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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 P = kP | 1– M m. dt P 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 k(M-m)t M(Po – m)e M k(M-m)t — т(Ро — М) P(t) = (Ро — тје - (Ро — М) 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).

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 P = kP | 1– M m. dt P 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 k(M-m)t M(Po – m)e M k(M-m)t — т(Ро — М) P(t) = (Ро — тје - (Ро — М) 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).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 28EQ
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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 ( 1 dt P m 1 M (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 = it to sketch several solutions for various initial populations. What are the equilibrium solutions? 10, 000, draw a direction field and use (c) One can show that k(M-m). M(Po – m)e M (Ро — т)е - m(Ро — М) - (Ро — М) P(t) = k(M-m)t 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 ( 1 dt P m 1 M (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 = it to sketch several solutions for various initial populations. What are the equilibrium solutions? 10, 000, draw a direction field and use (c) One can show that k(M-m). M(Po – m)e M (Ро — т)е - m(Ро — М) - (Ро — М) P(t) = k(M-m)t 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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