1. Two species X and Y are represented by the following modelXP= ri X(1 – aX – bY),dTdY= r2Y(1 – cY +dX).dTHere all parameters are positive.(a) What is the nature of the interaction beteen X and Y?(b) Show that the suitable nondimensionlization leads todr= r(1 -r - y),dtdy= 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 equilibriumand stability analysis for this question.)2. The following mathematical model is proposed for the interaction of two species:dN1N1b12 N2IPdN2K1K1b21 N1LPHere 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 toduu1 (1 - uj - u2)dtduzruz(1 – buj).dt(c) Determine the steady states of this system, and their stability. .

Answered: Image /qna-images/answer/8df81c3f-97c5-4156-82ed-670f0769d5b3.jpg

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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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

Question

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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