Graphics Programs Reference
In-Depth Information
where x denotes the derivative of x withrespect to t . We think of x as
a fraction of some maximal possible population. One advantage of this
continuous model over the discrete model in Chapter 9 is that we can get
a “reading” of the population at any point in time (not just on integer
intervals).
(a) The differential equation (3) is solved in any beginning course in
ordinary differential equations, but you can do it easily withthe
MATLAB command dsolve . (Look up the syntax via online help.)
(b) Now find the solution assuming an initial value x 0 = x (0) of x . Use
the values x 0 = 0 , 0 . 25 ,..., 2 . 0. Graphthe solutions and use your
picture to justify the statement: “Regardless of x 0 > 0, the solution
of (3) tends to the constant solution x ( t ) 1 in the long term.”
The logistic model presumes two underlying features of population
growth:(i)thatideallythepopulationexpandsatarateproportional
to its current total (that is, exponential growth — this corresponds
to the x term on the right side of (3)) and (ii) because of interactions
between members of the species and natural limits to growth, unfet-
tered exponential growth is held in check by the logistic term, given
by the x 2 expression in (3). Now assume there are two species
x ( t ) and y ( t ), competing for the same resources to survive. Then
there will be another negative term in the differential equation that
reflects the interaction between the species. The usual model pre-
sumes it to be proportional to the product of the two populations,
and the larger the constant of proportionality, the more severe the
interaction, as well as the resulting check on population growth.
(c) Hereisatypicalpairofdifferentialequationsthatmodelthegrowth
in population of two competing species x ( t ) and y ( t ):
x ( t ) = x x 2
0 . 5 xy
(4)
˙ y ( t ) = y y 2
0 . 5 xy .
The command dsolve can solve many pairs of ordinary differential
equations — especially linear ones. But the mixture of quadratic
termsin(4)makesitunsolvablesymbolically,andsoweneedtousea
numerical ODE solver as we did in the pendulum application. Using
the commands in that application as a template, graph numerical
Search WWH ::




Custom Search