SIMbyom, byom_sim_logistic.m, logistic growth

This is the classical ODE for logistic growth with predation. More background in

Model system:

$$ \frac{dN}{dt} = r N \left(1-\frac{N}{K} \right) -p $$


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.

% scenarios specify different starting conditions
X0mat(1,:) = [1 2 3 4 5];
X0mat(2,:) = [10 12 50 100 150];

Values for the model parameters

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

% syntax: = value
par.r   = 0.1;  % growth rate
par.K   = 100;  % carrying capacity
par.p   = 1;    % predation rate

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(0,100,1000); % time vector for the model curves

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

glo.leglab1 = 'Scen. '; % 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 = 30; % interval for plotting (how many time points to do in one go), default: 10
opt_sim.axrange  = [0 160]; % axis ranges for states, default: autoscale

opt_sim.plottype = 3; % 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

% equilibrium points (optional; comment out if not known)
opt_sim.Xeq(1,:) = [par.K*(0.5+sqrt(0.25-par.p/(par.r*par.K)))];
opt_sim.Xeq(2,:) = [par.K*(0.5-sqrt(0.25-par.p/(par.r*par.K)))];

Calculations and plotting

Here, the script is called that will do the calculation and the plotting. The yellow points in the graph indicate the equilibrium points, and the squares 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