Question

Analytical solution of the predator-prey equations The solution of the predator-prey equations $$x^{\prime}(t)=-a x+b x y, y^{\prime}(t)=c y-d x y$$ 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 \(\frac{d y}{d x}=\frac{y^{\prime}(t)}{x^{\prime}(t)},\) divide the first equation by the second equation to obtain a separable differential equation in terms of \(x\) and \(y\) b. Show that the general solution can be written in the implicit form \(e^{d x+b y}=C x^{r} y^{a},\) where \(C\) is an arbitrary constant. c. Let \(a=0.8, b=0.4, c=0.9,\) and \(d=0.3 .\) Plot the solution curves for \(C=1.5,2,\) and \(2.5,\) and confirm that they are, in fact, closed curves. Use the graphing window \([0,9] \times[0,9]\)

Short answer

In this problem, we first obtain a separable differential equation by dividing the given equations, and then show that the general solution can be written in the implicit form \(e^{dx+by}=Cx^ry^a\), where \(C\) is an arbitrary constant. We then plot the solution curves for specific values of constants and observe that they are closed curves. This confirms our initial analysis that the problem exhibits the mentioned properties.

Step-by-step solution

Divide the first equation by the second equation

First, divide both sides of the first equation by \(x':\) $$\frac{dy}{dx} = \frac{y'(t)}{x'(t)} = \frac{cy - dxy}{-ax + bxy}$$ #b. Convert the equation into separable form#

Rearrange the terms to separate x and y variables

Manipulate the resulting equation from step 1 to separate x and y variables: $$\frac{dy}{y(c - xd)} = \frac{dx}{x(-a + by)}$$

Integrate both sides of the equation

Now integrate both sides of the equation: $$\int \frac{dy}{y(c - xd)} = \int \frac{dx}{x(-a + by)}$$ Let \(u=-a+by\) and \(v=c-dx\). Then \(\frac{du}{y}=-b \, dy\) and \(\frac{dv}{x}=-d \, dx\). Thus, we have $$-b \int \frac{dy}{yv} = -d \int \frac{dx}{ux}$$ After integration, we get: $$-b\ln|yv|+K_1= -d\ln|ux|+K_2$$ where \(K_1\) and \(K_2\) are constants of integration.

Solve for the general solution

Combine the constants \(K_1\) and \(K_2\) into a new constant, \(K\), and solve for the general solution of the differential equation: $$-b\ln|yv|+K= -d\ln|ux|$$ $$\ln|yv|^d = K\ln|ux|^b$$ $$|yv|^d = C|xu|^b$$ $$e^{(d\ln|yv|)} = C e^{(b\ln|ux|)}$$ Where \(C=e^{K}\) is an arbitrary constant. Substituting \(u\) and \(v\) back into the equation: $$e^{dy + bx} = C x^{-a} y^{-b}$$ This is the implicit form of the general solution. #c. Plot the solution curves#

Substitute the given values for constants

Substitute the given values of \(a=0.8\), \(b=0.4\), \(c=0.9\), and \(d=0.3,\) into the solution and plot the solution curves for \(C=1.5,2,\) and \(2.5.\) For each value of \(C\), plot \(e^{0.3x + 0.4y}=Cx^{-0.8}y^{-0.4}\) using the graphing window \([0,9]\times[0,9].\)

Analyze closed curves

By observing the plotted curves, we can confirm that they are indeed closed curves.

Key concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. These equations describe how quantities change over time. They are essential in modeling real-world phenomena such as population growth, heat transfer, and motion. For predator-prey models, differential equations help us understand the dynamic interactions between species.
The predator-prey system is expressed as two linked differential equations: \(x^{\prime}(t) = -ax + bxy\) and \(y^{\prime}(t) = cy - dxy\). Here, \(x(t)\) and \(y(t)\) represent the population sizes of prey and predators, respectively. The rates \(a, b, c,\) and \(d\) are constants that characterize the interaction effects, such as predation rate and reproduction rate. By solving these equations, we can predict how populations change over time. Whether populations reach stability or exhibit cycles often depends on the specific values of these constants.

Separable Equations

Separable equations are a type of differential equation where variables can be separated onto different sides of the equation, making them easier to solve. To be separable, an equation should allow us to express it in the form \(g(y) \, dy = f(x) \, dx\).
In the predator-prey model, the first step to finding a solution is separating variables. We take the steps to express it as: \(\frac{dy}{y(c - xd)} = \frac{dx}{x(-a + by)}\). This formulation allows us to integrate each side separately, which is crucial for finding the general solution. The ability to separate variables simplifies the otherwise complex system into manageable parts, facilitating analytical or numerical solutions.

Implicit Solutions

An implicit solution is a form where the solution to a differential equation is given without explicitly solving for one variable in terms of another. Instead, the relationship between the variables is presented as an equation.
For the predator-prey model, the solution can be expressed as \(e^{dy + bx} = C x^{-a} y^{-b}\). This equation represents a complex relationship between predator and prey populations without directly isolating one variable. Implicit solutions are often necessary when the mathematical complexity makes solving explicitly impossible or overly complicated. By working with implicit forms, we can derive valuable insights about system behavior, such as identifying equilibrium points or closed curves around specific conditions.

Solution Curves

Solution curves, often called trajectories or orbits, graphically represent the set of all points that satisfy a differential equation. In a predator-prey model, these curves illustrate how populations of predator and prey interact over time.
After solving the differential equation implicitly, the next step is to plot solution curves for various values of the constant \(C\). The expression \(e^{0.3x + 0.4y} = C x^{-0.8} y^{-0.4}\) generates different curves, depending on \(C\). By plotting these over the specified range, we visualize the cycles of rising and falling population sizes, a classic characteristic of predator-prey interactions. Observing closed curves implies that the populations return to a previous state, an indication of periodicity or stability in the ecosystem dynamics.

Parametric Equations

Parametric equations are a set of equations where the variables are expressed as functions of one or more independent parameters. This representation is particularly useful in describing curves and surfaces.
In predator-prey equations, parametric representations express \(x\) and \(y\) as functions of the parameter \(t\), the time variable, which is not shown explicitly in our implicit solution. The parametric form helps in tracing the time evolution of predator and prey populations. By using parametric equations, we can more accurately plot solution curves based on realistic data or simulation outcomes, observing how populations fluctuate over time. This technique provides a dynamic and comprehensive perspective on changing ecosystem conditions.