Have a look at J.D. Murray, mathematical biology, 2007, vol II, Banks R.R. Growth and diffusion phenomena, Springer Verlag 1991, Okubo A. 1981, diffusion and ecological problems, and Shigesada N, & Kawasaki K (1997): Biological invasions: theory and practice. Oxford University Press, For numerical schemes see "Numerical recipes", the chapter on partial differential equations. As you refer to a constant source for preys, it seems that you intend to work in a single dimension of space, with a Dirichlet condition for preys. Functional response can only concern the prey equation. The interaction term of the predator is governed by its numerical response, which has much more variants as the three Holling types of functional response. Logistic growth for the predator is quite unusual, in most of models, it is supposed to die in the absence of preys. Okubo, in the book I referred above, studied diffusion-reaction predator-prey models. I don't remember precisely what cases he considered. For diffusion problems, implicit schemes are generally considered as the best solutions. In the case of two dimensions of space, operator splitting are convenient.

For the reaction term, the diffusion implicit scheme may be coupled with a Runge-Kutta reaction term.

Good luck !

Have a look at J.D. Murray, mathematical biology, 2007, vol II, Banks R.R. Growth and diffusion phenomena, Springer Verlag 1991, Okubo A. 1981, diffusion and ecological problems, and Shigesada N, & Kawasaki K (1997): Biological invasions: theory and practice. Oxford University Press, For numerical schemes see "Numerical recipes", the chapter on partial differential equations. As you refer to a constant source for preys, it seems that you intend to work in a single dimension of space, with a Dirichlet condition for preys. Functional response can only concern the prey equation. The interaction term of the predator is governed by its numerical response, which has much more variants as the three Holling types of functional response. Logistic growth for the predator is quite unusual, in most of models, it is supposed to die in the absence of preys. Okubo, in the book I referred above, studied diffusion-reaction predator-prey models. I don't remember precisely what cases he considered. For diffusion problems, implicit schemes are generally considered as the best solutions. In the case of two dimensions of space, operator splitting are convenient.

For the reaction term, the diffusion implicit scheme may be coupled with a Runge-Kutta reaction term.

Good luck !

Just using Sundials (http://computation.llnl.gov/casc/sundials/main.html ) to solve this should work.

See the documentation for references and example problems: http://computation.llnl.gov/casc/sundials/documentation/documentation.html

I have also looked at Sundials, but it is entirely for ODEs and DAEs. I concur with Marc on that. Malengier, Sundials is a software i have not been aware of although I have worked with systems of ODEs, I appreciate that information.

As far as I know, at the contrary of ODEs, there is no automatic solver available for PDEs, probably because the combinations of equation types and border conditions are too specific to each given problem. Some good libraries exist, aniway. Marc gave you the track of finite elements packages, an interesting alternative to finite difference methods, especially performing when you work on a space with complicated geometry. Some elements of PDEs solving exist also in formal calculus softwares as Mathematica or Maple, Useful libraries also exist in Matlab and Scilab (Scilab is open source and free). In any cases, some skill in programming is required. If you have no experience in PDEs, do not hesitate to spend time in programming small and simple problems on small grids or few finite elements, to get an insight on the problems which may arise (many). But please, precise the kind of geometry you need (1d, 2d,3d finite, infinite, straight or plane borders, curved ones, etc...) all this is of prominent importance.

You use the method of lines to convert whatever complicated PDE into a system of nonlinear ODE. As long as the grid is 1D, sundials will have no problems with the nonlinearities. 2D or 3D quickly become computationally too expensive, but have been done (eg using moving gridpoint techniques to reduce the number of gridpoints).

The nicety of using an ODE system is that ODE solvers have no problems with nonlinearities, doing automatic step refinement and having error control. Using a good grid is important though, as that is a fixed error you have. Going to PDE with nonlinearities, you need to linearize or do very expensive iterations for convergence (eg Picard). If the solver chokes on the solution, you actually know that the solution just diverged too much of the correct solution, and it is up to you to allow higher errors.

I solve convection-diffusion with nonlinear absorption like that, see my papers. Well, I'm no expert with predator-prey, but looking at the equations in a reference, it seems feasible to me. Having the actual equations as John sees them would allow to be certain.

I endorse the suggestions given by B. Malenger, Jean-Sebastien, Marc and John; collectively they should be helpful as you start your adventure. Solving nonlinear reaction diffusion equation can get very complicated depending on the nonlinearities. Issues of multiple solutions, bifurcations, and selecting parameter values can be very challenging. If you have a simple domain I would suggest using a pseudo-spectral method, with a suitable time integration method. To get your feet wet, I would suggest looking at Turing patterns ( See Murray's book) and then move on from there.

For fun, I have attached a PDF of a Mathematica notebook showing how to simulate the "spots on a leopard" using Turing patterns with a very simple finite difference method. Enjoy

I recommend you google FlexPDE, it's a very easy to use general-purpose pde solver; they have a free student version which is usually sufficient for these types of problems. I use it all the time for precisely these types of things. You can even grab a sample flexPDE script which solves exactly this type of pde's from my website, see these links:

http://www.mathstat.dal.ca/~tkolokol/classes/patterns/

http://www.mathstat.dal.ca/~tkolokol/classes/patterns/gs-split.pde

Thanks to Theodore ! FlexPDE seems very nice ! exactly what you need. And I'll use it willingly. Theodore's web pages are very nice too.

You could like to have a look at Simflowny as well

New to me. Thanks.

Ill try it out!

... AnyLogic

You can refer to my paper, It will surely help you.

Arya, R. K., 2013, “Finite Element Solution of Coupled-Partial Differential and ordinary Equations in Multicomponent Polymeric Coatings”, Computers and Chemical Engineering, 50, 152-183, DOI: 10.1016/j.compchemeng.2012.10.015 .

I refer you to the paper:

Manaa, S.A. and Qasem A.F., (2007), “Numerical Solution of Non-Linear Prey-Predator System Using Finite Elements Method”, Raf. J. of Comp. & Maths. , 4(2), 113-133.

http://www.iasj.net/iasj?func=fulltext&aId=8002

You can have some ideas also from my papers: http://www.sciencedirect.com/science/article/pii/S0898122111002963 and http://www.sciencedirect.com/science/article/pii/S0377221710007666 .

Traveling Wave Analysis of Partial Differential Equations :Numerical and Analytical Methods with MATLABr and Maple, This book written by GrahamW. Griffiths and William E. Schiesser will be help you.

I think the following link might help you:

1. http://www.macs.hw.ac.uk/~jas/paperpdfs/sherratt2016siap.pdf

2. http://www.macs.hw.ac.uk/~jas/paperpdfs/sherratt2001.pdf

3. http://www.macs.hw.ac.uk/~jas/paperpdfs/ecology.pdf

4. http://www.macs.hw.ac.uk/~jas/paperpdfs/taylorsherrattwhite2013.pdf

5. http://www.macs.hw.ac.uk/~jas/paperpdfs/sherrattlambinsherratt2002.pdf

And if your interest in periodic traveling waves (wavetrains) in predator-prey systems, I can help you.

With the SUNDIALS solvers you can readily solve this utilizing method of lines approach.

Python bindings of SUNDIALS exists, once installed, you can call it easily then.

Example: https://github.com/bmcage/odes/blob/master/ipython_examples/Planar%20Pendulum%20as%20DAE.ipynb

Hi John. I am sure you managed to solve this... Finite difference methods should do the trick.