Animal Kinematics & Dynamics: Pursuit of Prey and
Locomoting Listeria
Peter Love and Suzanne Amador Kane, Physics Department, Haverford College © 2011
Introduction
For simplicity, in introductory physics we often solve kinematics problems in which
either velocity or acceleration is constant in time. However, in the real world there are
relatively few instances of constant velocity or acceleration for long periods of time. No
real vehicle, for example, can accelerate indefinitely. In this lab we will investigate two
examples in which velocity and acceleration vary with time. The first example is the
interaction of predators and prey, and the second is the motion of Listeria bacteria. In the
first case drag forces create a scenario where accelerations are limited to extremely short
times. In the second case accelerations vary periodically. We will investigate both these
cases using numerical modeling – you will use a computer program to investigate models
of complicated motion.
Computers can approximately solve very complicated problems in Newtonian mechanics
in which the forces (and hence accelerations) are changing as a function of time. They do
this by using techniques which reduce a problem with complicated forces varying with
time into a sequence of problems in which the forces are constant. To understand what
the computer is doing, you only need to understand how to solve the constant force and
constant acceleration problems which you are assigned in your problem sets. All the
computer does is to solve hundreds or thousands (or millions or billions in the case of
very large simulations) of these problems one after the other.
In the simulations you will run in this lab you will always see a parameter called dt in the
code. Of course, this is not like the dt’s one writes down in calculus class – it’s an actual
small interval of time. The computer treats the force, which is varying with time, as
constant in time for the interval dt, and solves a constant force problem to advance the
system from time t to time t+dt. This is called one timestep. Then the computer does this
again. And again. And again. Computer simulation is often like this, reducing an
insoluble problem into a sequence of solvable problems and letting the computer go.
Remember, computers are not intelligent, so they can never solve problems in the way
that you can. They can only be told to do very simple things, and then do very many of
them very fast. However, this gives us tremendous power to investigate the properties of
models where we cannot solve the equations analytically. However, it is very important
to remember that a computer simulation is theoretical – it can only tell us about a model
of the real world, unlike experiments which probe reality directly. The results of
simulations must always be compared to experimental results to validate them.
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Predator/Prey questions
Predator-prey preamble
We call exact solutions for position r(t), velocity v(t) and acceleration a(t) analytical
solutions. Most of the time you cannot find such analytical solutions, and you need to
measure them or use a computer to obtain approximate solutions numerically. However,
there is one case of animal motion that can be usefully analyzed using a relatively simple
analytical model-- the case when:
(1)
This equation describes the case where there are two forces on an object of mass m. The
first, Fo, is a constant, positive force provided by the muscles of a moving animal. The
second force, -Bv, is a force proportional to the velocity but opposite in direction; B is a
constant that depends upon details of the objects shape and the medium through which it
moves. Drag is one example of such a force. We can also write down the corresponding
velocity as a function of time, v(t), for this equation.
(2)
This equation describes the motion of objects subject to a drag and is an approximation to
the equation of motion for a running animal. Today, we will apply it to the case of pursuit
of a prey by a predator. In that case, the animal typically starts with zero initial velocity,
undergoes a period of rapid acceleration, then decreases its acceleration as time increases
due to a combination of drag and muscle exhaustion, reaching a final top speed which it
can maintain for a certain period of time. We can summarize all this using the plot in
Fig. 1, which show just such behavior measured from videos of African lions pursuing
fleeing Thompson’s gazelles [1].
Figure 1: From Elliot et al. (1977) reproduced from Ref [1]
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Predator/Prey questions
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Predator/Prey questions
Learning about Listeria
Listeria monocytogenes is a short rod-like bacterium which is responsible for the rare but
dangerous food borne disease Listeriosis, which can have a fatality rate of up to 25%.
These organisms move themselves around in cells by inducing a chemical reaction called
polymerization in a protein called Actin. This reaction is induced by chemicals attached
to the surface of the bacterium and so occurs at one end of the bacterium. The reaction
leaves a ``comet trail’’ of polymerized actin behind the Listeria, which can be observed
using a microscope as shown in Figure 2.
Figure 2: Listeria motion imaged using
phase-contrast microcopy. The scale
bars correspond to 5 μm. The shapes of
trajectories are:
(a) small-amplitude sine,
(b) ”S”-curve or large-amplitude sine,
(c) serpentine,
(d) translating figure”8”,
(e) figure-”8”,
(f) spiral,
(g) circle and
(h) straight-line.
Reproduced from [2]
The data shown in Figure 2 presents us with a problem similar to that faced by Isaac
Newton when trying to explain the trajectory data for planets obtained by Tycho Brahe’s
careful observations. Proceeding from his three laws of motion and his law of universal
gravitation, Newton was able to explain the shapes of the trajectories (orbits) followed by
bodies in the solar system. In the second part of the lab you will numerically investigate a
simple model for the propulsion of Listeria to see if it can really explain all the shapes of
the trajectories shown in Figure 2.
Figure 3: The listeria bacterium is the green ellipse,
and we will only consider motion in two dimensions.
The force on the bacterium is applied at a particular
spot at one end marked by the cross in the insert. This
is a distance d from the long axis of the bacterium.
The long axis makes an angle θ with the x axis. The
bacterium rotates around its long axis with an
angular frequency ω so that ψ= ωt. The combination
of the fact that the force is offset from the long axis
and the bacterium is rotating can produce
complicated trajectories.
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Predator/Prey questions
Visual Python Prelab Exercise: “Thornton Vaseltarp’s first visual python graph”
Run this program on any of the computers in H204, the KINSC computer lab more or
less directly above the introductory lab where we meet. You must follow theses
instructions carefully
1. Download the python program. Go to Blackboard and download the file
Prelab.py by right clicking on it and using “Save file as”. In H204 you cannot
save files to the Desktop – save it in Student/Documents folder so you can find it
later.
2. Open VIDLE. Go to Start -> All Programs-> Academic Programs ->VIDLE for
Vpython and open VIDLE. You get a window called “Untitled” with a menu bar
across the top with “File”, “Edit”, “Format”, “Run”, “Options”, “Windows”,
“Help”
3. Open the python program in VIDLE. In this new VIDLE window go to “File > Open” in the VIDLE menu and navigate to the file Prelab.py It is easiest to
click on Desktop, then got to the folder Student, then to Documents (or wherever
you saved this program from Blackboard in step 1). When the file is open you
should see a bunch of red text in the vpython window, the first few lines of which
are:
#Visual Python Prelab Exercise September 24 2009
#
# Thornton Vaseltarps first visual python graph
4. Run the python program. In the VIDLE menu go to Run -> Run Module to run
the program You should see a window with a plot in labelled ``Thornton
Vaseltarp's first vpython graph''
5. Modify the graph. Now read through the program text which appears in the
VIDLE window and change the variable graphtitle so that Thornton Vaseltarp is
replaced by your name - make sure you leave the quote marks " " alone. Change
the value of the variable named parameter to any number between 4 and 10
Now run the program again and you will see a window labeled with your name,
and a slightly different graph.
6. Print the graph Click on this window and press Alt-PrintScreen together to
perform a screengrab of the window (that it, the computer makes a copy of your
window that you can later paste into another application). Open a blank Microsoft
word document and hit Ctrl-V to paste your screengrab into the document. Adjust
the size of the image in Word so it fits on the page. Print out your graph and bring
it to lab with you.
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Predator/Prey questions
Laboratory Exercise 1: Pursuit of Prey
Download the file Exercise1.py from Blackboard.
Open VIDLE (“VIDLE for Python”--it’s an icon on your computer’s desktop) now and
open the file “Exercise1.py “) (go to File menu, then Open… and select “Exercise1.py”)
To run it, go to the Run menu and select Run Module. You ought to get a display
showing a little animation. The red sphere stands in for the fleeing gazelle, while the
yellow sphere stands in for the pursuing lion. Red and yellow lines show their pathways.
After the animation has ended, you get a graph of velocity vs. time and position vs time.
Elliot et al. (1977) describe two types of typical hunting behavior by the African lion [1].
In case 1, “crouching stalks”, the lion approaches the prey and tries to get as close as
possible, sometimes using cover. Since this entails creeping slowly and pausing
frequently, it attacks starting from zero velocity. In Case 2: “running stalks”, the lion is
trotting at constant speed before attacking. We will consider only crouching stalks in this
lab.
Use your Vpython simulations to answer the following questions. In each case, use the
program to produce a graph exactly as you did for the prelab. You can print out and write
on your graph to illustrate the answers you give on the report form. Remember to write in
complete sentences and be clear about which graph you are referring to.
1. Produce plots of velocity vs time (like figure 1) and position vs time for the three
types of prey, Gazelle, Zebra and Wildebeast. Run the program Exercise1.py
three times setting the variable preyanimal = 1, then 2 then 3 (Gazelle, Zebra,
Wildebeast). Set prey_reaction_time=0.0 and lion_ambush_distance=7 for these
plots. Print the plots and indicate on them when (if ever) the lion catches the prey,
and when (if it does not) the lion should give up the chase.
2. The program has two parameters, gazelle_reaction_time and
lion_ambush_distance. The gazelle_reaction_time parameter allows the gazelle
to pause for a time after the lion begins to run before itself starting to run away.
The lion_ambush_distance parameter sets how close the lion can get before
attacking. Vary these two parameters over a reasonable range and make a table
showing when the lion catches the gazelle and when it does not. What is the
closest distance beyond which the gazelle cannot escape capture for a reaction
time of 0.5 seconds?
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Predator/Prey questions
Laboratory exercise 2: Listeria locomotion
Download the file Exercise2.py from Blackboard.
Open VIDLE (“VIDLE for Python”--it’s an icon on your computer’s desktop) now and
open the file “Exercise2.py “) (go to File menu, then Open… and select “Exercise2.py”)
To run it, go to the Run menu and select Run Module. You ought to get a display
showing an animation of 2D motion. The cone represents the listeria bacterium and the
red line represents the Actin comet trail which it leaves behind and which is shown in
Figure 1. While the motion that we model here is confined to two dimensions, you can
rotate and zoom your view of the trajectory using the mouse. Click the right button of the
mouse and move the mouse to rotate your view. Press both the left and the right button
together and move the mouse to zoom in and out.
In this model there is just one parameter that determines the shape of the trajectory. This
parameter is called wOmega in the program and is initially set to 1.0. Vary this parameter
and see if you can produce all the types of trajectories shown in Figure 1. For each type
of trajectory you produce, print out a graph showing the trajectory. Write on your printout
which of the types of trajectory you think this is from the set identified in Figure 2.
Footnotes and further reading
Getting started with Vpython
There is an introduction at: http://vpython.org/ , in particular:
http://vpython.org/contents/doc.html
Listeria motion:
Also see Juliet Theriot’s website for more on listeria locomotion:
http://cmgm.stanford.edu/theriot/movies.htm
References
[1] J. P. Elliot Cowan, I.M., and Holling, C.S. "Prey capture by the African lion"
Canadian Journal of Zoology vol. 55(11)pp 1811-28 (1977; Also see: J. P. Elliot, Lions
of the Serengeti
[2]``A kinematic description of the trajectories of Listeria monocytogenes propelled by
Actin comet trails’’, V. B. Shenoy, D. T. Tambe, A. Prasad, J. A. , Proceedings of the
National Academy of Sciences, vol. 104 (20), pp.8229-8234 (2007).
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