Direction Fields and Phase Portraits

Topics: Differential equation, Population, Plot Pages: 3 (713 words) Published: March 5, 2014
MTH 256–Applied Differential Equations
Lab 1: Qualitative Analysis of Differential Equations
Direction Fields and Phase Portraits
(borrowed from MTH 399H–Introduction to Mathematical Ecology by Prof. Bokil) You will need to use DFIELD for this Lab. Either use the online JAVA version or download the current version to your computer. DFIELD Setup Instructions: Go to the course website and click ODE Software for MATLAB: DFIELD, PPLANE and ODESOLVE in order to download the file “dfield8.m” and the corresponding manual “Dfield.pdf ”. Save these files in a folder of your choice. Start MATLAB and change directory to the folder containing “dfield8.m”.

A Harvesting Problem and a Geometric Approach
Goal: To study a model for harvesting of a fish species by plotting its equilibria via a phase plot and direction field.
Recall the logistic differential equation from class
dP
= rP
dt

1−

P
K

where r is the per capita growth rate and K is the carrying capacity. Suppose we model the population of a particular species of fish in a lake with a logistic differential equation with r = 0.25, K = 4. Here P (t) represents the number of fish in tens of thousands at time t in years. So, K = 4 implies that the carrying capacity of the fish species is 40, 000 fish.

dP
1. In MATLAB plot the derivative
for logistic growth versus P . Lets call this plot a
dt
“phase plot”. Use the range [0:0.01:5] for P . What are equilibria and corresponding stability for this model?
2. Direction field of the logistic differential equation Use either the online JAVA version of dfield (in which case you need to enter the model into the equation window and the other instructions below are slightly different), or at the MATLAB prompt enter

>> dfield8
This command will start an interactive GUI. The window that opens is called the “dfield8 setup”. Under Gallery choose “logistic equation”. Change the parameters in this window to r = 0.25, K = 4, input minimum and maximum value of t to be 0, and 20,...
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