Chapter 4: Problem 13
.Derive the first integral of the Lotka-Volterra system, \(a \ln y+d \ln x-\) \(c x-b y=C\).
Chapter 4: Problem 13
.Derive the first integral of the Lotka-Volterra system, \(a \ln y+d \ln x-\) \(c x-b y=C\).
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Get started for freeConsider a model of a food chain of three species. Assume that each population on its own can be modeled by logistic growth. Let the species be labeled by \(x(t), y(t)\), and \(z(t)\). Assume that population \(x\) is at the bottom of the chain. That population will be depleted by population \(y\). Population \(y\) is sustained by \(x^{\prime}\) s, but eaten by \(z^{\prime}\) s. A simple, but scaled, model for this system can be given by the system $$ \begin{aligned} x^{\prime} &=x(1-x)-x y \\ y^{\prime} &=y(1-y)+x y-y z \\ z^{\prime} &=z(1-z)+y z \end{aligned} $$ a. Find the equilibrium points of the system. b. Find the Jacobian matrix for the system and evaluate it at the equilibrium points. c. Find the eigenvalues and eigenvectors. d. Describe the solution behavior near each equilibrium point. e. Which of these equilibria are important in the study of the population model and describe the interactions of the species in the neighborhood of these point(s).
In Equation (3.153), we saw a linear version of an epidemic model. The commonly used nonlinear SIR model is given by $$ \begin{aligned} \frac{d S}{d t} &=-\beta S I \\ \frac{d I}{d t} &=\beta S I-\gamma I \\ \frac{d R}{d t} &=\gamma I \end{aligned} $$ where \(S\) is the number of susceptible individuals, \(I\) is the number of infected individuals, and \(R\) is the number who have been removed from the other groups, either by recovering or dying. a. Let \(N=S+I+R\) be the total population. Prove that \(N=\) constant. Thus, one need only solve the first two equations and find \(R=N-S-I\) afterward. b. Find and classify the equilibria. Describe the equilibria in terms of the population behavior. c. Let \(\beta=0.05\) and \(\gamma=0.2\). Assume that in a population of 100 there is one infected person. Numerically solve the system of equations for \(S(t)\) and \(I(t)\) and describe the solution being careful to determine the units of population and the constants. d. The equations can be modified by adding constant birth and death rates. Assuming these rates are the same, one would have a new system. $$ \begin{aligned} \frac{d S}{d t} &=-\beta S I+\mu(N-S) \\ \frac{d I}{d t} &=\beta S I-\gamma I-\mu I \\ \frac{d R}{d t} &=-\gamma I-\mu R \end{aligned} $$ How does this affect any equilibrium solutions? e. Again, let \(\beta=0.05\) and \(\gamma=0.2\). Let \(\mu=0.1\) For a population of 100 with one infected person, numerically solve the system of equations for \(S(t)\) and \(I(t)\) and describe the solution being careful to determine the units of populations and the constants.
Find the equilibrium solutions and determine their stability for the following systems. For each case, draw representative solutions and phase lines. a. \(y^{\prime}=y^{2}-6 y-16\). b. \(y^{\prime}=\cos y\). c. \(y^{\prime}=y(y-2)(y+3)\). d. \(y^{\prime}=y^{2}(y+1)(y-4)\).
For each problem, determine equilibrium points, bifurcation points, and construct a bifurcation diagram. Discuss the different behaviors in each system. a. \(y^{\prime}=y-\mu y^{2}\) b. \(y^{\prime}=y(\mu-y)(\mu-2 y)\) c. \(x^{\prime}=\mu-x^{3}\) d. \(x^{\prime}=x-\frac{\mu x}{1+x^{2}}\).
Evaluate the following in terms of elliptic integrals, and compute the values to four decimal places. a. \(\int_{0}^{\pi / 4} \frac{d \theta}{\sqrt{1-\frac{1}{2} \sin ^{2} \theta}}\) b. \(\int_{0}^{\pi / 2} \frac{d \theta}{\sqrt{1-\frac{1}{4} \sin ^{2} \theta}}\) c. \(\int_{0}^{2} \frac{d x}{\sqrt{\left(9-x^{2}\right)\left(4-x^{2}\right)}}\) d. \(\int_{0}^{\pi / 2} \frac{d \theta}{\sqrt{\cos \theta}}\). e. \(\int_{1}^{\infty} \frac{d x}{\sqrt{x^{4}-1}}\).
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