NOT INTENDED FOR REDISTRIBUTION

Posted for midterm review and PERSONAL use

ACTIVITY#08 ? Math 151 ? Calculus II ? Fall 2016

Professor/TA: Sec/Time: RedID:

NAME (printed): Partners:

(Family Name) (First Name)

Qualitative analysis of Differential Equations (DEs): The purpose of this activity is to learn how to analyze important

features of the dynamics of first order (autonomous) differential equations without the need to integrate (solve) them!.

Note: a little birdy told me that this material will be included in midterm#2 and the final…

(1) Consider the following differential equation (DE):

dx

dt = f(x), (1)

where the sketch of f(x) is given on the figure to the

right and where x(t) describes the evolution of x in time

(i.e., a trajectory). As the figure shows, f(x) has three

zeros (roots): x1, x2, and x3.

x1 x2 x3

x

f(x)

(a) Discuss with your partner(s) what is the solution to the DE (1) if the initial condition x(t= 0) is one of the zeros of f(x).

These points are called fixed points. Why?

(b) Find the intervals where f(x) > 0 and f(x) < 0. What happens on intervals where f(x) > 0 in terms of x(t)?

What happens on intervals where f(x) < 0 in terms of x(t)?

(c) Using the answers on the previous question, draw arrows on the x axis indicating whether x(t) is increasing or decreasing.

If increasing: draw a right arrow and if decreasing: draw a left arrow.

(d) What happens to the solution if one starts slightly to the right of x1? And if one starts slightly to the left of x1? Such a

point is called an UNSTABLE fixed point. Why?

What happens to the solution if one starts slightly to the right of x2? And if one starts slightly to the left of x2? Such a

point is called a STABLE fixed point. Why?

(e) Can you characterize the stability of the fixed points x1, x2, and x3 by thinking about the slope of f at those points?

Be as precise as possible.

The final punchline: Fixed points and their stability for dx/dt = f(x)

? A fixed point x

*

is a point such that: f(x

*

) = .

? A fixed point x

*

is STABLE if: f

‘

(x

*

) 0.

? A fixed point x

*

is UNSTABLE if: f

‘

(x

*

) 0.

1

NOT INTENDED FOR REDISTRIBUTION

Posted for midterm review and PERSONAL use

(2) Equipped with the experience you gained with the previous

problem, let us analyze the so-called logistic growth population

model:

dP

dt = f(P) = rP

1 –

P

M

, (2)

where r and M are positive constants.

(a) Draw the polynomial f(P). Be sure to find and label its roots.

Roots: P

f(P )

(b) Draw arrows on indicating where P is increasing/decreasing.

(c) Why is logistic growth equivalent to exponential (Malthusian) growth for small populations? Small with respect to what?

(d) What is the behavior of the logistic model in the long run (t ? +8) if one starts with an initial P(t= 0) = P0 such that:

(i) P0 is precisely at one of the roots:

(ii) P0 is just to the right of the first root:

(iii) P0 is to the right of the last root:

(e) (i) What happens to initial conditions very close to each one the fixed points? Do they get ?attracted? or ?repelled?

away from them?

(ii) Why is M called the carrying capacity?

(3) Application: In many ecological systems, when the population drops below a certain level, it cannot support itself anymore.

For example this happens when mates cannot be found, lack of cooperation, being too exposed to predators, etc. This effect

is commonly referred to as the Allee Effect. A simple model for the Allee effect can be cast in the following form:

dP

dt = g(P) where g(P) = -a P (P – Pm)(P – M), (3)

where a > 0 is a constant [related to r in Eq. (2)], M is the carrying capacity as before, and Pm is another special population

size such that Pm < M (as depicted in the plot).

(a) (i) What are the roots of g(P)?

, ,

(ii) lim

P?8

g(P) = , lim

P?-8

g(P) =

(iii) Sketch g(P). (Use the limits!)

(iv) Draw, as before, the direction in which

the trajectories move on the P-axis.

(v) What can you say, from the sketch,

about the stability of the fixed points:

P

*

1 = 0 is .

P

*

2 = Pm is .

P

*

3 = M is .

P

g(P )

Pm M

(b) Now, use the punchline in (1.e) together with the graph of g(P) to corroborate the stability of the fixed points:

g

‘

(P

*

1

) 0 ? P

*

1

is . g

‘

(P

*

2

) 0 ? P

*

2

is . g

‘

(P

*

3

) 0 ? P

*

3

is .

(c) Write an interpretation of the model for this Allee effect in terms of Pm and M. What is so special about Pm?

2

NOT INTENDED FOR REDISTRIBUTION

Posted for midterm review and PERSONAL use

ACTIVITY#08 ? Math 151 ? Calculus II ? Fall 2016

Professor/TA: Sec/Time: RedID:

NAME (printed): Partners:

(Family Name) (First Name)

Qualitative analysis of Differential Equations (DEs): The purpose of this activity is to learn how to analyze important

features of the dynamics of first order (autonomous) differential equations without the need to integrate (solve) them!.

Note: a little birdy told me that this material will be included in midterm#2 and the final…

(1) Consider the following differential equation (DE):

dx

dt = f(x), (1)

where the sketch of f(x) is given on the figure to the

right and where x(t) describes the evolution of x in time

(i.e., a trajectory). As the figure shows, f(x) has three

zeros (roots): x1, x2, and x3.

x1 x2 x3

x

f(x)

(a) Discuss with your partner(s) what is the solution to the DE (1) if the initial condition x(t= 0) is one of the zeros of f(x).

These points are called fixed points. Why?

(b) Find the intervals where f(x) > 0 and f(x) < 0. What happens on intervals where f(x) > 0 in terms of x(t)?

What happens on intervals where f(x) < 0 in terms of x(t)?

(c) Using the answers on the previous question, draw arrows on the x axis indicating whether x(t) is increasing or decreasing.

If increasing: draw a right arrow and if decreasing: draw a left arrow.

(d) What happens to the solution if one starts slightly to the right of x1? And if one starts slightly to the left of x1? Such a

point is called an UNSTABLE fixed point. Why?

What happens to the solution if one starts slightly to the right of x2? And if one starts slightly to the left of x2? Such a

point is called a STABLE fixed point. Why?

(e) Can you characterize the stability of the fixed points x1, x2, and x3 by thinking about the slope of f at those points?

Be as precise as possible.

The final punchline: Fixed points and their stability for dx/dt = f(x)

? A fixed point x

*

is a point such that: f(x

*

) = .

? A fixed point x

*

is STABLE if: f

‘

(x

*

) 0.

? A fixed point x

*

is UNSTABLE if: f

‘

(x

*

) 0.

1

NOT INTENDED FOR REDISTRIBUTION

Posted for midterm review and PERSONAL use

(2) Equipped with the experience you gained with the previous

problem, let us analyze the so-called logistic growth population

model:

dP

dt = f(P) = rP

1 –

P

M

, (2)

where r and M are positive constants.

(a) Draw the polynomial f(P). Be sure to find and label its roots.

Roots: P

f(P )

(b) Draw arrows on indicating where P is increasing/decreasing.

(c) Why is logistic growth equivalent to exponential (Malthusian) growth for small populations? Small with respect to what?

(d) What is the behavior of the logistic model in the long run (t ? +8) if one starts with an initial P(t= 0) = P0 such that:

(i) P0 is precisely at one of the roots:

(ii) P0 is just to the right of the first root:

(iii) P0 is to the right of the last root:

(e) (i) What happens to initial conditions very close to each one the fixed points? Do they get ?attracted? or ?repelled?

away from them?

(ii) Why is M called the carrying capacity?

(3) Application: In many ecological systems, when the population drops below a certain level, it cannot support itself anymore.

For example this happens when mates cannot be found, lack of cooperation, being too exposed to predators, etc. This effect

is commonly referred to as the Allee Effect. A simple model for the Allee effect can be cast in the following form:

dP

dt = g(P) where g(P) = -a P (P – Pm)(P – M), (3)

where a > 0 is a constant [related to r in Eq. (2)], M is the carrying capacity as before, and Pm is another special population

size such that Pm < M (as depicted in the plot).

(a) (i) What are the roots of g(P)?

, ,

(ii) lim

P?8

g(P) = , lim

P?-8

g(P) =

(iii) Sketch g(P). (Use the limits!)

(iv) Draw, as before, the direction in which

the trajectories move on the P-axis.

(v) What can you say, from the sketch,

about the stability of the fixed points:

P

*

1 = 0 is .

P

*

2 = Pm is .

P

*

3 = M is .

P

g(P )

Pm M

(b) Now, use the punchline in (1.e) together with the graph of g(P) to corroborate the stability of the fixed points:

g

‘

(P

*

1

) 0 ? P

*

1

is . g

‘

(P

*

2

) 0 ? P

*

2

is . g

‘

(P

*

3

) 0 ? P

*

3

is .

(c) Write an interpretation of the model for this Allee effect in terms of Pm and M. What is so special about Pm?

2