{Use of Tech} Analytical solution of the predator-prey equations The solution of the predator-prey equations
X'(t) = -ax + bxy,y’(t) = cy - dxy
can be viewed as parametric equations that describe the solution curves. Assume a, b, c, and d are positive constants and consider solutions in the first quadrant.


a. Recalling that dy/dx = y(t)/x′(t), divide the first equation by the second equation to obtain a separable differential equation in terms of x and y.

Verified step by step guidance

1. Start with the given system of differential equations: \(X'(t) = -aX + bXY\) and \(Y'(t) = cY - dXY\), where \(a\), \(b\), \(c\), and \(d\) are positive constants.
2. Recall that the derivative \(\frac{dy}{dx}\) can be expressed as \(\frac{dy/dt}{dx/dt} = \frac{Y'(t)}{X'(t)}\). Substitute the given expressions to get \(\frac{dy}{dx} = \frac{cY - dXY}{-aX + bXY}\).
3. Rewrite the expression for \(\frac{dy}{dx}\) as \(\frac{dy}{dx} = \frac{Y(c - dX)}{X(-a + bY)}\) by factoring \(Y\) in the numerator and \(X\) in the denominator.
4. Separate variables by rearranging terms to isolate \(y\) and \(x\) on opposite sides: \(\frac{dy}{y(c - dX)} = \frac{dx}{x(-a + bY)}\). This sets up the equation for integration.
5. Integrate both sides with respect to their variables to find an implicit relationship between \(x\) and \(y\), which describes the solution curves of the predator-prey system.

Key Concepts

Predator-Prey Model and System of Differential Equations

The predator-prey model describes interactions between two species using coupled differential equations. Here, x(t) and y(t) represent populations of prey and predators, respectively, with parameters a, b, c, and d governing growth and interaction rates. Understanding this system is essential to analyze population dynamics over time.

Parametric Representation and Derivative dy/dx

The solution curves of the system can be viewed parametrically with time t as the parameter. To find dy/dx, the derivative of y with respect to x, we use the chain rule: dy/dx = (dy/dt) / (dx/dt). This allows converting the system into a single differential equation relating y and x directly.

Separable Differential Equations

A separable differential equation can be written so that all terms involving y are on one side and all terms involving x on the other. This form allows integration of each side independently. Dividing the given system's equations to express dy/dx leads to a separable equation, facilitating analytical solutions.