Abstract
A discrete population model integrated using the forward Euler method is investigated. The qualitative bifurcation analysis indicates that the model exhibits rich dynamical behaviors including the existence of the equilibrium state, the flip bifurcation, the Neimark-Sacker bifurcation, and two invariant closed curves. The conditions for existence of these bifurcations are derived by using the center manifold and bifurcation theory. Numerical simulations and bifurcation diagrams exhibit the complex dynamical behaviors, especially the occurrence of two invariant closed curves.
Highlights
Differential equations and difference equations are generally applied in mathematical biology modeling
The researches on discrete models are still focused on the dynamical behaviors, and most of the scholars have studied the NeimarkSacker bifurcation, a codimension-1 bifurcation, which has shown one invariant closed curve bifurcating from an equilibrium point
They do not consider further, the same as the phenomenon of limit cycles bifurcation in continuous models, the fact that the discrete models can exhibit the interesting dynamic behavior of two invariant closed curves bifurcating from an equilibrium point
Summary
Differential equations and difference equations are generally applied in mathematical biology modeling. One major reason is that discrete models as computational models are more efficient for numerical simulations and research results exhibit more richer dynamics than continuous ones. Another reason is that difference equations are more realistic and intelligible to characterize the population size of a single species when it has a fixed interval between generations or measurements. The researches on discrete models are still focused on the dynamical behaviors (including stability, periodic solutions, bifurcations, chaos, and chaotic control; see [1,2,3,4,5,6]), and most of the scholars have studied the NeimarkSacker bifurcation, a codimension-1 bifurcation, which has shown one invariant closed curve bifurcating from an equilibrium point.
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