Stochastic Nicholson’s blowflies delayed differential equations

Accepted Manuscript Stochastic Nicholson’s blowflies delayed differential equations

Wentao Wang, Liqing Wang, Wei Chen

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S0893-9659(18)30241-6 https://doi.org/10.1016/j.aml.2018.07.020 AML 5592

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Applied Mathematics Letters

Received date : 30 May 2018 Accepted date : 15 July 2018 Please cite this article as: W. Wang, L. Wang, W. Chen, Stochastic Nicholson’s blowflies delayed differential equations, Appl. Math. Lett. (2018), https://doi.org/10.1016/j.aml.2018.07.020 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Stochastic Nicholson’s blowflies delayed differential equations ∗ a

Wentao Wang a , Liqing Wang b , Wei Chen b †, School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai, 201620, People’s Republic of China

b

School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai, 201209, People’s Republic of China

Abstract: In this paper, we consider a class of stochastic Nicholson’s blowflies delayed differential equations. Firstly, we obtain the existence and uniqueness of the global positive solution with nonnegative initial conditions. Then the ultimate boundedness in mean of solution is derived under the same condition. Moreover, we estimate the sample Lyapunov exponent of the solution, which is less than a positive constant. In the end, an example with its numerical simulations is carried out to validate the analytical results. Keywords: Stochastic delayed differential equation; Brownian motion; ultimate boundedness; Itˆ o formula

1. Introduction The famous Nicholson’s blowflies model with time delay can be described as follows: x′ (t) = −αx(t) + px(t − τ )e−γx(t−τ ) ,

(1.1)

x(s) = ϕ(s), for s ∈ [−τ, 0], ϕ ∈ C([−τ, 0], R+ ), ϕ(0) > 0.

(1.2)

with initial conditions

Here R+ = [0, +∞), p > 0 is the maximum per capita daily egg production rate,

1 γ

> 0 is the size at

which the population reproduces at its maximum rate, α > 0 is per capita daily adult mortality rate and τ is the generation time, or the time taken from birth to maturity. This equation was first introduced by Gurney et al. [1] who created the population model based on the work of Nicholson [2]. In the past forty years, there have been plenty of papers written about the classical Nicholson’s model and its generalizations (specially, to variable coefficients and delays), see, for example, [3-21]). In 2010, Berezansky et al. [22] have collected above results and presented seven open problems which are all concerned with deterministic model. †

Corresponding author. Tel.:+8602167705382, fax: +8602167705382. E-mail:[email protected] work was supported by the Development Fund for Shanghai Talents (No. 2017128).

∗ This

1

However, the Nicholson’s blowflies model is always affected by environmental noises. As pointed out by May [23] that due to environmental noises, in the population model the growth rates, environmental capacity, competition coefficient and others parameters should be stochastic. Therefore, the stochastic Nicholson’s blowflies delayed differential equation is more suitable to model the data of Nicholson [2]. Assume that the parameter α is affected by environmental noises, with α → α + σdB(t), where

B(t) is a one-dimensional Brownian motion with B(0) = 0 defined on a complete probability space

(Ω, {Ft }t≥0 , P), σ 2 denotes the intensity of the noise. Then corresponding to model (1.1), we obtain the following stochastic model:

dx(t) = [−αx(t) + px(t − τ )e−γx(t−τ ) ]dt + σx(t)dB(t),

(1.3)

It is easy to see that (1.1) has the trivial equilibrium and the nontrivial positive equilibrium ∗

x =

1 γ

ln αp , which exists for p > α. The results on the stability of the trivial equilibrium and the

nontrivial positive equilibrium of (1.1) have been collected in [22]. Moreover, So and Yu [3] have proved that the unique global solution of equation (1.1) with initial condition (1.2) is positive. Obviously, the trivial equilibrium satisfies the stochastic model (1.3), but the positive equilibrium x∗ no longer satisfies it. Then a question appears natural: Find the criteria to guarantee the unique global positive solution and the ultimate boundedness for (1.3). The paper is organized as follows. In section 2, we discuss the existence and uniqueness of a global almost surely positive solution of (1.3) with initial values (1.2). In section 3, we study the ultimate boundedness and the sample Lyapunov exponent of (1.3). We carry out an example and several numerical simulations to illustrate theoretical results in section 4. Finally, we provide a brief conclusion to summary and evaluate our work in section 5.

2. Preliminary results In this section, we introduce some basic definitions and lemmas which are important to prove the main result in the next section. Definition 2.1. Equation (1.3) is said to be ultimate bounded in mean if there is a positive constant L independent of initial conditions (1.2) such that lim sup E|x(t)| ≤ L. t→∞

Lemma 2.1. If α >

2

σ 2

, then for any x ∈ R, p x ≤ K(1 + x2 ), γe

(2.1)

p p2 x≤ 2 2 , γe γ e (2α − σ 2 )

(2.2)

−(2α − σ 2 )x2 + 2 −(2α − σ 2 )x2 + 2 2

p p where K = min{ γ 2 e2 (2α−σ 2 ) , γe }.

Proof. It is easy to analyze the property of the quadratic function, so we omit the proof.

2

Lemma 2.2. If α >

σ2 2 ,

then for any given initial condition (1.2), (1.3) has a unique solution x(t)

on [0, +∞) and x(t) is positive almost surely for t ≥ 0.

Proof. Because the constant coefficients of (1.3) are locally Lipschitz continuous, for any given

initial condition (1.2), there is a unique max local solution x(t) on [−τ, τe ), where τe is explosion time. To show this solution is global, it is sufficient to prove τe = ∞ a.s. Let k0 > 0 be sufficient large such

that max |x(t)| < k0 . For each integer k ≥ k0 , define the stopping time −τ ≤t≤0

τk = inf{t ∈ [0, τe ) : |x(t)| ≥ k}, where throughout this paper we set inf φ = ∞ (as usual φ denotes the empty set). Clearly, τk is

increasing as k → ∞. Set τ∞ = lim τk , where τ∞ ≤ τe a.s. If we can show that τ∞ = ∞ a.s. then k→∞

τe = ∞ a.s. for all t ≥ 0.

Define a C 2 -function V (x) = x2 . For 0 ≤ t ≤ τ∞ , it is to show by the Itˆ o formula that dV (x(t)) = LV (x(t), x(t − τ ))dt + 2σV (x(t))dB(t),

(2.3)

where LV : R × R → R is defined by LV (x, y) = −(2α − σ 2 )x2 + 2pxye−γy . For t ∈ [0, τ ], in view of

α>

σ2 2 ,

x(t − τ ) = ϕ(t − τ ) ≥ 0 and noting the fact that sup xe−x = 1e , it follows from (2.2) that x∈R+

LV (x(t), x(t − τ )) ≤ −(2α − σ 2 )x2 (t) + 2

p p2 |x(t)| ≤ 2 2 , t ∈ [0, τ ]. γe γ e (2α − σ 2 )

(2.4)

In view of (2.4), we obtain from (2.3) that dV (x(t)) ≤

p2 dt + 2σV (x(t))dB(t), t ∈ [0, τ ]. γ 2 e2 (2α − σ 2 )

(2.5)

For any k ≥ k0 and t1 ∈ [0, τ ], integrating both sides of (2.5) from 0 to tk ∧ t1 , and then taking the

expectations, yields

EV (x(τk ∧ t1 )) ≤ V (x(0)) + E e = V (x(0)) + where K

τ p2 γ 2 e2 (2α−σ2 ) .

show τ∞ ≥ τ a.s.

Z

τk ∧t1

0

p2 e 0 ≤ t1 ≤ τ, k ≥ k0 , dt ≤ K, γ 2 e2 (2α − σ 2 )

(2.6)

e for all k ≥ k0 . From this we will Specially, EV (x(τk ∧ τ )) ≤ K

Since for every ω ∈ {τk < τ }, it is obvious that V (x(τk , ω)) ≥ k 2 . It then follows from (2.6) that e ≥ EV (x(τk ∧ τ )) ≥ E[I{τ <τ } (ω)V (x(τk , ω))] ≥ P {τk < τ }k 2 , K k

where I{τk <ω} is the indicator function of {τk < ω}. Letting k → ∞ gives lim P {τk < τ } = 0, and k→∞

hence P {τ∞ < τ } = 0, i.e. P {τ∞ ≥ τ } = 1 as required. Repeating this procedure, we can then show that τ∞ ≥ 2τ a.s. and then τ∞ ≥ mτ a.s. for any integer m ≥ 1. Thus we must have τ∞ = ∞ a.s. as required.

Now we claim that x(t) is positive almost surely on [0, ∞). We will prove this step by step. On

t ∈ [0, τ ], the initial problem of (1.3) with (1.2) becomes the following linear stochastic differential equation:

(

dx(t) = [−αx(t) + a1 (t)]dt + σx(t)dB(t) x0 = ϕ(0) > 0 3

,

(2.7)

where a1 (t) = pϕ(t − τ )e−γϕ(t−τ ) ≥ 0 a.s., t ∈ [0, τ ]. Clearly, (2.7) has the explicit solution x(t) = Rt σ2 σ2 e−(α− 2 )t+σB(t) [x(0) + 0 e(α− 2 )s−σB(s) a1 (s)ds] > 0 a.s. for t ∈ [0, τ ]. Next, on t ∈ [τ, 2τ ], (1.3) becomes the following linear stochastic differential equation: ( dx(t) = [−αx(t) + a2 (t)]dt + σx(t)dB(t) xτ = x(τ ) > 0 a.s.

,

(2.8)

where a2 (t) = px(t − τ )e−γx(t−τ ) > 0 a.s., t ∈ [τ, 2τ ]. Obviously, (2.8) has the explicit solution Rt σ2 σ2 x(t) = e−(α− 2 )(t−τ )+σ(B(t)−B(τ )) [x(τ ) + τ e(α− 2 )s−σB(s) a2 (s)ds] > 0 a.s. for t ∈ [τ, 2τ ]. Repeating

this procedure, we can then show that x(t) > 0 a.s. on [mτ, (m + 1)τ ] a.s. for any integer m ≥ 1. So

(1.3) with (1.2) has the unique global solution x(t) and it is positive almost surely for t ∈ [0, ∞).

3. Main results One of the important properties in stochastic population model is the ultimate boundedness in mean. The following theorem provides a criterion for this property of (1.3). Theorem 3.1. Let α >

σ2 2

hold. Then for any given initial value (1.2), the global solution of

(1.3) x(t) is positive almost surely on t ≥ 0 and it has the properties that lim sup Ex(t) ≤ t→∞

and lim sup t→∞

1 t

Z

t

0

Ex2 (s)ds ≤

In particular, (1.3) is ultimately bounded in mean.

p αγe

(3.1) 4p2 . − σ 2 )2

γ 2 e2 (2α

(3.2)

Proof. From Lemma 2.2 the global solution x(t) of (1.3) is positive almost surely on t ≥ 0. It 1 e

follows from (1.3) and the fact sup xe−x = x∈R+

that

dx(t) ≤ (−αx(t) +

p )dt + σx(t)dB(t), γe

(3.3)

which, together the Itˆ o formula, implies that d[eαt x(t)] = eαt [αx(t)dt + dx(t)] ≤

p αt e dt + σeαt x(t)dB(t). γe

(3.4)

Integrating both sides of (3.4) from 0 to t, and then taking the expectations, we have eαt Ex(t) ≤ x(0) + This yields lim sup Ex(t) ≤ t→∞

Z

0

t

p p αs e ds = x(0) + (eαt − 1). γe αγe

p αγe .

By the Itˆ o formula and the fact sup xe−x = x∈R+

d[x2 (t)] = ≤

1 e

again, it follows from (1.3) that

[−(2α − σ 2 )x2 (t) + 2px(t)x(t − τ )e−γx(t−τ ) ]dt + 2σx2 (t)dB(t) 2p [−(2α − σ 2 )x2 (t) + |x(t)|]dt + 2σx2 (t)dB(t). γe 4

(3.5)

Integrating both sides of (3.5) from 0 to t, and then taking the expectations, we obtain Z t 2p 2 2 |x(s)|]dt. 0 ≤ Ex (t) ≤ x (0) + [−(2α − σ 2 )x2 (s) + γe 0 Rt 2 2p 2p2 2p2 σ2 2 2 Noting −(α− σ2 )x2 (s)+ γe |x(s)| ≤ γ 2 e2 (2α−σ 2 ) , we obtain (α− 2 ) 0 Ex (s)ds ≤ x (0)+ γ 2 e2 (2α−σ 2 ) t. Rt 4p2 This implies immediately that lim sup 1t 0 Ex2 (s)ds ≤ γ 2 e2 (2α−σ 2 )2 , which is the desired assertion t→∞

(3.2). The proof is now complete. Theorem 3.2. Let α >

σ2 2

hold. Then the sample Lyapunov exponent of the solution of (1.3)

with (1.2) should not be greater than

K 2,

that is

lim sup t→∞

1 K ln x(t) ≤ , a.s. t 2

Proof. By the Itˆ o formula and the fact sup xe−x = x∈R+ 2

ln(1 + x (t))

=

≤

≤ Rt

Z

1 e

, it follows from (1.3) and (2.1) that

t

1 [−(2α − σ 2 )x2 (s) 2 (s) 1 + x 0 Z t σ 2 x4 (s) ds + M (t) +2px(s)x(s − τ )e−γx(s−τ ) ]ds − 2 2 2 0 (1 + x (s)) Z t 1 ln(1 + x2 (0)) + [−(2α − σ 2 )x2 (s) 2 0 1 + x (s) Z t σ 2 x4 (s) p ds + M (t) +2 x(s)]ds − 2 2 2 γe 0 (1 + x (s)) Z t σ 2 x4 (s) ds + M (t), ln(1 + x2 (0)) + Kt − 2 2 2 0 (1 + x (s)) 2

ln(1 + x (0)) +

(3.6)

(3.7)

σx2 (s) dB(s). 0 1+x2 (s)

On the other hand, for every n ≥ 0, using the exponential martingale R t σ2 x4 (s) 1 inequality (i.e. Theorem 1.7.4 of [23]), one sees that P { sup [M (t) − 2 0 (1+x 2 (s))2 ds] > 2 ln n} ≤ n2 . where M (t) = 2

0≤t≤n

An application of the Borel-Cantelli lemma then yields that for almost all ω ∈ Ω there is a random R t σ2 x4 (s) integer n0 = n0 (ω) ≥ 1 such that sup [M (t) − 2 0 (1+x 2 (s))2 ds] ≤ 2 ln n if n ≥ n0 . That is 0≤t≤n

M (t) ≤ 2

Z

0

t

σ 2 x4 (s) ds + 2 ln n (1 + x2 (s))2

for all 0 ≤ t ≤ n, n ≥ n0 almost surely. Substituting (3.8) into (3.7) deduces that ln(1 + x2 (t)) ≤ ln(1 + x2 (0)) + Kt + 2 ln n for all 0 ≤ t ≤ n, n ≥ n0 almost surely. So, for almost all ω ∈ Ω, if n ≥ n0 , n − 1 ≤ t ≤ n, 1 1 ln(1 + x2 (t)) ≤ [ln(1 + x2 (0)) + Kn + 2 ln n]. t n−1

This implies lim sup t→∞

1 ln x(t) t

≤ ≤

1 ln(1 + x2 (t)) 2t K 1 lim sup [ln(1 + x2 (0)) + Kn + 2 ln n] = a.s. 2 n→∞ 2(n − 1)

lim sup t→∞

5

(3.8)

0.7

0.65

x(t)

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0

2

4

6

8

10

12

14

16

18

20

Figure 1: Numerical solutions of (4.1) for initial value 0.3, 0.5, 0.7. The proof is complete. Remark 3.1. It is interesting to find that the condition α >

σ2 2

in Lemma 2.2, Theorems 3.1 and

3.2 is dependent of noise intensity σ but the assertion (3.1) is no longer. That is to say, the property of the ultimate boundedness in mean of (1.3) will not change when the environmental noise is small. In other words, the property of this boundedness is robust under small noises.

4. An example and its numerical simulations In this section, we introduce an example and figures to illustrate our main result. Example 4.1. Consider the following stochastic Nicholson’s blowflies delayed differential equation: dx(t) = [−2x(t) + 3x(t − 1)e−x(t−1) ]dt + x(t)dB(t). Clearly, α = 2, p = 3, γ = σ = τ = 1 and α ≥

that the solution of (4.1) obeys lim sup Ex(t) ≤ 3 e2 ,

t→∞

(4.1)

σ2 2 holds. Then it follows from Theorems 3.1 and 3.2 R 1 t 1 3 4 2 2e , lim sup t 0 Ex (s)ds ≤ e2 and lim sup t ln x(t) ≤ t→∞ t→∞

a.s. This fact is verified by the numerical simulations in Figure 1 which are based on Milstein’s

numerical method [24] .

5. Conclusions This paper is concerned with stochastic Nicholson’s blowflies delayed differential equation. Using Itˆ o formula and some inequalities, we deduce the sufficient condition α >

σ2 2

for the uniqueness and

global existence of positive solutions and ultimate boundedness in mean. Moreover, we estimate the sample Lyapunov exponent of (1.3). It is not difficult to see that the sufficient condition in Lemma 2.2, Theorems 3.1 and 3.2 depends on the environmental noises.

References 6

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