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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 - 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).

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 24EQ: Suppose the coal and steel industries form a closed economy. Every $1 produced by the coal industry...
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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 - 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).
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