1. Two species X and Y are represented by the following model dX = rị X(1 – aX – bY), r2Y(1 – cY +dX). %3D Here all parameters are positive. (a) What is the nature of the interaction beteen X and Y? (b) Show that the suitable nondimensionlization leads to dr r(1 -1- y), dt dy ry(1 – 7y+ổx). dt (c) Write down the scalings for z, y, t, r,7 and o and check they are correct by sub- stituting them into the original equation. (You do not need to do an equilibrium and stability analysis for this question.)
1. Two species X and Y are represented by the following model dX = rị X(1 – aX – bY), r2Y(1 – cY +dX). %3D Here all parameters are positive. (a) What is the nature of the interaction beteen X and Y? (b) Show that the suitable nondimensionlization leads to dr r(1 -1- y), dt dy ry(1 – 7y+ổx). dt (c) Write down the scalings for z, y, t, r,7 and o and check they are correct by sub- stituting them into the original equation. (You do not need to do an equilibrium and stability analysis for this question.)
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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Transcribed Image Text:1. Two species X and Y are represented by the following model
XP
= ri X(1 – aX – bY),
dT
dY
= r2Y(1 – cY +dX).
dT
Here all parameters are positive.
(a) What is the nature of the interaction beteen X and Y?
(b) Show that the suitable nondimensionlization leads to
dr
= r(1 -r - y),
dt
dy
= ry(1 –1y+ 8 x).
dt
(c) Write down the scalings for r,y, t, r, y and d and check they are correct by sub-
stituting them into the original equation. (You do not need to do an equilibrium
and stability analysis for this question.)
2. The following mathematical model is proposed for the interaction of two species:
dN1
N1
b12 N2
IP
dN2
K1
K1
b21 N1
LP
Here all parameters are positive.
(a) Which of the following terms best describes the nature of the interaction be-
tween these species: predator-prey, competition, symbiosis?
(b) Show that a suitable nondimensionalisation leads to the systemy.
The model is modified to
du
u1 (1 - uj - u2)
dt
duz
ruz(1 – buj).
dt
(c) Determine the steady states of this system, and their stability. .
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