1 A population of giant tortoises is modelled by the delay differential dN (t)=bN(t-T)-c[N(t-T)]² - m[N(t)]², where m, b and c are positive parameters and T is the time delay between egg laying and hatching. (a) Give a biological interpretation for each of the three terms in the right hand side of the equation. Explain how we can see that the tortoises have a naturally long lifespan when the population is small. (b) Show that there is a unique positive equilibrium N., and find a linear differential equation for h(t) = N(t)- N. that approximately holds when h is small. Find the characteristic equation. (c) Using parameter continuation, show that as T is increased from zero, instability of the equilibrium N, will first be observed (if at all) when the following system has a real solution for w and T: (cm) cos(wT) = -2m (cm) sin(wT) = (c + m)w/b. (d) What is the meaning of the parameter w? Find an expression for w in terms of b, c and m only. Show that N, is in fact locally stable for all T20 if c/m is not too large.
1 A population of giant tortoises is modelled by the delay differential dN (t)=bN(t-T)-c[N(t-T)]² - m[N(t)]², where m, b and c are positive parameters and T is the time delay between egg laying and hatching. (a) Give a biological interpretation for each of the three terms in the right hand side of the equation. Explain how we can see that the tortoises have a naturally long lifespan when the population is small. (b) Show that there is a unique positive equilibrium N., and find a linear differential equation for h(t) = N(t)- N. that approximately holds when h is small. Find the characteristic equation. (c) Using parameter continuation, show that as T is increased from zero, instability of the equilibrium N, will first be observed (if at all) when the following system has a real solution for w and T: (cm) cos(wT) = -2m (cm) sin(wT) = (c + m)w/b. (d) What is the meaning of the parameter w? Find an expression for w in terms of b, c and m only. Show that N, is in fact locally stable for all T20 if c/m is not too large.
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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![A population of giant tortoises is modelled by the delay differential
dN
dt
-(t)=bN(t-T) - c[N(t - T)]² - m[N(t)]²,
where m, b and c are positive parameters and T is the time delay between egg laying and hatching.
(a) Give a biological interpretation for each of the three terms in the right hand side of the
equation. Explain how we can see that the tortoises have a naturally long lifespan when
the population is small.
(b) Show that there is a unique positive equilibrium N., and find a linear differential equation
for h(t) = N(t). - N, that approximately holds when h is small. Find the characteristic
equation.
(c) Using parameter continuation, show that as T is increased from zero, instability of the
equilibrium N, will first be observed (if at all) when the following system has a real solution
for w and T:
(cm) cos(wT) = -2m
(cm) sin(wT) = (c+m)w/b.
(d) What is the meaning of the parameter w? Find an expression for w in terms of b, c and m
only. Show that N, is in fact locally stable for all T≥ 0 if c/m is not too large.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb6f0fc2b-ed06-42f6-aea2-7d26d8b1334d%2Ffd99e1f5-fcf9-48e3-a8a6-33043d7eb6b7%2Fffeklel_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A population of giant tortoises is modelled by the delay differential
dN
dt
-(t)=bN(t-T) - c[N(t - T)]² - m[N(t)]²,
where m, b and c are positive parameters and T is the time delay between egg laying and hatching.
(a) Give a biological interpretation for each of the three terms in the right hand side of the
equation. Explain how we can see that the tortoises have a naturally long lifespan when
the population is small.
(b) Show that there is a unique positive equilibrium N., and find a linear differential equation
for h(t) = N(t). - N, that approximately holds when h is small. Find the characteristic
equation.
(c) Using parameter continuation, show that as T is increased from zero, instability of the
equilibrium N, will first be observed (if at all) when the following system has a real solution
for w and T:
(cm) cos(wT) = -2m
(cm) sin(wT) = (c+m)w/b.
(d) What is the meaning of the parameter w? Find an expression for w in terms of b, c and m
only. Show that N, is in fact locally stable for all T≥ 0 if c/m is not too large.
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