By the first assumption, a dextral snail is twice as likely to
choose a dextral snail than a sinistral snail
Could use real experimental verification of the assumptions
[2] C. H. Taubes,
Modeling Differential Equations in Biology
, Prentice Hall, 2001.
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Direction Fields and Phase Port
— (36/50)
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Mathematical Modeling
Introduction to MatLab
Qualitative Behavior of Differential Equations
More Examples
Maple  Direction Fields
Left Snail Model
Allee Effect
Left Snail Model
2
Taubes Snail Model
Let
p
(
t
) be the probability that a snail is dextral
A model that qualitatively exhibits the behavior described
on previous slide:
dp
dt
=
αp
(1

p
)
p

1
2
,
0
≤
p
≤
1
,
where
α
is some positive constant
What is the behavior of this differential equation?
What does its solutions predict about the chirality of
populations of snails?
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Direction Fields and Phase Portraits  1D
— (37/50)
Mathematical Modeling
Introduction to MatLab
Qualitative Behavior of Differential Equations
More Examples
Maple  Direction Fields
Left Snail Model
Allee Effect
Left Snail Model
3
Taubes Snail Model
This differential equation is not easy to solve exactly
Qualitative analysis
techniques for this differential
equation are relatively easily to show why snails are likely
to be in either the dextral or sinistral forms
The snail model:
dp
dt
=
f
(
p
) =
αp
(1

p
)
p

1
2
,
0
≤
p
≤
1
,
Equilibria
are
p
e
= 0
,
1
2
,
1
f
(
p
)
<
0 for 0
< p <
1
2
, so solutions decrease
f
(
p
)
>
0 for
1
2
< p <
1, so solutions increase
The equilibrium at
p
e
=
1
2
is
unstable
The equilibria at
p
e
= 0 and 1 are
stable
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Direction Fields and Phase Port
— (38/50)
Mathematical Modeling
Introduction to MatLab
Qualitative Behavior of Differential Equations
More Examples
Maple  Direction Fields
Left Snail Model
Allee Effect
Left Snail Model
4
Phase Portrait
:
dp
dt
=
αp
(1

p
)
p

1
2
0
0.2
0.4
0.6
0.8
1
0.1
0.05
0
0.05
0.1
>
>
>
<
<
<
p
α
p
(1

p
)(
p

1
/
2)
Phase Portrait for Snail Model (
α
=1
.
5)
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Direction Fields and Phase Portraits  1D
— (39/50)
Mathematical Modeling
Introduction to MatLab
Qualitative Behavior of Differential Equations
More Examples
Maple  Direction Fields
Left Snail Model
Allee Effect
Left Snail Model
4
Diagram of Solutions for Snail Model
Snail Model
0
0.2
0.4
0.6
0.8
1
p(t)
0
2
4
6
8
10
t
Joseph M. Mahaffy,
h
[email protected]
i
Lecture Notes – Direction Fields and Phase Port
— (40/50)
Mathematical Modeling
Introduction to MatLab
Qualitative Behavior of Differential Equations
More Examples
Maple  Direction Fields
Left Snail Model
Allee Effect
Left Snail Model
5
Snail Model  Summary
Figures show the solutions tend toward one of the
stable
equilibria
,
p
e
= 0 or 1
When the solution tends toward
p
e
= 0, then the
probability of a dextral snail being found drops to zero, so
the population of snails all have the sinistral form
When the solution tends toward
p
e
= 1, then the
population of snails virtually all have the dextral form
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