I asked my Fisheries Dynamics students to work on a homework assignment...

So in most typical universities in Japan, homework is often something that is not provided. Oftentimes, as long as you show up to class and pass the final exams, you will earn the credits.

Lucky kids. I remember getting a boat-load of problems to solve when I was in the College of Engineering... Not to mention the exams.

So in my class, I have decided to give homework. Only 6 times, by the way, but that's more than most courses they will take.

Anyways, we are working on examining coupled predator-prey models, similar to the Lotka-Volterra equations. More precisely, I want them to do some stability analysis on the coupled differential equations. So, for home work, I assigned them a problem involving logistic growth for the prey ($V$) and a constant death rate for the predator ($P$), where $r$ is a growth rate coefficient of the prey, $K$ is the carrying capacity, $a$ is the predation rate, and $d$ is the death rate coefficient of the predator.

$$\frac{\partial f_1}{dt}=\frac{dv}{dt} = \frac{rV}{K}(K-P) - aVP$$
$$\frac{\partial f_2}{dt}=\frac{dp}{dt} = P(-d+kaV)$$

After taking the Taylor expansion of these two equations, setting them to zero and taking their first-order approximations, we get the following in matrix form:

$$\begin{bmatrix}\frac{dv}{dt}\\ \frac{dp}{dt}\end{bmatrix}=\begin{bmatrix}\frac{\partial f_1}{\partial V} & \frac{\partial f_1}{\partial P}\\ \frac{\partial f_2}{\partial V} & \frac{\partial f_2}{\partial P}\end{bmatrix}\begin{bmatrix}v \\ p \end{bmatrix}$$

All they need to do is solve for the characteristic equation of the 2x2 matrix, and get the eigenvalues.

So far, I've gotten a few emails from the students regarding the homework. This is amazing in itself, since they almost never try to interact with me!





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