# fd2d_predator_prey

fd2d_predator_prey, a FORTRAN90 code which solves a predator-prey system in a two dimensional region. The code requires both some interactive input from the user, and two simple routines that define the initial values.

The nondimensional problem has the form

du/dt =         del u + ( 1 - u ) * u        - v * h(u/alpha)

dv/dt = delta * del v     - gamma * v + beta * v * h(u/alpha)

in a square [A,B]x[A,B], with initial conditions:
u(x,y,0) = u0(x,y)
v(x,y,0) = v0(x,y)

and Neumann boundary conditions along the boundary of the square:
du/dn = 0
dv/dn = 0

The Type II functional response employed here is
h(eta) = eta / ( 1 + eta )

The parameters ALPHA, BETA, GAMMA and DELTA are strictly positive.

The user must input a value H specifying the desired space step to be used in discretizing the space dimension.

A finite difference scheme is employed to integrate the problem from time 0 to a maximum time T. The user must input the value T, as well as an appropriate time step DELT.

A typical input for this problem is:

ALPHA =   0.4
BETA  =   2.0
GAMMA =   0.6
DELTA =  10.0
A     =   0.0
B     = 500.0
H     =   1.0
T     = 150.0
DELT  =   0.041666666666666
SOLVE =   0

with the following initial values of U and V supplied in auxiliary subroutines:
ustar = gamma * alpha / ( beta - gamma )

u0(i,j) = ustar - 2.0E-07 * ( x(i,j) - 0.1 * y(i,j) - 225.0 )
* ( x(i,j) - 0.1 * y(i,j) - 675.0 )

vstar = ( 1.0 - ustar ) * ( alpha + ustar )

v0(i,j) = vstar - 3.0E-05 * ( x(i,j) - 450.0 )
- 1.2E-04 * ( y(i,j) - 150.0 )

### Licensing:

The computer code and data files made available on this web page are distributed under the MIT license

### Languages:

fd2d_predator_prey is available in a FORTRAN90 version and a MATLAB version.

### Related Data and codes:

FD1D_PREDATOR_PREY, a FORTRAN90 code which uses finite differences to solve a 1D predator prey problem.

### Reference:

1. Marcus Garvie,
Finite-Difference Schemes for Reaction-Diffusion Equations Modeling Predator-Prey Interactions in MATLAB,
Bulletin of Mathematical Biology,
Volume 69, Number 3, 2007, pages 931-956.

### Source Code:

Last revised on 30 June 2020.