Permanence and stability of an Ivlev-type predator–prey system with impulsive control strategies

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Mathematical and Computer Modelling

journal homepage:www.elsevier.com/locate/mcm

Permanence and stability of an Ivlev-type predator–prey system with impulsive control strategies

^I

Hun Ki Baek

^∗

, Sang Dong Kim, Philsu Kim

Department of Mathematics, Kyungpook National University, Daegu 702-701, South Korea

a r t i c l e i n f o

Article history:

Received 28 October 2008

Received in revised form 19 June 2009 Accepted 2 July 2009

Keywords:

Predator–prey model Ivlev-type functional response Impulsive differential equation Floquet theory

a b s t r a c t

In this paper, we study a predator–prey system with an Ivlev-type functional response and impulsive control strategies containing a biological control (periodic impulsive immigration of the predator) and a chemical control (periodic pesticide spraying) with the same period, but not simultaneously. We find conditions for the local stability of the prey-free periodic solution by applying the Floquet theory of an impulsive differential equation and small amplitude perturbation techniques to the system. In addition, it is shown that the system is permanent under some conditions by using comparison results of impulsive differential inequalities. Moreover, we add a forcing term into the prey population’s intrinsic growth rate and find the conditions for the stability and for the permanence of this system.

©2009 Elsevier Ltd. All rights reserved.

1. Introduction

Understanding the dynamical relationship between predator and prey is one of the central goals in population dynamics.

One important component of the predator–prey relationship is the predator’s rate of feeding on the prey, i.e., the so-called predator’s functional response. Functional response refers to the change in the density of prey attached per unit time per predator as the prey density changes. Holling [1] gave three different kinds of functional response for different kinds of species to model the phenomena of predation, which made the standard Lotka–Volterra system more realistic. These functional responses are not only monotonically increasing, but also uniformly bounded functions in the first quadratic.

Another functional response, also both monotonically increasing and uniformly bounded, was suggested by [2]:

p

(

^x

) =

h

(

¹

−

exp

( −

cx

))(

^c

,

^h

>

⁰

),

which is called the Ivlev-type functional response, wherehrepresents the maximum rate of predation andcis a constant representing the decrease in motivation to hunt. With this functional response, we can write a predator–prey model as follows:







x⁰

(

^t

) =

ax

(

^t

)

1

−

^x

(

^t

)

b

−

h

(

¹

−

exp

( −

cx

(

^t

)))

^y

(

^t

),

y⁰

(

^t

) = −

dy

(

^t

) +

e

(

¹

−

exp

( −

cx

(

^t

)))

^y

(

^t

),

(

^x

(

⁰⁺

),

^y

(

⁰⁺

)) = (

^x0

,

^y0

).

(1.1)

I This work was supported by KRF-2005-070-C00017.

∗Corresponding author.

E-mail addresses:[email protected](H.K. Baek),[email protected](S.D. Kim),[email protected](P. Kim).

0895-7177/$ – see front matter©2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mcm.2009.07.007

There are a number of factors in the environment to be considered in predator–prey models[3]. One of the important factors is an impulsive perturbation such as fire, flood, etc., that are not suitable to be considered continuously. Generally, the impulsive perturbations that bring sudden changes to system can be classified into two cases. One is the perturbations made by nature. For example, consider the interaction between crops and locusts in a local region. Once a year or once every several years, a large number of locusts may invade the region and cause damage to crops together with the local locusts. The other is the perturbations that occur by human artificial activities to control the prey density. For example, let us consider pest outbreaks. There are many ways to beat agricultural pests, such as biological or chemical tactics. Biological control is to reduce the pest population using the actions of other living organisms, often called natural enemies or beneficial species.

Virtually all pests have some natural enemies, and the key to successful pest control is to identify the pest and its natural enemy and release the beneficial insect at fixed times for pest control. Another important method for pest control is chemical control. Pesticides are useful because they quickly kill a significant portion of a pest population and they sometimes provide the only feasible method for preventing economic loss. Consequently, it is natural to assume that these perturbations act instantaneously, that is, in the form of an impulse. With the idea of impulsive perturbations, in this paper, we consider the following predator–prey system with periodic pesticide spraying on all species and periodic impulsive immigration of the predator at fixed different times.























x⁰

(

^t

) =

ax

(

^t

)

1

−

^x

(

^t

)

b

−

h

(

¹

−

exp

( −

cx

(

^t

)))

^y

(

^t

),

y⁰

(

^t

) = −

dy

(

^t

) +

e

(

¹

−

exp

( −

cx

(

^t

)))

^y

(

^t

),

)

t

6=

nT

,

^t

6= (

ⁿ

+ τ −

1

)

^T

,

x

(

^t+

) = (

¹

−

p₁

)

^x

(

^t

),

y

(

^t+

) = (

¹

−

p₂

)

^y

(

^t

),

t

= (

ⁿ

+ τ −

1

)

^T

,

x

(

^t+

) =

x

(

^t

),

y

(

^t+

) =

y

(

^t

) +

q

,

t

=

nT

, (

^x

(

⁰⁺

),

^y

(

⁰⁺

)) = (

^x0

,

^y0

),

(1.2)

wherex

(

^t

)

^andy

(

^t

)

are functions of time representing the population densities of the prey and predator, respectively, 0

≤ τ,

^p1

,

^p2

<

1, with all parameters being positive constants,T is the impulsive immigration of the predator, andqis the amount of immigration or stock of the predator.ddenotes the death rate of the predator, andeis the rate of conversion of a consumed prey to a predator. This system is called an impulsive differential equation, whose theory and applications were greatly developed by the efforts of Bainov and Lakshmikantham et al. [4,5]. Moreover, the theory of impulsive differential equations is being recognized to be not only richer than the corresponding theory of differential equations without impulses, but it also represents a more natural framework for mathematical modeling of real world phenomena.

In recent years, population models with impulsive perturbations have been intensively researched, such as the Lotka–Volterra model [6–11], Holling-type[12–26], Beddington-type[27–33] and Ivlev-type [34,35]. Although much progress has been made in the study of population models with the Ivlev-type functional response [36,2,37,38], research of impulsive Ivelv-type differential equations seems not yet to be as prevalent considering its importance. Thus, the main purpose of this paper is to investigate the dynamics of system(1.2).

In the next section, we introduce some notations which are used in this paper. We study the qualitative properties of system(1.2)in Section3. So, we show the local stability of the prey-free periodic solution and give a sufficient condition for the permanence of system(1.2)by applying the Floquet theory and the comparison theorem, respectively. Finally, we take into account seasonal effects on the prey as a forcing term of system(1.2)in Section4and find the conditions for the stability and for the permanence of this system.

2. Preliminaries

In this section, we give some notations, definitions and Lemmas which will be useful for our main results.

LetR⁺

= [

0

, ∞ )

^andR²+

= {

x

= (

^x

(

^t

),

^y

(

^t

)) ∈

_R²

:

x

(

^t

),

^y

(

^t

) ≥

0

}

. Denote byNthe set of all of non-negative integers and byf

= (

^f1

,

^f2

)

^Tthe right-hand side of system(1.2). LetV

:

_R+

×

_R²₊

→

_R+; thenVis said to be in a classV₀if

(

¹

)

^V is continuous in

((

ⁿ

−

1

)

^T

, (

ⁿ

+ τ −

1

)

^T

] ×

_R²₊ and

((

ⁿ

+ τ −

1

)

^T

,

^nT

] ×

_R²₊

,

(t,y)→((nlim+τ−1)T+,x)V

(

^t

,

^y

) =

V

((

ⁿ

+ τ −

1

)

^T+

,

^x

)

and lim

(^t,^y)→(^nT+,^x)V

(

^t

,

^y

) =

V

(

^nT+

,

^x

)

exists for each x

∈

_R²₊andn

∈

_N

. (

²

)

^Vis locally Lipschitzian inx

.

Definition 2.1. LetV

∈

V₀

, (

^t

,

^x

) ∈ ((

ⁿ

−

1

)

^T

, (

ⁿ

+ τ −

1

)

^T

] ×

_R²₊and

((

ⁿ

+ τ −

1

)

^T

,

^nT

] ×

_R²₊. The upper right derivatives ofV

(

^t

,

^x

)

with respect to the impulsive differential system(1.2)are defined as

D⁺V

(

^t

,

^x

) =

lim sup

l→0+ 1

l

[

V

(

^t

+

h

,

^x

+

lf

(

^t

,

^x

)) −

V

(

^t

,

^x

) ] .

Definition 2.2. The system(1.2)is permanent if there existM

≥

m

>

0 such that, for any solution

(

^x

(

^t

),

^y

(

^t

))

^{of system} (1.2)with

(

^x0

,

^y0

) >

^0,

m

≤

lim

t→∞infx

(

^t

) ≤

lim

t→∞supx

(

^t

) ≤

M and m

≤

lim

t→∞infy

(

^t

) ≤

lim

t→∞supy

(

^t

) ≤

M

.

Remark 2.3. The smoothness properties offguarantee the global existence and uniqueness of the solutions of system(1.2).

(See [4,5] for the details.)

Lemma 2.4. Letx

(

^t

) = (

^x

(

^t

),

^y

(

^t

))

be a solution of system(1.2). Then we obtain that (1) If x

(

⁰⁺

) ≥

0thenx

(

^t

) ≥

0for all t

≥

0, and

(2) If x

(

⁰⁺

) >

^0thenx

(

^t

) >

^{0for all t}

≥

0.

We will use the following important comparison theorem on an impulsive differential equation [4,5].

Lemma 2.5 ([5]).Suppose V

∈

V₀and







D⁺V

(

^t

,

^x

) ≤

g

(

^t

,

^V

(

^t

,

^x

)),

^t

6= (

ⁿ

+ τ −

1

)

^T

,

^nT

,

V

(

^t

,

^x

(

^t+

)) ≤ ψ

n¹

(

^V

(

^t

,

^x

)),

^t

= (

ⁿ

+ τ −

1

)

^T

,

V

(

^t

,

^x

(

^t+

)) ≤ ψ

n²

(

^V

(

^t

,

^x

)),

^t

=

nT

,

(2.1)

where g

:

_R+

×

_R+

→

_Rsatisfies

(

^H

)

^and

ψ

n¹

, ψ

n²

:

_R+

→

_R+are non-decreasing for all n

∈

_N. Let r

(

^t

)

be the maximal solution for the impulsive Cauchy problem











u⁰

(

^t

) =

g

(

^t

,

^u

(

^t

)),

^t

6= (

ⁿ

+ τ −

1

)

^T

,

^nT

,

u

(

^t+

) = ψ

n¹

(

^u

(

^t

)),

^t

= (

ⁿ

+ τ −

1

)

^T

,

u

(

^t+

) = ψ

n²

(

^u

(

^t

)),

^t

=

nT

,

u

(

⁰⁺

) =

u₀

≥

0

,

(2.2)

defined on

[

0

, ∞ )

^{. Then V}

(

⁰⁺

,

^x0

) ≤

u₀implies that V

(

^t

,

^x

(

^t

)) ≤

r

(

^t

),

^t

≥

0, wherex

(

^t

)

is any solution of(2.1).

Similar result can be obtained when all conditions of the inequalities in the Lemma are reversed. Note that, if we have some smoothness conditions ofg

(

^t

,

^u

(

^t

))

to guarantee the existence and uniqueness of the solutions for(2.2), thenr

(

^t

)

^{is exactly} the unique solution of(2.2). Now, we give the basic properties of the following impulsive differential equation.







y⁰

(

^t

) = −

dy

(

^t

),

^t

6=

nT

,

^t

6= (

ⁿ

+ τ −

1

)

^T

,

y

(

^t+

) = (

¹

−

p₂

)

^y

(

^t

),

^t

= (

ⁿ

+ τ −

1

)

^T

,

y

(

^t+

) =

y

(

^t

) +

q

,

^t

=

nT

,

y

(

⁰⁺

) =

y₀

.

(2.3)

System(2.3)is a periodically forced linear system. So it is easy to obtain that

y^∗

(

^t

) =











qexp

( −

d

(

^t

− (

ⁿ

−

1

)

^T

))

1

− (

¹

−

p₂

)

^exp

( −

dT

) , (

ⁿ

−

1

)

^T

<

^t

≤ (

ⁿ

+ τ −

1

)

^T

,

q

(

¹

−

p₂

)

^exp

( −

d

(

^t

− (

ⁿ

−

1

)

^T

))

1

− (

¹

−

p₂

)

^exp

( −

dT

) , (

ⁿ

+ τ −

1

)

^T

<

^t

≤

nT

,

(2.4)

y^∗

(

⁰⁺

) =

y^∗

(

^nT+

) =

^q

1−(1−p2)exp(−dT),y^∗

(τ

^T+

) =

^{q(1−p2)exp(−dτT)}

1−(1−p2)exp(−dT)is a positive periodic solution of system(2.3). Moreover, we can figure out that

y

(

^t

) =



















(

¹

−

p₂

)

ⁿ⁻¹

y

(

⁰⁺

) −

^q

(

¹

−

p₂

)

^e−T 1

− (

¹

−

p₂

)

^exp

( −

dT

)

exp

( −

dt

) +

y^∗

(

^t

), (

ⁿ

−

1

)

^T

<

^t

≤ (

ⁿ

+ τ −

1

)

^T

,

(

¹

−

p₂

)

ⁿ

y

(

⁰⁺

) −

^q

(

¹

−

p₂

)

^e−T 1

− (

¹

−

p₂

)

^exp

( −

dT

)

exp

( −

dt

) +

y^∗

(

^t

), (

ⁿ

+ τ −

1

)

^T

<

^t

≤

nT

(2.5)

is the solution of(2.3). From(2.4)and(2.5), we easily get the following results.

Lemma 2.6. All solutions y

(

^t

)

^{of system(1.2)tend to y∗}

(

^t

)

^{; i.e.,}

|

y

(

^t

) −

y^∗

(

^t

) | →

0as t

→ ∞

.

It is fromLemma 2.6that the general solutiony

(

^t

)

^{of system(2.3)}can be synchronized with the positive periodic solution y^∗

(

^t

)

^{of system}(2.3), and we can obtain the complete expression for the prey-free periodic solution of system(1.2):

(

⁰

,

^y∗

(

^t

)),

where, fort

= ((

ⁿ

−

1

)

^T

,

^nT

]

,

y^∗

(

^t

) =











qexp

( −

d

(

^t

− (

ⁿ

−

1

)

^T

))

1

− (

¹

−

p₂

)

^exp

( −

dT

) , (

ⁿ

−

1

)

^T

<

^t

≤ (

ⁿ

+ τ −

1

)

^T

,

q

(

¹

−

p₂

)

^exp

( −

d

(

^t

− (

ⁿ

−

1

)

^T

))

1

− (

¹

−

p₂

)

^exp

( −

dT

) , (

ⁿ

+ τ −

1

)

^T

<

^t

≤

nT

.

To study the stability of the prey-free periodic solution as a solution of system(1.2)we present the Floquet theory for a linearT-periodic impulsive equation:

(_dx

dt

=

A

(

^t

)

^x

(

^t

),

^t

6= τ

k

,

^t

∈

_R

,

x

(

^t+

) =

x

(

^t

) +

B_kx

(

^t

),

^t

= τ

k

,

^k

∈

_Z

.

^(2.6)

Then we introduce the following conditions:

(H₁) A

( · ) ∈

PC

(

R

,

^Cn×n

)

^andA

(

^t

+

T

) =

A

(

^t

)(

^t

∈

_R

)

^{, wherePC}

(

R

,

^Cn×n

)

is the set of all piecewise continuous matrix functions which is left continuous att

= τ

k, andC^n×nis the set of alln

×

nmatrices.

(H₂) B_k

∈

C^n×n, det

(

^E

+

B_k

) 6=

0

, τ

k

< τ

k+1

(

^k

∈

_Z

)

^,

(H₃) There exists aq

∈

_Nsuch thatB_k+q

=

B_k

, τ

k+q

= τ

k

+

T

(

^k

∈

_Z

)

^.

LetΦ

(

^t

)

be a fundamental matrix of(2.6); then there exists a unique non-singular matrixM

∈

C^n×nsuch that

Φ

(

^t

+

T

) =

_Φ

(

^t

)

^M

(

^t

∈

_R

).

^(2.7)

By equality(2.7)there corresponds to the fundamental matrixΦ

(

^t

)

the constant matrixMwhich we call the monodromy matrix of(2.6)(corresponding to the fundamental matrix ofΦ

(

^t

)

). All monodromy matrices of(2.6)are similar and have the same eigenvalues. The eigenvalues

µ

1

, . . . , µ

nof the monodromy matrices are called the Floquet multipliers of(2.6).

Lemma 2.7 (Bainov et al., 1993).[Floquet theory] Let conditions

(

^H1

)

^–

(

^H3

)

hold. Then the linear T -periodic impulsive Eq.(2.6) is

(1) stable if and only if all multipliers

µ

j

(

^j

=

1

, . . . ,

ⁿ

)

^{of (2.6)}satisfy the inequality

| µ

j

| ≤

1, and moreover, to those

µ

jfor which

| µ

j

| =

1, there correspond simple elementary divisors;

(2) asymptotically stable if and only if all multipliers

µ

j

(

^j

=

1

, . . . ,

ⁿ

)

^{of (2.6)}satisfy the inequality

| µ

j

| <

^1;

(3) unstable if

| µ

j

| >

^{1for some j}

=

1

, . . . ,

^n.

3. Main results

We present a condition which guarantees the local stability of the prey-free periodic solution

(

⁰

,

^y∗

(

^t

))

^. Theorem 3.1. If

aT

−

^hcq

(

¹

+ (

^p2

−

1

)

^exp

( −

dT

) −

p₂exp

( −

d

τ

^T

))

d

(

¹

− (

¹

−

p₂

)

^exp

( −

dT

)) <

^{ln 1} 1

−

p₁

.

then

(

⁰

,

^y∗

(

^t

))

is locally asymptotically stable.

Proof. The local stability for the periodic solution

(

⁰

,

^y∗

(

^t

))

^{of system(1.2)}may be determined by considering the behavior of small amplitude perturbations of the solution. Let

(

^x

(

^t

),

^y

(

^t

))

be any solution of system(1.2). Defineu

(

^t

) =

x

(

^t

), v(

^t

) =

y

(

^t

) −

y^∗

(

^t

)

. Then they may be written as

u

(

^t

) v(

^t

)

=

_Φ

(

^t

)

u

(

⁰

)

v(

⁰

),

whereΦ

(

^t

)

^satisfies dΦ

dt

=

a

−

hcy^∗

(

^t

)

⁰ ecy^∗

(

^t

) −

d

Φ

(

^t

)

andΦ

(

⁰

) =

I, the identity matrix. So the fundamental solution matrix is

Φ

(

^t

) =







 exp

Z t 0

a

−

hcy^∗

(

^s

)

^ds

0 exp

ec

Z t 0

y^∗

(

^s

)

^ds

exp

( −

dt

)









.

The resetting impulsive condition of system(1.2)becomes u

((

ⁿ

+ τ −

1

)

^T+

)

v((

ⁿ

+ τ −

1

)

^T+

)

=

1

−

p₁ 0 0 1

−

p₂

u

((

ⁿ

+ τ −

1

)

^T

) v((

ⁿ

+ τ −

1

)

^T

)

and

u

(

^nT+

) v(

^nT+

)

=

1 0

0 1

u

(

^nT

) v(

^nT

)

.

Note that all eigenvalues of S

=

1

−

p₁ 0 0 1

−

p₂

1 0

0 1

Φ

(

^T

)

are

µ

1

= (

¹

−

p₂

)

^exp

( −

dT

) <

^{1 and}

µ

2

= (

¹

−

p₁

)

^exp

(

RT

0 a

−

hcy^∗

(

^t

)

^dt

)

^{. Since} Z T

0

y^∗

(

^t

)

^dt

=

^q

(

¹

+ (

^p2

−

1

)

^exp

( −

dT

) −

p₂exp

( −

d

τ

^T

))

d

(

¹

− (

¹

−

p₂

)

^exp

( −

dT

)) ,

the condition

| µ

2

| <

1 is equivalent to the equation

aT

−

^hcq

(

¹

+ (

^p2

−

1

)

^exp

( −

dT

) −

p₂exp

( −

d

τ

^T

))

d

(

¹

− (

¹

−

p₂

)

^exp

( −

dT

)) <

^{ln 1} 1

−

p₁

.

According toLemma 2.7,

(

⁰

,

^y∗

(

^t

))

is locally asymptotically stable.

Next, we show that all solutions of system(1.2)are uniformly ultimately bounded.

Proposition 3.2. There is an M

>

⁰such that x

(

^t

),

^y

(

^t

) ≤

M for all t large enough, where

(

^x

(

^t

),

^y

(

^t

))

is a solution of system (1.2).

Proof. Letx

(

^t

) = (

^x

(

^t

),

^y

(

^t

))

be a solution of system(1.2)and letV

(

^t

,

^x

) =

ex

(

^t

) +

y

(

^t

)

^{. ThenV}

∈

V₀. Ift

6= (

ⁿ

+ τ −

1

)

^T andt

6= (

ⁿ

+ τ)

^T, then we obtain

D⁺V

+ β

^V

= −

^ea

bx

(

^t

)

²

+ (

^ea

+

e

β)

^x

(

^t

) + (β −

d

)

^y

(

^t

).

^(3.1)

Whent

= (

ⁿ

+ τ −

1

)

^T,V

((

ⁿ

+ τ −

1

)

^T+

) ≤

V

((

ⁿ

+ τ −

1

)

^T

)

^{and whent}

=

nT,V

(

^nT+

) ≤

V

(

^nT

) +

q. Clearly, the right-hand side of Eq.(3.1)is bounded when 0

< β <

d. So we can choose 0

< β

0

<

^dandM0

>

0 such that







D⁺V

≤ − β

0V

+

M₀

,

^t

6= (

ⁿ

+ τ −

1

)

^T

,

^t

6=

nT

,

V

(

^t+

) ≤

V

(

^t

),

^t

= (

ⁿ

+ τ −

1

)

^T

,

V

(

^t+

) ≤

V

(

^t

) +

q

,

^t

=

nT

.

^(3.2)

ByLemma 2.5, we can obtain that V

(

^t

) ≤

V

(

⁰⁺

)

^exp

( − β

0t

) +

^M0

β

0

(

¹

−

exp

( − β

0t

)) +

^q

(

^exp

( − (β

0

+

1

)

^T

) −

exp

( − β

0

(

^t

− (

ⁿ

−

1

)

^T

)))

1

−

exp

( − β

0T

)

fort

∈ ((

ⁿ

−

1

)

^T

,

^nT

]

. Therefore,V

(

^t

)

is bounded by a constant for sufficiently larget. Hence there is anM

>

0 such that x

(

^t

) ≤

M

,

^y

(

^t

) ≤

Mfor a solution

(

^x

(

^t

),

^y

(

^t

))

^{with allt}large enough.

Theorem 3.3. System(1.2)is permanent if

aT

−

^hcq

(

¹

+ (

^p2

−

1

)

^exp

( −

dT

) −

p₂exp

( −

d

τ

^T

))

d

(

¹

− (

¹

−

p₂

)

^exp

( −

dT

)) >

^{ln 1} 1

−

p₁

.

Proof. Suppose

(

^x

(

^t

),

^y

(

^x

))

is any solution of system(1.2)with

(

^x

(

⁰⁺

),

^y

(

⁰⁺

)) >

^{0. From}Proposition 4.2, we may assume thatx

(

^t

) ≤

M,y

(

^t

) ≤

M,t

≥

0 andM

>

^a_c^{. Letm}2

=

^{q(1−p2)exp(−dT)}

1−(1−p2)exp(−dT)

−

2,

2

>

0. So, it is easily induced fromLemma 2.6 thaty

(

^t

) ≥

m₂for alltlarge enough. Now we shall find anm₁

>

0 such thatx

(

^t

) ≥

m₁for alltlarge enough. We will do this in the following two steps.

(Step 1) Since

aT

−

^hcq

(

¹

+ (

^p2

−

1

)

^exp

( −

dT

) −

p₂exp

( −

d

τ

^T

))

d

(

¹

− (

¹

−

p₂

)

^exp

( −

dT

)) >

^{ln 1} 1

−

p₁

,

we can choose m₃

>

^0,

1

>

0 small enough such that

δ =

e

(

¹

−

exp

( −

cm₃

)) <

^{d and R}

= (

¹

−

p₁

)

^exp

−

^hcq(1+(p₍^{2−1)exp(−d+δ)T−p2exp((−d+δ)τT))}

d−δ)(1−(1−p2)exp((−d+δ)T))

+

aT

−

^a

bTm₃

−

c

1T

>

1. Suppose thatx

(

^t

) <

^m3for allt. Then we gety⁰

(

^t

) ≤ ( −

d

+ δ)

^y

(

^t

)

from the above assumptions. By the comparison theorem, we havey

(

^t

) ≤

u

(

^t

)

^andu

(

^t

) →

u^∗

(

^t

)

^, t

→ ∞

, whereu

(

^t

)

is the solution of







u⁰

(

^t

) = ( −

d

+ δ)

^u

(

^t

),

^t

6= (

ⁿ

+ τ −

1

)

^T

,

^t

6=

nT

,

u

(

^t+

) = (

¹

−

p₂

)

^u

(

^t

),

^t

= (

ⁿ

+ τ −

1

)

^T

,

u

(

^t+

) =

u

(

^t

) +

q

,

^t

=

nT

u

(

⁰⁺

) =

y

(

⁰⁺

),

(3.3)

and

u^∗

(

^t

) =











qexp

(( −

d

+ δ)(

^t

− (

ⁿ

−

1

)

^T

))

1

− (

¹

−

p₂

)

^exp

(( −

d

+ δ)

^T

) , (

ⁿ

−

1

)

^T

<

^t

≤ (

ⁿ

+ τ −

1

)

^T

,

q

(

¹

−

p₂

)

^exp

(( −

d

+ δ)(

^t

− (

ⁿ

−

1

)

^T

))

1

− (

¹

−

p₂

)

^exp

(( −

d

+ δ)

^T

) , (

ⁿ

+ τ −

1

)

^T

<

^t

≤

nT

.

(3.4)

Then there existsT₁

>

0 such thaty

(

^t

) ≤

u

(

^t

) ≤

u^∗

(

^t

) +

1. Since 1

−

exp

( −

cx

(

^t

)) ≤

cx

(

^t

)

, we get that x⁰

(

^t

) =

x

(

^t

)(

^a

−

^a

bx

(

^t

)) − (

¹

−

exp

( −

cx

(

^t

))

^y

(

^t

))

≥

x

(

^t

)(

^a

−

^a

bm₃

−

cy

(

^t

))

≥

x

(

^t

)(

^a

−

^a

bm₃

−

c

(

^u∗

(

^t

) +

1

)),

^t

6= (

ⁿ

+ τ −

1

)

^T

,

x

(

^t+

) = (

¹

−

p₁

)

^x

(

^t

),

^t

= (

ⁿ

+ τ −

1

)

^T

(3.5)

fort

≥

T₁. LetN₁

∈

_Nand

(

^N1

+ τ −

1

)

^T

≥

T₁. Integrating(3.5)on

((

ⁿ

+ τ −

1

)

^T

, (

ⁿ

+ τ)

^T

]

,n

≥

N₁, we havex

((

ⁿ

+ τ)

^T

) ≥

x

((

ⁿ

+ τ −

1

)

^T

)(

¹

−

p₁

)

^exp

(

R₍n+τ)T

(n+τ−1)Ta

−

^a

bm₃

−

c

(

^u∗

(

^t

) +

1

)

^dt

) =

R. Then we havex

((

^N1

+ τ +

n

)

^T

) ≥

x

((

^N1

+ τ)

^T

)

^Rn

→ ∞

asn

→ ∞

, which is a contradiction. Hence there exists at₁

>

0 such thatx

(

^t1

) ≥

m₃.

(Step 2) Ifx

(

^t

) ≥

m₃for allt

≥

t₁, then we are done. If not, we may lett^∗

=

inf

{

t

>

^t1

:

x

(

^t

) <

^m3

}

. Thenx

(

^t

) ≥

m₃for t

∈ [

t₁

,

^t∗

]

and, by the continuity ofx

(

^t

)

^{, we havex}

(

^t∗

) =

m₃. In this step, we have only to consider two possible cases.

(Case 1)t^∗

= (

ⁿ1

+ τ −

1

)

^{Tfor somen}1

∈

_N. Then

(

¹

−

p₁

)

^m3

≤

x

(

^t∗+

) = (

¹

−

p₁

)

^x

(

^t∗

) <

^m3. Selectn₂

,

ⁿ3

∈

_Nsuch that

(

ⁿ2

−

1

)

^T

>

^ln−^(Md+⁺¹δ^q)and

(

¹

−

p₁

)

^n2Rn3exp

(

ⁿ2

σ

^T

) > (

¹

−

p₁

)

^n2Rn3exp

((

ⁿ2

+

1

)σ

^T

) >

^{1, where}

σ =

a

−

^a

bm₃

−

cM

<

^0.

LetT⁰

=

n₂T

+

n₃T. In this case we will show that there existst₂

∈ (

^t∗

,

^t∗

+

T⁰

]

such thatx

(

^t2

) ≥

m₃. Otherwise, by(3.3) withu

(

ⁿ1T⁺

) =

y

(

ⁿ1T⁺

)

^{, we have}

u

(

^t

) =



















(

¹

−

p₂

)

^n−(n1+1)

u

(

ⁿ1T⁺

) −

^q

(

¹

−

p₂

)

^exp

( −

T

)

1

− (

¹

−

p₂

)

^exp

(( −

d

+ δ)

^T

)

exp

(( −

d

+ δ)(

^t

−

n₁T

)) +

u^∗

(

^t

), (

ⁿ

−

1

)

^T

<

^t

≤ (

ⁿ

+ τ −

1

)

^T

, (

¹

−

p₂

)

⁽ⁿ⁻ⁿ¹⁾

u

(

ⁿ1T⁺

) −

^q

(

¹

−

p₂

)

^exp

( −

T

)

1

− (

¹

−

_p2

)

^exp

(( −

_d

+ δ)

^T

)

exp

(( −

d

+ δ)(

^t

−

n₁T

)) +

u^∗

(

^t

), (

ⁿ

+ τ −

1

)

^T

<

^t

≤

nT

,

andn₁

+

1

≤

n

≤

n₁

+

n₂

+

n₃

+

1. So we get

|

u

(

^t

) −

u^∗

(

^t

) | ≤ (

^M

+

q

)

^exp

(( −

d

+ δ)(

^t

−

n₁T

)) <

1andy

(

^t

) ≤

u

(

^t

) ≤

u^∗

(

^t

) +

1

forn₁T

+ (

ⁿ2

−

1

)

^T

≤

t

≤

t^∗

+

T⁰. Also we discover that x⁰

(

^t

) ≥

x

(

^t

)

a

−

^a

bm₃

−

c

(

^u∗

+

1

)

,

^t

6= (

ⁿ

+ τ −

1

)

^T

,

x

(

^t+

) = (

¹

−

p₁

)

^x

(

^t

),

^t

= (

ⁿ

+ τ −

1

)

^T

fort

∈ [

t^∗

+

n₂T

,

^t∗

+

T⁰

]

. As in Step 1, we have x

(

^t∗

+

T⁰

) ≥

x

(

^t∗

+

n₂T

)

^Rn3

.

Sincey

(

^t

) ≤

Mand 1

−

exp

( −

cx

(

^t

)) ≤

cx

(

^t

)

^{, we have} x⁰

(

^t

) ≥

x

(

^t

)

a

−

^a

bm₃

−

cM

= σ

^x

(

^t

),

^t

6=

nT

,

x

(

^t+

) = (

¹

−

p₁

)

^x

(

^t

),

^t

=

nT

,

^(3.6)

fort

∈ [

t^∗

,

^t∗

+

n₂T

]

. Integrating(3.6)on

[

t^∗

,

^t∗

+

n₂T

]

, we have x

((

^t∗

+

n₂T

)) ≥

m₃exp

(σ

ⁿ2T

)

≥

m₃

(

¹

−

p₁

)

^n2exp

(σ

ⁿ2T

) >

^m3

.

^(3.7)