Equilibrium Problem Solver

Description

This program is made for solving the equilibrium problem that appears in the Ulf Dieckmann and Richard Law's population dynamics model. The solution to this problem can be described as a function that satisfies an integral equation. The second order closure of the third spatial moment is used in this equation: $$ T^{(2)}_{\alpha\beta\gamma}(x, y) = \dfrac{1}{\alpha + \beta} \left( \alpha \dfrac{C(x)C(y)}{N} + \beta \dfrac{C(x)C(y - x)}{N} + \gamma \dfrac{C(y)C(y - x)}{N} - \beta N^3 \right). $$

See the papers to get more information about the model and the problem:

• https://user.iiasa.ac.at/~dieckman/reprints/DieckmannLaw2000.pdf
• https://user.iiasa.ac.at/~dieckman/reprints/LawDieckmann2000a.pdf
• https://user.iiasa.ac.at/~dieckman/reprints/MurrellEtal2004.pdf

Compiling:

There is a makefile that processes the project compilation. You can use the following compilation flags for g++ to change some special behaviour. These flags just define preprocess variables:

Flag	Description	Default value (if present)
-DSHOUT	Turn off full info about iterations and turn on a progress bar
-DASCETIC	Turn off all output information (even a progress bar) except the first moment value
-DBAR_WIDTH=	Set a progress bar width in character number	70
-DBAR_CHAR=	Set a progress bar character	'#'

Building:

make

Cleaning:

make clean

CMD arguments:

You can give the program command line arguments. Each argument has a name that starts with '-' and usually contains only one letter. The argument value follows its name. All the arguments, you don't specify, will be set in the default values.

General arguments

Argument	Description	Type	Default value
-D	Environment dimesionality	1, 2 or 3	1
-r	Integration area size (segment length or ball radius in 1D, 2D and 3D respectively). Set it with n if you want the size to be calculated automatically	Positive real or n	20.0
-i	Iterations count	Positive integer	1000
-n	Grid nodes count	Positive integer	5000
-A	Alpha parameter of the closure	Real	1.0
-B	Beta parameter of the closure	Real	1.0
-G	Gamma parameter of the closure	Real	1.0
-d	Environmental death rating	Real	0.0
-s	Competition rating (d')	Real	1.0
-b	Birth rating	Real	1.0
-p	Path to store a calculated second moment (set it with n if you don't want to save results in the file)	String or n	"graph.plt"
-e	Accurancy of the output in decimal places	Positive integer	5
-h	Show the reference about command line parameters	None	None

The kernel setting

You can also specify the dispersal and competiton kernels. The kernel argument has the following form:

-k* [parameters]

where * is a letter that defines a kernel type, [parameters] is a list of the kernel parameters (real numbers). The dispersal kernel parameters are defined before the competiton ones. The following table describes possible kernel types:

Letter	Kernel description	Parameters
n	Gausian kernels	The standart deviations for the dispersal and competiton kernels
k	Equal kurtic kernels	s0m = s0w and s1m = s1w (two numbers)
K	General kurtic kernels	s0m, s1m, s0w, and s1w
e	Danchenko's exponential kerenels	A and B
r	Roughgarden kernels	sm, gamma_m, sw, and gamma_w
p	Exponent polynomial kernels	am, bm, aw, and bw
c	Constant kernels	Dispersal radius and competition radius

Default kernels are the equal kurtic kernels with the parameters 1.0 1.0.

Numerical method choosing

By default the program uses nonlinear Neuman's method. You can change the method, but only in the 1D or 3D case. When using linear methods the program automatically sets the arguments -A 1 -B 0 -G 0, making the equation linear (the user-specified values for these arguments are ignored in that case). Using linear methods in 2D case is ignored.

To set the method use the following argument:

-m <method name>

where <method name> can be one of the folowing:

Name	Description
neuman	Nonlinear Neuman's method. This value is default and can be used for all dimensions
lneuman	Neuman's method modification for the linear case
nystrom	Nystrom's method that uses an algebraic equation system approximation

Example:

./neuman -i 2000 -r 2.2 -kn 1.3 3.45 -p solution.txt -r n -D 3 -s 2.0 -A 0.3

Output:

The program saves a calculated second moment in the file, defined by the -p argument value (if it doesn't equal to n), in two columns (for x and for C(x)). It also prints the first moment value in the standart output if -DASCETIC is set. If -DSHOUT is set the program prints the progress bar and, after completion, the first moment value and the second moment value in zero. If neither -DASCETIC nor -DSHOUT is set then the program prints the value of the approximate first moment on each iteration.

Dependencies:

Programm uses fftw-3.3.7 library for performing convolutions using Fourier method.

Author isn't responsible for the mental health that can be damaged during reading this code