91Ó°ÊÓ

Host-Parasitoid Interactions Find all biologically relevant equilibria of the negative binomial host-parasitoid model $$ \begin{array}{l} N_{t+1}=4 N_{t}\left(1+\frac{0.01 P_{t}}{2}\right)^{-2} \\ P_{t+1}=N_{t}\left[1-\left(1+\frac{0.01 P_{t}}{2}\right)^{-2}\right] \end{array} $$ and analyze their stability.

Determine the equation of the level curves \(f(x, y)=c\) and sketch the level curves for the specified values of \(c\). \(f(x, y)=y-x^{2} ; c=0,1,2\)

Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. $$ f(x, y)=x^{2}-y^{2} ; 2 x+y=1 $$

Find the indicated partial derivatives. \(f(x, y)=x e^{y} ; \frac{\partial^{2} f}{\partial x \partial y}\)

Host-Parasitoid Interactions Find all biologically relevant equilibria of the negative binomial host-parasitoid model $$ \begin{array}{l} N_{t+1}=4 N_{t}\left(1+\frac{0.01 P_{t}}{0.5}\right)^{-0.5} \\ P_{t+1}=N_{t}\left[1-\left(1+\frac{0.01 P_{t}}{0.5}\right)^{-0.5}\right] \end{array} $$ and analyze their stability.

Find a linear approximation to each func\mathrm{tion } \(f(x, y)\) at the indicated point. \(\mathbf{f}(x, y)=\left[\begin{array}{l}\frac{x}{y} \\\ \frac{y}{x}\end{array}\right]\) at \((1,2)\)

Determine the equation of the level curves \(f(x, y)=c\) and sketch the level curves for the specified values of \(c\). \(f(x, y)=x y ; c=0,1,2\)

Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. $$ f(x, y)=x^{2}+y^{2} ; 3 x-2 y=4 $$

Use Lagrange multipliers to find the maxima and minima of the functions under the given constraints. $$ f(x, y)=x y^{2} ; x^{2}-y=0 $$

Find a linear approximation to each func\mathrm{tion } \(f(x, y)\) at the indicated point. \(\mathbf{f}(x, y)=\left[\begin{array}{c}(x+y)^{2} \\ x y\end{array}\right]\) at \((-1,1)\)