# Hopf and Zero-Hopf Bifurcation of the Four-Dimensional Lotka-Volterra Systems

## Authors

• Sirwan A. Mustafa Department of Mathematics, Faculty of Science, Soran University, Erbil, Iraq.
• Niazy H. Hussen Department of Mathematics, Faculty of Science, Soran University, Erbil, Iraq & Department of Mathematics, College of Education, Salahaddin University, Erbil, Iraq.

## Keywords:

Averaging theory, Lotka-Volterra system, Periodic solutions, Quadratic polynomial differential system, Zero-Hopf bifurcation

## Abstract

In this work, the four-dimensional Lotka-Volterra model (4DLV) involving four species in a constant environment is considered. The objective of this investigation is to study the local bifurcations occurring in the system. This system has at most sixteen possible equilibrium points. One of the equilibrium points is considered in order to investigate the periodic solutions that bifurcate from the Hopf and the zero-Hopf equilibrium points, respectively. It has been proven that, five families of sufficient conditions exist on the parameters of the system in which the Jacobian matrix at equilibrium point has a pair of purely imaginary    ,  > 0 and two non-positive eigenvalues. Moreover, eight families of sufficient conditions exist on the parameters in which the Jacobian matrix at the equilibrium point has a pair of purely imaginary eigenvalues  and at least one of the other eigenvalues is zero. Next, this investigation reveals that certain four-dimensional Lotka-Volterra subsystems exhibit one periodic solution bifurcating from the Hopf equilibrium point and three periodic solutions bifurcating from the zero-Hopf equilibrium point respectively. The averaging method in any order for computing periodic solutions consists of providing sufficient conditions for the existence of periodic solutions in polynomial differential systems by studying the equilibrium points of their associated averaged systems. Then, the main tool utilized is the first-order averaging method to compute periodic solutions that bifurcate from the Hopf and zero-Hopf singular points of the four-dimensional Lotka-Volterra system under certain conditions. Finally, the obtained theoretical results are supported and verified by numerical examples.

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