On a population model with Allee effects and environmental perturbations
- PDF / 365,097 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 104 Downloads / 224 Views
On a population model with Allee effects and environmental perturbations Weiming Ji1 Received: 23 April 2020 / Revised: 31 May 2020 / Accepted: 5 June 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract The present study takes advantage of the white noise and the Lévy noise to portray the small and sudden environmental perturbations respectively, and puts forward a stochastic population model with Allee effects and Lévy jumps. First, it is confirmed that the model possesses a unique, global and positive solution. Then, permanence and extinction of the species are examined. Afterwards, the trajectory of population abundance is tested. The findings uncover that both the white noise and the Lévy noise have significant functions on the rate of population change. Finally, the theoretical findings are applied to dissect the rate of population change of the African hunting dog Lycaon pictus. Keywords Allee effects · Environmental perturbations · Extinction · Permanence Mathematics Subject Classification 60H10 · 60H30 · 92D25
1 Introduction The Allee effect [1], which implies that the per capita growth rate of a species attains its maximum at medium density, is fairly common in natural populations [8]. As a matter of fact, many species in the real world frequently cooperate with the same populations to seek prey or to oppose predators. For example, the African hunting dog (Lycaon pictus) frequently cooperates to capture prey [5]. As an example of cooperation, sardines cooperate to resist predators [8]. In addition, lots of social populations (for instance, ants, bees and humans) have evolved complicated cooperative behaviors, such as mating, nest building, raising the young, etc. These aggregations and cooperative behaviors offer individuals an increasing opportunity to subsist and reproduce
B 1
Weiming Ji [email protected] School of Mathematical Science, Huaiyin Normal University, Huai’an 223300, People’s Republic of China
123
W. Ji
as population size increases [8]. However, when the population size exceeds a certain value, the negative density dependent impacts may become dominant because of excessive crowding and intraspecific competition [8]. To dissect these phenomena, Takeuchi [23] proposed the following differential equation to depict the growth of a species population with Allee effects: θ M(t) M 2 (t) dM(t) = M(t) r + − , dt 1 + α M(t) 1 + α M(t)
(1)
where M(t) is the size of the population. The intrinsic growth rate r , the Allee threshold θ , the environmental carrying capacity α are all positive constants. Model (1) is deterministic, which ignores the environmental stochasticity. As a matter of fact, environmental stochasticity is ubiquitous in nature, and it may be a main reason for the extinction or the persistence of many species [16]. Accordingly, one should involve the environmental stochasticity in model (1) to be more realistic. A traditional means to involve the environmental stochasticity in deterministic models is to hypothesize that the environmental
Data Loading...
