Contents

Generate system files for the Bazykin’s predator-prey system

This Jupyter Notebook generates the system files for the following predator-prey ecosystem

\[\begin{split} \begin{cases} \begin{aligned} \dot x_1 &= x_1 - \frac{x_1 x_2}{1+\alpha x_1} - \epsilon x_1^2, \\ \dot x_2 &= -\gamma x_2 + \frac{x_1 x_2}{1+\alpha x_1} - \delta x_2^2. \\ \end{aligned} \end{cases} \end{split}\]

These are used in the Bazykin’s predator-prey demo.

Add MatCont path and load sym package if GNU Octave is used

matcontpath = '../';
addpath(matcontpath);
addpath([matcontpath, '/Utilities']);
if isOctave
  pkg load symbolic % for GNU Octave
end

Define the system name

system_name = 'Bazykin';

Create coordinates and parameter names as strings

coordsnames = {'x1', 'x2'};
parnames = {'alpha', 'delta'};

Create symbols for coordinates and parameters

The array par is the array of symbols in the same order as parnames. Due to the following two lines we may, for example, use either alpha or par(1). There should no changes be need of this code.

syms(parnames{:});       % create symbol for alpha and delta
par=cell2sym(parnames);  % now alpha1 is par(1) etc
syms(coordsnames{:});    % create symbol for alpha and delta
coords=cell2sym(coordsnames); % create 1 x n vector for coordinates

Define fixed parameters

gamma = 1;
epsilon = 0.01;

Define the system

dx1_dt = x1 - x1*x2/(1+alpha*x1) - epsilon*x1^2;
dx2_dt = -gamma*x2 + x1*x2/(1+alpha*x1) - delta*x2^2;
system = [dx1_dt; dx2_dt];

In general there are no modifications needed after this line.

Differentiate and generate code (directional derivatives)

Exporting it to <system_name>.m. This method uses directional derivatives. Then using polarization identities derivatives can be calculated in arbitrary direction.

suc = generate_directional_derivatives(...
  system,...   % n x 1 array of derivative symbolic expressions
  coords,... % 1 x n array of symbols for states
  par,...      % 1 x np array of symbols used for parameters
  system_name,... % argument specifying the system name
  [matcontpath, 'Systems/']... % directory to save to file
);

Higher-order parameter-dependent multi-linear form.

Exporting it to <system_name>_multilinearforms.m. These multi-linear forms are currently only used in the computation of the parameter-dependent center manifold for the codimension two Bogdanov-Takens bifurcation.

order = 3;
suc = generate_multilinear_forms(system_name, system, coords, par, order, ...
        [matcontpath, 'Systems/']);