Survival analysis of single-species population di ﬀ usion models with chemotaxis in polluted environment

: In this paper, single-species population di ﬀ usion models with chemotaxis in polluted environment are proposed and studied. For the deterministic single-species population di ﬀ usion model, the su ﬃ cient conditions for the extinction and strong persistence of the single-species population are established. For the stochastic single-species population di ﬀ usion model. First, we show that system has unique global positive solution. And then, the su ﬃ cient conditions for extinction and strongly persistent in the mean of the single-species are obtained. Numerical simulations are used to conﬁrm the e ﬃ ciency of the main results.


Introduction
Habitat fragmentation is usually observed in nature related with heterogeneity in the distribution of resources. For example, food, water, shelter sites, physical factors such as temperature, light, moisture, and any feature be able to affect the growth rate of the population of a given species [1]. These fragments, also known as patches, are not completely isolated because they are coupled by the motion of individuals in space. Therefore, mathematicians and ecologists apply diffusion models to explain many ecological problems [1][2][3][4][5][6][7]. One of the classical population diffusion model [7]: The above diffusion processes are all based on the random movement of matter in space. However, many ecologists have found that there are many practical phenomena that cannot be explained by simple diffusion population models, such as, tripping and killing pests. In general, an important feature of many biological individuals is that they can perceive external signals and cues from a specific stimulus, especially vertebrates. Due to the attraction of some external signals, species may move in specific directions, which is called chemotaxis [8][9][10][11]. Colombo and Anteneodo proposed a model to consider the interplay between spatial dispersal and environment spatiotemporal fluctuations in meta-population dynamics [1]. Li and Guo studied a reaction-diffusion model with chemotaxis and nonlocal delay effect [9]. In [12,13], they showed that vertebrates have better sensory and differentiated nervous systems than invertebrates, which can transmit sensory information in the polluted environment to the region of brain where it can analysis and make corresponding processing, either bear the concentration of toxins in the habitat or escape from the area. Wei and Chen [12] proposed a single-speices population model with psychological effects in the polluted environment: ) c 0 (t) = kc e (t) − (g + m)c 0 (t) c e (t) = −hc e (t) + u e (t) (1.2) where c e (t) and c 0 (t) denote the concentration of toxicant in the environment and organism at time t respectively, u e (t) represents the input rate of external toxins to the habitat at time t, and, it is a continuous and bounded non-negative function. Coefficients r, r 0 , a, k, g, m, h, λ and α are positive constants, and their biological significance has been given in [12]. As we all know, with the influence of human economic activities, not only habitats of population are destroyed, but also the environment of habitats are polluted. The survival of those unprotected populations will be seriously threatened, even human beings, therefore, it is necessary to consider the effect of toxins in polluted patches on the population [14][15][16][17]. The "psychological effect" mentioned in [12,13] is also due to the response of biological individuals to the stimulation of environmental toxins in polluted environment, in other words, it is "chemotaxis". Considering the chemotaxis of biological individuals, the single-species population in heavily polluted patches will increase their diffusion to other nonpolluting or lightly polluted patches, while the populations of lightly polluted or nonpolluting patches will slow down their diffusion to heavily polluted patches under the influence of chemotaxis. In order to understand the effect of chemotaxis on population survival, we suppose that patch 1 is heavily polluted patch, and patch 2 is nonpolluting patch. On the basis of previous studies, we propose a single-species population diffusion model with chemotaxis in polluted environment: where λ i (0 ≤ λ i ≤ 1) denotes the contact rate between the single-species population and the environment toxicant. The initial value satisfies However, in nature, the population will be more or less disturbed by various random factors, which usually composed of many tiny and independent random disturbances, such as temperature, weather and climate change. May [18] has pointed out that even the smallest environmental randomness resulted in a qualitatively different result from the deterministic one. In recent years, stochastic population models have received a lot of attention [19][20][21][22][23][24][25]. Zou and Fan studied a single-species stochastic linear diffusion system [23]. Zu and Jiang focused on the extinction, stochastic persistence and stationary distribution of a single-species stochastic model with directed diffusion [24]. Liu and Bai considered a stochastic logistic population with biased diffusion [25]. Studies of single-species stochastic population models with migrations between the nature preserve and natural environment had received increasing attention in recent works [26][27][28][29]. But, few studies discuss the single-species population diffusion model with chemotaxis in population environment.
In this paper, we assume that the white noise mainly affects the intrinsic growth rate r i of system (1.3), we thus model the single-species population diffusion system by replacing the intrinsic growth rate r i of system (1.3) by a stochastic process dt denotes white noise, σ 2 i represents the density of white noise. We therefore derive a single-species stochastic diffusion system with chemotaxis in polluted environment as follows: Remark 1.1. [17]. Since c 0 (t) and c e (t) denote the concentration of toxicant, thus, 0 ≤ c e (t) ≤ 1, 0 ≤ c 0 (t) ≤ 1, with this end in view, we need the following constraints f ≤ g + m, 0 ≤ u(t) ≤ u < h.

Preliminaries
In this paper, unless otherwise noted, let (Ω, F, P) is a complete probability space with a filtration {F} t≥0 satisfying the usual conditions. (i.e., it is right continuous and contains all P-null sets) For the convenience of later discussion, some notations, lemma and theorem are given in this section: ), Lemma 2.1. (see [22]) Suppose that x(t) ∈ C(Ω × [0, +∞), R + ).

By standard comparison theorem obtains that
Thus, it easily obtain that

Survival analysis of deterministic population model (1.3)
Assumption H1 : ) be the solution of the first two equations of (3) with the initial value x(0) ∈ R 2 + , (1) Suppose Assumption H1 and H2 hold simultaneously, single-species x will be extinct.
(2) Suppose Assumption H3 or H4 are not true, single-species x is strongly persistent.
Proof. It follows from the first two equations of (3) that, Comparison system and (1) If Assumption H1 and H2 hold simultaneously, it is easy to see that the eigenvalue of system (6) at equilibrium point (0, 0) has negative real part and is quasi-monotone non-decreasing. Since (2) If Assumption H3 or H4 are not true, the proof is similar to [6]. we know that system (3.2) have positive equilibrium point z and zero equilibrium point. According to the conclusion and proof of Theorem 1 (see Allen [6]), system (7) is unstable at zero equilibrium, but stable at positive equilibrium point z, then lim The proof is completes. According to the Theorem 3.1's (1), if Assumption H1 and H2 simultaneously true, population x will die out. Byr 1 < 0 andr 2 < 0, we get If (r 1 − r 0 (c 0 ) * )r 2 ≥ 0, by virtue of (3.3) and (3.4), one can imply that (r 1 − r 0 (c 0 ) * )r 2 < 0, it is contradiction with (r 1 − r 0 (c 0 ) * )r 2 ≥ 0. Thus, (r 1 − r 0 (c 0 ) * )r 2 < 0 is a necessary condition of Assumption H1 and H2 holding at the same time.

Survival analysis of stochastic population model (1.4)
In order to analysis the long-time behaviors of single-species of system (1.5), first of all, we shall show that system (1.5) has unique global positive solution x(t) = (x 1 (t), x 2 (t)).
Lemma 4.1. For any given initial value x(0) ∈ R 2 + , there is a unique positive solution x(t) to system (1.5), and the solution will remain R 2 + with probability 1. Proof. Because the coefficients of system (1.5) is locally Lipschitz continuous for any given initial value where τ e is the explosion time(see [23]). In order to proof the solution is global, we only need to prove τ e = +∞, a.s..

Extinction
Let (θ, ρ) be the solution of the following equations where b > 0 and d > 0. By virtue of (4.5), it easily observe that θ = d ρ−c , where ρ is the solution of equation Because a and c are the solutions of equation ρ 2 − (a + c)ρ + ac = 0, obviously, Eq (4.6) has two solutions, and there must be a solution ρ = (a + c) + (a − c) 2 + 4bd 2 which is greater than c, thus θ > 0. Remark 4.3. We next come to analyze the following possible cases of the solution of Eq (4.6).
(a) If a and c are negative constants, when bd − ac < 0, all solutions of Eq (4.6) are negative. However, when bd − ac ≥ 0, there must be a nonnegative solution of Eq (4.6).
(b) If a or c aren't both negative, we can imply that there must be a positive solution of Eq (4.6).
let be sufficient small such that ρ < 0.5σ 2 . Because lim t→+∞ N(t) t−t 1 = 0, a.s., it follows from (4.10) that lim sup t→+∞ (x 1 (t) + θx 2 (t)) ≤ 0, a.s.   is strongly persistent, but Theorem 4.4 shows that the single-species x of stochastic model will die out when white noises large enough, which means that the white noises in the environment will affect the sustainable survival of the species, especially the endangered species.
We next discuss the persistence in the mean of the population of each patch. Theorem 4.10. Let (x 1 (t), x 2 (t)) be the solution of system (1.5) with initial value (x 1 (0), the population x i in the patch i is strongly persistent in the mean, and Proof. It follows from (1.5) that Integrating both sides of above inequalities (4.15) and (4.16) from 0 to t, For sufficiently small > 0, such that R i (t) l − > 0, i = 1, 2. It follows from (4.17) and (4.18) that The proof of Theorem 4.10 is completes.

Examples and numerical simulations
In this section, we will show the numerical simulation results to illustrate the accuracy of analytical results in above section by using the famous Milstein's method [30]. It is very hard to choose parameters of the model from realistic estimation, which needs to apply many methods of statistical, therefore, we will only use some hypothetical parameters to simulate the theoretical effects in this section. In order to simulate the influence of chemotaxis on the survival of single-species, we change the values of λ 1 , λ 2 , and α. We firstly adopt λ 1 = 0.5, λ 2 = 0.2, α = 1.5, by simple calculation, we know that it satisfy Assumption H1 and H2, by virtue of the Theorem 4.4, one can see that the single-species population x will die out, see Figure 1(a). If λ 1 = 0.5, λ 2 = 0.5, α = 0.1, by computing, Assumption H4 is not true, by virtue of the Theorem 4.4's (2), we can observe that the single-species x is strongly persistent, see Figure 1   We next focus on the effect of the intensity of white noises on the survival of population x. we adopt σ 1 = 0.2, σ 2 = 0.2, computing shows that R 1 + R 2 + (R 1 − R 2 ) 2 + 4D 12 D 21 −σ 2 = 0.2298 − 0.04 = 0.1898 > 0, it follows from the Corollary 4.9 that the population x is strongly persistent in the mean, see Figure 2(a). Suppose σ 1 = 0.7, σ 2 = 0.8, and other parameters are the same as Figure   2(a), by computing, one can know that R 1 + R 2 + (R 1 − R 2 ) 2 + 4D 12 D 21 −σ 2 = 0.2298 − 0.2775 = −0.0477 < 0, according to Corollary 4.9, one can find that the population x will die out (see Figure  2(b)). Therefore, from Figure 2, we can observe that the single-species x will be extinct when the densities of white noises larger enough. Case a: Suppose that d 12 = 0, d 21 = 0, the population x live in two independent patches. Simple calculation shows that r 1 − r 0 c 0 (t) < 0.5σ 2 1 and r 2 > 0.5σ 2 2 . According to the Remark 3 in [22] and Lemma 2.3, we can get that the population x 1 goes to extinction, and the population x 2 is strongly persistent in the mean, see Figure 3(a).

Conclusion
It is a pretty active topic to consider spatial information affects population dynamics, when the habitat of species is polluted, the species will be stimulated by the toxins in the habitat and increase diffusion to other patch. Thus, single-species population diffusion models with chemotaxis in polluted environment are proposed and studied. For the deterministic model, sufficient conditions for persistent and extinction of population are obtain. And then, considering the influence of environmental noise, a single-species population diffusion model with chemotaxis in polluted environment is proposed. Firstly, we discussed that the model (1.4) has unique global positive solution. Secondly, we investigated the persistence in the mean and extinction of system (1.4), if R u 1 + R u 2 + (R u 1 − R u 2 ) 2 + 4D u 12 D u 21 < σ 2 , the single-species population will extinction; if R l 1 + R l 2 + (R l 1 − R l 2 ) 2 + 4D l 12 D l 21 >σ 2 , the single-species population is strongly persistent in the mean. Finally, numerical simulations are used to confirm the efficiency of the main results. Figure 2(a) and (b) show that the single-species x will die out when the densities of white noises large enough, therefore, it is significance to consider the effect of stochastic perturbation.
If we set d 12 = d 21 = 0, that is to say, the single-species population live in two independent environments, respectively. Literature [22] shows that, when r 1 − r 0 c e < 0.5σ 2 1 , the population x 1 will tend to extinct, when r 1 − r 0 c e > 0.5σ 2 1 , the population x 1 is persistent in the mean, see Figure 3(a). However, by virtue of Theorem 4.4 and Theorem 4.8, we can obtain that population diffusion would affect the survival of the population x, see Figure 3(a) and (c).