differential equation3

MATH 355-002 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

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