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We´re modeling the distribution of a population on a 2-dimensional plane with the reaction-diffusion equation:

$$\frac{\partial P}{\partial t} = \nabla (D(x,y)\nabla P) + rP(1-\frac{P}{k(x,y)})$$

Where $P$ is a function of space $(x,y)$ and time($t$), $D$ and $K$ depend on space only, and $r$ is a constant

I was trying to find out if this equation has any stationary points, if/how I can find them, and if there is an analytical way to solve it, but I'm actually really new in differential equations.

Does anyone know how to do this or could you please recommend a book or paper?! Thanks a lot


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For stationary solutions, you need to look for the case $\frac{\partial P}{\partial t} = 0$. For existence and uniqueness of both the elliptic and parabolic PDE, you could try the method of upper and lower solutions. A fantastic reference is Pao's Nonlinear Parabolic and Elliptic Equations. – Pragabhava Oct 20 '12 at 2:10

You could try sepatarion of variables, but this method is for linear PDE's. So the mos effective method to solve non-linear PDE is to use similarity method where you are going to propose a elongation as follows:

$\bar{x}=\epsilon^{a}x, \bar{t}=\epsilon^{b}t, \bar{u}=\epsilon^{c}u$

and then subsitute this equalities in the original PDE. The you are going to find another PDE (or on most of the cases you'll reduce a PDE to and ODE) more easily to solve.

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