The Dynamic Properties Of Synergy Between Non-Gaussian Noise And Delay In The Ant Foraging Colony System

Firstly,in this paper some related background knowledge that the autocatalytic mechanism of the the population model under Gaussian white noise driving were intro-duced.In nonlinear dynamics,some random delay theories related to this paper are described,including the analytical approximation and numerical simulation of complex systems under the action of non-Gaussian noise and time delay.Then we use site ap-proximation and apply random delay theory to study a typical biological system（i.e.,the population foraging system）,and to explore the effects of noise and delay time on the dy-namics of the biological system.Some physical phenomena are found and some reasonable explanations explanations are given.Secondly,we consider the non-Gaussian noise and time delay synergy in population foraging systems.Based on the population model,we construct stochastic delay differ-ential equations and obtain the corresponding Fokker-Planck equations.We apply the uniform color noise approximation and the small time delay approximation theory,etc.To investigate the steady-state changes and dynamic properties of noise and delay time for the foraging model of the population.These results show:（1）the time delay and non-Gaussian noise can induce transition from a single peak to double peaks in the SPD,i.e.,a type of bistability occurring in a foraging colony system where time delay and non-Gaussian noise not only cause transitions between stable states,but also construct the states themselves.Numerical simulations are presented and are in good agreement with the approximate theoretical results;（2）there exists a maximum in the ART as a function of the noise intensity,this maximum for ART is identified as the characteristic of the non-Gaussian noise-weakened stability of the foraging colonies in the steady state;（3）the ART as a function of the noise correlation time exhibits a maximum and a minimum,where the minimum for ART is identified as the signature of the non-Gaussian noise enhanced stability of the foraging colonies;and（4）the departure from Gaussian noise q enhances the related activity between two states at different times,decreases the evolution velocity,delays the evolution time of the system from an arbitrary initial state to the stable state,and weakens the stability of the bistable system in the steady state.In contrast,the time delayτweakens the related activity between two states at different times,accelerates the evolution velocity,shortens the evolution time of the system from an arbitrary initial state to the stable state,and enhances the stability of the bistable system in the steady state,namely,the time delay and departure from Gaussian noise play opposite roles in ART or NCF.Finally,the foraging model with non-Gaussian noise and time-delay feedback was established to investigate the mechanism of non-Gaussian noise and time delay feedback on SPD,variance β₂,skewness β₃ and kurtosis β₄.The results show that:（1）as the value of K is increased,we can see that the structure of the SPD is changed from double peaks to one peak,namely,the strength K of the feedback loop can induce a phase transition phenomenon.This result indicates that the process of aggregation can be induced by the strength K of the feedback loop.In biology,the strength of the feedback loop causes the ant population spread from the densely populated side to another side;（2）we consider the effect of time delayτ,noise intensity D,non-Gaussian noise deviation parameter q and noise correlation time τ₀ on SPD in both cases of positive feedback（K>0）and negative feedback（K<0）.In the case of positive feedback,it is seen that there only appears a single peak in the SPD curve.However in the case of negative feedback,the time delayτand non-Gaussian noise parameters D and q can induce a phase transition phenomenon,and noise correlation time τ₀ can not induce it;（3）it is found that the curves in β₂,β₃ and β₄ exhibit an optimum strength of feedback where β₂,β₃ and β₄ have a maximum.This maximum indicates the large deviations in β₂,β₃ and β₄.