key: cord-0676791-q62v0u51
authors: Sun, Fei
title: Dynamics of an imprecise stochastic Holling II one-predator two-prey system with jumps
date: 2020-06-25
journal: nan
DOI: nan
sha: 64cb842e5bd3a4a092007b2bcff4284e8c9028da
doc_id: 676791
cord_uid: q62v0u51

Groups in ecology are often affected by sudden environmental perturbations. Parameters of stochastic models are often imprecise due to various uncertainties. In this paper, we formulate a stochastic Holling II one-predator two-prey system with jumps and interval parameters. Firstly, we prove the existence and uniqueness of the positive solution. Moreover, the sufficient conditions for the extinction and persistence in the mean of the solution are obtained.

In ecology and mathematical ecology, the study of interrelationship between species has become one of the main topics. And there have been growing interests on the dynamical behavior of the population species living in groups, such as Holling type I, II, and III functional response. For a better review of Holling II functional response and its extension, see [1] - [9] as well as there references.

However, the sudden environmental perturbations may bring substantial social and economic losses. For example, the recent COVID-19 has a serious impact on the world. It is more realistic to study the population dynamics with imprecise parameters. Panja et al. [14] studied a cholera epidemic model with imprecise numbers and discussed the stability condition of equilibrium points of the system. Das and Pal [15] analyzed the stability of the system and solved the optimal control problem by introducing an imprecise SIR model. Other studies on imprecise parameters include those of [10] - [13] , and the references therein.

The main focus of this paper is dynamics of an imprecise stochastic Holling II one-predator two-prey model with jumps. To this end, we first introduce the imprecise stochastic Holling II one-predator two-prey model. With the help of Lyapunov functions, we prove the existence and uniqueness of the positive solution. Further, the sufficient conditions for the extinction and persistence in the mean of the solution are obtained.

The remainder of this paper is organized as follows. In Sect. 2, we introduce the basic models. In Sect. 3, the unique global positive solution of the system is proved. The sufficient conditions for the extinction and persistence in the mean of the solution are derived in Sect. 4.

In this section, we introduce the imprecise stochastic system. Let x i (t) (i = 1, 2) and y(t) denote the population sizes of prey species and the population size of predator species at time t, respectively. Then a stochastic Holling II one-predator two-prey system takes the following form [18] .

and y(t − ) are the left limits of x 1 (t), x 2 (t) and y(t), respectively. r i > 0 (i = 1, 2, 3) are the intrinsic growth rates or death rate, a ii (i = 1, 2, 3) stand for the intraspecies interaction, a ij (i = j) represent the effect of species j upon the growth rate of species i. B i (t), (i = 1, 2, 3) are mutually independent Brownian motion defined on a complete probability space (Ω, F , F t≥0 , P). σ 2 i represent the intensities of B i (t). Let λ be the characteristic measure of N which is defined on a finite measurable subset Y of (0, +∞) with λ(Y) < ∞. Define the compensated random measure by N (dt, du) = N (dt, du)λ(du)dt.

Before we state the imprecise stochastic Holling II one-predator two-prey system, definitions of Intervalvalued function should recalled (Pal [15] ).

where R is the set of real numbers and g l , g u are the lower and upper limits of the interval numbers, respectively. The interval number [g, g] represents a real number g. The arithmetic operations for any two interval numbers A = [g l , g u ] and B = [h l , h u ] are as follows:

Addition Letr i ,â ij ,σ i represent the interval numbers of r i , a ij , σ i (i, j = 1, 2, 3), respectively. The system (2.1) with imprecise parameters becomes:

According to the Theorem 1 in Pal et al. [10] and considering the interval-valued function

, we can prove that system (2.2) is equivalent to the following system:

for p ∈ [0, 1]. Throughout this paper, let

For convenience in the following investigation, we require that

The following theorem will prove that system (2.3) admits a unique global positive solution.

Proof. Since t ≥ 0, by system (2.3), we can construct the following system

Because the coefficients of this system are local Lipschitz continuous (Mao [16] ), for any given initial value (u 1 (0), u 2 (0), v(0)) = (ln x 1 (0), ln x 2 (0), ln y(0)) ∈ R 3 + , there is a unique local solution (u 1 (t), u 2 (t), v(t)) on t ∈ [0, τ ), where τ is the explosion time (see Mao [16] ). Hence, system (2.3) admits unique positive local solution (x 1 (t), x 2 (t), y(t)) = (e u1(t) , e u2(t) , e v(t) ). The proof of global existence of this local solution to system (2.3) is rather standard. Therefore, we omit the proof here.

By Theorem 3.1, system (2.3) admits a unique global positive solution. Next, we will show that the solution of system (2.3) is stochastically bounded. 

where K is a generic positive constant.

Proof. The Itǒs formula (Situ [17] ) shows that

Thus,

This completes the proof.

When studying mathematical ecology, two of the most interesting issues are persistence and extinction. In this section, we discuss the extinction of populations in system (2.3) and leave its persistence to the next section. 

We now establish sufficient conditions for extinction of populations in system (2.3). Proof. The Itǒs formula of system (2.3) yields

(4.1) from 0 to t and then dividing by t on both sides, we obtain

Similarly, we have

and ln(y(t)/y(0)) t =b 3 − (a l 33 ) 1−p (a u 33 ) p y(t) + (a l 31 x 1 (s)ds = b 1 (a l 11 ) 1−p (a u 11 ) p a.s.

Proof. b 2 < 0 toghter with (4.3) and (4.5) yields

which means lim t→∞ x 2 (t) = 0. Combining this with (4.4) and (4.5), we know that

which also implies lim t→∞ y(t) = 0. It is easy to check that, for any 0 < ǫ < b 1 2 , there exist a positive constant t 0 and a set Ω ǫ such that P(Ω ǫ ) ≥ 1 − ǫ, and for t ≥ t 0 we get (a l 12 ) 1−p (a u 12 ) p x 2 < ǫ, (a l 13 ) 1−p (a u 13 ) p y < ǫ. Thus, for any ω ∈ Ω ǫ ,

By Lemma 4.2 and stochastic comparison theorem, for b 1 > 0, x 1 (s)ds = b 1 (a l 11 ) 1−p (a u 11 ) p a.s., when ǫ → 0. This completes the proof.

Next, we establish sufficient conditions for persistence in the mean of system (2.3) . Before that, we need to consider several stochastic differential equations with jumps.

Thus, we know from stochastic comparison theorem that

) p a.s., and

which means all the populations in system (2.3) are strongly persistent in the mean.

Proof x 2 (s)ds ≥b 2 − (a l 21 ) 1−p (a u 21 ) p b 1 (a l 11 ) 1−p (a u 11 ) p − (a l 23 ) 1−p (a u 23 ) p b 3 + (a l 31 ) 1−p (a u 31 ) p + (a l 32 ) 1−p (a u 32 ) p (a l 33 ) 1−p (a u 33 ) p a.s.

This completes the proof.

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