Reference no: EM13960495
A model for the spruce budworm population u(t) is governed by
du/dt = ru (1-u/q)-u2/(1+u2), (1)
where q and r are positive dimensionless parameters. The non-zero steady states are thus given by the intersection of the two curves
U(u) = r(1-u/q), V(u) - u/(1+u2)
(a) Show, using the conditions for a double root, that the curve in q, r space which divides it into regions where there are 1 or 3 positive steady states is given parametrically by
q(a) = 2a3/(a2-1), r(a) =2a3/(1+a2)2
[Hint: find a cubic H(u) such that at the double root u = a, H(a) = H'(a) = 0 and solve these equations for q and r.]
(b) Show that the two curves meet in a cusp, that is where q'(a) = r'(a) = 0, at a =√3
(c) Use MAPLE to plot the curves in (q(a),r(a)) space, for 1.01 < a < 20 noting in particular the behaviours of q(a) and r(a) as a→∞ and a→1.
(d) Consider equation (1) in the case when r = 0.55. Use MAPLE to plot the solution for various choices of the initial condition u(0) in the cases when q = 6, q = 10 and q = 20. Discuss the change in behaviour as q increases.
|
What would be my over all grade in percentage
: What would be my over all grade in percentage? Basic arithmetic problem - What is the overall percentage of five test grades: 96%, 98%, 89%, 95% & 97%? I need the work shown.
|
|
Find the only way to construct a length from the parameters
: Find the only way to construct a length from the parameters in the Schrodinger equation for the harmonic oscillator. Compare the result with the classical amplitude of oscillation when the energy is hw/2.
|
|
Problem regarding the optimization of a box
: The problem is looking find the minimum material and dimensions needed to build a box that has a volume of 2 cubic ft. The box has to have a volume of 2ft3. The box is folded and has 4 flaps on the top and bottom as shown below. The price of mater..
|
|
What does the given passage mean
: What does the passage mean? Using your own words, convey the literal meaning of Marx's claims in this passage as simply and clearly as you can. Refer to the surrounding text to support your interpretation
|
|
Discuss the change in behaviour as q increases
: A model for the spruce budworm population u(t) is governed by du/dt = ru (1-u/q)-u2/(1+u2), Discuss the change in behaviour as q increases.
|
|
What are some of the objections to standard costs
: What are some of the objections to standard costs and variance analysis, especially overhead volume variances?
|
|
Write a formula for the mass of a planet according to kepler
: Given the above problem, provide written responses to address the following points as precisely and thoroughly as possible. Explain the problem in your own words. What mathematical concepts learned in this module apply to this problem?
|
|
Annual fixed and production costs
: The management has decided to undertake a preliminary analysis in order to evaluate the minimum cost for the network configuration under the assumption that the network will be designed from scratch (i.e., as if no plants currently exist) The annu..
|
|
What is the probability of choosing a ball numbered 30
: A lottery game contains 29 balls numbered 1 through 29. What is the probability of choosing a ball numbered 30?
|