FD1D
Predator Prey Simulation
in 1D
by Marcus Garvie

FD1D is a MATLAB program using finite difference methods for the dynamics of predator-prey interactions in 1 spatial dimension and time.

The MATLAB code is mostly self explanatory, with the names of variables and parameters corresponding to the symbols used in the finite difference methods described in the paper. Copies of both sets of MATLAB codes can be downloaded from http://www.csit.fsu.edu/~burkardt/m_src/pred_prey_sim/.

The code employs the sparse matrix facilities of MATLAB when solving the linear systems, which provides advantages in both matrix storage and computation time. The code is vectorized to minimize the number of "for-loops" and conditional "if-then-else" statements, which again helps speed up the computations. There is just one "for-loop" and no "if-then-else" statements in fd1d.m, and one "for-loop" and two "if-then-else" statements in fd2d.m.

The linear systems are solved using MATLAB's built in function lu.m. We remark that a pure C or FORTRAN code is likely to be faster than our codes, but with the disadvantage of much greater complexity and length.

The user is prompted for all the necessary parameters, time and space-steps, and initial data. Due to a limitation in MATLAB, vector indices cannot be equal to zero; thus the nodal indices 0,...,J are shifted up one unit to give 1,...,(J+1) so x_i=(i-1)*h + a. See Line 28 in fd1d.m and fd2d.m.

Program fd1d.m, is structured as follows:

• Lines 4-12: User prompted for model parameters.
• Lines 14-15: User prompted for initial data as a string (allowable formats discussed below).
• Lines 17-20: Calculation of some constants.
• Lines 22-25: Initialization of matrices.
• Lines 27-29: Initial data assigned numerically.
• Lines 31-38: Assembly of matrices L, B₁, and B₂.
• Lines 40-41: LU factorization of B₁ and B₂.
• Lines 43-59: Solving the linear system repeatedly up-to time level t_N=T using forward and backward substitution.
• Line 61: Numerical solution plotted for u and v at time T.

The initial data functions are entered by the user as a string, which can take several different formats. Functions are evaluated on an element by element basis, where x=(x₁,...,x_J+1) is a vector of grid points, and so a "." must precede each arithmetic operation between vectors. The exception to this rule is when applying MATLAB's intrinsic functions where there is no ambiguity. Some arbitrary examples with an acceptable format include the following:

        >> Enter initial prey function u0(x)  0.2*exp(-(x-100).^2)
        >> Enter initial predator function v0(x)  0.4*x./(1+x)
      

        >> Enter initial prey function u0(x)  0.3+(x-1200).*(x-2800)
        >> Enter initial predator function v0(x)  0.4
      

        >> Enter initial prey function u0(x)  0.4*((x>90)&(x<110))+0.1*((x<=90)|(x>=110))
      

Related Data and Programs:

FD1D is also available in a FORTRAN77 version and a FORTRAN90 version.

FD1D_PLOT is a MATLAB program which displays the solution components computed by FD1D.

FD2D is a similar program for the 2D case.

FEM1D, is an executable FORTRAN90 program which applies the finite element method, with piecewise linear basis functions, to a two point boundary value problem;

Reference:

1. Marcus Garvie,
   Computational Algorithms for Spatially Extended Predator-Prey Systems with the Holling Type II Functional Response,
   (to appear).
   

Source Code:

• fd1d.m, the code for the one dimensional problem.
• fd1d.inp, a text file containing the input that a user might enter to define a simple problem.
• fd1d.out, printed output from a run of the program with the sample input.
• fd1d.eps, an Encapsulated PostScript file containing an image of the u and v components of the solution.
• fd1d.jpg, a JPEG file containing an image of the u and v components of the solution.

You can go up one level to the MATLAB source codes.