Page 360 - Advanced engineering mathematics

340 CHAPTER 10 Systems of Linear Differential Equations

Bay Company has kept records of pelts traded at their stations since the middle of the nineteenth
century when trappers worked these areas. Assuming that the actual populations of lynx and hare
were proportional to the number of pelts obtained by trappers, the records for 1845 through 1935
exhibit cyclical variations as seen in the predator/prey model, with about a ten year cycle.
Another predator/prey setting occurred in Michigan’s Isle Royale, an untamed island having
a length of about 45 miles. At one time, moose abounded on the island, having no natural enemy.
However, the harsh winter of 1949 caused wolves from Canada to cross over a frozen stretch
of Lake Superior, searching for food. The resulting behavior of the moose/wolf populations was
studied by Purdue biologist Durward Allen, who observed cyclic variations in the populations
over the 1957–1993 period. In this case, the predator wolf population that came to the island had
two problems which would alter the model—a very narrow genetic base coupled with the spread
of a canine virus that destroyed many wolves.
More complex predator/prey models have been used in many contexts, including research
into the behavior of the HIV virus. In one such model, the predators consist of the invading
viruses and the prey consists of their target cells within the body. The model is complicated by
the fact that the viruses mutate over time, presenting the immune system with many different
predators. One study along these lines is given in a November 15, 1991 paper in the journal
Science, entitled Antigenic Diversity Thresholds and the Development of AIDS and authored by
Martin A. Nowak, A.R. McLean, and R.M. May of the Department of Zoology, University of
Oxford; T. Wolfe and J. Goudsmit of the Human Retrovirus Laboratory, Department of Virol-
ogy, Amsterdam, the Netherlands; and R.M. Anderson of the Department of Biology, Imperial
College of London University.

A Competing Species Model
A competing species model offers a different type of population dynamic. In this model, we
have some environment in which two species compete for a common resource, but neither preys
on the other. In this case, it seems reasonable that an increase in either population decreases
the availability of this resource for both, causing a decline in both populations. Assuming no
restriction on the needed resource, one possible competing species model is given by

x = ax − bxy

y = ky − cxy,
in which a,b,c, and k are positive constants. Now a term proportional to the product of the
populations is subtracted in both equations.
As with the predator/prey model, we can obtain an implicitly defined solution of this system.
Divide the differential equations to obtain
dy/dt dy y k − cx
= = .
dx/dt dx x a − by
Separate variables to obtain
a − by k − cx
dy = dx.
y x
Integrate and rearrange terms to obtain
a −by
k −cx
y e = Kx e
in which K is the constant of integration and can be any positive number.
Typical trajectories of this model are shown in Figure 10.27. Asymptotes of these trajectories
pass through (k/c,a/b) and subdivide the first octant into four regions, I, II, III, and IV. If the
initial population (x(0), y(0)) is in regions I or IV, then the x population increases with time,

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 20:32 THM/NEIL Page-340 27410_10_ch10_p295-342