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(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 10, 000, draw a direction field and use solutions? (c) One can show that k(M-m)t - m(Po – M) M(Po – m)eM P(t) k(М-т) (Ро — т)е м - (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).

(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 10, 000, draw a direction field and use solutions? (c) One can show that k(M-m)t - m(Po – M) M(Po – m)eM P(t) k(М-т) (Ро — т)е м - (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 24EQ: Suppose the coal and steel industries form a closed economy. Every $1 produced by the coal industry...
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(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)+ M(Po – m)eª M т(Ро — М) P(t) k(М-т) (Ро — т)е м - (Ро — М) M - is a solution with initial population P(0) = there is a time t at which P(t) = 0 (and so the population will be extinct). Po. Use this to show that, if P(0) < m, then
Transcribed Image Text:(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)+ M(Po – m)eª M т(Ро — М) P(t) k(М-т) (Ро — т)е м - (Ро — М) M - is a solution with initial population P(0) = there is a time t at which P(t) = 0 (and so the population will be extinct). Po. Use this to show that, if P(0) < m, then
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 m 1 M
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 m 1 M
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