The Lotka-Volterra equations, which model the populations of predators (say foxes) and prey (say rabbits) [3]. These are a pair of nonlinear, first order differential equations, and exhibit the behaviour that in the absence of predators, the prey population grows exponentially, while the predator population shrinks if the prey population is too small. The equations are given by
\begin{equation} \label{EqLotka.1}
\dot{x} = a\,x - b\,xy , \qquad \dot{y} = -c\,y + d\,xy ,
\end{equation}
where 𝑎,
b,
c, and
d are real parameters that describe the interaction of the two species.
N = chebop(@(t,u,v) [diff(u)-u+u.*v; diff(v)+v-u.*v], [0 10]);
quiver(N, [0 5 0 5],'xpts',30,'ypts',30,'normalize',true,'scale',.4)
hold on
for rabbits = 0.1:.2:1.9
N.lbc = @(u,v) [u - rabbits; v - 1]; % Initial populations
[u, v] = N\0;
arrowplot(u, v)
end
hold off
title('Phase portrait for Lotka-Volterra equations')
xlabel('Rabbits'), ylabel('Foxes')
The cyclical behaviour of the populations is evident. What happens if we increase the reproduction rate of the rabbits by 50%?
N = chebop(@(t,u,v) [diff(u)-1.5*u+u.*v; diff(v)+v-u.*v], [0 10]);
quiver(N, [0 5 0 5],'xpts',30,'ypts',30,'normalize',true,'scale',.4)
hold on
for rabbits = 0.1:.2:1.9
N.lbc = @(u,v) [u - rabbits; v - 1]; % Initial populations
[u, v] = N\0;
arrowplot(u, v)
end
xlim([0 5]), ylim([0 5])
hold off
title('Phase portrait for L-V eqns., increased rabbit reproduction')
xlabel('Rabbits'), ylabel('Foxes')