Modeling Coral Growth
Biology of Corals  Modeling Assumptions
Main modeling simplification: a polyp buds when it encounters a nutrient in the water.
Simplified Coral Biology
corals are sessile, filter-feeding
a coral is made up of genetically identical
polyps a few millimeters in diameter
Model Assumption
passive filtering of diffusing nutrients
every node of a coral
is an identical ‘polyp’ unit
every edge node of a coral
can grow into a neighboring node
corals grows asexually by polyp budding
polyps bud when they reach maturity
coral nodes grow
in response to a threshold level
of local nutrient
corals are a very simple organism/colony
no communication or interaction assumed
between polyps
Canonical Model: Diffusion Limited Aggregation
Eden model, 1961: seed with random budding
Witten and Sander DLA model, 1981: seed with diffusion limited budding
Kaandorp, Lowe, Frenkel and Sloot, 1996: nutrient diffuses and flows
Discrete Agent-Based Modeling Components
Nutrient Source
point source from one direction or homogeneous source from all directions?
do nutrients spontaneously decay or disappear off-lattice?
will the equilibrium density of nutrient be small or large?
Nutrient Diffusion (and diffusion boundary conditions)
random walk
rate of diffusion?
do nutrients spontaneously decay or disappear off-lattice?
Nutrient Flow (and flow boundary conditions)
do nutrients flow as an ensemble? (current-like)
do nutrients flow independently in a biased direction? (biased random walk)
Coral Growth
nodes of potential growth need to be identified
Nutrient and coral need to be initialized, and the rules for nutrient source, flow and
diffusion need to be integrated and coupled with rules for coral growth.
Developing the Octave Code
(This worksheet can be downloaded electronically from the course web page
so that code sections can be cut and paste into Octave.)
Compose the Function In a Text Editor
The function coral_growth(INITIALIZE, T) will model coral growth.
If you type > coral_growth(1,5) into the command line, the algorithm will initialize and
proceed for 5 time steps. If you type > coral_growth(0,5) into the command line, the
algorithm will continue from where it left off and proceed for 5 time steps.
function [CORAL,polyps,radius]=coral_growth(INITIALIZE, T);
global CORAL
% global makes these matrices …
global NUTRIENT
% … available outside the function
N = 25;
% size of the lattice
f = 0;
% flow rate of nutrient
source=1;
%number of particles added per time step
Save the file as coral_growth.m .
Initial Conditions
We will model the ‘sea’ as an NN matrix. We will need to keep track of the amount of
nutrient in each node of the sea and which nodes are occupied by coral as it grows.
%%% INITIAL CONDITIONS %%%
if INITIALIZE == 1
'initialize algorithm'
NUTRIENT = zeros(N,N);
CORAL = zeros(N,N);
CORAL(fix(N/2),fix(N/2)) = 5; %one ‘seed’ node
imagesc(CORAL+NUTRIENT)
polyps=sum(sum(CORAL>0))
radius=1
colorbar
end
Type > coral_growth(1,5) and check the output.
Try moving the initial seed position to different locations on the lattice.
Try replacing ‘zeros’ with ‘rand’ in the line defining the nutrient.
Main Algorithm
During each time step, several things need to happen. They will happen independently,
each once per time step. Cut and paste the following and we’ll address each component
one at a time.
%%% MAIN BODY ALGORITHM %%%
if T>0
for timesteps=1:T
%NUTRIENT SOURCE
%NUTRIENT DIFFUSION
%NUTRIENT FLOW
%IDENTIFY POTENTIAL CORAL GROWTH
potential_growth=zeros(N,N);
%CORAL GROWTH
end
%%% OUTPUT %%%
imagesc(CORAL+NUTRIENT + 5*potential_growth)
polyps=sum(sum(CORAL>0))
[I,J]=find(CORAL>0);
radius=max(max(abs(J-fix(N/2)),abs(I-fix(N/2))))+1
end
Nutrient Source
%NUTRIENT SOURCE
%courtesy of Michael
%adding nutrient at edge nodes in N by N lattice
edges=ones(N,N);
edges(2:N-1,2:N-1)=0;
edge_nodes=find(edges==1);
%determine the size of the list
s=size(edge_nodes,1);
%pick 5 of the edge nodes at random as a nutrient source
nutrient_source=fix(rand(source,1)*s)+1;
%node occupied as nutrient source
added_nutrient=zeros(N,N);
added_nutrient(edge_nodes(nutrient_source))=1;
NUTRIENT = NUTRIENT + added_nutrient;
Nutrient Diffusion (and diffusion boundary conditions)
%NUTRIENT DIFFUSION
new_nutrient=zeros(N,N);
for i=1:N
for j=1:N
number_particles=NUTRIENT(i,j);
for b=1:number_particles
direction=fix(rand(1)*4)+1;
%add particle if new location is on lattice,
%otherwise, the particle disappears
if direction==1
if (i+1)<(N+1) %moves up
new_nutrient(i+1,j)=new_nutrient(i+1,j)+1;
end
end
if direction==2
if (j+1)<(N+1) %moves right
new_nutrient(i,j+1)=new_nutrient(i,j+1)+1;
end
end
if direction==3
if (i-1)>0 %moves down
new_nutrient(i-1,j)=new_nutrient(i-1,j)+1;
end
end
if direction==4
if (j-1)>0 %moves left
new_nutrient(i,j-1)=new_nutrient(i,j-1)+1;
end
end
end
end
end
NUTRIENT=new_nutrient;
%save nutrient positions after diffusion algorithm
Nutrient Flow (and flow boundary conditions)
%NUTRIENT FLOW
%nutrient has a small probability f of flowing to the right
new_nutrient=zeros(N,N);
for i=1:N
for j=1:N
number_particles=NUTRIENT(i,j);
for b=1:number_particles
%with probability f, each particle moves right
if rand(1)<f
%add particle if new location is on lattice,
%otherwise, the particle disappears
if (j+1)<(N+1) %moves right
new_nutrient(i,j+1)=new_nutrient(i,j+1)+1;
end
else
new_nutrient(i,j)=new_nutrient(i,j)+1;
end
end
end
end
NUTRIENT=new_nutrient; %save nutrient positions after diffusion algorithm
Identify Potential Coral Growth
%IDENTIFY POTENTIAL CORAL GROWTH
%every node in contact with coral is a potential growth node
potential_growth=zeros(N,N);
for i=2:(N-1)
for j=2:(N-1)
neighborhood=CORAL(i-1:i+1,j-1:j+1);
if sum(sum(neighborhood))>0
potential_growth(i,j)=1;
end
end
end
Coral Growth
%CORAL GROWTH
new_growth=(potential_growth==1).*(NUTRIENT>0).*(CORAL==0);
NUTRIENT=NUTRIENT-new_growth;
CORAL=CORAL+5*new_growth;
Using the Octave Code To Study Diffusion Limited Aggregation
Generally, a more branching structure will result if the lattice is large, the source is
small and the flow is small. Also, the coral will grow more slowly.
(1) Set the flow rate equal to 0, the source equal to 1 and the lattice size to 25. What
does the coral look like after 500 time steps? (Run one time and we’ll compare for
different students in the class.)
polyps =
radius =
(2) Set the flow rate equal to 1. What does the coral look like after 500 time steps?
(Run one time and we’ll compare for different students in the class.)
polyps =
radius =
(3) Set the flow rate equal to 1 and increase the nutrient source to 2 particles per time
step. What does the coral look like after 500 time steps? (Run one time and we’ll
compare for different students in the class.)
polyps =
radius =