Model Analysis
Introduction
 To investigate the behaviours of a given 2-D ODE systems
 Use Lotka-Volterra Equation to illustrate the analysis method.
Lotka-Volterra Equation
 Earliest known model proposed for a predator-prey system
 Natural oscillation of population of predator and prey
 Modelled by a pair of first order, non-linear, differential equations, describing the dynamics of
biological systems in which the predator and prey interact
Objective:
 To study and understand the dynamics of Lotka-Volterra system
 To define an approach to study a given set of parameters
Assumption made by Lotka-Volterra system
 The prey has unlimited exponential growth
 The prey does not die itself, its death is only caused by predator
 The predator’s growth is upon its killing of prey, and it will have its natural decay
ODE (Ordinary Differential Equation):
dx
 ax  bxy
dt
dy
 cxy  dy
dt
x
The population of prey at time t
y
The population of predator at time t
a
Rate constant on how fast prey reproduce themselves
b
Rate constant on how fast prey killed by predator
c
Rate constant on how fast predator reproduce when fed from prey
d
Rate constant on how fast predator die
Stationary Points
 Stationary Points are points when the system reach a steady state (equilibrium), and the overall
system are ceased to change anymore over the time.
 A stable stationary point will attract all the points around it towards itself, while the unstable
stationary point will repel the neighborhood points away.
 Stationary Points can be found by setting ODEs to 0, i.e. dx/dt=0 & dy/dt=0, since it means x and y
are not changing anymore.


In this Lotka-Volterra model, there are two stationary points. (0, 0) & (d/c, a/b)
Jacobian matrix is used to determine whether these points are stable.
Jacobian Analysis
 Jacobian matrix is the best linear approximation to a differentiable function near a given point.
 Jacobian matrix is formed by the partial derivatives of the ODE functions. In this way, we could
approximately decouple the interfering parameters; such linearization allows us to study each
parameter independently.
a  by  bx 
cx  d 
 cy

In our Lotka-Volterra model, the Jacobian matrix is 

In dynamic systems, the behaviour of the system near a stationary point can be determined by the
eigenvalues of the Jacobian matrix at the stationary points.
Stability analysis (eigenvalue)

 a11 a12 
 the eigenvalues can be found by
a21 a22 
For a typical matrix A= 
1 
2 

a11  a 22   a11  a 22 2  4a11a 22  a 21a12 
2
a11  a 22   a11  a 22 2  4a11a 22  a 21a12 
2
In x-y two dimensional systems, the eigenvalues are related to the local change of the point, i.e. it
t
can be approximated that x  e 1 x and y  e


2t
y
The sign of the real parts of eigenvalues will determine the behaviours of surrounding points.
 A negative value means they will be attracted towards the stationary point, because the change
of x or y has an exponential decay.
 A positive value means they will be repelled from the stationary point, since the change of x or
y has an exponential increase.
 A zero value, or eigenvalues appear as conjugate pairs, then they will oscillate around the
stationary point, since there is not increase of decrease of x or y.
 The magnitude of the eigenvalues will determine the strength of the change.
Hence, the stationary points can be characterized as follow:
 If both the real parts of eigenvalues are negative, it is stable.
 If one of the real parts of eigenvalues is positive, it is unstable.
 If both the real parts of eigenvalues are zero, it is a center.

a 0 
 , the
0  d 
In Lotka-Volterra system, for the stationary point (0, 0) the Jacobian matrix is 
eigenvalue are “a” & “-d”, hence it is a saddle point.


0
For the stationary point (d/c, a/b), the Jacobian matrix is 
ac

b

bd 
c  , the eigenvalues are i ca

0 

&  i ca , hence it is a center.
Another approach for stability
 For more complicated systems, eigenvalues might be difficult to compute, but if we are only
interested in the sign of the eigenvalues, we could investigate the sign of the trace and the
determinant instead. From the above eigenvalues function:


a11  a 22   a11  a 22 2  4a11a 22  a 21a12 
2
Trace is defined as a11  a22  , and determinant is defined as
a11a22  a21a12 


Unless the trace is equal to zero, the stationary point is stable only when both eigenvalues are
negative and unstable otherwise.
 Hence the trace must be negative in order to achieve stability (negative eigenvalues), since the
square root terms cannot be negative.
 When the trace is negative, the magnitude of square root terms must be smaller than the trace
for stability; hence the determinant must be positive.
In conclusion, to determine the nature of a stationary point:
 If the trace is zero, it is a center
 If the trace is negative AND the determinant is positive, it is stable
 Otherwise it is unstable
Vector Field Representation
 Vector Field Representation is a powerful numerical method to analyze the likely change of a single
point in the given plane.
dy/dt Vector field
 In our 2-D analysis of points on X-Y plane, the vector is
formed by two vectors dx/dt and dy/dt in the x and y direction
respectively.
dx/dt
 The direction and magnitude of resultant vector field is totally
dependant on the direction and magnitude of the two vectors dx/dt and dy/dt.
 With a set of the vector fields on the plane, we could easily identify the behaviours of the whole
system, e.g. contours, stationary points, nature of the stationary points etc.
 Different parameters of the ODEs will result in different patterns of vector fields.
 We are more interested in the behaviours of points around the stationary points.
 By using the sign of the dx/dt and dy/dt, we could identify the general trend of vector fields.


The general trend is easier to identify by group points into regions bounded by the dx/dt=0 line and
the dy/dt=0 line (Nulclines).
In the Lotka-Volterra system, dx/dt=0 when x=0 or a-by=0, dy/dt=0 when y=0 or –c+dx=0, through
analysis of the region bounded by these nulclines (by calculating a few sampled points), we could
define the general direction of the vector field.
y=0
a-by=0
dx-c=0
x=0



From the vector field, we could identify the intersection of the two dashed lines (second stationary
point) is a center.
A Matlab programme has been created to generate Vector Field plots with a given set of ODEs and
parameters.
With parameters a = 1 b = 0.1 c = 0.02 d = 0.5, the following graph is plotted
22
20
18
16
14
12
10
8
6
4
-5

0
5
10
15
20
25
30
35
The general trend of “center” characteristic stationary point is indeed the same as we predicted

The contour of the Predator against Prey can also be plotted by a Matlab programme. The following
graph, which uses the same values of parameter given above with the initial population of predator
of 2, is generated by increasing the initial prey population from 0 to 2 by 0.2 each time.
population of predator against prey
60
50
y = predator
40
30
20
10
0
0
50
100
150
200
250
x = prey

From the graph above, we could see that the contour is clustered together near the axis and the
origin; this is due to the saddle nature of the stationary point (origin).
Conclusion from Lotka-Volterra system
 As we could see from the graph above, each contour is an enclosed encirclement, i.e. it is a periodic
oscillation on its own.
 Any perturbation can drive the oscillation into a different cycle.
 It is especially chaotic if the perturbation is near the origin.
 Hence we should set our second stationary point and initial condition far away from the axis and
origin to avoid the drastic change in the output waveform.
Post stability analysis
 With a given set of possible parameters from experiments, we will use the analysis above and
determine which sets of parameters best suit our interest.
 With the proposed set of parameters and initial conditions, we will run a sensitivity test to
characterize each parameter and the expected output.