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

From OpenWetWare
Jump to navigationJump to search
mNo edit summary
Line 2: Line 2:
*Lotka Volterra is the most famous and fundamental model to describe the interaction between the prey and predator
*Lotka Volterra is the most famous and fundamental model to describe the interaction between the prey and predator
= '''Introduction''' =
= '''Introduction''' =
<b>Lotka-Volterra Equation</b>
*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


<b> Basic Assumptions of the Model </b>
Lotka Volterra is the earliest (and most famous) model describing the interaction between a population of preys and a population of predators.
* The prey has unlimited exponential growth
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
* 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
'''Basic Assumptions of the Model'''
<b>Relevance of the Model</b>
* 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
*Assume 2-D assumption is valid, i.e. the production and death of AHL and LuxR behave the same way
* 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 prey is AHL, and the predator is aiiA
* 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.
**AHL will bind with the promoter in the prey cell, and hence more AHL will be produced, thus AHL is capable of reproducing itself
* Finally the population of the predators is controlled by its natural death (proportional to the population)
**AHL will be degraded by the enzyme aiiA, thus the degradation rate depends on the encounter rate with aiiA
 
**AHL is quite stable, the self-degradation rate is much smaller than the degradation rate caused bt aiiA, hence we assume aiiA does not degrade itself
'''Relevance of Lotka-Volterra to the 2D Model'''
**AHL will bind with LuxR and promote the production of aiiA and LuxR, hence the production rate of aiiA depends on the concentration of AHL
 
**The aiiA and LuxR will have the same degradation rate according to our assumption
: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.
*The Lotka-Volterra model is a good representation of our system
: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.
<b>ODE (Ordinary Differential Equation)</b>
Furthermore, as we shall see, this model has a critical flaw regarding stability of its oscillations.
:::[[Image:2d model 1a.PNG]]
<b>Physical interpretation of the equations</b>
*The production rate of prey is directly proportional to the prey population, i.e. prey will have exponential growth itself. The more number of preys, the more production of preys.
*The death rate of prey is proportional to the product of prey and predator population, i.e. it modeled the death of prey during the encounter of predator
*The production rate of the predator is proportional to the product of prey and predator population, i.e. it modeled the growth of predator when it is fed upon prey.
*The death rate of predator is directly proportional to the predator population, i.e. predator will have exponential decay itself. The more number of predators, the more death of predators occurred.
<b>Comment</b>
*The Lotka-Volterra ODE is probably the simplest prey-predator equations. It allows us to understand the basic feature of prey-predator interaction
*However, this model is too ideal to represent our real system. Further modification might be needed.


= '''Basic Results on the Steady Points and Vector Field''' =
= '''Basic Results on the Steady Points and Vector Field''' =

Revision as of 09:27, 27 October 2006

2D Model 1 : Lotka volterra

  • Lotka Volterra is the most famous and fundamental model to describe the interaction between the prey and predator

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

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

  • After analysis, there are two stationary points. (0, 0) & (d/c, a/b)
  • The existence of both stationary points is independent of the parameters
  • After Jacobian analysis, the stationary point (0, 0) is a saddle point, while (d/c, a/b) is a center

Vector Field

Limit cycles

  • Trajectories will be remain bounded for all the different parameters

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>

Retrieved from "https://openwetware.org/mediawiki/index.php?title=IGEM:IMPERIAL/2006/project/Oscillator/Theoretical_Analyses/Results/2D_Model1&oldid=82282"