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Consider a cooperating species model dx dt dy dt x(1-x) + Bxy = y(1-y) + Bxy, = where the species interact weakly, say = 1/2. (a) Find the x and y-nullelines for this system. (b) Find all equilibrium points for this system. (c) Sketch the phase plane for this system. (d) If the initial populations are given by (0) what happens to the two populations as t→→∞o? = 2.4 and y(0) 0.3,

Consider a cooperating species model dx dt dy dt x(1-x) + Bxy = y(1-y) + Bxy, = where the species interact weakly, say = 1/2. (a) Find the x and y-nullelines for this system. (b) Find all equilibrium points for this system. (c) Sketch the phase plane for this system. (d) If the initial populations are given by (0) what happens to the two populations as t→→∞o? = 2.4 and y(0) 0.3,

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.3: Maxima And Minima
Problem 20E
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I have parts a and b completed but was still needing some clarification on if I did them correctly. So help on parts a through d would be very helpful thanks!

**Cooperating Species Model** Consider a cooperating species model: \[ \begin{aligned} \frac{dx}{dt} &= x(1 - x) + \beta xy \\ \frac{dy}{dt} &= y(1 - y) + \beta xy, \end{aligned} \] where the species interact weakly, say \(\beta = \frac{1}{2}\). 1. **Find the \(x\) and \(y\)-nullclines for this system.** 2. **Find all equilibrium points for this system.** 3. **Sketch the phase plane for this system.** 4. **If the initial populations are given by \(x(0) = 2.4\) and \(y(0) = 0.3\), what happens to the two populations as \(t \to \infty\)?** --- For visual aids such as graphs or diagrams: - **Nullclines**: - The \(x\)-nullcline is found where \(\frac{dx}{dt} = 0\). - The \(y\)-nullcline is found where \(\frac{dy}{dt} = 0\). - **Equilibrium Points**: - These are the points where both \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\). - **Phase Plane**: - A sketch showing trajectories of the system in the \(xy\)-plane, illustrating the behavior of the system over time. **Note**: Include the phase plane sketch and specific calculations if available for clarity.
Transcribed Image Text:**Cooperating Species Model** Consider a cooperating species model: \[ \begin{aligned} \frac{dx}{dt} &= x(1 - x) + \beta xy \\ \frac{dy}{dt} &= y(1 - y) + \beta xy, \end{aligned} \] where the species interact weakly, say \(\beta = \frac{1}{2}\). 1. **Find the \(x\) and \(y\)-nullclines for this system.** 2. **Find all equilibrium points for this system.** 3. **Sketch the phase plane for this system.** 4. **If the initial populations are given by \(x(0) = 2.4\) and \(y(0) = 0.3\), what happens to the two populations as \(t \to \infty\)?** --- For visual aids such as graphs or diagrams: - **Nullclines**: - The \(x\)-nullcline is found where \(\frac{dx}{dt} = 0\). - The \(y\)-nullcline is found where \(\frac{dy}{dt} = 0\). - **Equilibrium Points**: - These are the points where both \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\). - **Phase Plane**: - A sketch showing trajectories of the system in the \(xy\)-plane, illustrating the behavior of the system over time. **Note**: Include the phase plane sketch and specific calculations if available for clarity.
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