Abstract
We investigate the stochastic dynamics of a prey-predator type ecosystem with time delay and the discrete random environmental fluctuations. In this model, the delay effect is represented by a time delay parameter and the effect of the environmental randomness is modeled as Poisson white noise. The stochastic averaging method and the perturbation method are applied to calculate the approximate stationary probability density functions for both predator and prey populations. The influences of system parameters and the Poisson white noises are investigated in detail based on the approximate stationary probability density functions. It is found that, increasing time delay parameter as well as the mean arrival rate and the variance of the amplitude of the Poisson white noise will enhance the fluctuations of the prey and predator population. While the larger value of self-competition parameter will reduce the fluctuation of the system. Furthermore, the results from Monte Carlo simulation are also obtained to show the effectiveness of the results from averaging method.
Highlights
In the real-world ecosystem, a time delay effect in the ecosystems is inevitable
One can see that the approximate stationary probability density functions (PDFs) for time-delayed ecosystem under Poisson white noise are higher than those for system under Gaussian white noise
This paper investigates the stochastic behaviors of a time-delayed predator-prey ecosystem
Summary
In the real-world ecosystem, a time delay effect in the ecosystems is inevitable. Such as, in the predator-prey type ecosystem, it takes time for the predator population to adjust any change of the prey population. There are some unavoidable sparse, drastic changes in the environment, for instance sudden natural disasters, earthquakes, forest fires, floods [25,26,27] These changes are pulse-type perturbations, which cannot be characterized by the continuous stochastic processes properly. It is necessary to study the stochastic dynamics of time-delayed ecosystems driven by stochastic jump processes. We study the stochastic behaviors of a time-delayed predator-prey ecosystem model under discrete random environmental fluctuations. The influences of the time delay, self-competition parameter and the Poisson white noises on the stochastic behaviors of the ecosystem are discussed based on the stationary PDFs. In addition, the results from Monte Carlo simulation are carried out to show the effectiveness of proposed method
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