6. A population of fish is living in an environment with limited resources. This environment canonly support the population if it contains no more than M fish (otherwise some fish wouldstarve due to an inadequate supply of food, etc.). There is considerable evidence to supportthe theory that, for some fish species, there is a minimum population m such that the specieswill become extinct if the size of the population falls below m. Such a population can bemodelled using a modified logistic equation:dPmkP 1dt%D-M(a) Use the differential equation to show that any solution is increasing if m < P< M anddecreasing if 0 < P < m.(b) For the case where k = 1, M = 100, 000 and m = 10,000, draw a direction field and useit to sketch several solutions for various initial populations. What are the equilibriumsolutions?(c) One can show thatk(M-m)tMM(Po – m)e(Ро — т)е м-m(Po – M)t- (Po – M)P(t) =k(М-т)M|is a solution with initial population P(0) = Po. Use this to show that, if P(0) < m, thenthere is a time t at which P(t) = 0 (and so the population will be extinct).

The differential equation given is as follows: dPdt=kP1-PM1-mP Also given that the maximum…

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 P 1- т 1- (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 М (Ро — т)е м -m(Po- M) E(M=m), (Po- M) P(t) = (Po – m)eM 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 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 P 1- т 1- (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 М (Ро — т)е м -m(Po- M) E(M=m), (Po- M) P(t) = (Po – m)eM 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

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 28EQ

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