Part A – Stochastic and Deterministic Chemical Reactions
Consider the following system of reactions:
c1
X  Y 

2Y
c2
2Y 

Z

The bar over the X denotes that the number of molecules in state X remains constant. We
can write the following system of ODE’s to describe the dynamics of this system of
reactions:
c 
dY
 c1 XY  2 2 Y 2
 2 
dt
dZ c 2  2
  Y
dt  2 
1) Simply look at the first ODE since dY/dt does not depend on Z:

c 
dY
 c1 XY  2 2 Y 2
 2 
dt
Solve for all possible fixed points, Y*. Characterize the stability of the fixed points.

2) Now let c1X  5 and c2  0.005, and set (Y(0), Z(0)) = (10, 0) and (3000, 0). Run the
code in final_q1.zip to show the deterministic and stochastic solutions of this system.
Allow the simulations to run long enough to see Y approach a steady state value. Show
the graphs of Y vs. time for each case.


3) Now let c1X  5 and c2  0.00125, and set (Y(0), Z(0)) = (10, 0) and (3000, 0). For
each case, choose some time t = t* so that the number of Y molecules is close to the
steady state value. Run each set of simulations 100 times and save the values for Ys (t*) ,
where Ys (t*) is the number of Y molecules in the stochastic simulation at time t*. Make a

 these values of Y for both initial conditions.
histogram
showing
 #2. What is the
4) Do the same thing that you did in #3 for the parameter values given in
 mean Y (t*) for each case, and what is the standard deviation? How does the standard
s
deviation for Ys (t*) with parameter values c1X  5 and c2  0.005 compare to Ys (t*) with
parameter values c1X  5 and c2  0.00125?
 5) Suppose that you are designing a chemical reaction with a small number of molecules
 stochastic effects are important.

 reaction has approached
 a steady state,
so that
After the


c X to minimize the
you want to minimize the noise in Y. How would you choose 1
c2
noise?

Part B – The Lotka Reactions
Consider the following system of reactions:
c1
X  Y1 

2Y1
c2
Y1  Y2 

2Y2
c3
Y2 

Z

As before, the bar over the X denotes that the number of molecules in state X remains
constant. We can write the following system of ODE’s to describe the dynamics of this
system of reactions:
dY1
 c1 XY1  c 2Y1Y2
dt
dY2
 c 2Y1Y2  c 3Y2
dt
dZ
 c 3Y2
dt

Since the dynamics of Z do not affect the dynamics of Y1 and Y2, you can ignore the last
equation for the time being.
1) Solve for all steady state values Y1* and Y2* . Characterize the stability of each of the
fixed points (Hint: you can use pplane to check this).
2) Open the folder final_q2.zip and run the simulation for c1X 10, c2  0.01, c3 10 with


initial conditions Y1 1000, Y2 1000 and 0  t  30 . Show the graphs of Y1 and Y2 vs.
time. Explain the reason for the differences between the stochastic and deterministic
simulations.


 same parameter values, but now set Y 1010, Y 1010 . Compare the plots of
3) Use the
1
2
Y1 vs. Y2 for the stochastic and deterministic cases and explain the differences.
4) Now allow X to be depleted. You will need to rewrite the code so that X is no longer
one each time the first interaction takes place.
constant, and be sure to decrease X by
Please use the following choice of parameters and initial conditions:
c1  0.0001, c2  0.01, c3 10 , X 100,000, Y1 1000, Y2 1000.

Run the simulation for 0  t  30 . Describe how this system is different from the case
where X is held constant. What happens to Y1 for t > 20 and why? (Hint: look at the
graphs of X and Y2). 

5) Continue to allow X to be depleted and add the following reaction to the system:
c4
Y1 

Z

Use the following parameter values and initial conditions:
c1  0.0002, c2  0.01, c3  c4 10
X(0)  10 5 , Y1(0)  Y2 (0)  10 3, Z(0)  0


When do Y1 and Y2 become extinct? Is this what you would expect given that
c3  c4 10 ?
Part C – Lotka-Volterra (Predator-Prey)

Now assume that Y1 represents a prey species and Y2 represents a predator species. In
this case, c2 is not necessarily equal to c3.
dY1
 c1Y1  c 2Y1Y2
dt
dY2
 c 3Y1Y2  c 4Y2
dt
Open the folder final_q3.zip to find the code associated with this problem.



1) Solve for the fixed points and characterize their stability. What do the constants
c1, c2, c3, c4 mean in terms of the predator prey system?
2) Run the stochastic and deterministic simulations about 10 times for
c1  0.04, c2  0.0005, c3  0.00005, c4  0.1, Y1(0) 1000,Y2 (0) 100. You should notice
that sometimes the predators and prey oscillate out of phase during this time frame. In
other simulations, something strange happens. Show the graphs for examples of each and
explain what happens and why in the case where the oscillations don’t persist. What does
this mean in the context of the application?
3) Consider the following predator-prey system with three species:
dY1
 c1Y1  c 2Y1Y2  c 3Y1Y3
dt
dY2
 c 4Y1Y2  c 5Y2
dt
dY3
 c 6Y1Y3  c 7Y3
dt

Please describe in words the type of interaction between the three species (which are the
predators, the prey, etc).
4) For what parameter values will Y3 go extinct? Hint: set Y3* = 0, and solve for Y1*
and Y2*. Solve for the eigenvalues of det J Y1* ,Y2* ,0 I . For what parameter values


will Y2 go extinct?
5) Write a system of differential equations that describe a predator prey system consisting
of the following four species:



Two species of prey competing for the same resources with logistic growth in the
absence of other species.
One predator that eats both of the prey
One predator that only eats one species of prey
Describe each of the terms in the system of equations.
Part D – Discrete Predator Prey
Now consider a simple discrete predator prey model with logistic growth:
xn 1  rxn (1 xn )  aPxn

In this model, r is a growth rate, a is a predation rate, P is a constant number of predators,
and x is a normalized population size. Please note that 0  xn 1, r  0, a,P  0 .
Open the folder final_q4.zip.
1) Solve for the fixed points ( x n  xn 1). Howdo the fixed points compare to the logistic
map?
2) Let a = 0.1, P = 0.2, x0 = 0.3, and vary r. Construct a bifurcation diagram similar to the
one we constructed 
for the logistic map.
3) Now consider the coupled system where two species compete for resources:
x n 1  r1 x n (1 x n )  a1 x n y n
y n 1  r 2 y n (1 y n )  a2 x n y n

Describe each term in this system. Choose biologically reasonable values for r1, r2, a1,
a2, x(0), and y(0). Write a short Matlab code that simulates the system. You might wish
to look at the breathing code on the class homepage. Run the simulations for 50 time
steps and graph your results.
4) Construct a discrete model of a predator-prey system where both the number of
predators and prey change in time. Explain your choice of model and describe all terms.
5) Select a couple sets of biologically appropriate parameters for the model you made in
#4 and modify the code from #3 for this system. Justify your choice of parameter values.
Run the simulation and show your graphs of predators and prey vs. time.