Hopf Bifurcation - Biological Interpretation - SIR Model - MATLAB Assignment Help

Assignment Task

Hopf Bifurcation - Biological Interpretation - SIR Model - MATLAB Assignment Help

1. Consider two competing populations governed by the equations x0 = x(5−x)−xy, y0 = y(5−y)(y−1)−axy where x,y ≥ 0 are the dimensionless populations and a > 0 is a parameter. (a) Give a biological interpretation of the terms in the equations. (b) Find the equilibria when x = 0 and use (only) 1-dimensional graphical techniques to analyze their stability. (c) Find the equilibria when y = 0 and use (only) 1-dimensional graphical techniques to analyze their stability. (d) Find the non-trival equilibrium of the system and classify its stability. (e) For any equilibria with real eigenvalues, ?nd the corresponding eigenvectors. (f) Find the parameter values at which any “λ = 0” and Hopf bifurcations occur and state the type of bifurcation corresponding to each. Be sure to con?rm any analytical results with numerical/graphical evidence of the bifurcation. (g) Either with the computer or by hand, sketch the qualitatively distinct phase portraits. (h) Use the results of the previous parts to draw the bifurcation diagrams in the (a,x∗) plane.
2. Consider the SIR model with vital dynamics where newborns are born with immunity that later disappears at a rate p. The modi?ed SIR model is then
dS/dt = pR−β I/N S −µS
dI/dt = β I/N S −αI −µI
dR/dt = αI −µR−pR + µN
(a) Show that the population level remains constant. (b) Compute the DFE and EE. (c) Determine the stability of the DFE. If it undergoes/is involved in any bifurcations, state what these are and the parameter relations that give them. (d) Show that no Hopf bifurcation is possible in this system. (e) Either analytically or with numerical evidence, determine the stability of the EE. If it undergoes/is involved in any bifurcations, state what these are and the parameter relations that give them. (f) Either with the computer or by hand, sketch the qualitatively distinct phase portraits. (g) Find the basic reproductive number and interpret it. Compare the results of this model with the typical “born susceptible” model and explain any di?erences.
3. Consider the 1-D map xn+1 =rx, 0 ≤ xn ≤ 1 2, r(1−x), 1 2 < xn ≤ 1, 0 ≤ r ≤ 2.

(a) Determine the ?xed points and their stability.

(b) Verify your results above with a cobweb picture, either carefully by hand or using MATLAB. (c) For the parameter value r = 2, show there is a 2-cycle and determine its stability. (d) Use MATLAB to draw the orbit (or bifurcation) diagram.
4. Consider the system

? x = −y−z ? y = x + ay ? z = 2 + z(x−4),

where x,y,z ∈R and a ≥ 0 is a parameter.

(a) Find the equilibria of the system and give conditions for when they exist.

(b) By examining the characteristic equation, show that no “λ = 0” bifurcations occur.

(c) Find the a-value where a Hopf bifurcation occurs.

(d) Based on MATLAB plots, what type of Hopf bifurcation is it? Show these plots from which you draw your conclusions. As with the Lorenz system (in class), you may use whatever projection may be most helpful to observe the dynamics (or remain in 3-d phase space).

(e) Experiment with additional a-values and submit plots that suggest the system is chaotic.

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