I'm trying to set up a predator-prey model, where the predator affects the distribution of the prey. Effectively, over time the prey has moved from areas of high concentration of the predator to areas of low concentration. I can set up a simple predator-prey model like:
$$\frac{\partial B}{\partial t} = r\left(1-\frac{B}{K}\right)B - \alpha B M $$
Where $B$ is the prey, $r$ is the intrinsic growth rate, $K$ is the carrying capacity, $M$ is the predator and $\alpha$ the predation rate. And obviously an equation for the predator too.
With diffusion I was thinking of having the equations:
$$\frac{\partial B}{\partial t} = r_B\left(1-\frac{B}{K_B}\right)B - \alpha B M + D_B\frac{\partial ^2 B}{\partial x^2}$$
$$\frac{\partial M}{\partial t} = r_M\left(1-\frac{M}{K_M}\right)M + \beta B M + D_M\frac{\partial ^2 M}{\partial x^2}$$
Firstly, do I have the right idea with the equations I've set up, and secondly, what exactly should I have as my diffusion constants?
I'd really appreciate any help on this because I've been trying to teach it all to myself and a lot of the stuff online is a bit too complicated for me!
Thanks!