LECTURE 2

10.2 More about direction fields

Direction field of the logistic equation

Consider the following equation:

dP/dt = P ( 1 - P/5 )

where t=time=the independent variable; P=P(t)=population at time t; 5=K=carrying capacity.

> with(DEtools):

> DE := diff(P(t),t)=P*(1-P/5):

Its direction field looks as follows:

> DEplot( DE ,P(t), t=-7..7, P=-1..6,
[[P(0)=0], [P(0)=5]], arrows=SMALL,
dirgrid=[30, 30], stepsize=.1,
linecolor=black, color=BLUE);

Warning, P is present as both a dependent variable and a name. Inconsistent specification of the dependent variable is deprecated, and will be removed in the next release.

Note. The black curves on the picture represent the "critical points"=constant solutions.

One can describe the above direction field, or, if the direction field cannot be obtained, one can predict

qualitatively the behaviour of the solutions by looking at the graph of the right hand side of the equation,

F(P)=P ( 1 - P/5 ):

> plot(P*(1-P/5),P=-1..6,title=`Graph of F(P)`);

Keep in mind that the important information here is the sign of F(P) (and consequently that of dP/dt).

10.3 Separable equations and more modelling

Mixing problems

Exercise 34, page 635.

We obtained the IVP: dQ/dt = 0.65 - 15 Q / 1000, Q(0) = 0.

Its direction field and solution(in blue) are:

> DEplot( diff(Q(t),t)=0.65-15*Q(t)/1000, Q(t), t=-3..200, Q=-3..50,
[[Q(0)=0],[Q(0)=650/15]], arrows=SMALL,
dirgrid=[30, 30], stepsize=.1,
linecolor=[green,black], color=BLUE);