Environmental Engineering Reference
In-Depth Information
The following M-file explores the situation in a phase diagram. As example
a situation is studied in which the parameters are non-dimensionalized (Murray
2002 ), i.e. in which rates and equilibria are set to unity.
T = 1000;
% maximum time
r = [1; 1];
% rates
e = [1; 1];
% equilibria
lambda = 0.2;
% lambda parameter
c0 = [0.1; 0.1];
% initial concentrations
%----------------------execution-----------------------------------
options = odeset('AbsTol',1e-20);
[t,c] = ode15s(@CS,[0 T],c0,options,r,e,lambda);
%---------------------- graphical output --------------------------
plot (c(:,1)',c(:,2)'); hold on;
plot (e(1),0,'s'); plot (0,e(2),'s');
legend ('trajectory');
xlabel ('species 1'); ylabel ('species 2');
title ('competing species');
%---------------------- function ----------------------------------
function dydt = CS(t,y,r,e,lambda)
dydt = zeros(2,1);
k = [e(1)/(1+lambda*y(2)/y(1)); e(2)/(1+y(1)/y(2)/lambda)];
dydt = r.*y.*(1-y./k);
The complete code is included in the accompanying software under the name
“compspec.m”
Various trajectories (see Chap. 18.3) for the same starting populations, but with
varying parameter l , are shown in Fig. 19.3 . Obviously, for almost all l -values the
solutions at coordinates (0,1) or (1,0) are approached.
In both equilibria cases one of the species becomes extinct. As both equilibria
for single species are identical, the marginal parameter value is
1
species two uses resources more efficiently and species 1 becomes extinct.
For
l ¼
1. For
l >
1 the fate of the species is reversed. In Fig. 19.3 the two positions in the
phase space, which represent these two situations, are marked by 'species 1' and
'species 2'.
In fact the two mentioned states are equilibria , because they fulfil the system
( 19.4 ) for vanishing left hand side, i.e. for zero time derivatives. A refined exami-
nation of the two-equations- system ( 19.4 ) reveals that there are three equilibria in
the competing species model, which are given by:
l <
!
;
!
;
r 1
a 1 h 1
0
r 2
a 2 h 2
0
0
(19.5)
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