I hope help me in this problem. I like find the fixes point in this system.
I wrote a code in Matlab as follow:
clear all;
close all;
clc;
%
tic;
rand('state',sum(100*clock)); % seed
%
numreps=2; % Number of iterations
for j=1:numreps
options = odeset('RelTol',1e-6,'Stats','on');
% Parameters values of adults and tadpoles
% Fecundity (number of new individuals)
aH = 800; % minimum number offsprings
% per each host adults per year
% Mitchell 2008
% Metamorphosis rate
ah = 0.16; % Metamorphosis rate per year
% Mitchell 2008
% Natural dead hosts whitout infection
bH = 0.73; % Natural mortality rate in adults
% per year Mitchell 2008
bh = 7.55; % Natural mortality rate in adults
% per year Mitchell 2008
% Host mortality due to infection
alphaH = 0.001; % Mortality rate per adult host
% per unit time due to infection
% this work
alphah = 3.25; % Mortality rate per adult host
% per unit time due to infection
% this work
% Natural mortality zoospores within hosts
muH = 117*52; % Natural mortality in zoospores
% in adults Woodhams 2008 at an
% average temperature 17.38 per year
muh = 117*52; % Natural mortality in zoospores
% in adults Woodhams 2008 at an
% average temperature 17.38 per year
%zoospore release rate
lambdaH = 1*10^5; % Release rate of new zoospores
% within the body of the tadpole
% host to the pool per year Mitchel 2008
lambdah = 6.6*10^5; % Release rate of new zoospores
% within the body of the tadpole
% host to the pool per year Mitchel 2008
% Zoospores recruitment rate within host
rH = 124*52; % Maximum birth rate of new zoospores
% within the pre-adult host body
% per unit time Woodhams et al 2008
rh = 124*52; % Maximum birth rate of new zoospores
% within the pre-adult host body
% per unit time Woodhams et al 2008
% Zoospores transmission rate since pool
% Mitchell 2008
betaH = 6*10^-9; % Rate of transmission of a spore
% to an individual per year
% Mitchell pre-adult 2008
betah = 6*10^-9; % Rate of transmission of a spore
% to an individual per year
% Mitchell pre-adult 2008
% Recapture factor of spores that are released by
% sporangium from the skin of the hosts
varrhoH1 = 10^-2;
varrhoH = varrhoH1*lambdaH; % Proportion of zoospores
% that are immediately absorbed into
% the skin of adult hopederos,
% who depend on the zoospores released
% by sporangiumos sporangios
varrhoh1 = 10^-2;
varrhoh = varrhoh1*lambdah; % Proportion of zoospores
% that are immediately absorbed into
% the skin of adult hopederos,
% who depend on the zoospores released
% by sporangiumos sporangios
% Maximum absorption factor of
% zoospores by host values taken
% from Woodhams 2008
phiH = 10^4; % Inverse absorption factor
% per host, if low
% absorption is high and viceversa
phih = 10^4; % Inverse absorption factor
% per host, if low
% absorption is high and viceversa
% Natural mortality of zoospores into the pool
muZ = 45*52; % Tasa de mortalidad natural
% de la zoosporas Castro 2015
% por año
% Carrying capacity of species
KH = 10^5; % adults host
Kh = 10^5; % tadpoles host
%x(1) = H; x(2) = h; x(3) = ZH;
%x(4) = Zh; x(5) = Z;
G = @(t, x, ah, aH, bh, bH, muh, muH, alphah, ...
alphaH, lambdah, lambdaH, rh, rH, phih, phiH, ...
varrhoh, varrhoH, muZ, betaH, betah, Kh, KH) ...
[ah * x(2) * exp(-Kh * x(2)) - bH * x(1) - alphaH * x(3); ...
aH * x(1) - (bh + ah * exp(-Kh * x(2))) * x(2) - alphah * x(4); ...
rH * x(3) * exp(-phiH * x(3)) + lambdaH * x(3) * (x(1)./(varrhoH + x(1))) + x(5) * betaH * (x(1)./(x(1) + x(2))) - x(3) * (bH + muH) - alphaH * x(1) * (x(3)./x(1) + (x(3)./x(1))^2 * ((phiH + 1)./phiH)); ...
rh * x(4) * exp(-phih * x(4)) + lambdah * x(4) * (x(2)./(varrhoh + x(2))) + x(5) * betah * (x(2)./(x(1) + x(2))) - x(4) * (bh + ah * exp(-Kh * x(2)) + muh) - alphah * x(2) * (x(4)./x(2) + (x(4)./x(2))^2 * ((phih + 1)./phih)); ...
lambdaH * x(3) + lambdah * x(4) - x(5) * (muZ + betaH * (x(1)./(x(1) + x(2))) + betah * (x(2)./(x(1) + x(2))))-(lambdaH * x(3) * (x(1)./(varrhoH + x(1))) + lambdah * x(4) * (x(2)./(varrhoh + x(2))))];
%tspan = [0:0.001:50];
x0 = [100 100 10 10 500];
[t,xa] = ode45(@(t,x) G(t, x, ah, aH, bh, bH, muh, muH, ...
alphah, alphaH, lambdah, lambdaH, rh, rH, phih, phiH, ...
varrhoh, varrhoH, muZ, betaH, betah, Kh, KH), ...
[0 50], x0, options);
H = xa(:,1)';
h = xa(:,2)';
Zh = xa(:,3)';
ZH = xa(:,4)';
Z = xa(:,5)';
end
The solutions of system above when I make the graphic with this code:
%
figure1 = figure;
% Create axes
axes1 = axes('Parent',figure1);
hold(axes1,'on');
% Create multiple lines using matrix input to plot
plot1 = plot(t,xa,'LineWidth',3,'Parent',axes1);
set(plot1(1),'DisplayName','Adults host, H(t)');
set(plot1(2),'DisplayName','Tadpoles host, h(t)');
set(plot1(3),'DisplayName','Spores into the adults, ZH(t)','LineStyle','--');
set(plot1(4),'DisplayName','Spores into the tadpole, Zh(t)','LineStyle',':');
set(plot1(5),'DisplayName','External Pool of spores, Z(t)',...
'LineStyle','-.');
% Uncomment the following line to preserve the X-limits of the axes
% xlim(axes1,[0 5]);
% Uncomment the following line to preserve the Y-limits of the axes
% ylim(axes1,[-500 3000]);
box(axes1,'on');
grid(axes1,'on');
% Create xlabel
xlabel('Time in years');
% Create ylabel
ylabel('Population size');
% Set the remaining axes properties
set(axes1,'FontSize',12);
% Create legend
legend1 = legend(axes1,'show');
set(legend1,'FontSize',10);
%}
and this is the graphic
according to the previous graph, the solutions of the system converge to fixed points. Is there any way that the fixed points of the previous system can be calculated numerically using Matlab?