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Chapter 4: Problem 13

.Derive the first integral of the Lotka-Volterra system, \(a \ln y+d \ln x-\) \(c x-b y=C\).

Short Answer

Expert verified
The derivative of the first integral of the Lotka-Volterra system is \(\frac{a}{y}+\frac{d}{x}-c-b=0\).

Step by step solution

01

Identify the constants and variables

In this equation, \(x\) and \(y\) are variables, while \(a,b,c,d\) and \(C\) are constants.
02

Derive the constants and logarithms

The derivative of a constant is zero and the derivative of the natural logarithm function, \(\ln x\), is \(1/x\). Therefore, the derivative of \(a \ln y\) is \(\frac{a}{y}\). Similarly, the derivative of \(d \ln x\) is \(\frac{d}{x}\).
03

Derive the product of variables

The derivative of \(c x\) is \(c\) and the derivative of \(-b y\) is \(-b\).
04

Combine all the derivatives

Combining all these, the derivative of the whole expression \(a \ln y+d \ln x-c x-b y\) is \(\frac{a}{y}+\frac{d}{x}-c-b\)
05

Set the equation

Set the derivative of the whole equation equal to 0, because the derivative of the constant \(C\) on the right-hand side of the original equation is 0.
06

Final Equation

The derivative of the first integral of the Lotka-Volterra system is therefore: \(\frac{a}{y}+\frac{d}{x}-c-b=0\)

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Most popular questions from this chapter

Another population model is one in which species compete for resources, such as a limited food supply. Such a model is given by $$ \begin{aligned} &x^{\prime}=a x-b x^{2}-c x y \\ &y^{\prime}=d y-e y^{2}-f x y \end{aligned} $$ In this case, assume that all constants are positive. a. Describe the effects/purpose of each terms. b. Find the fixed points of the model. c. Linearize the system about each fixed point and determine the stability. d. From the above, describe the types of solution behavior you might expect, in terms of the model.

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. \(x^{\prime}=x-\frac{\mu x}{1+x^{2}}\).

Solve the general logistic problem, $$ \frac{d y}{d t}=k y-c y^{2}, \quad y(0)=y_{0} $$ using separation of variables.

Show that the system \(x^{\prime}=x-y-x^{3}, y^{\prime}=x+y-y^{3}\), has at least one limit cycle by picking an appropriate \(\psi(x, y)\) in Dulac's Criteria.

Consider 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).
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