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:dP= kPdtPmM(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, Mit to sketch several solutions for various initial populations. What are the equilibrium100, 000 and m = 10,000, draw a direction field and usesolutions?(c) One can show thatk(М—т)М (Ро — т)е мk(M-m)t-т(Ро — М)— (Ро — М)P(t) =(Ро — т)еis a solution with initial population P(0) = Po. Use this to show that, if P(0) < m, thenthere is a timet at which P(t) = 0 (and so the population will be extinct).

Given is differential equation in P (a) In this part by just looking at the differential equation we…

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: (1-4) (-). dP P = kP | dt m 1 - P (a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P

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: (1-4) (-). dP P = kP | dt m 1 - P (a) Use the differential equation to show that any solution is increasing if m < P < M and decreasing if 0 < P

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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