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. 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 dt 3 (a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P

. 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 dt 3 (a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P

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
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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 3 (a) Use the differential equation to show that any solution is incaeasing 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 M(P, – m)em - m(Po – M) P(t) M-m) (Po – m)e (Po – 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 3 (a) Use the differential equation to show that any solution is incaeasing 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 M(P, – m)em - m(Po – M) P(t) M-m) (Po – m)e (Po – 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).
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