MATH MODELING(NEED THE CODE) In this lab you will explore the Lotka-Volterra mod

Question

MATH MODELING(NEED THE CODE)

In this lab you will explore the Lotka-Volterra model of competition between two species. Suppose that rabbits and sheep compete for the same food supply (grass) and the amount available is limited. Let x(t) be the population of rabbits and y(t) be the population of sheep, where x, y 0. For each of the models given below: (a) Plot the nullclines (b) Determine the number of fixed points (c) Determine the type and stability of each fixed point (d) Draw a phase portrait, showing trajectories starting from enough different initial conditions to demonstrate all the qualitatively different solutions (e) Provide a brief biological interpretation of the solutions

1. dx /dt = x(3 x 2y)

dy /dt = y(2 x y)

2. dx /dt = x(3 x y)

dy /dt = y(2 x y)

3. dx /dt = x(3 2x y)

dy /dt = y(2 x y)

4. dx/dt = x(3 2x 2y)

dy /dt = y(2 x y)

Explanation / Answer

You can use a technique known as Separation of Variables.
Take all the y to one side and the t on the other...
You get:

dy2y10=dt

Now you can integrate both sides with respect to the correspondent variables:

12y10dy=dt
12(y5)dy=dt

And finally
12ln(y5)=t+c

Now you can express y as:
ln(y5)=2t+c
y5=c1e2t where c1=ec
y=c1e2t+5

You can substitute back to check your result (calculating dydt) remembering that now it is: y=c1e2t+5

Whenever it exists, the limit
J(y0, v) := limt0
J(y0 + tv) J(y0)
t
is called the Gâteaux derivative (or first variation) of J at y0 in the direction v E.
This defines a mapping J(y0, ·) : E R. If this mapping is linear and continuous,
we denote it by J

(y0) and say that J is Gâteaux differentiable at y0. Thus, under
these conditions, J(y0, v) = J

(y0)(v). Another common notation is J(y0, v) =
y0 J(v).
The y0 D such that J

(y0) = 0 are called critical points of the functional J.
The extension to higher-order derivatives is immediate. If, for a fixed v E,
J(z, v) exists for every z D, we have a mapping D : R and we can compute its
Gâteaux derivative. Given an y0 D and z, v E, the second Gâteaux derivative
(or second variation) of J at y0 in the directions v and z (in that order) is

2
J(y0, v, z) := limt0
Jy0+tz(v) y0 J(v)

Related Questions

Navigate

• Browse (All)
• Browse M
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.