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The research investigates synchronizing dynamics of three nonidentical ecological systems, one of which is invaded by an exterior top predator. For studying the effects of species invasion in the synchronizing dynamics, combination synchronization of one drive system and two response systems is designed: the ecological system with species invasion taken as drive system, and the other ecological systems as response systems. Using active backstepping design, the conditions for achievement of combination synchronization are derived. Under the conditions, numerical simulations are performed to demonstrate the synchronizing dynamics of ecological systems under species invasion. The results suggest that mechanism of synchronization plays an important role in determining chaos when ecological systems are interacting with each other.

Chaos phenomena have been observed in many natural systems. As the research on chaos continues, great efforts were devoted to chaos synchronization of chaotic systems, a very important subfield in nonlinear science [

Most of the research works of chaos synchronization are limited to the synchronization of two chaotic systems, i.e., one drive system and one response system. Recently, Runzi et al. [

The combination synchronization (or combination-combination synchronization) can be possibly applied in the field of secure communication and information processing [

In recent years, the complex dynamics of ecological systems, as an interesting and significant topic, is more and more concerned by the mathematicians and biologist. Via analyzing the nonlinear characteristics, many research works have played an important role in understanding the properties of natural ecosystems, such as basic structure, productivity, resource availability, interactions, stability, ecosystem size, and network connection. Systematical and detailed analysis on these characteristics can contribute to profound comprehension on population regulation, trophic relationships, energy transformation, and community behaviors in the ecological systems. Moreover, it can also promote the improvement as well as practical application of ecological theory. Emmerson and Dave provided a biological mechanism that explains the skewed distributions of interaction strengths in real communities, where emergent features of food webs reflect patterns of predator-prey body size [

Since natural ecological systems are usually not isolated and closely connected with each other, the coupled networks of ecological systems can be more accurate to reflect the characteristics of nature and deserve further investigations. On the other hand, the coupled networks of ecological systems may exhibit more complex dynamic behaviors. Synchronization dynamics is one of the most interesting topics in the research where ecological systems are considered in connection. For example, Blasius and Stone studied the phase synchronization of two nonidentical diffusively predator-prey-resource systems and found that the synchronization patterns and travelling wave structures exhibited can correspond to those observed in natural systems [

Many evidences in field observation confirm the wide existence of synchronization between natural ecological systems. A classical example is the synchronization of Canada’s hare-lynx cycle over millions of square kilometres [

Some researchers suggested that the achievement of ecological synchronization results from migration of species between ecological systems [

In comparison with previous research works, the present study may exhibit three aspects of improvement in investigating the synchronization of chaotic ecological systems. Firstly, different from the former investigations which mainly focused on the complex dynamics and synchronization of two chaotic ecological systems, this research further tries to comprehend the synchronization among three chaotic ecological systems. Secondly, the three chaotic ecological systems considered are nonidentical and have different dimensions; therefore, the synchronization of the three systems is achieved through the interactions of complex dynamics induced by different ecological mechanisms. Thirdly, with the application of the method of combination synchronization, the case of exterior predator species simultaneously invading into two types of local environments is investigated. This may provide a new insight for studying the ecological dynamics of species invasion.

In this research, we apply active backstepping design to achieve the combination synchronization of the three ecological systems and numerical simulations of the combination synchronization are then performed, verifying the theoretical calculations. Following the former synchronization works of ecological systems, the combination synchronization in this research also demonstrates that one of the intrinsic mechanisms, predation (although the predation results from an exterior top predator), is a significant cause for population synchrony. Moreover, we find that, with the achievement of the combination synchronization, the invasion of predator species can force the extinction of local predator or dynamical transition between periodic and chaotic population oscillations. These nonlinear characteristics may help to guide the application in ecological engineering, such as harvesting or reduction of the invasive species.

Because of the importance of food chain in ecological systems, the food chain models are often employed in the approaches of synchronization of ecological systems [

Hastings and Powell [

In order to investigate synchronizing dynamics of ecological systems under the influences of species invasion, an exterior top predator

Many researchers took into consideration the different functional responses for the tritrophic level food chain described by the Hastings-Powell model. In Naji and Balasim [

For the tritrophic level food chain which has the prey−intermediate predator−top predator structure, Upadhyay et al. [

The combination synchronization of the systems (

According to the description of ecological systems in the above section, the invasive species

Combination synchronization of the drive system (

Let the response systems be controlled, and the controlled systems (

According to Runzi et al. [

We need to find the combination controllers

(1) First let

(2) Let

(3) Let

If the combination control laws are chosen as follows:

Numerical simulations are performed to demonstrate the combination synchronization of the systems (

It is necessary to describe the dynamic behaviors with given feasible parameter values for the three systems involved in the combination synchronization. The dynamics of the three systems has been studied a lot in literature. According to Naji and Balasim [

The system (

As for the system (

The initial states of populations

The numerical results of combination synchronization are depicted in Figures

Combination synchronization of systems (

Combination synchronization of systems (

Figure

Figure ^{−8} or smaller. Therefore, it suggests the dynamical behavior in Figure

Dynamics of systems (^{−5}, are applied. The parametric conditions are the same as those in Figure

Nowadays, many ecological systems suffer from the problem of species invasion [

To study ecological mechanisms for such phenomenon, the combination synchronization of three ecological systems is investigated in this research. Each ecological system is described by a food chain and one of the three systems is considered to suffer from species invasion of an exterior top predator. Taking ecological system with species invasion as drive system, and the other two systems as response systems, the conditions for combination synchronization of the three systems are derived using active backstepping design. Under the conditions, numerical simulations of combination synchronization are performed. With the results obtained from numerical simulations, the following should be addressed:

The achievement of combination synchronization makes the variation of population dynamics, which results from species invasion, expand from drive system to response systems.

Under species invasion and achievement of combination synchronization, the frequencies of population oscillations of the three ecological systems can be greatly changed.

For the dynamics of response systems, the achievement of combination synchronization can lead to the transition from periodic behaviors to chaotic behaviors. For the case where the two response systems exhibit periodic oscillation, the achievement of combination synchronization can force the chaotic dynamics of the drive system extending to the response systems (see Figure

The invasion of exterior top predator and achievement of combination synchronization can bring about the extinction of local predator in all three ecological systems.

The combination synchronization research of three ecological systems provides an insight into the complex dynamics due to interactions of ecological systems. Moreover, the conclusions described above may be able to extend to the synchronizing dynamics of more than three ecological systems, in which one is drive system and three or more are response systems. For the case where the numbers of drive system and response system are both larger than two, combination-combination synchronization can be further considered to study the complex dynamics based on the results obtained here.

The data of numerical results are generated during the study.

The authors declare no conflicts of interest.

This research was financed by the Fundamental Research Funds for the Central Non-profit Research Institution of CAF (No. CAFINT2014C15).