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):
x / The population of prey at time ty / 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.
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)
For a typical matrix A= the eigenvalues can be found by
In x-y two dimensional systems, the eigenvalues are related to the local change of the point, i.e. it can be approximated that and
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.
In Lotka-Volterra system, for the stationary point (0, 0) the Jacobian matrix is , the eigenvalue are “a” “-d”, hence it is a saddle point.
For the stationary point (d/c, a/b), the Jacobian matrix is, the eigenvalues are , 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:
Trace is defined as , and determinant is defined as
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.
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.
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 patternsof 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.
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 = 0.1c = 0.02 d = 0.5, the following graph is plotted
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.
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.