key: cord-0219132-xdph25ld
authors: Sun, Fei
title: Dynamics of an imprecise stochastic multimolecular biochemical reaction model with L'{e}vy jumps
date: 2020-04-26
journal: nan
DOI: nan
sha: 29bc90c6aca9bbd9936824486bf00d851f7dac2a
doc_id: 219132
cord_uid: xdph25ld

Population dynamics 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 multimolecular biochemical reaction model that includes L'{e}vy jumps and interval parameters. Firstly, we prove the existence and uniqueness of the positive solution. Moreover, the threshold between extinction and persistence of the reaction is obtained. Finally, some simulations are carried out to demonstrate our theoretical results.

In recent decades, the study of biochemical reaction model has become one of the famous topics in mathematical biology and catalytic enzyme research. For a better review of mathematical models on the theory of biochemical reaction, see Kwek and Zhang [1] and Tang and Zhang [2] .

Considering the biochemical reaction is inevitably affected by environmental noise. Kim and Sauro [3] studied the sensitivity summation theorems for stochastic biochemical reaction models. In order to capture essential feature of stochastic biochemical reaction systems, some researchers have used different methods to add random terms into the deterministic chemical reaction or epidemic models and studied the dynamical behavior of the corresponding stochastic models driven by white noise (see e.g. [4] - [11] as well as there references).

Most population systems assume that model parameters are accurately known. 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. [16] studied a cholera epidemic model with imprecise numbers and discussed the stability condition of equilibrium points of the system. Das and Pal [17] 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 [12] - [15] , and the references therein.

The main focus of this paper is Dynamics of an imprecise stochastic multimolecular biochemical reaction model with Lévy jumps. To this end, we first introduce the imprecise stochastic multimolecular biochemical reaction model. With the help of Lyapunov functions, we prove the existence and uniqueness of the positive solution. Further, the threshold between extinction and persistence of the reaction is 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 threshold between extinction and persistence of the reaction are derived in Sect. 4 and Sect. 5.

In this section, we introduce the imprecise stochastic multimolecular biochemical reaction model. The multimolecular reactions described by the following reaction formulas (Selkov [18] ),

Let x(t) and y(t) denote the concentrations of Ξ 1 and Ξ 2 at time t, respectively, and using x 0 to denote the concentration of Ξ 0 . Then a stochastic multimolecular biochemical reaction model with Lévy jumps takes the following form (Gao and Jiang [19] ).

where p ≥ 1, and B(t) is standard Brownian motion with B(0) = 0. σ 2 > 0 represent the intensity of white noise. γ(u) : Y × Ω → R is the bounded and continuous functions satisfying |γ(u)| < z with z > 0 is a constant. The x(t − ) and y(t − ) are the left limits of x(t) and y(t), respectively. N denotes the compensated random measure defined by N (dt, du) = N (dt, du)λ(du)dt, where N is the Poisson counting measure and λ is the characteristic measure of N which is defined on a finite measurable subset Y of (0, +∞) with λ(Y) < ∞. We assume B and N are independent throughout the paper and denote

Before we state the imprecise stochastic multimolecular biochemical reaction model, definitions of Intervalvalued function should recalled (Pal [20] ). , the interval-valued function is take as h(π) = a (1−π) b π for π ∈ [0, 1]. Letk 1 ,k 2 ,k 3 ,k 4 ,p,σ represent the interval numbers of k 1 , k 2 , k 3 , k 4 , p, σ, respectively. The system (2.1) with imprecise parameters becomes,

According to the Theorem 1 in Pal et al. [12] and considering the interval-valued function

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

for υ ∈ [0, 1]. For convenience in the following investigation, let (Ω, F , {F t } t≥0 , P) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions and define

We also need the following assumption.

To research the dynamical behavior of an imprecise stochastic multimolecular biochemical reaction model, our first concern is whether the solution is global and positive. In this section, with the help of Lyapunov function, we show that system (2.3) has a unique global positive solution with any given initial value.

Because the coefficients of system (2.3) are local Lipschitz continuous (Mao [21] ), for any given initial value (x(0), y(0)) ∈ R 2 + , there is a unique local solution (x(t), y(t)) on t ∈ [0, τ e ), where τ e is the explosion time (see Mao [21] ). In order to show that the local solution is global, we only need to prove that τ e = ∞ a.s. In this context, choosing a sufficiently large number m 0 ≥ 1 such that (x(0), y(0)) lie within the interval

For each integer m ≥ m 0 , we define the stopping time as

where inf ∅ = ∞ (∅ being empty set). By the definition, τ m increases as m → ∞. Set τ ∞ = lim m→∞ τ m . Hence τ ∞ ≤ τ e a.s. If τ ∞ = ∞ a.s. is true, then τ e = ∞ a.s. for all t > 0. In other words, we need to verify τ ∞ = ∞ a.s. If this claim is wrong, then there exist a constant T > 0 and an ǫ ∈ (0, 1) such that

Hence there is an interger m 1 ≥ m 0 such that

Consider the Lyapunov function V :

Let m ≥ m 1 and T > 0. Then, for any 0 ≤ t ≤ min τ m , T , the Itǒs formula (Situ [22] ) shows that

where L is a differential operator, and

where

By use of (H) and Taylor formula, we know that

with θ ∈ (0, 1). Similarly, we have

Taking the expectations on both sides of (3.2), we obtain that

Let Ω m = {τ m ≤ T } for m ≥ m 1 . Then, by (3.1), we know that P(Ω m ) ≥ ǫ. Noting that for every ω ∈ Ω m , there exist x(τ m , ω) or y(τ m , ω), all of which equal either m or 1 m . Hence x(τ m , ω), y(τ m , ω)) is no less than

where 1 Ωm(ω) represents the indicator function of Ω m (ω). Setting m → ∞ leads to the contradiction

Therefore, we have τ ∞ = ∞ a.s. The proof is complete.

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

Theorem 4.1. Let Assumption (H) hold. For any initial value (x(0), y(0)) ∈ Υ, there is a unique positive solution (x(t), y(t)) to system (2.3) . If one of the following two conditions holds

That is to say the reaction will become extinct exponentially with probability one.

Proof. Integrating from 0 to t on both sides of (2.3), yields

Clearly, we can derive that

Applying Ito formula to (2.3) we can conclude that

Integrating (4.3) from 0 to t and then dividing by t on both sides, we obtain

] N (ds, du) are all martingale terms and h : (0, (

(4.5)

Thus, by (4.4), we have

(4.6)

Moreover, The quadratic variation can be calculated Clearly, taking the superior limit on both sides of (4.6), we know that

Similarly, we get

which yields lim t→∞ y(t) = 0 a.s.

This completes the proof.

In this section, we establish sufficient conditions for persistence in the mean of system (2.3). This together with (4.1) implies ln y(t) − ln y(0) t ≥(p l ) 1−υ (p u ) υ (k 3

(5.4)

Taking the inferior limit on both sides of (5.4) and combining with Lemma , from (4.2) and (4.7) we have lim inf

Therefore, by Assumption (L), we can easily obtain (5.1).

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Proof. According to (4.4) , we know that ln y(t) − ln y(0) t =(p l ) 1−υ (p u ) υ (k 3