Bifurcation analysis in a toxic-phytoplankton and zooplankton ecosystem
with double delays and Monod-Haldane type functional response
Abstract
In this paper, we structure a phytoplankton zooplankton interaction
system with two delays and Monod-Haldane type functional response, and
mainly discuss the affect of $\tau$ and
$\tau_1$ to the dynamic behavior of system. Firstly,
we give the existence of equilibrium and property of solution. The
sufficient conditions ensuring the globally asymptotical stability of
the boundary equilibrium are given. The nonexistence of the positive
equilibrium ensures the global stability of the boundary equilibrium.
Secondly, let $\tau_1=0$ and dynamic behavior of
system with one delay ($\tau$) is investigated. The
stability switches phenomenon can occur as $\tau$
varying. Then fixed $\tau$ in stable interval, using
$\tau_1$ as parameter, it can investigate the effect
of $\tau_1$ and find $\tau_1$ can
also cause the oscillation of system. Specially, when
$\tau=\tau_1$, the system can also
occur the stable switching phenomenon, and, under certain conditions,
the periodic solution will exist with the wide range as delay away from
critical value. Furthermore, using the crossing curve methods, it can
obtain the stable changes of positive equilibrium in
$(\tau,\tau_1)$ plane. When choosing
$\tau$ in the unstable interval, the system still can
occur Hopf bifurcation as delays varying. Some numerical simulations are
given to indicate the correction of the theoretical analyses. At last,
some conclusions are given.