Question

consider the following systems of rate of change equations system A : dx/dt=3x(1-x/10)-1/20xy , dy/dt=-5y+xy/20, system B: dx/dt=3x-xy/100, dy/dt=15y(1-y/17)+25xy. in both of these systems,x and y refer to the number of two different species at time t.In particular, in one of these systems, the prey is large animals and the predators are small animals, such as piranhas and humans. Thus it takes many predators to eat one prey, but each prey eaten is a tremendous benefit for the predator population. The other system has very large predators and very small prey.(a) For both systems of differential equations, what does x represent? the predator or the prey? explain. (b) what system represents predator and prey that are relatively the same size? explain. (c) for the system (a0 plot all nullclines and use this plot to determine all equilibrium solutions, verify your equilibrium solutions algebraically

Answer #1

The solution is as follows

The system of differential equations
dx/dy = 0.4x − 0.004x2 − 0.001xy
dy/dt = 0.6y − 0.001y2 − 0.008xy
is a model for the populations of two species.
(a) Does the model describe cooperation, or competition, or a
predator-prey relationship?
predator-prey relationship
competition
cooperation
(b) Find the equilibrium solutions. (Enter solutions from smallest
to largest value of x. If solutions have the same value of
x, enter them from smallest to largest y.)
(x, y) = (___)
(x, y) =...

Solve the following system of ordinary differential
equations.
2 dx/dt − 2 dy /dt − 3x = e (n+1)t
2 dx/dt + 2 dy/dt + 3x + 8y = 2

Consider the following system of differential equations dx/dt =
(x^2 + 2x + 1)(x^2 − 4x + 4) dy/dt = xy − 1
Which of the following is not an equilibrium point of the above
system? (A) (3, 1/3 ) (B) (−1, −1) (C) (1, 1) (D) (1, 3)

Consider the dynamical system below.
dx/dt = -x+2y-4
dy/dt = -x-2y
a.Sketch the nullclines for this system
b. Find the equilibrium point
c. From the initial point (0,2), and using a Δt
value of 0.2, compute the current position after one iteration.

Solve the following system of differential equations:
dx/dt =x+2y
dy/dt =−x+3y

Solve the system of differential equations using laplace
transformation
dy/dt-x=0,dx/dt+y=1,x(0)=-1,y(0)=1

Use the Laplace transform to solve the given system of
differential equations. dx/dt=x-2y dy/dt=5x-y x(0) = -1, y(0) =
6

Solve the following:
(3x^2 - y^2)dx + (xy - x^3y^-1)dy = 0

Consider the following linear system (with real eigenvalue)
dx/dt=-2x+7y
dy/dt=x+4y
find the specific solution coresponding to the initial values
(x(0),y(0))=(-5,3)

Use the method for solving homogeneous equations to solve the
following differential equation.
(9x^2-y^2)dx+(xy-x^3y^-1)dy=0
solution is F(x,y)=C, Where C= ?

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