Asymptotic properties of a population model with Allee effects in random environments

In this research, we probe the influences of two common types of stochastic noises in the environment, namely, white noise and telephone noise, and put forward a stochastic differential equation population model with Allee effects. We analyze some asymptotic behaviors of the model, including extermination, persistence and invariant measure. Some vital functions of white noise and telephone noise on these asymptotic behaviors of the model are discovered and numerically expounded.


Introduction
In the nature, species often forms various degrees of aggregations [17]. Aggregation may enhance intraspecific competition, but this influence is often offset by the increased survival ability of the species, because aggregation could help the individual to protect itself, discover resources, or adapt to the environmental stresses [17]. The degree of aggregation, which varies with species and circumstances, promotes optimum population growth and survival. A group often has an optimal population size at some intermediate level. Both undercrowding and overcrowding have limitations [16]. This aggregation principle is the famous Allee principle [16].
To mathematically depict the Allee principle, several models have been put forward (see, e.g., [4,5,8]). Especially, Jacobs [5] has put forward the following deterministic differential equation model and probed its stability: where N(t) is the population abundance; r > 0, ρ > 0 and β > 0 measure the growth rate, the positive influence of aggregation and the negative influence of overcrowding, respectively. a > 0 and b > 0 are flexible constants to fit the real data. Nevertheless, environmental noises should be thought over owing to their universal existence [15]. In fact, ignoring environmental noises may be one of the main causes for the vanishing of many species [10]. As a result, one should concentrate on model (1) with environmental noises to exploit the Allee principle more accurately. However, no findings of model (1) with environmental noises have been reported.
The goal of this research is to put forward a stochastic form of model (1) and to probe its dynamical behaviors. We put forward the stochastic differential equation model in Sect. 2. Then we probe the extermination-persistence and invariant measure of the model in Sects. 3 and 4, respectively. Afterwards, we discuss and numerically expound the impacts of environmental noises on the properties of model (3) in Sect. 5. Finally, we summarize the research in Sect. 6.

The stochastic model
Two common types of stochastic noises in the environment are white noise and telephone noise [19]. White noise could influence one or more parameters in a population model [15]; and the growth rate might be the most sensitive parameter [2]. There are several means to bring white noise to the growth rate. One widely adopted means (see, e.g., [6,7,13]) is to presume that whereω(t) is white noise, namely, {ω(t)} t≥0 is a standard Brownian motion, 2 represents the intensity. Therefore, model (1) turns into Now we move to telephone noise which can cause the changes of parameters [12]. Mathematically, one can recur to a right continuous irreducible Markov chain κ(t) with a finite state space ϒ = {1, . . . , N * } to describe telephone noise [20]. Adhering to this means, model (2) turns into where min i∈ϒ {r(i), ρ(i), θ (i)} > 0. Model (3) could depict the reality better than model (2). As a matter of fact, assume initially, κ(t) = i 0 , then model (3) is as follows: for a random time, until κ(t) jumps to a new state, i 1 , then model (3) is as follows: until κ(t) jumps again. Clearly, if κ(t) does not jump, then model (3) reduces to model (2). As a result, model (3) could depict the random changes of the environment, which cannot be described by model (2).

Extinction and persistence
Let κ(t) be independent with ω(t), and let M = (m il ) N * ×N * and φ = (φ 1 , . . . , φ N * ) represent the generator and the invariant measure of κ(t), respectively. Then, for i = l, m il > 0, and, for i ∈ ϒ, l∈ϒ m il = 0. Define For the scalar equation define an operator L as follows: Proof The proof is standard and therefore is omitted (see, e.g., [19]). Now, let us probe the extermination of (3).

Stability
In the investigation of stochastic population models, the existence and uniqueness of an invariant measure are key topics [18]. Now let us give a sufficient condition for the existence and uniqueness of an invariant measure of model (3).

Conclusions
In the field of mathematical biology, it is meaningful to probe the functions of the environmental noises on the features of population models [11]. In the present research, we have developed a stochastic differential equation population model with Allee effects. We have analyzed the extermination, persistence and invariant measure of the model. The findings uncover that the features of the model are closely related to the environmental noises. First, if the Markov chain κ(t) spends more time on the desired states   Hybrid system (16) with parameter values given in Fig. 1 and Fig. 2, and φ = (0.2, 0.8). (a) is a trajectory of the system which suggests that the system is persistent; (b) is the density function of the solution of the system t = 3000 (the states where r(·) -2 (·)/2 > 0), then the species is persistent (see Fig. 4); conversely, if the Markov chain κ(t) spends more time on the undesired states (the states where r(·) + ρ(·) -2 (·)/2 < 0), then the species vanishes (see Fig. 3). Second, sufficiently large white noise on the growth rate could lead to the extermination of the species (compare Fig. 4 with Fig. 5).
Some issues are not analyzed in this research. First, the case that¯ +ρ > 0 >¯ is not settled yet. Second, model (3) presumes that white noise only affects r; the case that other parameters are also affected by white noise is not settled yet. Finally, there are some sudden environmental noises which can be described by Lévy jumps (see, e.g., [1]). It is interesting to consider model (3) with Lévy jumps.