# Direction Fields and Phase Portraits

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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