This study examines the complexity of a discretetime predatorprey system with ratiodependent functional response. We establish algebraically the conditions for existence of fixed points and their stability. We show that under some parametric conditions the system passes through a bifurcation (flip or NeimarkSacker). Numerical simulations are presented not only to justify theoretical results but also to exhibit new complex behaviors which include phase portraits, orbits of periods 9, 19, and 26, invariant closed circle, and attracting chaotic sets. Moreover, we measure numerically the Lyapunov exponents and fractal dimension to confirm the chaotic dynamics of the system. Finally, a state feedback control method is applied to control chaos which exists in the system.
The interaction between predator and prey is one of the most studied topics in ecology and mathematical biology. The LotkaVolterra model [
In recent times, there are a few number of articles discussing the dynamics of ratiodependent discretetime predatorprey systems [
A ratiodependent predatorprey system takes the form:
If
This paper is organized as follows. Section
The fixed points of (
A simple algebraic computation yields the following result.
System (
(i) the predator free fixed point
(ii) the interior fixed point
To show the region in the space
Graphical depiction for positive fixed point.
Next, we investigate stability of (
Then the characteristic equation of (
For all permissible parameters values, there are four different topological types of
sink if
source if
nonhyperbolic if
saddle if otherwise.
Condition (iii) can be written as
It follows that if parameters
If
sink if either of the following conditions holds:
source if either of the following conditions holds:
nonhyperbolic if either of the following conditions holds:
saddle if otherwise,
where
From Proposition
Also two eigenvalues
and if the parameters lie in
In this section, attention is paid to recapitulate bifurcations (flip and/or NeimarkSacker) of system (
We consider system (
Let
Using the transformation
Therefore, we get the following symmetric multilinear vector functions of
Let
We set
The above discussion leads to the following result.
If (
We first discuss flip bifurcation of system (
Then the eigenvalues of positive fixed point
Using the transformation
where
It follows that
Therefore, we get the following symmetric multilinear vector functions of
Let
We set
Then by the standard scalar product in
The above discussion leads to the following result.
If (
We next discuss a NeimarkSacker bifurcation of system (
We first take parameter
Since the parameters belong to
Moreover, if
Let
Then by algebraic calculation, we obtain
We set
Then it is obvious that
According to multilinear symmetric vector functions, we can express the coefficients
We calculate the coefficient
It is clear that two conditions (
If
Here, diagrams for bifurcation, phase portraits, Lyapunov exponents, and fractal dimension of system (
Now we consider bifurcation parameters in the following cases.
For case (i): by taking parameters as in Table
Set of parameter values.
Varying parameter  Fixed parameters  

Case (i) 


Case (ii) 


Case (iii) 


Figure
Flip bifurcation and Lyapunov exponent of system (
For case (ii): by taking parameters as in Table
Figures
NeimarkSacker bifurcation and Lyapunov exponent of system (
The phase portraits of bifurcation diagrams related to Figures
Phase portraits of bifurcation diagrams Figures
For case (iii): the dynamics of map (
Numerical diagnostic of system (
Sensitivity to initial values of a system demonstrates that at the beginning the two trajectories are arbitrarily closely overlapped but a significant difference builds up for future trajectories. For numerical simulation, two perturbed trajectories for state variable
Initial perturbation for system (
The fractal dimension (FD) [
Taking parameter values given as in case (i) and by computer simulation, two Lyapunov exponents are
Fractal dimension of system (
We shall apply a state feedback control method [
The Jacobian matrix
The solution of the equations
Now, suppose that
Next, assume that
Control of chaotic trajectories of controlled system (
In order to observe how feedback method works and controls chaos at unstable state, we have presented some numerical simulations. We set
We investigated the dynamic complexities of discrete predatorprey system with ratiodependent functional response (
The author declares that there are no conflicts of interest.