5. Find all the physically relevant steady-states of equations (1)-(2), stating their bio- logical interpretation. Find the lower (62) and upper (6) bounds of 5, for which the coexistence state (u, v) is physical. These bounds will depend on a.
5. Find all the physically relevant steady-states of equations (1)-(2), stating their bio- logical interpretation. Find the lower (62) and upper (6) bounds of 5, for which the coexistence state (u, v) is physical. These bounds will depend on a.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Q5 solution needed
Transcribed Image Text:5. Find all the physically relevant steady-states of equations (1)-(2), stating their bio-
logical interpretation.
Find the lower (5;) and upper (6;) bounds of 6, for which the coexistence state
(u, v) is physical. These bounds will depend on a.
6. Write down all the null-clines of equations (1)-(2). For each of the cases
i 8i < 6
ii d; < 6 < đị;
ii 6 < ó,
sketch the mull-clines in the phase-plane (recalling a > 1) and indicate clearly the
general direction of the trajectories in each of the regions bounded by the null-clines.
On each phase-plane sketch representative trajectories.
For each case:
• Where possible, deduce from your diagrams the type and stability of the steady-
states; if such a deduction cannot be made purely from the diagram, state all
the possible types that the steady-states could be.
• State the possible behaviour of the solutions (u(t), v(t)) as t 0o.
Transcribed Image Text:Consider the following interaction between two species of animals, denoted by variables
X and Y:
dX
XY
(1-*)- "Y + eX'
dr
K
AP
dr
XY
bY +eX
XP
where r,K, a, b, c, d and e are all non-negative constants.
1. What kind of interaction does this model represent. Provide your reasoning.
2. Give a biological interpretation of each of the terms on the right-hand sides.
3. Given that X and Y are in "population/mile?" and z is in "months", write down
the units for each of the constants.
4. Show by suitable non-dimensionalisation that the model is equivalent to
du
uv
u(1 — и) - а-
f(u, v),
(1)
%3!
v+u
dv
dt
= Bv ( a
u + v
(2)
where u and v are the non-dimensionalised animal populations, and a, B and o are
non-negative constants. Express the new variables and constants in terms of the
original variables and constants.
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