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This paper is concerned with chaos in a discrete delay population model. The map of the model is proved to be chaotic in the sense of both Devaney and Li-Yorke under some conditions, by employing the snap-back repeller theory. Some computer simulations are provided to visualize the theoretical result.

Delay differential equations have been largely used to model phenomena in economics, biology, medicine, ecology, and other sciences. The studies on delay differential equations in population dynamics not only focus on the discussion of stability, attractivity, and persistence, but also involve many other dynamical behaviors such as periodic phenomenon, bifurcation, and chaos, see [

As we well know, the discrete time population models governed by difference equations are more appropriate than the continuous time population models governed by differential equations when the populations have nonoverlapping generations or the size of the population is rarely small. Moreover, some qualitative properties of the difference equations can also provide a lot of useful information for analyzing the properties of the original differential equations. In addition, discrete time models can also provide efficient computational models of continuous time models for numerical simulations. Therefore, many researchers studied the complex behaviors of the discrete population model, see, for example, [

Recently, some researchers used the Euler discretization to explore the complex dynamical behaviors of nonlinear differential systems, such as determining the bifurcation diagrams with Hopf bifurcation, observing stable or unstable orbits, and chaotic behavior, see [

In this paper, we study the chaotic behavior of the following discrete delay population model

Equation (

When

To the best of our knowledge, the research works on the chaotic behavior of (

The rest of the paper is organized as follows. In Section

In this section, some basic concepts and lemmas are introduced.

Since Li and Yorke [

Let

There are three conditions in the original characterization of chaos in Li-Yorke's theorem [

Let

the set of the periodic points of

In Definition

Some researchers consider that condition (i) in Definition

For convenience, we present some definitions in [

Let

A point

Assume that

In 1978, Marotto [

We now present two lemmas which will be used to study chaos in the delay population model.

Let

Since

Let

The conclusions of Lemma

In this section, we will transform the delay population model (

Let

The map

(i) A point

(ii) The concept of snap-back repeller and its classifications of system (

(iii) The concepts of density of periodic points, topological transitivity, sensitive dependence on initial conditions, and the invariant set for system (

System (

In this section, we will investigate the chaotic behavior of system (

It is obvious that system (

There exists a positive constant

The idea in the proof is motivated by the proof of [

For convenience, we translate the fixed-point

First, we show that there exists a positive constant

Next, we show that

For convenience, Let

For

For

Take

Now, it is clear that

For

For

Therefore, all the assumptions in Lemma

From the proof of Theorem

In order to help better visualize the theoretical result, six computer simulations are done, which exhibit complicated dynamical behaviors of the induced system (

Bifurcation diagram of system (

Bifurcation diagram of system (

Zoom area of the rectangular box

Bifurcation diagram of system (

Bifurcation diagram of system (

Zoom area of the rectangular box

In this paper, we rigorously prove the existence of chaos in a discrete delay population model. The map of the system is proved to be chaotic in the sense of both Devaney and Li-Yorke under some conditions, by employing the snap-back repeller theory. Computer simulations confirm the theoretical analysis. The system (

This work was supported by the National Natural Science Foundation of China (Grant nos. 11101246 and 11101247).