Science 200 or 311 Lab #16 Differential Equations: Predator-Prey models using EXCEL.
In this lab we solve the Lotka-Volterra model for predator-prey relations. The governing equations for
Predator (P) Change Rate and Herbivore (H) Change Rate and the coefficients are given on the
worksheet Equations.
Solution Technique and Assignment:
1. Change Calculus to Algebra (i. e., Differential Equations to Difference Equations)
2. Multiply the equation by Dt (time step) and solve for the future value of P In Column B
3. Do the same for the Herbivore Difference Equation
4. Increment the time in Column A
5. Graph P and H for at least 3 cycles.
6. Calculate the average value of P and H for a full cycle (partial cycles give biased results)
7. Find the largest value of Dt that will not cause the model to crash (give negative # of animals).
8. In text box 1, explain why there is a limit to Dt.
9. In Worksheet, Impacts, find the impact of changing various constants.
y models using EXCEL.
s. The governing equations for
efficients are given on the
e Equations)
of P In Column B
give biased results)
ve negative # of animals).
dP dH
= A× B × P × H - D × P = RR× H - A× P× H
dt dt
dP P(t + Dt ) - P(t )
»
dt Dt
P(t + Dt) = P(t) + Dt × (A× B× P× H - D× P) You create the Prediction
Equation for the Herbivores
A = Capture Efficiency (Catch per hunt)
B = Ingestion Efficiency (Eat what you kill)
D = Predator Digestion Rate (Metabolic Rate)
RR = Herbivore Reproduction Rate
dH
= RR× H - A× P× H
dt
You create the Prediction
Equation for the Herbivores
Under Equilibrium the animal populations are balanced and don't change.
1. In a text box below, tell how the equation of change simplifies for equilibrium.
2. Find the equations for the populations at equilibrium.
Equation of Change for Equilibrium
n't change.
s for equilibrium.
t P H Pavg
0 Havg
1. Reason for limits to Dt
A 0.8
B 0.5
D 3
RR 8
dt ???
P0 6
H0 90
Period Havg Pavg
H0 = 45 Change only one factor at a time (restoring the
P0 = 3 initial value to all other factors). Find the
RR = 12 resulting cycle periods and average animal
A = .25 populations.
B = .1
D=9 Try to anticipate and then give a reason for
the changes that occur.
restoring the
ge animal
eason for