IGEM:IMPERIAL/2006/project/Oscillator/Theoretical Analyses/Results/2D Model1

Introduction

Lotka Volterra is the earliest (and most famous) model describing the interaction between a population of preys and a population of predators. It is built upon a pair of first order, non-linear, differential equations, describing the dynamics of ecological systems in which the predator and prey interact

insert equations

Basic Assumptions of the Model

• The growth of the prey population X is due to natural reproduction of the preys. In the case of Lotka-volterra the growth is proportional to the population
• The population of preys is controlled by the hunting carried out by the predators (the natural death rate of the prey is assumed negligible compared to the killing rate due to the predators). In the present case the population decay is proportional to the population of both preys and predators.
• The predator’s growth is the direct result of hunting the preys. In the present case it is proportional to the population of both preys and predators.
• Finally the population of the predators is controlled by its natural death (proportional to the population)

Relevance of Lotka-Volterra to the 2D Model

The Lotka-Volterra model is probably the simplest prey-predator model and is widely known to yield oscillating populations. It is therefore an ideal candidate as a starting point for our series of analyses.

However, it is straightforward to see that the assumptions it is based on are very different from the assumptions of our model. We therefore cannot expect to learn too much about our own system.

Furthermore, as we shall see, this model has a critical flaw regarding stability of its oscillations.

Basic Results on the Steady Points and Vector Field

Steady Points

• The Lotka-Volterra model has two steady points:

- The origin (0, 0)

- (d/c, a/b) (interestingly the steady population of the preys depends only on parameters associated to predators and vice-versa).

• After Jacobian analysis, it can be shown that the stationary point (0, 0) is a saddle point, while (d/c, a/b) is a centre (that is the vector field will rotate around the point without being attracted or repelled by it).

Vector Field

The previous results help sketch the general allure of the Vector Field :

Closed Trajectories and Oscillations

All trajectories are Closed

It is easy to prove a very strong result for Lotka-Volterra: once the parameters (a,b,c and d) are set, all trajectories are not only bounded but closed.This remarkable result instantly prove that if we start from initial conditions separate from the steady points of the system, then the population of both preys and predators will oscillate.

However, there is an unwelcome flip side to the result (called neutral stability in Dynamical System Theory).

Let's imagine that there is an 'accident' such as an unexpected large number of birth of preys at time t (such scenario is realistic in a system like ours due to the intrinsic noise of the system). With the Lotka-volterra dynamic we will then jump to a different (closed) orbit corresponding to the 'accidental' values of the populations X and Y at time t.

If there is no other accident we will stay on the new orbit, hence settle on a different oscillatory regime. From an experimental point of view this is unsatisfying since we have no way of predicting the regime when we started the experiment. Furthermore nothing ensures us that the system will not jump again to another orbit.

Lotka-Volterra is therefore too fatally flawed to be a desirable model for our oscillator

Control of the Oscillations

Theoretically, we have total control over the range of amplitude and frequency of the oscillations.

To see this , let's fix the initial conditions to (Xo,Yo). We can ensure the ratios (d/c and a/b) of the model parameters are such that the centre coincide with (Xo,Yo) and thus we have oscillations of zero amplitude. Now we increase these ratios steadily (the centre moves away from the origin and our initial point), which also increases the amplitude of the oscillations.

Once we have found an orbit with an amplitude that suits us, we can still modify the parameters so we get any frequency we want. It is straightforward to see that multiplying the parameters of the model (a,b,c and d) by a positive constant does not change the trajectories of the system, only the speed at which they are travelled. Consequently to obtain an oscillation of frequency f1 from an oscillation of frequency f0 it simply takes to multiply a,b,c and d by f1/f0

Jimmy: Insert simulations shown in presentation please

Results of Dynamic Analysis

• 2 stationary points, (0, 0) – saddle, (d/c, a/b) center

• From the graph above, we could see that the contours are clustered together near the axis and the

origin; this is due to the saddle nature of the stationary point (origin).

• the contours are also encirclements about a point, which is the second stationary point (center)
• Each encirclement is in fact a periodic oscillation

• Stress analysis with a given set of parameters
• Or refer to here Section 2 “Back to simplicity” for detail analysis

Conclusion

Summary of Results

• With a given set of parameters and initial conditions, the result on X-Y plane is a contour around the second stationary point, which is a center.
• 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 if possible 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.

Physical Interpretations of Results

• The population of prey and predator oscillated with a certain period and amplitude.
• It is indeed what we expect for the general behaviour of prey-predator interaction

Appendices

Related documents

• Stability analysis
  • Jacobian anaylsis of stationary point for stability of oscillation. click here --04/08/06
  • Introduction to analysis of the 2-D ODEs. click here --21/08/06
• Limit Cycle
  • Analysis report of the past progress --31/07/06
• Stress Analysis
  • click here for the first report --02/08/06
    
  • click here for the second report --04/08/06
    
  • click here for the third report --07/08/06
    
  • click here for the fourth report --07/08/06
    
  • click here for the fifth report --10/08/06
    
  • click here for the sixth report --10/08/06
    
• Java Applet

<html> <HEAD> <TITLE>Molecular Prey-Predator-System</TITLE>

       <LINK REL="stylesheet" TYPE="text/css" HREF="style.css">

</HEAD> <BODY> <H1>Molecular Prey-Predator using JOde</H1> <P><!-- Insert HTML here --><APPLET code="com/rychlik/jode/JOdeApplet.class" NAME="JOde" width=650 height=750 archive="http://openwetware.org/images/a/a3/JOdeApplet.jar" mayscript="true"> <PARAM NAME="isframed" VALUE="false"> <PARAM NAME="background" VALUE="c0c0c0"> <PARAM NAME="autonomous" VALUE="true"> <PARAM NAME="min1" VALUE="0"> <PARAM NAME="max1" VALUE="100"> <PARAM NAME="min2" VALUE="0"> <PARAM NAME="max2" VALUE="30"> <PARAM NAME="variable0" VALUE="t"> <PARAM NAME="equation0" VALUE=" a = 1;b = 0.1;c = 0.02;d = 0.5"> <PARAM NAME="variable1" VALUE="X"> <PARAM NAME="variable2" VALUE="Y"> <PARAM NAME="equation1" VALUE="a*X-b*X*Y "> <PARAM NAME="equation2" VALUE="c*X*Y-d*Y">

       <PARAM NAME="initconds" VALUE="40,15;10,5">

<PARAM NAME="showinitconditions" VALUE="true"> <PARAM NAME="showpoints" VALUE="false"> <PARAM NAME="showslopes" VALUE="true"> <PARAM NAME="showaxes" VALUE="true"> <PARAM NAME="minparameter" VALUE="0"> <PARAM NAME="maxparameter" VALUE="2000"> <PARAM NAME="parametersegments" VALUE="20000"> <PARAM NAME="algorithm" VALUE="RK4"> <PARAM NAME="label" VALUE="Molecular Prey-Predator"> </APPLET> </P>

a = 1;b = 0.1;c = 0.02;d = 0.5

Instructions on using the JOde Applet</A></EM></P> <P>Using the applet written by: <A HREF="http://alamos.math.arizona.edu/">Marek Rychlik</A> </BODY> </html>