function LV_competition_two_species_asym_curvature(W,H,D,P,delta,R)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Nick Lowery and Tristan Ursell
% 2018
%
% Structured environments fundamentally alter dynamics and stability of ecological communities
% https://www.biorxiv.org/content/early/2018/07/10/366559
% 
% This function runs a non-dimensionalized modified 2D Lotka-Volterra ecological
% simulation of two mutally killing species, for the specific purpose of 
% calculating the critical curvature of the interface between two species with
% asymmetric competitive fitness. It requires the image_save.m script to be in 
% the active directory. The simulation stops when dynamics ceases either due to
% stabilization of the competition interface or extinction of one of the species.
%
% Input parameters:
% W = width of the simulation box (pixels)
% H = height of the simulation box (pixels) - this determines the open space
%	between pillars
% N = number of simulation iterations (dt units, specified in the script)
% D = diffusion coefficient (dimensionless)
% P = average interaction strength (dimensionless)
% delta = degree of competition asymmetry (species A gets P + delta, B gets P - delta)
% R = pillar radius (pixels)
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%*******************************************
%********* INITIALIZATION ******************
%*******************************************

%output images
imageq=1;

%time step in ND units
dt=0.01;

%local concentration discrete cutoff (will be set to 0 below this value)
dsc=0.001;

%frequency to show results
showt=50;

%intialize the density matrices in blocks
A = zeros(H,W);
A(:,1:W/2) = 0.3;
B = zeros(H,W);
B(:,W/2:W) = 0.3;

%diffusion convolution filter
sigma = sqrt(4*D*dt);
if mod(ceil(3*sigma), 2) == 0
    win_sz = ceil(3*sigma) + 1;
else
    win_sz = ceil(3*sigma);
end
Gauss = fspecial('gaussian', [win_sz,win_sz], sigma);
rescale_filt = 1 ./ conv2(ones(size(A)), Gauss, 'same');

if min(H,W) < win_sz
    warning('Diffusion window size is larger than simulation box.')
end

%file IO
file1 = ['2sp_LV_comp_curve_H-' num2str(H) ...
       '_W-' num2str(W) ...
       '_D-' num2str(D) ...
       '_P-' num2str(P) ...
       '_delta-' num2str(delta) ...
       '_R-' num2str(R) ...
       '_dsc-' num2str(dsc)];
 
%*******************************************
%********* PILLARS *************************
%*******************************************

%create spatial matrices / filters
Xmat = ones(H,1)*(1:W);
Ymat = (ones(W,1)*(1:H))';

%pillar locations
x_cent = [W/2 W/2];
y_cent = [R H-R];    

%create pillars
filt_all = ones(size(Xmat));
for i = 1:length(x_cent)
    filt_temp = ((Xmat - x_cent(i)).^2 + (Ymat - y_cent(i)).^2) >= R^2;
    filt_all = filt_all .* filt_temp;
end

%prepare reflecting boundary conditions in grid
conv_filt = conv2(filt_all, Gauss, 'same');
rescale_filt = filt_all ./ conv_filt;
rescale_filt(isnan(rescale_filt)) = 0;

%*******************************************
%********* SIMULATION **********************
%*******************************************

q = 0;
arrest = 0;
tic
tnow = toc;
while arrest < 1
    %track image for comparison
    Aprev = A;
    Bprev = B;
    
    %convolutions
    A = conv2(A, Gauss, 'same') .* rescale_filt;
    B = conv2(B, Gauss, 'same') .* rescale_filt;
    
    %interspecies interactions    
    dA = dt * A .* (1 - (A+B)) .* (1 - B / (P + delta));
    dB = dt * B .* (1 - (A+B)) .* (1 - A / (P - delta));
    
    %deterministic update populations
    A = A + dA;
    B = B + dB;
    
    %hard upper bound
    A(A > 1) = 1;
    B(B > 1) = 1;
    
    %discrete cutoff (hard lower bound)
    A(A < dsc) = 0;
    B(B < dsc) = 0;
    
    if and(max(abs(Aprev - A)) < (100*eps), max(abs(Bprev - B)) < (100*eps))
        disp('Simulation terminated due to extinction or inactivity.')
        arrest = 1;
	break
    end
    
    q = q + 1;

    %output
    if mod(q,showt) == 1
        tnow_temp = toc;
        tnow = tnow_temp - tnow;
        disp(['step ' num2str(i) ' for ' file1 ' in ' num2str(tnow) 's'])
        tnow = toc;
    end
end

%***********************************************
%************ FINAL IMAGE **********************
%***********************************************

%grey filter 
Atemp = A;
Btemp = B;
Atemp(filt_all == 0) = 1/2;
Btemp(filt_all == 0) = 1/2;

im_out = zeros(H, W, 3, 'uint8');
im_out(:,:,1) = uint8(Atemp * 255);
im_out(:,:,2) = uint8(Btemp * 255);
im_out(:,:,3) = uint8(Atemp * 255);

%***********************************************
%************ CURVATURE ************************
%***********************************************

% hard boundary, based on A majority
im_bin = A > B;

if sum(im_bin(:)) == sum(filt_all(:))
    K = NaN;
    extinct = 1;
else
    extinct = 0;

    % extract curve
    im_curve = edge(im_bin, 'Sobel') - edge(filt_all, 'Sobel');
    curve = im_curve == 1;

    % define curve coordinates
    curve_props = regionprops(curve, 'Extrema');

    % top right corner
    x1 = curve_props.Extrema(2,1);
    y1 = curve_props.Extrema(2,2);
    % right side of midpoint
    x2 = max(curve_props.Extrema(:,1));
    y2 = 0.5*(max(curve_props.Extrema(:,2))+min(curve_props.Extrema(:,2)));
    % bottom right corner
    x3 = curve_props.Extrema(5,1);
    y3 = curve_props.Extrema(5,2);

    % calculate curvature
    a = sqrt((x1 - x2)^2 + (y1 - y2)^2);
    b = sqrt((x2 - x3)^2 + (y2 - y3)^2);
    c = sqrt((x3 - x1)^2 + (y3 - y1)^2);
    A = 1/2 * abs((x1 - x2) * (y3-y2) - (y1-y2) * (x3-x2)); 
    if a*b*c == 0
	K = 0;
    else 
        K = 4*A/(a*b*c); 
    end
end

disp(['extinction = ' num2str(extinct) ', K = ' num2str(K)])

%***********************************************
%************ METADATA *************************
%***********************************************

if imageq==1            
    image_save(im_out,[file1 '.png'])
end

save([file1 '.mat'], 'W','H','dt','dsc','D','P','delta','R','K','extinct')

disp(['Done with ' file1 '.'])

%exit

end