2 Existence of periodic solution

Periodic solutions for an impulsive

semi-ratio-dependent predator–prey model with patches and time delays

Ruixi Liang

^B

Department of Mathematics and Statistics, Central South University Changsha, 410073 Hunan, P.R. China

Received 30 January 2015, appeared 27 May 2015 Communicated by Leonid Berezansky

Abstract. In this paper, we study the existence of a positive periodic solution for a two-species semi-ratio-dependent predator-prey system with time delays and impulses in a two-patch environment. By using the method of coincidence degree theorem, a set of easily verifiable conditions are obtained for the existence of at least one strictly positive periodic solution for the system. In particular, our result generalizes some known criteria.

Keywords: predator-prey model, impulse, periodic solution, semi-ratio-dependent, patch, delay.

2010 Mathematics Subject Classification: 34C25, 34K13.

1 Introduction

In recent years, with the application of the theory of differential equations in mathematical ecology, a lot of mathematical models have been proposed in the study of population dy- namics [2–4,6,8–11,15–23]. One of the famous models for the dynamics of populations is the so-called semi-ratio-dependent predator-prey system with functional response [4,7,9,15,17], for example

x⁰ = x(a−bx)−g(x)y, y⁰ =y

d− fy x

, (1.1)

where x and y stand for the population of the prey and predator, respectively, g(x) is the predator functional response to prey.

In equation (1.1), it has been assumed that the prey grows logistically with growth ratea and carrying capacity a/bin the absence of predation. The predator consumes the prey ac- cording to the functional responseg(x)and grows logistically with growth ratedand carrying

BEmail: lrxcsu@163.com

capacityx(t)/f proportional to the population size of prey. The parameter f is a measure of the food quality that the prey provides for conversion into predator birth.

The form of the predator equation in (1.1) was first proposed by Leslie [9]. The functional responseg(x)in (1.1) can be classified into five types, including the Leslie–Gower model, the Holling–Tanner model, the Holling type III model, the Ivlev’s functional response and so on.

For more detail see reference [18].

We note that any biological or environmental parameters are naturally subject to fluctua- tion in time. Cushing [2] pointed out that it is necessary and important to consider models with periodic ecological parameters or perturbations which may be naturally exposed (for example, those due to seasonal effects of weather, food supply, mating habits, hunting or harvesting seasons, etc.). Thus, the assumption of periodicity of the parameters is a way of in- corporating the periodicity of the environment. On the other hand, dispersal between patches often occurs in natural ecological environments, and more realistic models should include the dispersal process [8,20,21].

We consider the following systems

x₁⁰(t) =x₁(t)(r₁(t)−a₁(t)x₁(t))−x₃(t)g(t,x₁(t−τ₁)) +D₁(t)(x₂(t)−x₁(t)), x₂⁰(t) =x2(t)(r2(t)−a2(t)x2(t)) +D2(t)(x₁(t)−x2(t)),

x₃⁰(t) =x₃(t)

r₃(t)−a₃(t)^x3(t−τ₂) x₁(t−τ₂)

,

(1.2)

with initial conditions

x_i(θ) =φ_i(θ)_, θ ∈[−_{τ, 0}]_,

φ_i(0)>0, φ_i ∈C([−τ, 0),R+), i=1, 2, 3,

wherex_i(t)represents the prey population in thei-th patch (i=1, 2), andx₃(t)represents the predator population. D_i(t)denotes the dispersal rate of the prey in the i-th patch (i = 1, 2).

τ=max{τ₁,τ₂}.

However, there are numerous examples of evolutionary systems which at certain instants in time are subject to rapid changes. In the simulations of such processes it is frequently convenient and valid to neglect the durations of rapid changes and to assume that the changes can be represented by state jumps. Appropriate mathematical models for processes of the type described above are so-called systems with impulsive effects, see [1]. One note that the research on theory and applications of impulsive differential equations have been many nice works [3,6,10,12–14,22,23]. Because harvest of many a populations are not continuous, the harvest is an annual harvest pulse. To describe a system more accurately, we should consider to use the impulsive differential equation. If we consider the regularly harvest, then (1.2) is revised as the following form:

x⁰₁(t) =x₁(t)(r₁(t)−a₁(t)x₁(t))−x₃(t)g(t,x₁(t−τ₁)) +D₁(t)(x₂(t)−x₁(t)), x⁰₂(t) =x₂(t)(r₂(t)−a₂(t)x₂(t)) +D₂(t)(x₁(t)−x₂(t)),

x⁰₃(t) =x₃(t)

r₃(t)−a₃(t)^x3(t−τ₂) x₁(t−τ₂)

,

∆xi(t_k) =b_ikx_i(t_k), i=1, 2, 3, k=1, 2,· · · ,

(1.3)

whereb_ikx_i(t_k)(i=1, 2, 3) represents the populationx_i(t)att_kregular harvest pulse. Through this paper, for system (1.3) the following conditions are assumed.

(C₁) r_i(t), a_i(t)(i = 1, 2, 3),D₁(t)andD₂(t)are positive continuousω-periodic functions; τ₁ andτ₂ are positive constants.

(C₂) g(t,x)is a continuousω-periodic function with respect to the first variable and is differ- entiable with respect to the second variable, and g(t; 0) = 0, g(t,x)> 0 for anyt ∈ _R, x>0.

(C₃) There exists a positive constantc₀ such thatg(t,x)≤c₀for any t∈_R, x>0.

(C₄) −1 < b_ik ≤ 0, i = 1, 2, 3 for all k ∈ N and there exists a positive integer q such that t_k+q=t_k+ω,b_i(k+q) =b_ik,i=1, 2, 3 andt_k−τ₁, t_k−τ₂6= t_m.

In the following, we shall use the notations.

f¯= ¹ ω

Z _ω

0 f(s)ds, f^L= min

t∈[0,ω] f(t), f^M = max

t∈[0,ω] f(t).

Without loss of generality, we shall assumet_k 6=_0,ω and[_0,ω]∩ {t_k}= {t₁,t₂, . . . ,t_q}_. The existence of positive periodic solution of (1.2) is investigated in [4], and the following result is obtained.

Theorem 1.1. In addition to(C₁)–(C₃), assume further that the following hold:

(H₂) r_i(t)−D_i(t)>0, i=1, 2, (H₃) a₃^L(r₁−D₁)−c₀r₃>0.

Then system(1.2)has at least one positive ω-periodic solution with strictly positive components.

The proof in [4] shows that Theorem1.1has room for improvement.

The organization of this paper is as follows. In the next section, we establish some simple criteria for the existence of a positive periodic solution of system (1.3). We also note that our results improve Theorem A as b_ik ≡ 0, because our results do not need the condition (H₂). Finally, we give some applications to show our results.

2 Existence of periodic solution

In this section, by using continuation theorem which was proposed in [5] by Gaines and Mawhin, we will establish the existence conditions of at least one positive periodic solution of system (1.3). To do so, we need to make some preparations.

LetX,Zbe real Banach spaces,L: DomL⊂X→Zbe a Fredholm mapping of index zero (indexL = dim KerL−codim ImL), and letP: X → X, Q: Z → Z be continuous projectors such that ImP = KerL, KerQ = ImLand X = KerL⊕KerP, Z = ImL⊕ImQ. Denote by LP the restriction of L to DomL∩KerP, KP: ImL→ KerP∩DomLthe inverse (to LP), and J: ImQ→KerLan isomorphism of ImQonto KerL.

For convenience, we first introduce Mawhin’s continuation theorem [5] as follows.

Lemma 2.1. LetΩ⊂X be an open bounded set. Let L be a Fredholm mapping of index zero and N be L-compact onΩ. Assume¯

(a) Lx6= λNx for eachλ∈ (_{0, 1}), x∈∂Ω∩DomL,

(b) for each x∈KerL∩_{∂Ω, QNx}6=0, (c) deg{JQN,Ω∩KerL, 0} 6=0.

Then Lx=Nx has at least one solution inΩ¯ ∩DomL.

To prove the main conclusion by means of the continuation theorem, we need to introduce some functional spaces.

Let PC(_R,R³) = {x: R → _R³ | x is continuous at t 6= t_k, x(t⁺_k ), x(t⁻_k )exist andx(t⁻_k ) = x(t_k), k=1, 2, . . . ,}, letX ={(u₁(t),u2(t),u3(t))^T ∈ PC(_R,R³):u_i(t+ω) =u_i(t), i=1, 2, 3} with the norm

k(u₁(t),u₂(t),u₃(t))k^T =

∑

3 i=1

sup

t∈[0,ω]

|u_i(t)|, and

Y =X×_R^3q with the norm kuk_Y =kxk+kyk, for u∈Y, x∈ X, y∈_R^3q, where| · |denotes the Euclidean norm. ThenX andYare Banach spaces.

Theorem 2.2. In addition to(C₁)–(C₄), assume further that the following hold:

(C₅) r₃ω+_∑^q_k=1ln(1+b_3k)>0,

(C6) a₃^L(r₁−D₁)ω+a^L₃∑^q_i=1ln(1+b_1k)>c0r3ω.

Then system(1.3)has at least one positiveω-periodic solution.

Proof. Let

u₁(t) =ln[x₁(t)], u₂(t) =ln[x₂(t)], u₃(t) =ln[x₃(t)], (2.1) then system (1.3) can be translated to

u⁰₁(t) =r₁(t)−D₁(t)−a₁(t)e^u1(t)−g(t,e^u1(t−τ1))e^{u3(t)−u1(t)}+D₁(t)e^{u2(t)−u1(t)}, u⁰₂(t) =r₂(t)−D₂(t)−a₂(t)e^u2(t)+D₂(t)e^{u1(t)−u2(t)},

u⁰₃(t) =r₃(t)−a₃(t)e^{u3(t−τ2)−u1(t−τ2)},

∆ui(t_k) =ln(1+b_ik), i=1, 2, 3, k=1, 2, . . .

(2.2)

It is easy to see that if system (2.2) has oneω-periodic solution (u₁^∗(t),u^∗₂(t),u₃^∗(t))^T, then (x₁^∗(t),x^∗₂(t),y^∗(t))^T = (exp[u^∗₁(t)], exp[u^∗₂(t)], exp[u^∗₃(t)])^T is a positiveω-periodic solution of (1.3). Therefore, to complete the proof, we need only to prove that (2.2) has one ω-periodic solution.

Let L: DomL⊂X →Y,u→(u⁰,∆u(t₁), . . . ,∆u(t_q)),

Nu=









u⁰₁(t) =r₁(t)−D₁(t)−a₁(t)e^u1(t)−g(t,e^u1(t−τ1))e^{u3(t)−u1(t)}+D₁(t)e^{u2(t)−u1(t)} u⁰₂(t) =r₂(t)−D₂(t)−a₂(t)e^u2(t)+D₂(t)e^{u1(t)−u2(t)},

u⁰₃(t) =r₃(t)−a₃(t)e^{u3(t−τ2)−u1(t−τ2)}



,





ln(1+b₁₁) ln(1+b₂₁) ln(1+b₃₁)



,





ln(1+b₁₂) ln(1+b₂₂) ln(1+b₃₂)



,· · · ,





ln(1+b_1q) ln(1+b2q) ln(1+b_3q)







.

Evidently

KerL={u:u(t) =c∈ R³, t ∈[0,ω]}, ImL=

(

z= (f,a₁, . . . ,a_q)∈Y: Z _ω

0 f(s)ds+

∑

q k=1

a_k =0 )

, and

dim KerL=3=codim ImL.

So ImLis closed inY,Lis a Fredholm mapping of index zero. Define Px= ¹

ω Z _ω

0 x(t)dt,

Qz =Q(f,a₁,a₂, . . . ,a_q) = ¹ ω

"

Z _ω

0 f(s)ds+

∑

q k=1

a_k

#

, 0, . . . , 0

! . It is easy to show thatPandQare continuous projectors satisfying

ImP=KerL, ImL=KerQ=Im(I−Q).

Furthermore, through an easy computation, we can find that the inverse K_P: ImL→KerP∩ DomLhas the form

K_P(z) =

Z _t

0 f(s)ds+

∑

tk<t

a_k− ¹ ω

Z _ω

0

Z _t

0 f(s)ds dt−

∑

q k=1

a_k. Thus

QNu =



































































 1 ω

Z _ω

0

h

r₁(t)−D₁(t)−a₁(t)e^u1(t)−g(t,e^u1(t−τ1))e^{u3(t)−u1(t)} +D₁(t)e^{u2(t)−u1(t)}i

dt+ ¹ ω

∑

q k=1

ln(1+b_1k), 1

ω Z _ω

0

h

r2(t)−D2(t)−a2(t)e^u2(t)+D2(t)e^{u1(t)−u2(t)}i dt +¹

ω

∑

q k=₁

ln(1+b_2k), 1

ω Z _ω

0

h

r₃(t)−a₃(t)e^{u3(t−τ2)−u1(t−τ2)}i dt+ ¹

ω

∑

q k=1

ln(1+b_3k),



































, 0, . . . , 0

































 ,

and

K_P(I−Q)Nu

=

























 Z _t

0

h

r₁(s)−D₁(s)−a₁(s)e^u1(s)−g(t,e^u1(s−τ1))e^{u3(s)−u1(s)} +D₁(s)e^{u2(s)−u1(s)}i

ds+

∑

t>t_k

ln(1+b_1k)

Z _t

0

h

r2(s)−D2(s)−a2(s)e^u2(s)+D2(s)e^{u1(s)−u2(s)}i

ds+

∑

t>t_k

ln(1+b_2k)

Z _t

0

h

r₃(s)−a₃(s)e^{u3(s−τ2)−u1(s−τ2)}i

ds+

∑

t>tk

ln(1+b_3k)



























−



























 1 ω

Z _ω

0

Z _t

0

h

r₁(s)−D₁(s)−a₁(s)e^u1(s)−g(t,e^u1(s−τ1))e^{u3(s)−u1(s)} +D₁(s)e^{u2(s)−u1(s)}i

ds+

∑

q k=1

ln(1+b_1k) 1

ω Z _ω

0

Z _t

0

h

r₂(s)−D₂(s)−a₂(s)e^u2(s)+D₂(s)e^{u1(s)−u2(s)}+ⁱds+

∑

q k=1

ln(1+b_2k) 1

ω Z _ω

0

Z _t

0

h

r3(_s)−a3(_s)_e^{u3(s−τ2)−u1(s−τ2)i}_ds+

∑

q k=1

ln(₁+_b3k)





























−



























 t

ω −¹ 2

_{Z t}

0

h

r₁(s)−D₁(s)−a₁(s)e^u1(s)−g(t,e^u1(s−τ1))e^{u3(s)−u1(s)}

+D₁(s)e^{u2(s)−u1(s)}i ds+

∑

q k=₁

ln(1+b_1k) t

ω −¹ 2

_{Z t}

0

h

r₂(s)−D₂(s)−a₂(s)e^u2(s)+D₂(s)e^{u1(s)−u2(s)}+ⁱds+

∑

q k=1

ln(1+b_2k) t

ω −¹ 2

_{Z t}

0

h

r₃(s)−a₃(s)e^{u3(s−τ2)−u1(s−τ2)}i ds+

∑

q k=1

ln(1+b_3k)



























 .

Clearly, QN and K_P(I−Q)N are continuous. Using Lemma 2.4 in [1], it is not difficult to show thatQN(_Ω^¯)_, K_p(I−Q)N(_Ω^¯)are relatively compact for any open bounded set Ω⊂X.

HenceNis L-compact on ¯Ωfor any open bounded setΩ⊂X.

Now we reach the position to search for an appropriate open, bounded subsetΩ for the application of the continuation theorem. Corresponding to equation Lu = λNu, λ ∈ (_{0, 1})_, we have

u⁰₁(t) =λ h

r₁(t)−D₁(t)−a₁(t)e^u1(t)−g(t,e^u1(t−τ1))e^{u3(t)−u1(t)}+D₁(t)e^{u2(t)−u1(t)}i , u⁰₂(t) =λ

h

r₂(t)−D₂(t)−a₂(t)e^u2(t)+D₂(t)e^{u1(t)−u2(t)}i , u⁰₃(t) =λ

h

r₃(t)−a₃(t)e^{u3(t−τ2)−u1(t−τ2)}i ,

∆ui(t_k) =λln(1+b_ik), i=1, 2, 3, k=1, 2, . . .

(2.3)

Sinceu_i(t)(i=1, 2, 3)areω-periodic functions, we need only to prove the result in the interval [0,ω]. Integrating (2.3) over the interval [0,ω] leads to

Z _ω

0 a₁(t)e^u1(t)dt+

Z _ω

0 g(t,e^u1(t−τ1))e^{u3(t)−u1(t)}dt

=

Z _ω

0

(r₁(t)−D₁(t))dt+

Z _ω

0 D₁(t)e^{u2(t)−u1(t)}dt+

∑

q k=1

ln(1+b_1k),

(2.4)

Z _ω

0 a2(t)e^u2(t)dt=

Z _ω

0

(r2(t)−D2(t))dt+

Z _ω

0 D2(t)e^{u1(t)−u2(t)}dt+

∑

q k=₁

ln(1+b_2k), (2.5) and

Z _ω

0 a₃(t)e^{u3(t−τ2)−u1(t−τ2)}dt=

Z _ω

0 r₃(t)dt+

∑

q k=1

ln(1+b_3k). (2.6) Noting that

Z _ω

0 e^{u3(t−τ2)−u1(t−τ2)}dt=

Z _ω

0 e^{u3(t)−u1(t)}dt,

and−1<b_3k ≤0, we derive from (2.6) that a^L₃

Z _ω

0 e^{u3(t)−u1(t)}dt= a₃^L Z _ω

0 e^{u3(t−τ2)−u1(t−τ2)}dt≤r₃ω, which implies

Z _ω

0 e^{u3(t)−u1(t)}dt≤ ^r3ω a^L₃ .

This together with the first equation of (2.3), (2.4), (C2) and (C3), yields Z _ω

0

|u⁰₁(t)|dt<

Z _ω

0

r₁(t) +D₁(t) +a₁(t)e^u1(t)+g(t,e^u1(t−τ1))e^{u3(t)−u1(t)} +D₁(t)e^{u2(t)−u1(t)}

dt

=₂

Z _ω

0 a1(_t)_e^u1(t)_dt+₂

Z _ω

0 g(_t,e^u1(t−τ1))_e^{u3(t)−u1(t)}_dt +₂

Z _ω

0

D₁(t)dt−

∑

q k=1

ln(₁+b_1k)

≤2 Z _ω

0 a₁(t)e^u1(t)dt+2c₀ Z _ω

0 e^{u3(t)−u1(t)}dt+2D₁ω−

∑

q k=1

ln(1+b_1k)

≤2 Z _ω

0 a₁(t)e^u1(t)dt+ ^2c0r3ω

a^L₃ +2D₁ω−

∑

q k=1

ln(1+b_1k).

(2.7)

From (2.3), (2.5) and (2.6), we also have Z _ω

0

|u⁰₂(t)|dt<

Z _ω

0

r2(t) +D2(t) +a2(t)e^u2(t)+D2(t)e^{u1(t)−u2(t)} dt

= 2 Z _ω

0 a2(t)e^u2(t)dt+2 Z _ω

0 D2(t)dt−

∑

q k=₁

ln(1+b_2k),

(2.8)

Z _ω

0

|u⁰₃(t)|dt<

Z _ω

0

r₃(t) +a₃(t)e^{u3(t−τ2)−u1(t−τ2)} dt

=2r₃ω+

∑

q k=1

ln(1+b_3k)≤2r₃ω.

(2.9)

Multiplying the first equation of (2.3) bye^u1(t) and integrating over[0,ω], we obtain

−

∑

p k=1

b_1ke^u1(tk)+

Z _ω

0 a₁(t)e^2u1(t)dt

=

Z _ω

0

(r₁(t)−D₁(t))e^u1(t)dt+

Z _ω

0 D₁(t)e^u2(t)dt−

Z _ω

0 g(t,e^u1(t−τ1)e^u3(t))dt

<

Z _ω

0

(r₁(t)−D₁(t))e^u1(t)dt+

Z _ω

0 D₁(t)e^u2(t)dt Since−1<_b1k ≤0, so we have

Z _ω

0 a₁(t)e^2u1(t)dt≤(r₁−D₁)^M

Z _ω

0 e^u1(t)dt+D₁^M Z _ω

0 e^u2(t)dt,

which yields

a^L₁ Z _ω

0 e^2u1(t)dt≤(r₁−D₁)^M

Z _ω

0 e^u1(t)dt+D₁^M Z _ω

0 e^u2(t)dt. (2.10) Similarly, multiplying the second equation in (2.3) bye^u2(t)and integrating over[0,ω]gives

−

∑

p k=1

b_2ke^u1(tk)+

Z _ω

0 a₂(t)e^2u2(t)dt=

Z _ω

0

(r₂(t)−D₂(t))e^u2(t)dt+

Z _ω

0 D₂(t)e^u1(t)dt, which implies

a^L₂ Z _ω

0 e^2u2(t)dt<(r₂−D₂)^M

Z _ω

0 e^u2(t)dt+D₂^M Z _ω

0 e^u1(t)dt. (2.11) By using the inequalities

Z _ω

0 e^ui(t)dt 2

≤ω Z _ω

0 e^2ui(t)dt, i=1, 2, it follows from (2.10) and (2.11) that

a^L_{1 Z ω}

0 e^u1(t)dt 2

< ω(r₁−D₁)^M

Z _ω

0 e^u1(t)dt+D₁^Mω Z _ω

0 e^u2(t)dt, (2.12) a^L₂

Z _ω

0 e^u2(t)dt 2

< ω(r2−D2)^M

Z _ω

0 e^u2(t)dt+D₂^Mω Z _ω

0 e^u1(t)dt. (2.13) IfRω

0 e^u2(t)dt≤Rω

0 e^u1(t)dt, then we derive from (2.12) that a^L₁

Z _ω

0

e^u1(t)dt 2

< ω(r₁−D₁)^M

Z _ω

0

e^u1(t)dt+D₁^Mω Z _ω

0

e^u1(t)dt,

which implies

Z _ω

0

e^u2(t)dt≤

Z _ω

0

e^u1(t)dt< ^ω(r₁−D₁)^M+ωD₁^M

a^L₁ . (2.14)

IfRω

0 e^u1(t)dt≤Rω

0 e^u2(t)dt, then we can conclude Z _ω

0

e^u1(t)dt≤

Z _ω

0

e^u2(t)dt< ^ω(r₂−D₂)^M+ωD₂^M

a^L₂ . (2.15)

Set

A=max

((r₁−D₁)^M+D₁^M

a₁^L ,(r₂−D₂)^M+D₂^M a^L₂

)

. (2.16)

Then it follows from (2.14)–(2.16) that Z _ω

0 e^ui(t)dt<ωA, i=1, 2. (2.17) This, together with (2.7) and (2.8), yields

Z _ω

0

|u₁⁰(t)|dt≤_2ω

a₁^MA+^c0r3 a^L₃

+2ωD₁−

∑

q k=1

ln(₁+b_1k) =_:c₁, Z _ω

0

|u₂⁰(t)|dt≤2ωa₂^MA+2ωD₂−

∑

q k=1

ln(1+b_2k) =:c₂.

(2.18)

Sinceu(t)∈X, there existξ_i,η_i ∈ [0,ω] (i=1, 2, 3)such that u_i(ξ_i) = min

t∈[0,ω]u_i(t), u_i(η_i) = max

t∈[0,ω]u_i(t), i=1, 2, 3. (2.19) From (2.17) and (2.19), we see that

u_i(ξ_i)<lnA, i=1, 2. (2.20) Thus, from (2.18) and (2.20) we have

u₁(t) =











u₁(ξ₁) +

Z _t

ξ₁

u⁰₁(s)ds+

∑

ξ1<t_k<t

ln(1+b_1k), t ∈(ξ₁,ω] u₁(ξ₁) +

Z _t

ξ1

u⁰₁(s)ds−

∑

t≤t_k≤ξ⁻₁

ln(1+b_1k), t∈[0,ξ₁]

≤u₁(ξ₁) +

Z _ω

0

|u⁰₁(t)|dt−

∑

q k=₁

ln(1+b_1k)

<lnA+c₁−

∑

q k=1

ln(1+b_1k),

(2.21)

u2(t)≤u2(ξ2) +

Z _ω

0

|u⁰₂(t)|dt−

∑

q k=₁

ln(1+b_2k)

<lnA+c₂−

∑

q k=1

ln(1+b_2k).

(2.22)

It follows from (2.4) that (r₁−D₁)ω<

Z _ω

0 a₁(t)e^u1(t)dt+

Z _ω

0 g(t,e^u1(t−τ1))e^{u3(t)−u1(t)}dt−

∑

q k=1

ln(1+b_1k)

≤

Z _ω

0 a₁(t)e^u1(t)dt+c₀ Z _ω

0 e^{u3(t)−u1(t)}dt−

∑

q k=1

ln(1+b_1k)

≤

Z _ω

0 a₁(t)e^u1(t)dt+^c0r3ω a₃^L −

∑

q k=₁

ln(1+b_1k) This, together with (2.19), deduces

e^u1(η1)a₁ω≥

Z _ω

0 a₁(t)e^u1(t)dt≥(r₁−D₁)ω−^c0r3ω a₃^L +

∑

q k=1

ln(1+b_1k), which implies

u₁(η₁)≥ln (r₁−D₁)ω−(c₀r₃/a₃^L)ω+_∑^q_k=1ln(1+b_1k) a₁ω

!

=:d₁.