Abstract
This paper is concerned with a stochastic population model with Allee effect and jumps. First, we show the global existence of almost surely positive solution to the model. Next, exponential extinction and persistence in mean are discussed. Then, we investigated the global attractivity and stability in distribution. At last, some numerical results are given. The results show that if attack rate a is in the intermediate range or very large, the population will go extinct. Under the premise that attack rate a is less than growth rate r, if the noise intensity or jump is relatively large, the population will become extinct; on the contrary, the population will be persistent in mean. The results in this paper generalize and improve the previous related results.
Highlights
The Allee effect represents the relationship between population growth and population density
For a population with Allee effect, if its population density is too sparse, it is so difficult to find a mate that reproduction does not compensate for mortality, its population number will be reduced
There is a threshold population level for the strong Allee effect such that the species become extinct below this threshold population density
Summary
The Allee effect represents the relationship between population growth and population density. It is more objective to modeling stochastic population models with white noise in mathematical biology (to name a few, see [8, 9, 11, 22, 25, 30]) In these papers the authors revealed how noise affects the population dynamics. [8, 9, 11, 25, 30] investigated the dynamics of stochastic population models with Allee effect. One can get the following stochastic population model with Allee effect a dx(t) = x(t) r − cx(t) −. From the boundedness of γ(z), there is K > 0 satisfying [ln(1 + γ(z))]2λ(dz) < K
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