A Study On The Bifurcations And Chaotic Dynamics In Four-Species Food Web Models

The complex dynamic behavior of food web systems is one of the frontier topics on population dynamics.It is of great significance for the revealing of evolution mechanisms of population in food web systems.In this paper,the complex dynamics of Beddington-DeAngelis type.Holling-? type,and Leslie-Gower type four-species food web system was studied.The stability,Hopf bifurcation,Hopf-Hopf bifurcation,period-doubling bifurcation and numerical simulation results of food web model were analyzed.We exhibited abundant complex dynamic behaviors and transitions between these behaviors.The main results of this paper can be shown as follows:(1)The study of Beddington-DeAngelis type four-species food chain system indicated that the system can generate Hopf bifurcation,period-doubling bifurcation,chaos,and other complex dynamic behaviors.Hopf bifurcation can lead to periodic oscillation behavior of the system,while period-doubling bifurcation can lead to period-doubling cascade and route to chaos.In the chaotic regions,the sudden periodic windows such as 5,6,7,18,25,27,29 can be found,as well as secondary period-doubling bifurcation,period-half bifurcation and intermittent chaos in periodic windows.When the system parameters change,the above dynamic behaviors can exhibit complex transformation phenomenon,presenting the dynamic complexity of the food chain system.(2)In four-spccics food web dynamic system model of Holling-? type,the system undergoes Hopf bifurcation when the stability conditions and Hopf bifurcation theorem arc satisfied.The Hopf bifurcation can induce the variation of stable state to periodic oscillation state.This shows the dynamic transition of population from stable equilibrium to periodic oscillation.The system undergoes Hopf-Hopf bifurcation when the Hopf-Hopf bifurcation condition is satisfied.The Hopf-Hopf bifurcation can induce the transition of periodic oscillation state to quasi-periodic oscillation state and invariant torus.This further shows that the dynamic behavior of the population from periodic oscillation to quasi-periodic oscillation.When the system experiences period-doubling bifurcation,the period of the system starts to increase exponentially until it reaches the chaotic state,and then the chaotic state deteriorates into a periodic oscillation state.These dynamic behaviors show that there are different types of bifurations in the food web system.The generation of these bifurcations reveal the distribution law of population evolution with time and changes of external parameters and conditions,which provides theoretical reference for predicting the dynamic balance of food web(3)In four-species food web dynamic system model of Leslie-Gower type,the populations verge on stable coexistence,and populations densities tend to stable value in the long-term behavior when the stability conditions are satisfied.The Hopf-Hopf bifurcation can cause the transition of periodic oscillation state to stable torus when the Hopf-Hopf bifurcation theorem is satisfied.Correspondingly,this dynamic behavior shows a transition from point to limit cycle in the Poincare map.On that basis,different parameters are chosen as bifurcation parameters.And it is found that the direction of bifurcation is different with different bifurcation parameters.Moreover,the period-doubling bifurcation can lead to chaos,while intermittent chaos can appear simultaneously,which both give rise to a change of complex dynamics behaviors.This also explains that the variations of the food web system with the parameters condition may show the transition of complex dynamics behaviors.(4)Bifurcation can alter the global topology behavior of the food web system.Different from the other bifurcation types(Hopf bifurcation and period-doubling bifurcation),Hopf-Hopf bifurcation of the food web system can not only change the characteristics of the stable system state,but also induce the new complex attractor of food web system,such as quasi-periodic oscillation and torus attractor.The change of global characteristics of the food web system has strong regularity.When Hopf-Hopf bifurcation occurred,the dynamics of food web system will start from the periodic oscillation,and after the Hopf-Hopf bifurcation critical point,the periodic oscillation will lose its periodically,and then the system will enter the quasi-periodic oscillation.When Hopf bifurcation occurred,the dynamics of the food web system will start from the stable state,and after the Hopf bifurcation critical point,the stable state will lose its stability,and then the system will enter the periodic oscillation.For the period-doubling bifurcation,the system will enter a period-doubling oscillation.However,through the continuous evolution of the attraction state,the system will eventually enter the chaotic oscillation state,which will cause the dynamic behavior unpredictability of the system.Nevertheless,when the food web system enters the chaotic oscillation state,it may occasionally return to the periodic oscillation state.Further,it is shown that the dynamic behavior of the four-species food web system is more complex than that of the three-species food web system,which mainly shows the dynamic behavior that the three-species food web system does not have,namely the intermittent chaos and Hopf-Hopf bifurcation.In addition,under the same parameter condition,after removing one population from the four-species food web,the dynamic complexity of the remaining three-species food web will decrease,which further indicates that the dynamic behavior of the four-species food web system is more complex.That is to say,the periodic windows appear in the chaotic region.The formation mechanism of Hopf bifurcation,Hopf-Hopf bifurcation,period-doubling bifurcation lead to chaos and interment chaotic are a key of understanding the food web dynamic complexity and stability.This article is based on Beddington-DeAngelis type,Holling-? type,and Leslie-Gower type studied dynamic complexity of four-species food web systems.The formation and transformation of complex dynamic behaviors induced by different bifurcation are explored.Reveal the common laws of the formation of bifurcations and the evolution of chaos in food web systems.It promotes the understanding of the bifurcation and chaos of food web systems.This provides a theoretical basis for predicting the dynamic complexity of food web systems.