Analysis of a stochastic predator–prey population model with Allee effect and jumps

Abstract

This paper is concerned with a stochastic predator-prey model with Allee effect and Lévy noise. First, by the comparison theorem of stochastic differential equations, we prove that the model has a unique global positive solution starting from the positive initial value. Then we investigate the asymptotic pathwise behavior of the model by the generalized exponential martingale inequality and the Borel-Cantelli lemma. Next, we establish the conditions under which predator and prey populations are extinct. Furthermore, we show that the global positive solution is stochastically ultimate bounded under some conditions by using the Bernoulli equation and Chebyshev's inequality. At last, we introduce some numerical simulations to support the main results obtained. The results in this paper generalize and improve the previous related results.

Dynamic Analysis of a Stochastic Delayed SEIRS Epidemic Model with Lévy Jumps and the Impact of Public Health Education

“…We integrate the inequality ( 22) from 0 to t and divided by t, then substituting (25) into it, thus we can derive that…”

“…In [18], a stochastic cholera model with a saturated recovery rate was discussed, showing that high levels of noise could result in disease extinction. Major environmental disturbances such as floods, earthquakes, volcanoes, tsunamis, and hurricanes suggest the need to introduce Lévy jumps into infectious disease models to account for such abrupt fluctuations [20][21][22][23][24][25]. Zhang Xuegui et al [20] studied a stochastic two predator-one prey system with Lévy jumps and mixed functional responses, observing that Lévy jumps negatively impacted species' existence.…”

Extinction and ergodic stationary distribution of a Markovian-switching prey-predator model with additional food for predator

“…The importance of studying predator-prey model is to gain more insight into the complex processes and solve ecological problems, e.g., the control of populations of economically damaging species, particularly of agricultural weed and insect pests [3]. It is well documented that biological processes like disease [4], Allee effects [5,6], migration [7] and alternative food sources [8], etc., have the potential to dramatically alter system dynamics.…”