differential equation3

typed answer in pdf. 

 

for the first part, 432, 436, 440, 452, 473, 475, 487, 489 should be compeleted.

 

part two is attached in pdf separately.

    

details required

MATH

3

55-00

2

HAND-IN HOMEWORK #4
(due Thursday, September 26)

PART 2

A biological population is said to grow “logistically” if its population size x = x(t) satisfies
the so-called “logistic” equation

x

′ = rx

(

1

−

x

K

)
.

Here r > 0 is a positive constant called the “inherent population growth rate” and K is a
positive constant called the environmental “carrying capcity”. The phase line portrait of this
nonlinear, autonomous differential equation consists of a repeller at x = 0 (the extinction
equilibrium) and an attractor at the carrying capacity x = K :

←− 0 −→ K ←− x

Of course, only the positive solutions & orbits are of interest in the biological application.
Suppose a logistically growing population is subjected to harvesting at a fixed, constant

rate h > 0. Then the dynamics of the harvested population satisfy the differential equation

x
′ = rx

(
1−
x
K

)
−h.

(a) Draw the phase line portrait of the harvested logistic equation. Note: there will be
different cases depending on the harvested rate h > 0 (in relation to r and K).

(b) Show that there is a bifurcation value h0 for h (using the definition of bifurcation
value). Find a formula for the bifurcation value h0.

(c) Draw a bifurcation diagram, using h as the bifurcation parameter.

(d) What kind of bifurcation occurs at h0?

(e) What important implication does this bifurcation analysis have with regard to the
long term fate of the population as the harvest rate increases above bifurcation value h0?

1

ANSWERS
MATH 355-002

HAND-IN HOMEWORK #4
(due Thursday, September 26)

PART 1

Pages 2, exercise.
ANSWER.

2

ANSWERS
MATH 355-002
HAND-IN HOMEWORK #4
(due Thursday, September 26)
PART 1
Pages 2, exercise.
ANSWER.
PART 2
3