Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus...
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Transcript of Math 3C Systems of Differential Equations Autonomous Examples Prepared by Vince Zaccone For Campus...
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
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
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
v-nullcline: y=2x
h-nullcline: y=x/4
Nullclines for this system
Here is the phase plane diagram. Notice the equilibrium point is where the nullclines intersect.
Looks stable - all the arrows point toward it.
A typical solution trajectory is shown in blue. Notice that is attracted to the equilibrium point, as expected.
)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
)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
)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
v-nullcline: y=1
h-nullcline: x=-1
Nullclines for this system
Here is the phase plane diagram. Notice the equilibrium point is where the nullclines intersect.
A typical solution trajectory.
circle centered at (-1,1)
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
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
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
h-nullcline: x=1
h-nullcline: y=0
v-nullcline: y=2.5
v-nullcline: x=0
Nullclines for this system
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)
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)
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)