1. Phase Plane Diagrams ENT 420 Biological System Modeling Lecturer
Engr. Mohd Yusof bin Baharuddin
MBiomedEng (Melbourne)
BBiomedEng (Malaya)

2. Objectives Motivation Graphical methods for exploring systems of ODEs
ENT 420
Biological System Modeling
Objectives
Motivation
Graphical methods for exploring systems of ODEs
Phase plane diagrams
Example/tutorial: Chemostats
Numerical software: xppaut
Limit cycles, oscillators
Introduce the FH-N class of models
The Hopf Bifurcation

3. The story so far…. Model type Difference equations Systems of ODEs
ENT 420
Biological System Modeling
The story so far….
Model type
Difference equations
Systems of ODEs
General form
Steady state condition
Stability condition
Graphical exploration
Phase plane diagrams
Cobweb graph

4. Graphical interpretation of ODEs
ENT 420
Biological System Modeling
Graphical interpretation of ODEs
Consider
When y=0 and y=1, t y 1

5. Graphical interpretation of ODEs
ENT 420
Biological System Modeling
Graphical interpretation of ODEs
When 0<y<1,
When 1<y,

6. Graphical interpretation of ODEs
ENT 420
Biological System Modeling
Graphical interpretation of ODEs t y 1 t y 1

7. What about systems of ODEs?
ENT 420
Biological System Modeling
What about systems of ODEs?
Too hard! y x
We could produce a 3D plot i.e. with axis (x,y,t) t y t x

8. A phase plane graph Graphical interpretation of ODEs
ENT 420
Biological System Modeling
Graphical interpretation of ODEs
Instead let’s look at how the slope and direction of the vector (dx/dt,dy/dt) varies within a plane (x-y 2D plot) e.g. y y x x
Slope of vector given by
A phase plane graph

9. ENT 420
Biological System Modeling
Phase plane graph
Example: Recall the Lotka-Volterra predator-prey model  1
prey
predator
Oscillatory neutral stable steady state

10. ENT 420
Biological System Modeling
Phase plane graph
Instead of graphing the behaviour of both predator and prey populations as a function of time, we instead can look at the change in prey populations versus change in predator populations on a phase plane plot.
That is…..

11. Phase plane graph Rearrange to and integrate rearrange ENT 420
Biological System Modeling
Phase plane graph
Rearrange to
and integrate
rearrange

12. Phase plane graph  v c1 c2 c3 c4 1
ENT 420
Biological System Modeling
Phase plane graph  1
prey
predator u v 1 c1 c2 c3 c4
Can’t always integrate to get exact solution like in this example.
Neutrally stable

13. A more systematic approach to phase plane diagrams
ENT 420
Biological System Modeling
A more systematic approach to phase plane diagrams
Draw the nullclines:
Let’s look at a previous example y
X - nullclines
y-nullclines
y=1
steady states
y - nullclines x
x=1
These define the nullclines
x-nullclines
Steady states (singularities) occur at the intersection between x nullclines and y nullclines

14. A more systematic approach
ENT 420
Biological System Modeling
A more systematic approach
Assign direction along the nullclines according to vector:
Note one component of the vector will be zero on a nullcline.
Arrows on y nullclines parallel to x axis
Arrows on x nullclines parallel to y axis

15. A more systematic approach
ENT 420
Biological System Modeling
A more systematic approach
We can start to see the flow patterns already
Nullclines separate the phase plane plot into regions in which the flow is in the same general direction
Dramatic local changes in the flow pattern can really only occur in the vicinity of steady states
Begin to see global behaviour i.e. behaviour not local to steady states

16. Flow patterns around (linear) steady states in phase plane diagrams
ENT 420
Biological System Modeling
Flow patterns around (linear) steady states in phase plane diagrams
Image from Fall et al. “Computational Cell Biology” (2002).

17. ENT 420
Biological System Modeling
Murray (2002)

18. Gorillas again H H=k/m0 H=rN/j N=0 These define the nullclines
ENT 420
Biological System Modeling
Gorillas again
Identify and draw nullclines H
H=k/m0
H=rN/j
N=0
These define the nullclines
Steady states occur at the intersection between x nullclines and y nullclines N
steady state

19. Gorillas again H H=k/m0 H=rN/j N=0 N
ENT 420
Biological System Modeling
Gorillas again
Assign flow direction arrows along nullclines
When N=0 H
H=k/m0
When H=k/m0
H=rN/j
N=0
When H=rN/j N
steady state

20. Gorillas again H H=k/m0 H=rN/j N=0 N
ENT 420
Biological System Modeling
Gorillas again
What kind of steady state do we have?
Can’t tell just from phase plane plot! H
H=k/m0
H=rN/j
N=0 N
steady state

21. Biological System Modeling
ENT 420
Biological System Modeling
k= at t=0 N=1000 and H=5
m0= r= j=0.1

22. Tutorial Example: Chemostat
ENT 420
Biological System Modeling
Tutorial Example: Chemostat
Bacterial growth in a Chemostat
A chemostat is a device for harvesting bacteria.
Stock nutrient of concentration c0 enters the bacterial culture chamber with a flowrate F. For mass conservation there is an equal flow rate F out of the culture chamber. V is the volume of the culture chamber and N is the number density (number per volume) of bacteria in the culture chamber F c0 V N c F

23. Bacterial growth in a Chemostat
ENT 420
Biological System Modeling
Bacterial growth in a Chemostat
The equations for the rate of change of the nutrient concentration and bacteria number density are as follows: F c0 c N V F
k(c) is the concentration dependent reproduction rate of the bacteria
α is units of nutrients consumed in producing one unit of population growth
Note F has units of volume/time,
FN/V has units number density of bacteria per time.

24. Example: Chemostat Bacterial growth in a Chemostat
ENT 420
Biological System Modeling
Example: Chemostat
Bacterial growth in a Chemostat
The rate of growth of bacteria increases with nutrient availability only up to some limiting value. (The poor little individual bacterium can only consume nutrient and reproduce at some limited rate.)
k(c)
kmax
0.5kmax
Michaelis-Menton kinetics (more next week or the week after) c
c1/2

25. Bacterial growth in a Chemostat Non-linear ODE:
ENT 420
Biological System Modeling
Bacterial growth in a Chemostat
Non-linear ODE:
We can non-dimensionalise these equations to make them look neater and to reduce the number of parameters.
Let t=t*V/F, c=c*c1/2 and N=N*c1/2F/(αVkmax), where N*, C* and t* denote dimensionless bacteria density, nutrient concentration and time. We can then write the ODEs as:
where α1=Vkmax/F and α2=c0/c1/2

26. ENT 420
Biological System Modeling
Example: Chemostat
For simplicity of mathematics let’s drop the superscript * and let’s assume that α1=2 and α2=3

27. Tutorial: Phase plane diagram of chemostat model
ENT 420
Biological System Modeling
Tutorial: Phase plane diagram of chemostat model c
Find nullclines:
N nullclines ->
c nullclines ->
0,0 N

28. Example: Chemostat Bacterial growth in a Chemostat Non-linear ODE:
ENT 420
Biological System Modeling
Example: Chemostat
Bacterial growth in a Chemostat
Non-linear ODE:
Steady states:

29. Example: Chemostat Bacterial growth in a Chemostat
ENT 420
Biological System Modeling
Example: Chemostat
Bacterial growth in a Chemostat
Stability of steady states:
Linearise our equation to obtain the A matrix:
Recall
where

30. Example: Chemostat Bacterial growth in a Chemostat
ENT 420
Biological System Modeling
Example: Chemostat
Bacterial growth in a Chemostat
Stability of steady states:
Linearise our equation to obtain the A matrix:
Find eigenvalues at each steady state by

31. Example: Chemostat Stability of steady states: When ENT 420
Biological System Modeling
Example: Chemostat
Stability of steady states:
When

32. ENT 420
Biological System Modeling
Example: Chemostat
When

33. ENT 420
Biological System Modeling
Example: Chemostat

34. ENT 420
Biological System Modeling
Software
XPPAUT (freeware) (
a tool for solving
differential equations, (up to 590 differential equations)
difference equations,
delay equations,
functional equations,
boundary value problems, and
stochastic equations.
Also creates phase plane diagrams, cobweb graphs does stability and bifurcation analysis etc (unlike Matlab)

35. Xppaut Program Write program in a text editor
ENT 420
Biological System Modeling
Xppaut Program
Write program in a text editor
The following is program for a linear system of 2 ODEs.
Minimum program
# equations
dx/dt=a*x+b*y
dy/dt=c*x+d*y
# parameters
par a=0, b=1, c=-1, d=0
# intial conditions
init x=1, y=0
# that's all folks
done
dx/dt=a*x+b*y
dy/dt=c*x+d*y
par a, b, c, d
done

36. Xppaut Program Write program in a text editor
ENT 420
Biological System Modeling
Xppaut Program
Write program in a text editor
The following is program for the Lotka-Volterra system  1
prey
predator
du/dt=u*(1-v)
dv/dt=a*v*(u-1)
par a=1
done u v 1

37. Xppaut Program Write program in a text editor
ENT 420
Biological System Modeling
Xppaut Program
Write program in a text editor
The following is program for the system considered at the start of this lecture
du/dt=u*v-v
dv/dt=v*u-u
done

38. Xppaut Program Write program in a text editor
ENT 420
Biological System Modeling
Xppaut Program
Write program in a text editor
The following is program for the chemostat N c
dN/dt=alpha1*(c/(1+c))*N-N
dc/dt=-(c/(1+c))*N-c+alpha2
par alpha1=5, alpha2=0.5
@ xp=N, yp=C, xlo=-.25, xhi=3, ylo=-.1, yhi=1, total=100
@maxstor=10000
done

39. Xppaut Program Write program in a text editor
ENT 420
Biological System Modeling
Xppaut Program
Write program in a text editor
The following is program for the cancer model in lecture 5
dc/dt=delta*c*(w*c/(1+I)-1)
dI/dt=sig+b*(gamma*c-I)
par delta=1, W=1, sig=0, b=1, gamma=0.2
@ xp=c, yp=I, xlo=0, xhi=3, ylo=0, yhi=3, total=100
done

40. Closed paths like C correspond to periodic solutions
ENT 420
Biological System Modeling
At critical points S, R, Q the vector V is zero, and corresponds to stationary points
Closed paths like C correspond to periodic solutions
Simmons, G. (1991) “Differential equations with applications and historical notes”

41. Poincaré-Bendixson theorem
ENT 420
Biological System Modeling
Poincaré-Bendixson theorem
If the phase plane contains a domain, enclosed by a boundary B on which the vector always points into the domain, and the domain contains a singular point P which is an unstable spiral or node then any phase trajectory cannot tend towards the singularity or leave the domain.
Murray (2002)
The Poincaré-Bendixson theorem says that as the trajectory will tend to a limit cycle

42. Oscillators, excitability and FitzHugh-Nagumo Models
ENT 420
Biological System Modeling
Oscillators, excitability and FitzHugh-Nagumo Models
FitzHugh and Nagumo independently proposed a simplified model of the Hodgkin-Huxley equations that are used to describe the variation in sodium and potassium ions across a cell membrane of a nerve cell (neuron).
Now, FH-N models are those with one linear nullcline for the “slow variable” and a cubic nullcline with an inverted “N” shape for the “fast variable”

43. Oscillators, excitability and FitzHugh-Nagumo Models
ENT 420
Biological System Modeling
Oscillators, excitability and FitzHugh-Nagumo Models
Example:
x represents the excitation variable “fast variable”
y represents the recovery variable “slow variable”
Parameters a, b and c in the model are assumed to be positive with the following restrictions
z is a stimulus intensity

44. Phase plane diagram of FH-N model
ENT 420
Biological System Modeling
Phase plane diagram of FH-N model y
z=0 x
Two Nullclines: y
z=-0.4 x
Slight shift in curve

45. Oscillators, excitability and FitzHugh-Nagumo Models
ENT 420
Biological System Modeling
Oscillators, excitability and FitzHugh-Nagumo Models
Stability?
For the trace to be negative and the determinant positive i.e. a stable steady state or

46. Oscillators, excitability and FitzHugh-Nagumo Models
ENT 420
Biological System Modeling
Oscillators, excitability and FitzHugh-Nagumo Models
That is the steady state is not stable if it falls within the region
Imaginary number
Since the requirement for a real value steady state not being stable is just
Note: The stability condition of the steady state does not change with z, but the steady state itself does.

47. Xppaut Program Write program in a text editor
ENT 420
Biological System Modeling
Xppaut Program
Write program in a text editor
The following is program for the FH-N model
du/dt=c*(v+u-(1/3)*u^3+zed)
dv/dt=-(1/c)*(u-a+b*v)
par a=0.7, b=0.8, c=3, zed
@ xp=u, yp=v, xlo=-3, xhi=3, ylo=-3, yhi=3, total=100
@maxstor=10000
done

48. ENT 420
Biological System Modeling

49. ENT 420
Biological System Modeling
The Hopf Bifurcation
Hopf Bifurcation Theorem predicts the appearance of a limits cycle about any steady state that undergoes a transition from a stable to an unstable focus (spiral) as some parameter is varied.

50. The Hopf Bifurcation Consider A steady state occurs at
ENT 420
Biological System Modeling
The Hopf Bifurcation
Consider
A steady state occurs at
At this steady state

51. The Hopf Bifurcation du/dt=v dv/dt=-v^3+r*v-u par r
ENT 420
Biological System Modeling
The Hopf Bifurcation
du/dt=v
dv/dt=-v^3+r*v-u
par r
@ xp=u, yp=v, xlo=-3, xhi=3, ylo=-3, yhi=3, total=100
@maxstor=10000
done

52. ENT 420
Biological System Modeling
Summary
Phase plane diagrams, nullclines, shapes of flow patterns near various linear steady states
Examples: Chemostat, gorillas
Xppaut
Limit cycles
Excitability
FH-N models

53. Somewhere people still eating once a day…. ~Yusof~
ENT 420
Biological System Modeling
Somewhere people still eating once a day… ~Yusof~