MTH108: HOMEWORK 5 – NONLINEAR MODELS
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MTH108: HOMEWORK 5 – NONLINEAR MODELS
Section A – Warm-up questions
A1. Find all equilibrium solutions (constant solutions) of the difference equation xn＋1 = 3xn _ 2xn(2) .
A2. Suppose xn＋1 = F (xn ) = xn(2) _ 2 for all n > 0 . For which values of x0 ∈ R will the next term x1 be: (i) greater than x0 , (ii) less than x0 , (iii) equal to x0 .
Section B – Main questions
B1. For the difference equation xn＋1 = F (xn ), find the equilibrium points and check their stability using eF\ e:
(i) F (x) = 3 _ x/2;
(ii) F (x) = x2 ;
(iii) F (x) = 9x2 (1 _ x)/2;
(iv) F (x) = x + sin(x) .
B2. For the difference equation xn＋1 = F (xn ), make a cobweb diagram for the given initial condition:
(i) F (x) = 1 _ x/2, x0 = 0
(ii) F (x) = sin(x), x0 = 1;
(iii) F (x) = x2 /5, x0 = 4;
(iv) F (x) = 2x _ 2x2 , x0 = 0 .1.
B3. For the given initial conditions and parameter values, describe in words the qualitative behaviour of the solution (xn ) of the logistic model: (i) R = 0 .6 and x0 = 0 .7, (ii) R = 1 .6 and x0 = 0 .3, (iii) R = 2 .5 and x0 = 0 .4, and (iv) R = 3 .05 and x0 = 0 .7.
B4. The population size xn in year n of a population of baboons is modelled by the difference equation
xn＋1 = F (xn ) = 75xn _ xn(2)
Show that there is a unique positive equilibrium and that it is stable . Make a cobweb diagram to show how the population will change in the future, given that the population size is initially x0 = 10 .
B5. Suppose that the population density xn of mackerel fish in the North Sea in year n follows the Ricker model xn＋1 = F (xn ) = Rxn eR2xn . Find the equilibria, and determine how the stability varies with R .
B6. Suppose that the model xn＋1 = F (xn ) has the point 0 as its only equilibrium solution and suppose F\ (0) = 1 . Show that there are several different possible types of behaviour for solutions starting near to 0 . Make a cobweb diagram to demonstrate each case . Do the same, now supposing that F\ (0) = _1.
Section C – Extra problems
C1. Consider the logistic map with the parameter value R = 4 . There are no stable equilibria, but can you find any initial conditions for which the solution converges . How many are there?
C2. Consider the population model xn＋1 = F (xn ), where F (x) = rx2 (1 _ x) . What ranges of values of x and r are biologically meaningful? Describe how the number and stability of the equilibrium solutions changes as r is increased from zero .
2023-04-21