Question

Consider the following difference equation for populations called the nonlinear logistic model ?t+1=?xt(1−xt) (a) Find the...

Consider the following difference equation for populations called the nonlinear logistic model

?t+1=?xt(1−xt)

(a) Find the equilibria. Here ?>0 is an unknown parameter (i.e., is a constant).

(b) Determine the values of r for which each equilibrium is stable.

Homework Answers

Answer #1

so solution is

And so

We have as it is stable and

Which is so we must have

So stability criteria are

Hope this was helpful. Please do leave a positive rating if you liked this answer. Thanks and have a good day!

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
DIFFERENTIAL EQUATIONS PROBLEM Consider a population model that is a generalization of the exponential model for...
DIFFERENTIAL EQUATIONS PROBLEM Consider a population model that is a generalization of the exponential model for population growth (dy/dt = ky). In the new model, the constant growth rate k is replaced by a growth rate r(1 - y/K). Note that the growth rate decreases linearly as the population increases. We then obtain the logistic growth model for population growth given by dy/dt = r(1-y/K)y. Here K is the max sustainable size of the population and is called the carrying...
3. Consider the nonlinear oscillator equation for x(t) given by ?13 ? x ̈+ε 3x ̇...
3. Consider the nonlinear oscillator equation for x(t) given by ?13 ? x ̈+ε 3x ̇ −x ̇ +x=0, x(0)=0, x ̇(0)=2a where a is a positive constant. If ε = 0 this is a simple harmonic oscillator with frequency 1. With non-zero ε this oscillator has a limit cycle, a sort of nonlinear center toward which all trajectories evolve: if you start with a small amplitude, it grows; if you start with a large amplitude, it decays. For ε...
Given a nonlinear system   x(t)" + cx(t)' + sin(x(t)) = 0                       (3-1) a) Consider its phase...
Given a nonlinear system   x(t)" + cx(t)' + sin(x(t)) = 0                       (3-1) a) Consider its phase plane by assuming proper parameters of c. Do you have singular points with the cases of CENTER FOCUS, NODE and SADDLE ? Are those stable? asymptotic stable or unstable? b) When c = 0, plot the phase plane and find these singular points
Consider the nonlinear second-order differential equation 4x"+4x'+2(k^2)(x^2)− 1/2 =0, where k > 0 is a constant....
Consider the nonlinear second-order differential equation 4x"+4x'+2(k^2)(x^2)− 1/2 =0, where k > 0 is a constant. Answer to the following questions. (a) Show that there is no periodic solution in a simply connected region R={(x,y) ∈ R2 | x <0}. (Hint: Use the corollary to Theorem 11.5.1>> If symply connected region R either contains no critical points of plane autonomous system or contains a single saddle point, then there are no periodic solutions. ) (b) Derive a plane autonomous system...
Consider the curve c(t) = (t – 2sin(t), 1 – 2cos(t)). (a) Find an equation for...
Consider the curve c(t) = (t – 2sin(t), 1 – 2cos(t)). (a) Find an equation for the tangent line to the curve at t=π/6 (b) For 0 ≤ t ≤ 2π, find the interval(s) of t-values that make dx/dt < 0.
The number of fish per square kilometer in a lake is determined by the DTDS xt+1=400xt200+0.1xt...
The number of fish per square kilometer in a lake is determined by the DTDS xt+1=400xt200+0.1xt , where t is the time in years since the beginning of the observation. The initial observation is of 2,100 fish per square kilometer. a) How many fish per square kilometer will there be after three years? Give your answer with an accuracy of two decimal places. Answer:       b) Find the updating function f. Answer: f(x)=     c) Find the inverse of the updating function....
Consider the following IS-LM model: (1 - b)Y + i1r - a - G = i0...
Consider the following IS-LM model: (1 - b)Y + i1r - a - G = i0 - bT (1) c1Y = Ms + c2r (2) where Y and r are the endogenous variables. Solve for Y and r using matrix algebra. How are these equilibrium values affected by increases in Ms? Increases in T? (i.e. nd the comparative statics)
The Hubble time is defined as t​H = 1/H​0. Consider a Closed model for the Universe...
The Hubble time is defined as t​H = 1/H​0. Consider a Closed model for the Universe without a cosmological constant, so that the mass density parameter is Ω​M > 1. Give an expression for the age t​max (in units of t​H) of a Closed model at the time when the Universe has reached its maximum size, is no longer expanding, and is about to start contracting.
Problem 2. Consider the following example of the IS-LM model: C = 340 + 0.5(Y–T) I...
Problem 2. Consider the following example of the IS-LM model: C = 340 + 0.5(Y–T) I = 400 – 1500(r + x) G = 150 T = 100 x=0.02 r = 0.04 ?e = 0.02 (1) Derive the IS equation. (2) Find the equilibrium value of Y. (3) Write down the zero lower bound constraint. Does the real interest rate of r=0.04 satisfy the constraint? (4) Suppose that the risk premium x has increased to x=0.09. (a) Derive the new...
Consider the lines ℓ1 = { (−1 + 2t, t, α − t) |t ∈ R}...
Consider the lines ℓ1 = { (−1 + 2t, t, α − t) |t ∈ R} and ℓ2 = { (2s, α + s, 3s) | s ∈ R }. (a) There is only one real value for the unknown α such that the lines ℓ1 and ℓ2 intersects each other at a point P ∈ R^3 . Calculate that value for α and determine the point P. (b) If α = 0, is there a plane π ⊂ R^3...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT