Consider the following systems of rate of change equations: System A System B dx dx = 3x- dt 1 ху = 3x(1-)-xy dt 10 20 100 dy -5y+ dt ху dy di = 15y(1-) + 25xy 20 In both of these systems, x(t) and y(t) represent the number of members of two different species at time t. In particular, in one of these systems the prey are large animals and the predators are small animals, such as humans and piranhas. Thus it takes many predators to eat one prey, but each prey eaten is a tremendous benefit for the predator population. The other system has very large predators and small prey, such as dragons and goats. (a) For both systems of differential equations, what does x represent, the predator or the prey? Explain. (b) For system (A), plot all nullclines and use this plot to determine all equilibrium solutions. Verify your equilibrium solutions algebraically. (c) Use your results from 6b to sketch in the long-term behavior of solutions with initial conditions any- where in the first quadrant of the phase plane. For example, describe the long-term behavior of so- lutions if the initial condition is in such and such region of the first quadrant. Provide a sketch of your analysis in the x-y plane and write a paragraph summarizing your conclusions and any conjec- tures that you have about the long-term outcome for the two populations depending on the initial conditions.
Consider the following systems of rate of change equations: System A System B dx dx = 3x- dt 1 ху = 3x(1-)-xy dt 10 20 100 dy -5y+ dt ху dy di = 15y(1-) + 25xy 20 In both of these systems, x(t) and y(t) represent the number of members of two different species at time t. In particular, in one of these systems the prey are large animals and the predators are small animals, such as humans and piranhas. Thus it takes many predators to eat one prey, but each prey eaten is a tremendous benefit for the predator population. The other system has very large predators and small prey, such as dragons and goats. (a) For both systems of differential equations, what does x represent, the predator or the prey? Explain. (b) For system (A), plot all nullclines and use this plot to determine all equilibrium solutions. Verify your equilibrium solutions algebraically. (c) Use your results from 6b to sketch in the long-term behavior of solutions with initial conditions any- where in the first quadrant of the phase plane. For example, describe the long-term behavior of so- lutions if the initial condition is in such and such region of the first quadrant. Provide a sketch of your analysis in the x-y plane and write a paragraph summarizing your conclusions and any conjec- tures that you have about the long-term outcome for the two populations depending on the initial conditions.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:6. Consider the following systems of rate of change equations:
System A
System B
= 3x(1-%)-20*y
dx
dx
= 3x–
dt
ху
dt
100
dy
= -5y+
dt
ху
di = 15y(1-)+ 25xy
20
In both of these systems, x(t) and y(t) represent the number of members of two different species at time
t. In particular, in one of these systems the prey are large animals and the predators are small animals,
such as humans and piranhas. Thus it takes many predators to eat one prey, but each prey eaten is a
tremendous benefit for the predator population. The other system has very large predators and small prey,
such as dragons and goats.
(a) For both systems of differential equations, what does x represent, the predator or the prey? Explain.
(b) For system (A), plot all nullclines and use this plot to determine all equilibrium solutions. Verify your
equilibrium solutions algebraically.
(c) Use
where in the first quadrant of the phase plane. For example, describe the long-term behavior of so-
lutions if the initial condition is in such and such region of the first quadrant. Provide a sketch of
your analysis in the x-y plane and write a paragraph summarizing your conclusions and any conjec-
tures that you have about the long-term outcome for the two populations depending on the initial
your
results from 6b to sketch in the long-term behavior of solutions with initial conditions any-
conditions.
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