectory shows periodic behavior when the delay time

τ

is zero and increasing

τ

shows damped oscillations leading to a steady state; both of these conditions are observed clinically [1].