SIMbyom, byom_sim_rosmac.m, Rosenzweig-MacArthur model

This is the classical Rosenzweig-MacArthur model. More background in http://www.jstor.org/stable/2458702. With low K, the predator (H) goes extinct, with medium K, there is a stable equilibrium, with high K, there is a limit cycle.

Model system:

$$ \frac{dR}{dt}=r R \left(1- \frac{R}{K} \right) - \frac{bRH}{1+bvR} $$

$$ \frac{dH}{dt}= c_o \frac{bRH}{1+bvR} - \mu H$$

Contents

Initial things

Make sure that this script is in a directory somewhere below the BYOM folder.

clear, clear global % clear the workspace and globals
global glo X0mat    % allow for global parameters in structure glo
diary off           % turn of the diary function (if it is accidentaly on)
% set(0,'DefaultFigureWindowStyle','docked'); % collect all figure into one window with tab controls
set(0,'DefaultFigureWindowStyle','normal'); % separate figure windows

pathdefine % set path to the BYOM/engine directory
glo.basenm  = mfilename; % remember the filename for THIS file for the plots

Initial values for the state variables

Initial states, scenarios in columns, states in rows. First row are the 'names' of all scenarios.

% Initial states, first row is different values for K
X0mat(1,:) = [1 5 7.5];
X0mat(2,:) = 6;
X0mat(3,:) = 1;

Values for the model parameters

Model parameters are part of a 'structure' for easy reference.

% syntax: par.name = value
par.r   = 1;    % growth rate of resource
% par.K   = 7.5;  % carrying capacity of resource (scenario name is now used)
par.b   = 1;    % searching rate of consumer
par.v   = 0.25; % handling time of consumer
par.co  = 1;    % conversion factor resource to consumer
par.mu  = 1;    % maintenance rate for consumer

Time vector and labels for plots

Specify what to plot. If time vector glo.t is not specified, a default is used, based on the data set

glo.t = linspace(0,100,5000); % time vector for the model curves

% specify the y-axis labels for each state variable
glo.ylab{1} = 'R';
glo.ylab{2} = 'H';
% specify the x-axis label (same for all states)
glo.xlab    = 'time';

glo.leglab1 = 'K = '; % legend label before the 'scenario' number
glo.leglab2 = ''; % legend label after the 'scenario' number

% If needed, a stiff solver or events function can be selected in call_deri

Optional settings for the simulations

opt_sim.plot_int = 100; % interval for plotting (how many time points to do in one go), default: 10
opt_sim.axrange  = [0 6.5;0 4.5]; % axis ranges for states, default: autoscale

opt_sim.plottype = 6; % default: 4
% 1) 3d plot
% 2) 2d plot
% 3) states in subplots
% 4) scenarios in subplots
% 5) dx/dt versus x
% 6) 2d plot for each scenario separate
% 7) 3d plot for each scenario separate

Calculations and plotting

Here, the script is called that will do the calculation and the plotting. The squares in the graph indicate the starting points. In the top of the file sim_and_plot.m, there are several more options to modify the plotting.

sim_and_plot(par,opt_sim) % call the script which calculates and plots