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 sorne fish species, there is a minimum population m snch that the species will bexome extinet if the size of the population falls below m. Such a population can be modelled using a modilied lugistic equatiou: kP dt (a) Use the differential equation to show that any solution is increasing if m < P< M and decreasing if 0 < P

Advanced Engineering Mathematics
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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 sorne fish species, there is a minimmum population m such that the species
will bexzome extinet if the size of the population falls below m. Such a populalion can be
modelled using a modilied logistic equatioa:
dP
dt
(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
M(R – m)ee – m(Pa – M)
(Po - m)e
is a solution with initial population P(0) = P. 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).
P(0)
A(M
(P - 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 sorne fish species, there is a minimmum population m such that the species will bexzome extinet if the size of the population falls below m. Such a populalion can be modelled using a modilied logistic equatioa: dP dt (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 M(R – m)ee – m(Pa – M) (Po - m)e is a solution with initial population P(0) = P. 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). P(0) A(M (P - M)
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