Describe in words how eutrophication (i.e., increasing K) and harvesting (i.e., increasing d) in this model affect the stability of the internal equilibrium.

MATH3070: Natural Resource Mathematics ASSIGNMENT 3 – SEMESTER 2, 2021
Attempt all questions. This assignment is due on Friday, September 17, 10:00 am. Make sure that you show clearly the reasoning you use to solve the problems. It is also advisable to keep a photocopy of the assignment you hand in. You are free to use any reference you wish as long as you cite your source. However, you must define and explain all notation or concepts used that were not covered in the lecture or a prerequisite course. Each question is worth 25 points.
Q1. Consider the Rosenzweig-MacArthur predator-prey model presented in class,
dN dt
= rN1− N K− aNP 1 + ahN
,
dP dt
=
caNP 1 + ahN −dP,
where N is the prey density and P is the predator density.
(a) What are nullclines of N and P?
(b) What are equilibria of N and P? (c) Show the parameter condition in which an internal equilibrium (i.e., ¯ N, ¯P > 0) is stable.
(d) Describe in words how eutrophication (i.e., increasing K) and harvesting (i.e., increasing d) in this model affect the stability of the internal equilibrium.
(e) Derive an analytical solution of dN dt when a = 0. (f) Consider the Lotka-Volterra predator-prey model (i.e., K →∞ and h = 0 in the Rosenzweig-MacArthur model). Using the model and R, plot trajectories (P vs. N) given parameter values r = 1, a = 2, c = 0.5, and d = 0.1 with the initial condition N(0) = 0.2 and P(0) = 0.5, from t = 0 to t = 200. Repeat for N(0) = 0.5 and N(0) = 1.
(g) Derive an analytical solution of dP dN of the Lotka-Volterra predator-prey model. How does the analytical solution help explain the behaviour in the graphs of (f)?
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Q2. Consider the Lotka-Volterra competition model presented in class,
dN1 dt
= r1N1(1−α11N1 −α12N2),
dN2 dt
= r2N2(1−α21N1 −α22N2),
where Ni is the density of competing species i (i = 1,2).
(a) What are nullclines of N1 and N2?
(b) What are equilibria of N1 and N2? (c) Show the parameter condition in which an internal equilibrium (i.e., ¯ N1, ¯ N2 > 0) is stable.
(d) Show the Jacobian matrix of the model.
(e) Calculate the Jacobian matrix of the model at the internal equilibrium.
(f) When r1 = r2 = 1,α11 = α22 = 1, and α12 = α21 = 0.5, calculate eigenvalues and eigenvectors.
(g) Explain the difference between the eigenvectors and nullclines.
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Q3. Consider the following model,
dx dt
= x2y−x + ay,
dy dt
= b−ay−x2y,
where x and y are concentrations of chemical substances.
(a) What are nullclines of x and y?
(b) What are equilibria of x and y?
(c) Show the Jacobian matrix of the model.
(d) Calculate the Jacobian matrix of the model at an internal equilibrium (i.e., ¯ x, ¯y > 0).
(e) Calculate the trace and determinant of the Jacobian matrix at the internal equilibrium.
(f) Show the parameter condition in which the internal equilibrium is unstable.
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Q4. Consider the following model,
dx dt
= σ(y−x),
dy dt
= rx−xz−y,
dz dt
= xy−bz,
where x, y, and z are scaled variables and can be negative. The parameters, σ,r, and b, are positive.
(a) What are nullclines of x, y, and z?
(b) What are equilibria of x, y, and z?
(c) Show the Jacobian matrix of the model.
(d) Show the characteristic polynomial.
(e) Calculate the Jacobian matrix of the model at an equilibrium at the origin (i.e., ¯ x = ¯ y = ¯ z = 0).
(f) Show the parameter condition in which the equilibrium at the origin is un- stable.
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