Answer the following questions:
Which variable will we represent as the prey and which variable will we represent as the predator in a predator-prey model?
What’s the difference between a predator-prey model and a competing species model?
According to how this online text, what do \(a_{11}\) and \(a_{22}\) represent? What do \(a_{12}\) and \(a_{21}\) represent?
For 2-11, identify whether the model is a predator-prey model or a competing species model. Sketch the phase portrait.
\(x' = x(2-y) \hspace{0.4in} y'=y(-3+x)\)
\(x' = x(3-x-y) \hspace{0.4in} y'=y(4-2y-\frac{3}{2}x)\)
\(x' = x(1-2y) \hspace{0.4in} y'=y(-1+x)\)
\(x' = 4x(1-x-y) \hspace{0.4in} y'=y(2-y-3x)\)
\(x' = x(3-2x-y) \hspace{0.4in} y'=y(4-y-3x)\)
\(x' = x(3-2x-y) \hspace{0.4in} y'=y(-1+2x)\)
\(x' = x(3-x-2y) \hspace{0.4in} y'=y(\frac{3}{2}-2y-\frac{1}{4}x)\)
\(x' = x(12-8x-4y) \hspace{0.4in} y'=y(-1+x)\)
\(x' = x(2-x-y) \hspace{0.4in} y'=y(-1+2x)\)
\(x' = x(2-2x-2y) \hspace{0.4in} y'=y(3-2y-2x)\)
Sketch the phase portrait for the following cooperating species model. How is it different from a competing species phase portrait and explain how that makes intuitive sense? \[x' = x(4-4x+2y) \hspace{0.4in} y'=y(10-6y+x)\]
For the following system, find the values of \(\alpha\) and \(\beta\) for which there is a counterclockwise center at the critical point that is NOT the origin. (Note: \(\alpha,\beta \ne 0\))
\[x' = x(\alpha - y) \hspace{0.4in} y'=y(x - \beta)\]
For the following system of differential equations where \(\alpha, \beta, \gamma, \delta > 0\): \[\begin{aligned} \frac{\dx}{\dt} &=& x (1-\alpha x - \beta y) \\ \frac{\dy}{\dt} &=& y (2 - \gamma y - \delta x) \\\end{aligned}\]
Find ALL of the stationary points.
Calculate the Jacobian, \({\bf \partial f}(0,0)\). (ie Only evaluate the Jacobian at the origin). What type of phase portrait is it? Where do the values on the diagonal of the evaluated Jacobian originate from (within the system of differential equations)?
If \(\gamma = \beta\) and \(\delta = 5\alpha\), describe the behavior (just whether it’s a nodal sink/source, spiral, etc.) and the eigenvalues at each of the stationary points.