Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus...

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Page 1: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

Math 3C

Systems of Differential Equations Autonomous Examples

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

Page 2: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

y4xdt

dy

yx2dt

dx

Find the nulllclines and equilibrium points for this system of differential equations.

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

Page 3: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

y4xdt

dy

yx2dt

dx

Find the nulllclines and equilibrium points for this system of differential equations.

Nullclines are the curves that result when x’=0 or y’=0:

x2yyx20dt

dx

nullclinev

xyy4x0dt

dy

nullclineh

41

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

Page 4: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

y4xdt

dy

yx2dt

dx

Find the nulllclines and equilibrium points for this system of differential equations.

Nullclines are the curves that result when x’=0 or y’=0:

x2yyx20dt

dx

nullclinev

xyy4x0dt

dy

nullclineh

41

Equilibrium points occur at the intersection of the nullclines.

In this case that is at x=0,y=0

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

The next slides show the nullclines and some solution curves graphed using the PPlane application which can be found online at http://math.rice.edu/~dfield/dfpp.html

Page 5: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

v-nullcline: y=2x

h-nullcline: y=x/4

Nullclines for this system

Page 6: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

Here is the phase plane diagram. Notice the equilibrium point is where the nullclines intersect.

Looks stable - all the arrows point toward it.

Page 7: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

A typical solution trajectory is shown in blue. Notice that is attracted to the equilibrium point, as expected.

Page 8: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

)1x(dt

dy

)y1(dt

dx

Find the nulllclines and equilibrium points for this system of differential equations.

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

Page 9: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

)1x(dt

dy

)y1(dt

dx

Find the nulllclines and equilibrium points for this system of differential equations.

Nullclines are the curves that result when x’=0 or y’=0:

1yy10dt

dx

nullclinev

1x1x0dt

dy

nullclineh

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

Page 10: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

)1x(dt

dy

)y1(dt

dx

Find the nulllclines and equilibrium points for this system of differential equations.

Nullclines are the curves that result when x’=0 or y’=0:

1yy10dt

dx

nullclinev

1x1x0dt

dy

nullclineh

Equilibrium points occur at the intersection of the nullclines.

In this case that is at x=-1,y=1

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

The next slides show the nullclines and some solution curves graphed using the PPlane application which can be found online at http://math.rice.edu/~dfield/dfpp.html

Page 11: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

v-nullcline: y=1

h-nullcline: x=-1

Nullclines for this system

Page 12: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

Here is the phase plane diagram. Notice the equilibrium point is where the nullclines intersect.

Page 13: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

A typical solution trajectory.

circle centered at (-1,1)

Page 14: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

xyydt

dy

xy2x5dt

dx

Find the nulllclines and equilibrium points for this system of differential equations.Note: This is a Lotka-Volterra Predator-Prey model.

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

Page 15: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

xyydt

dy

xy2x5dt

dx

Find the nulllclines and equilibrium points for this system of differential equations.Note: This is a Lotka-Volterra Predator-Prey model.

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

25y;0x)y25(x0

xy2x50dt

dx

nullclinesv

1x;0y)x1(y0

xyy0dt

dy

nullclinesh

Page 16: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

xyydt

dy

xy2x5dt

dx

Find the nulllclines and equilibrium points for this system of differential equations.Note: This is a Lotka-Volterra Predator-Prey model.

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

25y;0x)y25(x0

xy2x50dt

dx

nullclinesv

1x;0y)x1(y0

xyy0dt

dy

nullclinesh

Equilibrium points occur when the nullclines intersect.Here we get 2 points - (0,0) and (1,2.5)

The next slides show the nullclines and some solution curves graphed using the PPlane application which can be found online at http://math.rice.edu/~dfield/dfpp.html

Page 17: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

h-nullcline: x=1

h-nullcline: y=0

v-nullcline: y=2.5

v-nullcline: x=0

Nullclines for this system

Page 18: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

Here is the phase-plane diagram for this predator-prey model.

We are only concerned with solutions where x and y are positive or zero. (negative #s of rabbits don’t make sense)

Page 19: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

A typical solution curve is shown. Notice that it orbits around the equilibrium point.

We would say that the equilibrium point at (1,2.5) is stable because the solution trajectories do not lead away from there.

(even though the curves don’t go through the equilibirium point)

Equilibrium at (1,2.5)

Page 20: Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

Here is the same equation with “harvesting” (e.g., if we have rabbits and wolves, assume each species is hunted at the same rate).

Notice that the solution is very similar to the previous case, except the equilibrium point has moved.

New equilibrium at (1.1,2.45)