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

Model 1: Lotka-Volterra

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

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 analysis.
 * 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 Neutral Stability =


 * 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

=Typical Lotka-Volterra Simulations = 


 * Simulations using different initial conditions are assigned different colors (the open-end of the trajectories is the starting point). In the phase diagram, red dots symbolise a steady points.


 * We can see that the trajectories of Lotka-Volterra are - as expected- closed and spin around the centre steady-point. The nature of the steady point at the origin (it is a saddle point)is visible by the allure of the trajectories at its vicinity ( they are clustered together near the axis and the origin).
 * Furthermore the closer a trajectory is to an axis the larger the amplitude of the trajectory is. Simulations show that for trajectories very close to an axis (say X-Axis) a tiny variation in the initial conditions (or an accidental variation) results in a huge variation in amplitude (and frequency): the Lotka-Volterra is therefore extremely fragile in these regions.
 * Every closed trajectory corresponds to coupled oscillations of the prey and predator populations.

=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


 * Thus, we could have oscillations with different amplitudes and frequencies as shown below:



=Conclusion=

Lotka-Volterra is a very simple dynamical model for prey-predator interactions. It is very similar in s

Lotka-Volterra is endowed with properties that we hope our system replicates
 * - it generates oscillations
 * - by playing on its parameters we can in theory obtain any amplitude and frequency.

However, Lotka-Volterra is also fatally flawed for a system like the one we wish to build as it is far too sensitive to any perturbation/accident/noise. This problem (linked to the concept of neutral stability)can be simply remedied to if we manage to build a system with a unique limit-cycle.

=Java Applet: Molecular Prey-Predator-System=

 

Molecular Prey-Predator-System   

     <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>