Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
6. A population of fish is living in an environment with limited resources. This environment can only support the population if it contains no more than M fish (otherwise some fish would starve due to an inadequate supply of food, etc.). There is considerable evidence to support the theory that, for some fish species, there is a minimum population m such that the species will become extinct if the size of the population falls below m. Such a population can be modelled using a modified logistic equation: dP = kP - r (1-) (: M 3 (a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P< m. (b) For the case where k = 1, M = 100, 000 and m = 10,000, draw a direction field and use it to sketch several solutions for various initial populations. What are the equilibrium solutions? (c) One can show that KM - m(P - M) M(P - m)e P(t) = A(M + (Po - m)e - (P - M) is a solution with initial population P(0) = Po. Use this to show that, if P(0) < m, then there is a timet at which P(t)%= = 0 (and so the population will be extinct).

6. A population of fish is living in an environment with limited resources. This environment can only support the population if it contains no more than M fish (otherwise some fish would starve due to an inadequate supply of food, etc.). There is considerable evidence to support the theory that, for some fish species, there is a minimum population m such that the species will become extinct if the size of the population falls below m. Such a population can be modelled using a modified logistic equation: dP = kP - r (1-) (: M 3 (a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P< m. (b) For the case where k = 1, M = 100, 000 and m = 10,000, draw a direction field and use it to sketch several solutions for various initial populations. What are the equilibrium solutions? (c) One can show that KM - m(P - M) M(P - m)e P(t) = A(M + (Po - m)e - (P - M) is a solution with initial population P(0) = Po. Use this to show that, if P(0) < m, then there is a timet at which P(t)%= = 0 (and so the population will be extinct).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
6. A population of fish is living in an environment with limited resources. This environment can only support the population if it contains no more than M fish (otherwise some fish would starve due to an inadequate supply of food, etc.). There is considerable evidence to support the theory that, for some fish species, there is a minimum population m such that the species will become extinct if the size of the population falls below m. Such a population can be modelled using a modified logistic equation: dP = kP - r (1-) (: M 3 (a) Use the differential equation to show that any solution is increasing if m <P < M and decreasing if 0 < P<m. (b) For the case where k = 1, M = 100, 000 and m = 10,000, draw a direction field and use it to sketch several solutions for various initial populations. What are the equilibrium solutions? (c) One can show that (Mm - m(Po- M) M(P - m)eM P(() = (Po – m)e - (P – M) is a solution with initial population P(0) = Po. Use this to show that, if P(0) < m, then there is a time t at which P(t) = 0 (and so the population will be extinct).
Transcribed Image Text:6. A population of fish is living in an environment with limited resources. This environment can only support the population if it contains no more than M fish (otherwise some fish would starve due to an inadequate supply of food, etc.). There is considerable evidence to support the theory that, for some fish species, there is a minimum population m such that the species will become extinct if the size of the population falls below m. Such a population can be modelled using a modified logistic equation: dP = kP - r (1-) (: M 3 (a) Use the differential equation to show that any solution is increasing if m <P < M and decreasing if 0 < P<m. (b) For the case where k = 1, M = 100, 000 and m = 10,000, draw a direction field and use it to sketch several solutions for various initial populations. What are the equilibrium solutions? (c) One can show that (Mm - m(Po- M) M(P - m)eM P(() = (Po – m)e - (P – M) is a solution with initial population P(0) = Po. Use this to show that, if P(0) < m, then there is a time t at which P(t) = 0 (and so the population will be extinct).
Expert Solution
steps

Step by step

Solved in 3 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS