Open Access

Four periodic solutions for a food-limited two-species Gilpin-Ayala type predator-prey system with harvesting terms on time scales

Advances in Difference Equations20132013:278

https://doi.org/10.1186/1687-1847-2013-278

Received: 25 January 2013

Accepted: 23 August 2013

Published: 23 September 2013

Abstract

By using Mawhin’s coincidence degree theory, this paper establishes a new criterion on the existence of four periodic solutions for a food-limited two-species Gilpin-Ayala type predator-prey system with harvesting terms on time scales. An example is given to illustrate the effectiveness of the result.

1 Introduction

The theory of calculus on time scales was initiated by Hilger [1] in order to unify continuous and discrete analysis, and it has become an effective approach to the study of mathematical models involving the hybrid discrete-continuous processes. Since the population dynamics in the real world usually involves the hybrid discrete-continuous processes, it may be more realistic to consider population models on time scales [2].

In recent years, some researchers studied the existence of periodic solutions for some population models on time scales under the assumption of periodicity of the parameters by using Mawhin’s coincidence degree theory (see [37]). To our best knowledge, few papers deal with the existence of multiple periodic solutions for population models with harvesting terms on time scales. The main difficulty is that the techniques used in continuous population models with harvesting terms are generally not available to population models with harvesting terms on time scales. Indeed, almost all papers involving continuous population models with harvesting terms used Fermat’s theorem on local extrema of differentiable functions in real analysis; for example, see [811]. However, Fermat’s theorem is not true in time scales calculus.

In this paper, we consider a food-limited two-species Gilpin-Ayala type predator-prey system with harvesting terms on time scales:
{ u 1 Δ ( t ) = 1 k 1 ( t ) + c 1 ( t ) e u 1 ( t ) [ a 1 ( t ) a 11 ( t ) e θ 1 u 1 ( t ) a 12 ( t ) e θ 2 u 2 ( t ) ] H 1 ( t ) e u 1 ( t ) , u 2 Δ ( t ) = 1 k 2 ( t ) + c 2 ( t ) e u 2 ( t ) [ a 2 ( t ) + a 21 ( t ) e θ 1 u 1 ( t ) a 22 ( t ) e θ 2 u 2 ( t ) ] H 2 ( t ) e u 2 ( t ) .
(1.1)
In system (1.1), let x i ( t ) = exp { u i ( t ) } , i = 1 , 2 . If the time scale T = R (the set of all real numbers), then system (1.1) reduces to
{ x 1 ( t ) = x 1 ( t ) k 1 ( t ) + c 1 ( t ) x 1 ( t ) [ a 1 ( t ) a 11 ( t ) x 1 θ 1 ( t ) a 12 ( t ) x 2 θ 2 ( t ) ] H 1 ( t ) , x 2 ( t ) = x 2 ( t ) k 2 ( t ) + c 2 ( t ) x 2 ( t ) [ a 2 ( t ) + a 21 ( t ) x 1 θ 1 ( t ) a 22 ( t ) x 2 θ 2 ( t ) ] H 2 ( t ) ,
(1.2)

where x 1 and x 2 denote the prey and the predator, respectively; a i ( t ) ( i = 1 , 2 ), a i i ( t ) ( i = 1 , 2 ), H i ( t ) ( i = 1 , 2 ) are all positive continuous functions denoting the intrinsic growth rate, the intra-specific competition rates and the harvesting rates, respectively; a 12 ( t ) is the predation rate of the predator and a 21 ( t ) represents the conversion rate; k i ( t ) ( i = 1 , 2 ) are the population numbers of two species at saturation, respectively. θ i ( i = 1 , 2 ) represent a nonlinear measure of interspecific interference. When c i ( t ) 0 ( i = 1 , 2 ), a i ( t ) k i ( t ) c i ( t ) ( i = 1 , 2 ) are the rate of replacement of mass in the population at saturation (including the replacement of metabolic loss and of dead organisms). In this case, system (1.2) is a food-limited population model. For other food-limited population models, we refer to [1217] and the references cited therein. When θ i 1 ( i = 1 , 2 ), system (1.2) is a Gilpin-Ayala type population model. Gilpin-Ayala type population models were firstly proposed by Gilpin and Ayala in [18]. For some recent work, we refer to [17, 1922]. When c i ( t ) 0 , θ i = 1 ( i = 1 , 2 ), system (1.2) was consider by Zhao and Ye [9].

Motivated by the work of Bohner et al. [3] and Chen [8], we study the existence of multiple periodic solutions of (1.1) by using Mawhin’s coincidence degree.

2 Preliminaries from calculus on time scales

In this section, we briefly present some foundational definitions and results from the calculus on time scales so that the paper is self-contained. For more details, one can see [1, 23, 24].

Definition 2.1 [23]

A time scale is an arbitrary nonempty closed subset of the real numbers .

Let ω > 0 . Throughout this paper, the time scale is assumed to be ω-periodic, i.e., t T implies t + ω T . In particular, the time scale under consideration is unbounded above and below.

Definition 2.2 [23]

We define the forward jump operator σ : T T , the backward jump operator ρ : T T , and the graininess μ : T R + = [ 0 , + ) by
σ ( t ) : = inf { s T : s > t } , ρ ( t ) : = sup { s T : s < t } , μ ( t ) = σ ( t ) t for  t T ,

respectively. If σ ( t ) = t , then t is called right-dense (otherwise, right-scattered), and if ρ ( t ) = t , then t is called left-dense (otherwise, left-scattered).

Definition 2.3 [23]

Assume that f : T R is a function, and let t T . Then we define f Δ ( t ) to be the number (provided it exists) with the property that given any ε > 0 , there is a neighborhood U of t (i.e., U = ( t δ , t + δ ) T for some δ > 0 ) such that
| [ f ( σ ( t ) ) f ( s ) ] f Δ ( t ) [ σ ( t ) s ] | ε | σ ( t ) s | for all  s U .

In this case, f Δ ( t ) is called the delta (or Hilger) derivative of f at t. Moreover, f is said to be delta or Hilger differentiable on if f Δ ( t ) exists for all t T .

Definition 2.4 [23]

A function F : T R is called an antiderivative of f : T R provided F Δ ( t ) = f ( t ) for all t T . Then we define
r s f ( t ) Δ t = F ( s ) f ( r ) for  r , s T .

Definition 2.5 [23]

A function f : T R is said to be rd-continuous if it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in . The set of rd-continuous functions f : T R will be denoted by C rd ( T ) .

The following notation will be used throughout this paper.

Let
T κ = { T ( ρ ( sup T ) , sup T ] , if  sup T < + , T , if  sup T = + , κ = min { [ 0 , + ) T } , I ω = [ κ , κ + ω ] T , g u = sup t I ω g ( t ) , g l = inf t I ω g ( t ) , g ¯ = 1 ω I ω g ( s ) Δ s = 1 ω κ κ + ω g ( s ) Δ s ,

where g C rd ( T ) is a nonnegative ω-periodic real function, i.e., g ( t + ω ) = g ( t ) for all t T .

Lemma 2.1 [23]

Every rd-continuous function has an antiderivative.

Lemma 2.2 [23]

Assume that f : T R is a function, and let t T κ . Then we have:
  1. (i)
    If f is continuous at t and t is right-scattered, then f is differential at t with
    f Δ ( t ) = f ( σ ( t ) ) f ( t ) μ ( t ) .
     
  2. (ii)
    If f is right-dense, then f is differential at t iff the limit
    lim s t f ( t ) f ( s ) t s
     
exists as a finite number. In this case,
f Δ ( t ) = lim s t f ( t ) f ( s ) t s .

Lemma 2.3 [3]

Let t 1 , t 2 I ω and t T . If g : T R is ω-periodic, then
g ( t ) g ( t 1 ) + κ κ + ω | g ( s ) | Δ s and g ( t ) g ( t 2 ) κ κ + ω | g ( s ) | Δ s .

Lemma 2.4 [6]

Assume that { f n } n N is a function sequence on I ω such that
  1. (i)

    { f n } n N is uniformly bounded on I ω ;

     
  2. (ii)

    { f n Δ } n N is uniformly bounded on I ω .

     

Then there is a subsequence of { f n } n N converging uniformly on I ω .

3 Existence of multiple periodic solutions

We first briefly state Mawhin’s coincidence degree theory (see [25]).

Let X, Z be normed vector spaces, L : dom L X Z be a linear mapping, N : X Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim Ker L = codim Im L < + and ImL is closed in Z. If L is a Fredholm mapping of index zero, then there exist continuous projectors (i.e., linear and idempotent linear operators) P : X X and Q : Z Z such that Im P = Ker L , Im L = Ker Q = Im ( I Q ) . If we define L P : dom L Ker P Im L as the restriction L | dom L Ker P of L to dom L Ker P , then L P is invertible. We denote the inverse of that map by K P . If Ω is an open bounded subset of X, the mapping N will be called L-compact on Ω ¯ if Q N ( Ω ¯ ) is bounded and K P ( I Q ) N : Ω ¯ X is compact, i.e., continuous and such that K P ( I Q ) N ( Ω ¯ ) is relatively compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J : Im Q Ker L .

For convenience, we introduce Mawhin’s continuation theorem [25] as follows.

Lemma 3.1 Let L be a Fredholm mapping of index zero, and let N : Ω ¯ × [ 0 , 1 ] Z be L-compact on Ω ¯ × [ 0 , 1 ] . Suppose
  1. (a)

    L u λ N ( u , λ ) for every u dom L Ω and every λ ( 0 , 1 ) ;

     
  2. (b)

    Q N ( u , 0 ) 0 for every u Ω Ker L ;

     
  3. (c)

    Brouwer degree deg B ( J Q N ( , 0 ) | Ker L , Ω Ker L , 0 ) 0 .

     

Then L u = N ( u , 1 ) has at least one solution in dom L Ω ¯ .

Set
h ( x ) = b a x α c x , x ( 0 , + ) .

Lemma 3.2 [17]

Assume that a, b, c, α are positive constants and
b > ( 1 + α ) a 1 1 + α ( c α ) α 1 + α .
Then there exist 0 < x < x + such that
h ( x ) = h ( x + ) = 0 , h ( x ) > 0 for  x ( x , x + ) , h ( x ) < 0 for  x ( 0 , x ) ( x + , + )
and
h ( x ) > 0 , h ( x + ) < 0 .
Set
N 1 = ln ( a 1 a 11 ) u θ 1 + 2 ω ( a 1 k 1 ) u , N 2 = ln [ ( a 2 a 22 ) u + ( a 21 a 22 ) u e θ 1 N 1 ] θ 2 + 2 ω ( a 2 k 2 ) u + 2 ω ( a 21 k 2 ) u e θ 1 N 1 .

From now on, we always assume that:

(H1) c i ( t ) , k i ( t ) , a i ( t ) , H i ( t ) ( i = 1 , 2 ), a i j ( t ) ( i , j = 1 , 2 ) are positive continuous ω-periodic functions, θ i ( i = 1 , 2 ) are positive constants.

(H2) 1 1 + ( c 1 k 1 ) u e N 1 [ ( a 1 k 1 ) l ( a 12 k 1 ) u e θ 2 N 2 ] > ( 1 + θ 1 ) [ ( a 11 k 1 ) u ] 1 1 + θ 1 [ H 1 u θ 1 ] θ 1 1 + θ 1 .

(H3) ( a 2 k 2 ) l 1 + ( c 2 k 2 ) u e N 2 > ( 1 + θ 2 ) [ ( a 22 k 2 ) u ] 1 1 + θ 2 [ H 2 u θ 2 ] θ 2 1 + θ 2 .

Set
M 1 ( x ) = 1 1 + ( c 1 k 1 ) u e N 1 [ ( a 1 k 1 ) l ( a 12 k 1 ) u e θ 2 N 2 ] ( a 11 k 1 ) u x θ 1 H 1 u x , M 2 ( x ) = ( a 2 k 2 ) l 1 + ( c 2 k 2 ) u e N 2 ( a 22 k 2 ) u x θ 2 H 2 u x , p i ( x ) = ( a i k i ) ¯ ( a i i k i ) ¯ x θ i H ¯ i x ( i = 1 , 2 ) , m 1 ( x ) = ( a 1 k 1 ) u ( a 11 k 1 ) l 1 + ( c 1 k 1 ) u [ ( a 1 a 11 ) u ] 1 θ 1 x θ 1 H 1 l x , m 2 ( x ) = ( a 2 k 2 ) u + ( a 21 k 2 ) u e θ 1 N 1 ( a 22 k 2 ) l 1 + ( c 2 k 2 ) u [ ( a 2 a 22 ) u + ( a 21 a 22 ) u e θ 1 N 1 ] 1 θ 2 x θ 2 H 2 l x .
Lemma 3.3 Assume that (H1)-(H3) hold. Then the following assertions hold:
  1. (1)
    There exist 0 < u i < u i + such that
    M i ( u i ) = M i ( u i + ) = 0 , M i ( x ) > 0 for  x ( u i , u i + )
     
and
M i ( x ) < 0 for  x ( 0 , u i ) ( u i + , + ) , i = 1 , 2 .
  1. (2)
    There exist 0 < x i < x i + such that
    p i ( x i ) = p i ( x i + ) = 0 , p i ( x ) > 0 for  x ( x i , x i + ) , p i ( x ) < 0 for  x ( 0 , x i ) ( x i + , + )
     
and
p i ( x i ) > 0 , p i ( x i + ) < 0 , i = 1 , 2 .
  1. (3)
    There exist 0 < l i < l i + such that
    m i ( l i ) = m i ( l i + ) = 0 , m i ( x ) > 0 for  x ( l i , l i + )
     
and
m i ( x ) < 0 for  x ( 0 , l i ) ( l i + , + ) , i = 1 , 2 .
  1. (4)
    l i < x i < u i < u i + < x i + < l i + , i = 1 , 2 .
     
  2. (5)
    ln u i + < N i , i = 1 , 2 .
     
Proof It follows from (H1)-(H3) and Lemma 3.2 that assertions (1)-(3) hold. Noticing that
1 1 + ( c 1 k 1 ) u e N 1 [ ( a 1 k 1 ) l ( a 12 k 1 ) u e θ 2 N 2 ] < ( a 1 k 1 ) ¯ ( a 1 k 1 ) u , ( a 11 k 1 ) l 1 + ( c 1 k 1 ) u [ ( a 1 a 11 ) u ] 1 θ 1 < ( a 11 k 1 ) ¯ ( a 11 k 1 ) u , ( a 2 k 2 ) l 1 + ( c 2 k 2 ) u e N 2 < ( a 2 k 2 ) ¯ < ( a 2 k 2 ) u + ( a 21 k 2 ) u e θ 1 N 1 , ( a 22 k 2 ) l 1 + ( c 2 k 2 ) u [ ( a 2 a 22 ) u + ( a 21 a 22 ) u e θ 1 N 1 ] 1 θ 2 < ( a 22 k 2 ) ¯ ( a 22 k 2 ) u , H i l H ¯ i H i u ( i = 1 , 2 ) ,
we have
M i ( x ) < p i ( x ) < m i ( x ) , i = 1 , 2 .

By assertions (1)-(3), assertion (4) holds.

It follows from M i ( u i + ) = 0 ( i = 1 , 2 ) that
( a 11 k 1 ) u [ u 1 + ] θ 1 < 1 1 + ( c 1 k 1 ) u e N 1 [ ( a 1 k 1 ) l ( a 12 k 1 ) u e θ 2 N 2 ] < ( a 1 k 1 ) l , ( a 22 k 2 ) u [ u 2 + ] θ 2 < ( a 2 k 2 ) l 1 + ( c 2 k 2 ) u e N 2 < ( a 2 k 2 ) l .
Therefore, we have
u 1 + < [ ( a 1 k 1 ) l ( a 11 k 1 ) u ] 1 θ 1 [ ( a 1 a 11 ) u ] 1 θ 1 < e N 1 , u 2 + < [ ( a 2 k 2 ) l ( a 22 k 2 ) u ] 1 θ 2 [ ( a 2 a 22 ) u ] 1 θ 2 < e N 2 ,

which implies that assertion (5) also holds. □

Now, we are ready to state the main result of this paper.

Theorem 3.1 Assume that (H1)-(H3) hold. Then system (1.1) has at least four ω-periodic solutions.

Proof Take
X = Z = { u = ( u 1 , u 2 ) T : u i C rd ( T , R 2 )  and  u i ( t + ω ) = u i ( t )  for all  t T , i = 1 , 2 } ,
and define
u = [ i = 1 2 ( max t I ω | u i ( t ) | ) 2 ] 1 2 , u X ( or  Z ) .

Equipped with the above norm , it is easy to verify that X and Z are both Banach spaces.

Set
Δ 1 ( u , t , λ ) = [ k 1 ( t ) + ( 1 λ ) c 1 ( t ) e u 1 ( t ) k 1 ( t ) + c 1 ( t ) e u 1 ( t ) ] Δ 1 ( u , t , λ ) = × [ a 1 ( t ) k 1 ( t ) a 11 ( t ) e θ 1 u 1 ( t ) k 1 ( t ) λ a 12 ( t ) e θ 2 u 2 ( t ) k 1 ( t ) ] H 1 ( t ) e u 1 ( t ) , Δ 2 ( u , t , λ ) = [ k 2 ( t ) + ( 1 λ ) c 2 ( t ) e u 2 ( t ) k 2 ( t ) + c 2 ( t ) e u 2 ( t ) ] Δ 2 ( u , t , λ ) = × [ a 2 ( t ) k 2 ( t ) + λ a 21 ( t ) e θ 1 u 1 ( t ) k 2 ( t ) a 22 ( t ) e θ 2 u 2 ( t ) k 2 ( t ) ] H 2 ( t ) e u 2 ( t ) .
Define the mappings L : X Z , N : X × [ 0 , 1 ] Z , P : X X and Q : Z Z as follows:
N ( u , λ ) = ( Δ 1 ( u , t , λ ) Δ 2 ( u , t , λ ) ) , L u = ( u 1 Δ u 2 Δ ) , P u = 1 ω κ κ + ω u ( t ) Δ t = Q u , u X ( or  Z ) .

We first show that L is a Fredholm mapping of index zero and N is L-compact on Ω ¯ × [ 0 , 1 ] for any open bounded set Ω X . The argument is standard, one can see [35]. But for the sake of completeness, we give the details here.

It is easy to see that Ker L = { u X : ( u 1 ( t ) , u 2 ( t ) ) T = ( h 1 , h 2 ) T R 2  for  t T } , Im L = { u X : κ κ + ω u ( t ) Δ t = 0 } is closed in Z, and dim Ker L = codim Im L = 2 . Therefore, L is a Fredholm mapping of index zero. Clearly, P and Q are continuous projectors such that
Im P = Ker L , Ker Q = Im L = Im ( I Q ) .
On the other hand, K p : Im L dom L Ker P , the inverse to L, exists and is given by
K p ( u ) = κ t u ( s ) Δ s 1 ω κ κ + ω κ η u ( s ) Δ s Δ η .

Obviously, QN and K p ( I Q ) N are continuous. By Lemma 2.4, it is not difficult to show that K p ( I Q ) N ( Ω ¯ × [ 0 , 1 ] ) ¯ is compact for any open bounded set Ω X . Moreover, Q N ( Ω ¯ × [ 0 , 1 ] ) is bounded. Hence, N is L-compact on Ω ¯ × [ 0 , 1 ] for any open bounded set Ω X .

In order to apply Lemma 3.1, we need to find at least four appropriate open, bounded subsets Ω 1 , Ω 2 , Ω 3 , Ω 4 in X.

Corresponding to the operator equation L u = λ N ( u , λ ) , λ ( 0 , 1 ) , we have
u 1 Δ ( t ) = λ [ k 1 ( t ) + ( 1 λ ) c 1 ( t ) e u 1 ( t ) k 1 ( t ) + c 1 ( t ) e u 1 ( t ) ] u 1 Δ ( t ) = × [ a 1 ( t ) k 1 ( t ) a 11 ( t ) e θ 1 u 1 ( t ) k 1 ( t ) λ a 12 ( t ) e θ 2 u 2 ( t ) k 1 ( t ) ] λ H 1 ( t ) e u 1 ( t ) ,
(3.1)
u 2 Δ ( t ) = λ [ k 2 ( t ) + ( 1 λ ) c 2 ( t ) e u 2 ( t ) k 2 ( t ) + c 2 ( t ) e u 2 ( t ) ] u 2 Δ ( t ) = × [ a 2 ( t ) k 2 ( t ) + λ a 21 ( t ) e θ 1 u 1 ( t ) k 2 ( t ) a 22 ( t ) e θ 2 u 2 ( t ) k 2 ( t ) ] λ H 2 ( t ) e u 2 ( t ) .
(3.2)
Suppose that ( u 1 ( t ) , u 2 ( t ) ) T is an ω-periodic solution of (3.1), (3.2) for some λ ( 0 , 1 ) . Since u ( t ) = ( u 1 ( t ) , u 2 ( t ) ) X , there exist t i M , t i m [ κ , κ + ω ] T , i = 1 , 2 , such that
u i ( t i M ) = max t [ κ , κ + ω ] T u i ( t ) , u i ( t i m ) = min t [ κ , κ + ω ] T u i ( t ) , i = 1 , 2 .
By Lemma 2.2, it is easy to see that
u i Δ ( t i M ) 0 , u i Δ ( t i m ) 0 , i = 1 , 2 .
From this and (3.1), (3.2), we obtain that
0 [ k 1 ( t 1 M ) + ( 1 λ ) c 1 ( t 1 M ) e u 1 ( t 1 M ) k 1 ( t 1 M ) + c 1 ( t 1 M ) e u 1 ( t 1 M ) ] 0 × [ a 1 ( t 1 M ) k 1 ( t 1 M ) a 11 ( t 1 M ) e θ 1 u 1 ( t 1 M ) k 1 ( t 1 M ) λ a 12 ( t 1 M ) e θ 2 u 2 ( t 1 M ) k 1 ( t 1 M ) ] H 1 ( t 1 M ) e u 1 ( t 1 M ) ,
(3.3)
0 [ k 2 ( t 2 M ) + ( 1 λ ) c 2 ( t 2 M ) e u 2 ( t 2 M ) k 2 ( t 2 M ) + c 2 ( t 2 M ) e u 2 ( t 2 M ) ] 0 × [ a 2 ( t 2 M ) k 2 ( t 2 M ) + λ a 21 ( t 2 M ) e θ 1 u 1 ( t 2 M ) k 2 ( t 2 M ) a 22 ( t 2 M ) e θ 2 u 2 ( t 2 M ) k 2 ( t 2 M ) ] H 2 ( t 2 M ) e u 2 ( t 2 M )
(3.4)
and
0 [ k 1 ( t 1 m ) + ( 1 λ ) c 1 ( t 1 m ) e u 1 ( t 1 m ) k 1 ( t 1 m ) + c 1 ( t 1 m ) e u 1 ( t 1 m ) ] 0 × [ a 1 ( t 1 m ) k 1 ( t 1 m ) a 11 ( t 1 m ) e θ 1 u 1 ( t 1 m ) k 1 ( t 1 m ) λ a 12 ( t 1 m ) e θ 2 u 2 ( t 1 m ) k ( t 1 m ) ] H 1 ( t 1 m ) e u 1 ( t 1 m ) ,
(3.5)
0 [ k 2 ( t 2 m ) + ( 1 λ ) c 2 ( t 2 m ) e u 2 ( t 2 m ) k 2 ( t 2 m ) + c 2 ( t 2 m ) e u 2 ( t 2 m ) ] 0 × [ a 2 ( t 2 m ) k 2 ( t 2 m ) + λ a 21 ( t 2 m ) e θ 1 u 1 ( t 2 m ) k 2 ( t 2 m ) a 22 ( t 2 m ) e θ 2 u 2 ( t 2 m ) k 2 ( t 2 m ) ] H 2 ( t 2 m ) e u 2 ( t 2 m ) .
(3.6)
Claim A
u 1 ( t 1 M ) < N 1 : = ln ( a 1 a 11 ) u θ 1 + 2 ω ( a 1 k 1 ) u , u 2 ( t 2 M ) < N 2 : = ln [ ( a 2 a 22 ) u + ( a 21 a 22 ) u e θ 1 N 1 ] θ 2 + 2 ω ( a 2 k 2 ) u + 2 ω ( a 21 k 2 ) u e θ 1 N 1 .
From (3.1), we obtain that
κ κ + ω [ k 1 ( t ) + ( 1 λ ) c 1 ( t ) e u 1 ( t ) k 1 ( t ) + c 1 ( t ) e u 1 ( t ) ] a 1 ( t ) k 1 ( t ) Δ t = κ κ + ω [ k 1 ( t ) + ( 1 λ ) c 1 ( t ) e u 1 ( t ) k 1 ( t ) + c 1 ( t ) e u 1 ( t ) ] [ a 11 ( t ) e θ 1 u 1 ( t ) k 1 ( t ) + λ a 12 ( t ) e θ 2 u 2 ( t ) k 1 ( t ) ] Δ t + κ κ + ω H 1 ( t ) e u 1 ( t ) Δ t .
Therefore, we have
κ κ + ω | u 1 Δ ( t ) | Δ t < κ κ + ω [ k 1 ( t ) + ( 1 λ ) c 1 ( t ) e u 1 ( t ) k 1 ( t ) + c 1 ( t ) e u 1 ( t ) ] [ a 1 ( t ) k 1 ( t ) + a 11 ( t ) e θ 1 u 1 ( t ) k 1 ( t ) + λ a 12 ( t ) e θ 2 u 2 ( t ) k 1 ( t ) ] Δ t + κ κ + ω H 1 ( t ) e u 1 ( t ) Δ t = 2 κ κ + ω [ k 1 ( t ) + ( 1 λ ) c 1 ( t ) e u 1 ( t ) k 1 ( t ) + c 1 ( t ) e u 1 ( t ) ] a 1 ( t ) k 1 ( t ) Δ t < 2 κ κ + ω a 1 ( t ) k 1 ( t ) Δ t 2 ω ( a 1 k 1 ) u .
(3.7)
By (3.5), we have
a 11 ( t 1 m ) e θ 1 u 1 ( t 1 m ) k 1 ( t 1 m ) < a 1 ( t 1 m ) k 1 ( t 1 m ) ,
which implies
e θ 1 u 1 ( t 1 m ) < a 1 ( t 1 m ) a 11 ( t 1 m ) ( a 1 a 11 ) u .
Therefore, we have
u 1 ( t 1 m ) < ln ( a 1 a 11 ) u θ 1 .
(3.8)
From (3.7), (3.8) and Lemma 2.3, we have
u 1 ( t 1 M ) u 1 ( t 1 m ) + κ κ + ω | u 1 Δ ( t ) | Δ t < ln ( a 1 a 11 ) u θ 1 + 2 ω ( a 1 k 1 ) u : = N 1 .
(3.9)
From (3.2), we obtain that
κ κ + ω [ k 2 ( t ) + ( 1 λ ) c 2 ( t ) e u 2 ( t ) k 2 ( t ) + c 2 ( t ) e u 2 ( t ) ] [ a 2 ( t ) k 2 ( t ) + λ a 21 ( t ) e θ 1 u 1 ( t ) k 2 ( t ) ] Δ t = κ κ + ω [ k 2 ( t ) + ( 1 λ ) c 2 ( t ) e u 2 ( t ) k 2 ( t ) + c 2 ( t ) e u 2 ( t ) ] a 22 ( t ) e θ 2 u 2 ( t ) k 2 ( t ) Δ t + κ κ + ω H 2 ( t ) e u 2 ( t ) Δ t .
Therefore, we have
κ κ + ω | u 2 Δ ( t ) | Δ t < κ κ + ω [ k 2 ( t ) + ( 1 λ ) c 2 ( t ) e u 2 ( t ) k 2 ( t ) + c 2 ( t ) e u 2 ( t ) ] [ a 2 ( t ) k 2 ( t ) + λ a 21 ( t ) e θ 1 u 1 ( t ) k 2 ( t ) + a 22 ( t ) e θ 2 u 2 ( t ) k 2 ( t ) ] Δ t + κ κ + ω H 2 ( t ) e u 2 ( t ) Δ t = 2 κ κ + ω [ k 2 ( t ) + ( 1 λ ) c 2 ( t ) e u 2 ( t ) k 2 ( t ) + c 2 ( t ) e u 2 ( t ) ] [ a 2 ( t ) k 2 ( t ) + λ a 21 ( t ) e θ 1 u 1 ( t ) k 2 ( t ) ] Δ t < 2 κ κ + ω [ a 2 ( t ) k 2 ( t ) + λ a 21 ( t ) e θ 1 u 1 ( t ) k 2 ( t ) ] Δ t .
From this and (3.9), we have
κ κ + ω | u 2 Δ ( t ) | Δ t < 2 ω ( a 2 k 2 ) u + 2 ω ( a 21 k 2 ) u e θ 1 N 1 .
(3.10)
By (3.6), we have
a 22 ( t 2 m ) e θ 2 u 2 ( t 2 m ) k 2 ( t 2 m ) < a 2 ( t 2 m ) k 2 ( t 2 m ) + λ a 21 ( t 2 m ) e θ 1 u 1 ( t 2 m ) k 2 ( t 2 m ) ,
which implies
e θ 2 u 2 ( t 2 m ) < a 2 ( t 2 m ) a 22 ( t 2 m ) + a 21 ( t 2 m ) a 22 ( t 2 m ) e θ 1 N 1 ( a 2 a 22 ) u + ( a 21 a 22 ) u e θ 1 N 1 .
Therefore, we have
u 2 ( t 2 m ) < ln [ ( a 2 a 22 ) u + ( a 21 a 22 ) u e θ 1 N 1 ] θ 2 .
(3.11)
From (3.10), (3.11) and Lemma 2.3, we have
u 2 ( t 2 M ) u 2 ( t 2 m ) + κ κ + ω | u 2 Δ ( t ) | Δ t < ln [ ( a 2 a 22 ) u + ( a 21 a 22 ) u e θ 1 N 1 ] θ 2 + 2 ω ( a 2 k 2 ) u + 2 ω ( a 21 k 2 ) u e θ 1 N 1 : = N 2 .
(3.12)
Claim B
ln l i < u i ( t i m ) < ln l i + , i = 1 , 2 .
From (3.5) and noticing that
k 1 ( t 1 m ) k 1 ( t 1 m ) + c 1 ( t 1 m ) e u 1 ( t 1 m ) < k 1 ( t 1 m ) + ( 1 λ ) c 1 ( t 1 m ) e u 1 ( t 1 m ) k 1 ( t 1 m ) + c 1 ( t 1 m ) e u 1 ( t 1 m ) < 1 ,
we have
0 < ( a 1 k 1 ) u a 11 ( t 1 m ) k 1 ( t 1 m ) + c 1 ( t 1 m ) e u 1 ( t 1 m ) e θ 1 u 1 ( t 1 m ) H 1 l e u 1 ( t 1 m ) .
Therefore, by (3.8), we have
0 < ( a 1 k 1 ) u ( a 11 k 1 ) l 1 + ( c 1 k 1 ) u [ ( a 1 a 11 ) u ] 1 θ 1 e θ 1 u 1 ( t 1 m ) H 1 l e u 1 ( t 1 m ) .
From assertion (3) of Lemma 3.3 and the above inequality, we have
ln l 1 < u 1 ( t 1 m ) < ln l 1 + .
(3.13)
Similarly, from (3.6) and noticing that
k 2 ( t 2 m ) k 2 ( t 2 m ) + c 2 ( t 2 m ) e u 2 ( t 2 m ) < k 2 ( t 2 m ) + ( 1 λ ) c 2 ( t 2 m ) e u 2 ( t 2 m ) k 2 ( t 2 m ) + c 2 ( t 2 m ) e u 2 ( t 2 m ) < 1 ,
we have
0 < ( a 2 k 2 ) u + ( a 21 k 2 ) u e θ 1 N 1 a 22 ( t 2 m ) k 2 ( t 2 m ) + c 2 ( t 2 m ) e u 2 ( t 2 m ) e θ 2 u 2 ( t 2 m ) H 2 l e u 2 ( t 2 m ) .
Therefore, by (3.11), we have
0 < ( a 2 k 2 ) u + ( a 21 k 2 ) u e θ 1 N 1 ( a 22 k 2 ) l 1 + ( c 2 k 2 ) u [ ( a 2 a 22 ) u + ( a 21 a 22 ) u e θ 1 N 1 ] 1 θ 2 e θ 2 u 2 ( t 2 m ) H 2 l e u 2 ( t 2 m ) .
From assertion (3) of Lemma 3.3 and the above inequality, we have
ln l 2 < u 2 ( t 2 m ) < ln l 2 + .
(3.14)
Claim C
u i ( t i M ) > ln u i + or u i ( t i M ) < ln u i , i = 1 , 2 .
From (3.3), (H2) and noticing that
k 1 ( t 1 M ) k 1 ( t 1 M ) + c 1 ( t 1 M ) e u 1 ( t 1 M ) < k 1 ( t 1 M ) + ( 1 λ ) c 1 ( t 1 M ) e u 1 ( t 1 M ) k 1 ( t 1 M ) + c 1 ( t 1 M ) e u 1 ( t 1 M ) < 1 ,
we have
0 > k 1 ( t 1 M ) k 1 ( t 1 M ) + c 1 ( t 1 M ) e u 1 ( t 1 M ) [ ( a 1 k 1 ) l ( a 12 k 1 ) u e θ 2 N 2 ] ( a 11 k 1 ) u e θ 1 u 1 ( t 1 M ) H 1 u e u 1 ( t 1 M ) .
Therefore, we have
0 > 1 1 + ( c 1 k 1 ) u e N 1 [ ( a 1 k 1 ) l ( a 12 k 1 ) u e θ 2 N 2 ] ( a 11 k 1 ) u e θ 1 u 1 ( t 1 M ) H 1 u e u 1 ( t 1 M ) .
(3.15)
From assertion (1) of Lemma 3.3 and (3.15), we have
u 1 ( t 1 M ) > ln u 1 + or u 1 ( t 1 M ) < ln u 1 .
(3.16)
By a similar argument, it follows from (3.4) that
0 > ( a 2 k 2 ) l 1 + ( c 2 k 2 ) u e N 2 ( a 22 k 2 ) u e θ 2 u 2 ( t 2 M ) H 2 u e u 2 ( t 2 M ) .
(3.17)
From assertion (1) of Lemma 3.3 and (3.17), we have
u 2 ( t 2 M ) > ln u 2 + or u 2 ( t 2 M ) < ln u 2 .
(3.18)
It follows from (3.9), (3.13), (3.16) and assertions (4)-(5) of Lemma 3.3 that
u 1 ( t 1 M ) ( ln l 1 , ln u 1 ) ( ln u 1 + , N 1 ) ,
(3.19)
u 1 ( t 1 m ) ( ln l 1 , ln l 1 + ) .
(3.20)
It follows from (3.12), (3.14), (3.18) and assertions (4)-(5) of Lemma 3.3 that
u 2 ( t 2 M ) ( ln l 2 , ln u 2 ) ( ln u 2 + , N 2 ) ,
(3.21)
u 2 ( t 2 m ) ( ln l 2 , ln l 2 + ) .
(3.22)
Clearly, l i ± , u i ± ( i = 1 , 2 ) are independent of λ. Now, let us consider Q N ( u , 0 ) with u = ( u 1 , u 2 ) T R 2 . Note that
Q N ( u , 0 ) = ( ( a 1 k 1 ) ¯ ( a 11 k 1 ) ¯ e θ 1 u 1 H ¯ 1 e u 1 ( a 2 k 2 ) ¯ ( a 22 k 2 ) ¯ e θ 2 u 2 H ¯ 2 e u 2 ) .
Therefore, it follows from assertion (2) of Lemma 3.3 that Q N ( u , 0 ) = 0 has four distinct solutions:
u ˜ 1 = ( ln x 1 + , ln x 2 + ) T , u ˜ 2 = ( ln x 1 + , ln x 2 ) T ,
(3.23)
u ˜ 3 = ( ln x 1 , ln x 2 + ) T , u ˜ 4 = ( ln x 1 , ln x 2 ) T .
(3.24)
Let
Ω 1 = { u = ( u 1 , u 2 ) T X | max t I ω u 1 ( t ) ( ln u 1 + , max { ln l 1 + , N 1 } ) , min t I ω u 1 ( t ) ( ln l 1 , ln l 1 + ) , max t I ω u 2 ( t ) ( ln u 2 + , max { ln l 2 + , N 2 } ) , min t I ω u 2 ( t ) ( ln l 2 , ln l 2 + ) } , Ω 2 = { u = ( u 1 , u 2 ) T X | max t I ω u 1 ( t ) ( ln u 1 + , max { ln l 1 + , N 1 } ) , min t I ω u 1 ( t ) ( ln l 1 , ln l 1 + ) , max t I ω u 2 ( t ) ( ln l 2 , ln u 2 ) , min t I ω u 2 ( t ) ( ln l 2 , ln l 2 + ) } , Ω 3 = { u = ( u 1 , u 2 ) T X | max t I ω u 1 ( t ) ( ln l 1 , ln u 1 ) , min t I ω u 1 ( t ) ( ln l 1 , ln l 1 + ) , max t I ω u 2 ( t ) ( ln u 2 + , max { ln l 2 + , N 2 } ) , min t I ω u 2 ( t ) ( ln l 2 , ln l 2 + ) } , Ω 4 = { u = ( u 1 , u 2 ) T X | max t I ω u 1 ( t ) ( ln l 1 , ln u 1 ) , min t I ω u 1 ( t ) ( ln l 1 , ln l 1 + ) , max t I ω u 2 ( t ) ( ln l 2 , ln u 2 ) , min t I ω u 2 ( t ) ( ln l 2 , ln l 2 + ) } .
Then Ω 1 , Ω 2 , Ω 3 , Ω 4 are bounded open subsets of X. It follows from assertion (4) of Lemma 3.3, (3.23) and (3.24) that u ˜ i Ω i ( i = 1 , 2 , 3 , 4 ). From assertion (4) of Lemma 3.3, (3.19)-(3.22), it is easy to see that Ω ¯ i Ω ¯ j = ( i , j = 1 , 2 , 3 , 4 , i j ) and Ω i satisfies (a) in Lemma 3.1 for i = 1 , 2 , 3 , 4 . Moreover, Q N ( u , 0 ) 0 for u Ω i Ker L . By assertion (2) of Lemma 3.3, a direct computation gives
deg { J Q N ( , 0 ) , Ω 1 Ker L , 0 } = 1 , deg { J Q N ( , 0 ) , Ω 2 Ker L , 0 } = 1 , deg { J Q N ( , 0 ) , Ω 3 Ker L , 0 } = 1 , deg { J Q N ( , 0 ) , Ω 4 Ker L , 0 } = 1 .

Here, J is taken as the identity mapping since Im Q = Ker L . So far we have proved that Ω i satisfies all the assumptions in Lemma 3.1. Hence, (1.1) has at least four ω-periodic solutions ( u 1 i ( t ) , u 2 i ( t ) ) T ( i = 1 , 2 , 3 , 4 ) with ( u 1 i , u 2 i ) T dom L Ω ¯ i . Obviously, ( u 1 i ( t ) , u 2 i ( t ) ) T ( i = 1 , 2 , 3 , 4 ) are different.  □

Example 3.1 In system (1.1), take
ω = 2 , T 0 = { 0 , 0.5 } [ 1 , 2 ] , T = { t 0 + 2 k | t 0 T 0 , k Z } ,
where is the integer set. Clearly, the time scale is ω-periodic, i.e., t T implies t + ω T . In this case, we have
κ = min { [ 0 , + ) T } = 0 , I ω = [ κ , κ + ω ] T = T 0 = { 0 , 0.5 } [ 1 , 2 ] .
Take
θ 1 = θ 2 = 0.5 , H 1 ( t ) = 2 + sin ( π t ) 54 , 000 , H 2 ( t ) = 2 + sin ( π t ) 3 , 000 , k 1 ( t ) = 100 ( 2 + sin ( π t ) ) , k 2 ( t ) = 100 ( 3 + sin ( π t ) ) , a 1 ( t ) = [ 2 + sin ( π t ) ] 2 , a 11 ( t ) = [ 2 + sin ( π t ) ] 2 10 , a 12 ( t ) = [ 2 + sin ( π t ) ] 2 9 ( 20 + 0.5 e 0.06 ) e ( 0.08 + 0.002 e 0.06 ) , a 2 ( t ) = [ 3 + sin ( π t ) ] 2 , a 22 ( t ) = [ 3 + sin ( π t ) ] 2 20 , a 21 ( t ) = [ 3 + sin ( π t ) ] 2 400 , c 1 ( t ) = 2 + sin ( π t ) e 0.12 , c 2 ( t ) = 3 + sin ( π t ) [ 2 + 0.05 e 0.06 ] 2 e ( 0.16 + 0.004 e 0.06 ) .
Then
H 1 u = 1 18 , 000 , H 2 u = 1 1 , 000 , ( a 1 k 1 ) l = 1 100 , ( a 1 k 1 ) u = 3 100 , ( a 11 k 1 ) l = 1 1 , 000 , ( a 11 k 1 ) u = 3 1 , 000 , ( a 12 k 1 ) u = 1 300 ( 20 + 0.5 e 0.06 ) e ( 0.08 + 0.002 e 0.06 ) , ( a 2 k 2 ) l = 1 50 , ( a 2 k 2 ) u = 1 25 , ( a 22 k 2 ) l = 1 1 , 000 , ( a 22 k 2 ) u = 1 500 , ( a 21 k 2 ) u = 1 10 , 000 , ( a 1 a 11 ) u = 10 , ( a 2 a 22 ) u = 20 , ( a 21 a 22 ) u = 1 20 , ( c 1 k 1 ) u = 1 100 e 0.12 , ( c 2 k 2 ) u = 1 [ 20 + 0.5 e 0.06 ] 2 e ( 0.16 + 0.004 e 0.06 ) , N 1 = 2 ln 10 + 0.12 , N 2 = 2 ln [ 20 + 0.5 e 0.06 ] + 0.16 + 0.004 e 0.06 .
Therefore, we have
1 1 + ( c 1 k 1 ) u e N 1 [ ( a 1 k 1 ) l ( a 12 k 1 ) u e θ 2 N 2 ] = 1 300 , ( a 2 k 2 ) l 1 + ( c 2 k 2 ) u e N 2 = 1 100 , ( 1 + θ 1 ) [ ( a 11 k 1 ) u ] 1 1 + θ 1 [ H 1 u θ 1 ] θ 1 1 + θ 1 = 1.5 × ( 3 1 , 000 ) 2 3 × ( 1 9 , 000 ) 1 3 = 1.5 1 , 000 , ( 1 + θ 2 ) [ ( a 22 k 2 ) u ] 1 1 + θ 2 [ H 2 u θ 2 ] θ 2 1 + θ 2 = 1.5 × ( 1 500 ) 2 3 × ( 1 500 ) 1 3 = 3 1 , 000 .

Hence, all the conditions in Theorem 3.1 are satisfied. By Theorem 3.1, system (1.1) has at least four 2-periodic solutions.

Declarations

Acknowledgements

This research is supported by the National Natural Science Foundation of China (Grant No. 10971085).

Authors’ Affiliations

(1)
Department of Mathematics, Kunming University of Science and Technology

References

  1. Hilger S: Analysis on measure chains - a unified approach to continuous and discrete calculus. Results Math. 1990, 18: 8–56.MathSciNetView ArticleMATHGoogle Scholar
  2. Gamarra JGP, Solé RV: Complex discrete dynamics from simple continuous population models. Bull. Math. Biol. 2002, 64: 611–620. 10.1006/bulm.2002.0286View ArticleGoogle Scholar
  3. Bohner M, Fan M, Zhang J: Existence of periodic solutions in predator-prey and competition dynamic systems. Nonlinear Anal., Real World Appl. 2006, 7: 1193–1204. 10.1016/j.nonrwa.2005.11.002MathSciNetView ArticleMATHGoogle Scholar
  4. Fazly M, Hesaaraki M: Periodic solutions for predator-prey systems with Beddington-DeAngelis functional response on time scales. Nonlinear Anal., Real World Appl. 2008, 9: 1224–1235. 10.1016/j.nonrwa.2007.02.012MathSciNetView ArticleMATHGoogle Scholar
  5. Zhang WP, Bi P, Zhu DM: Periodicity in a ratio-dependent predator-prey system with stage-structured predator on time scales. Nonlinear Anal., Real World Appl. 2008, 9: 344–353. 10.1016/j.nonrwa.2006.11.011MathSciNetView ArticleMATHGoogle Scholar
  6. Xing Y, Han M, Zheng G: Initial value problem for first-order integro-differential equation of Volterra type on time scale. Nonlinear Anal. 2005, 60: 429–442.MathSciNetMATHGoogle Scholar
  7. Yu SB, Wu HH, Chen JB: Multiple periodic solutions of delayed predator-prey systems with type IV functional responses on time scales. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 271672Google Scholar
  8. Chen Y: Multiple periodic solutions of delayed predator-prey systems with type IV functional responses. Nonlinear Anal., Real World Appl. 2004, 59: 45–53.View ArticleMathSciNetMATHGoogle Scholar
  9. Zhao K, Ye Y: Four positive periodic solutions to a periodic Lotka-Volterra predatory-prey system with harvesting terms. Nonlinear Anal., Real World Appl. 2010, 11: 2448–2455. 10.1016/j.nonrwa.2009.08.001MathSciNetView ArticleMATHGoogle Scholar
  10. Zhang ZQ, Tian TS: Multiple positive periodic solutions for a generalized predator-prey system with exploited terms. Nonlinear Anal., Real World Appl. 2007, 9: 26–39.MathSciNetView ArticleMATHGoogle Scholar
  11. Li YK, Zhao KH, Ye Y: Multiple positive periodic solutions of n species delay competition systems with harvesting terms. Nonlinear Anal., Real World Appl. 2011, 12: 1013–1022. 10.1016/j.nonrwa.2010.08.024MathSciNetView ArticleMATHGoogle Scholar
  12. Gopalsamy K, Kulenovic MRS, Ladas G: Environmental periodicity and time delays in a food-limited population model. J. Math. Anal. Appl. 1990, 147: 545–555. 10.1016/0022-247X(90)90369-QMathSciNetView ArticleMATHGoogle Scholar
  13. Gourley SA, Chaplain MAJ: Travelling fronts in a food-limited population model with time delay. Proc. R. Soc. Edinb. A 2002, 132: 75–89.MathSciNetView ArticleMATHGoogle Scholar
  14. Gourley SA, So JW-H: Dynamics of a food-limited population model incorporating non-local delays on a finite domain. J. Math. Biol. 2002, 44: 49–78. 10.1007/s002850100109MathSciNetView ArticleMATHGoogle Scholar
  15. Chen FD, Sun DX, Shi JL: Periodicity in a food-limited population model with toxicants and state dependent delays. J. Math. Anal. Appl. 2003, 288: 136–146. 10.1016/S0022-247X(03)00586-9MathSciNetView ArticleMATHGoogle Scholar
  16. Li YK: Periodic solutions of periodic generalized food limited model. Int. J. Math. Math. Sci. 2001, 25: 265–271. 10.1155/S0161171201004215MathSciNetView ArticleMATHGoogle Scholar
  17. Fang H: Multiple positive periodic solutions for a food-limited two-species Gilpin-Ayala competition patch system with periodic harvesting terms. J. Inequal. Appl. 2012., 2012: Article ID 291Google Scholar
  18. Gilpin ME, Ayala FJ: Global models of growth and competition. Proc. Natl. Acad. Sci. USA 1973, 70: 3590–3593. 10.1073/pnas.70.12.3590View ArticleMATHGoogle Scholar
  19. Lian B, Hu S: Asymptotic behaviour of the stochastic Gilpin-Ayala competition models. J. Math. Anal. Appl. 2008, 339: 419–428. 10.1016/j.jmaa.2007.06.058MathSciNetView ArticleMATHGoogle Scholar
  20. Lian B, Hu S: Stochastic delay Gilpin-Ayala competition models. Stoch. Dyn. 2006, 6: 561–576. 10.1142/S0219493706001888MathSciNetView ArticleMATHGoogle Scholar
  21. He MX, Li Z, Chen FD: Permanence, extinction and global attractivity of the periodic Gilpin-Ayala competition system with impulses. Nonlinear Anal., Real World Appl. 2010, 11: 1537–1551. 10.1016/j.nonrwa.2009.03.007MathSciNetView ArticleMATHGoogle Scholar
  22. Zhang SW, Tan DJ: The dynamic of two-species impulsive delay Gilpin-Ayala competition system with periodic coefficients. J. Appl. Math. Inform. 2011, 29: 1381–1393.MathSciNetMATHGoogle Scholar
  23. Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston; 2001.View ArticleMATHGoogle Scholar
  24. Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston; 2003.View ArticleMATHGoogle Scholar
  25. Gaines RE, Mawhin JL: Coincidence Degree and Nonlinear Differential Equation. Springer, Berlin; 1997.MATHGoogle Scholar

Copyright

© Fang and Wang; licensee Springer 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.