Periodic solution of single population models on time scales

Mathematical and Computer Modelling 52 (2010) 515–521

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Periodic solution of single population models on time scalesI Jimin Zhang a,b , Meng Fan a,∗ , Huaiping Zhu c a

School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, Changchun, Jilin, 130024, PR China

b

School of Mathematical Sciences, Heilongjiang University, 74 Xuefu Street, Harbin, Heilongjiang, 150080, PR China

c

Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada

article

info

Article history: Received 16 May 2009 Accepted 22 March 2010 Keywords: Time scales Periodic solutions Coincidence degree theory Logistic equations

abstract By using the calculus on time scales, we study and establish criterion for the existence of periodic solutions of some scalar dynamical equations on time scales. The existence of periodic solutions for some concrete well-known single population models is obtained. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction Population dynamics is the study of how and why a population changes over time and space or the relationship between the population and its environment, which has been at the core of population ecology ever since the origination of the discipline during the 1920s [1]. Population dynamicists document the empirical patterns of population change and attempt to investigate and explain the mechanisms responsible for the observed patterns [2]. Mathematical modeling is a powerful tool which has been helping to understand such mechanisms. Ecologists can build mechanistic population models from a more natural starting point. There have been two basic mathematical frameworks to model the population dynamics: continuous time overlapping generation models (differential equations) and discrete time nonoverlapping generation models (difference equations). However, in terms of complex population systems both the continuous time and the discrete time theoretical models may not really catch some of the properties for certain specific population dynamics. For example, annual insect population, which is thought to be nonoverlapping [3], are usually continuous in season, die out in (say) winter, while their eggs incubate or dormant in cold winter, and then hatch when a new season starts. Similar situations occur for annual plants. These facts all suggest that models for a single population on continuous–discrete–mixed time scales should be more reasonable candidates which are closer to the biological reality. The theory of calculus on time scales (see [4] for more details) was initiated by Stefan Hilger [5] in order to unify continuous and discrete analysis, which has a tremendous potential for applications and has recently received much attention since his foundational work due to the fact that a dynamic equation on time scales is related not only to the set of real numbers (continuous time scale) and the set of integers (discrete time scale) but also to those pertaining to more general time scales. Therefore, it is more appropriate and more realistic to first establish single species model governed by dynamic equations on time scales. In addition, from the ecologists’ view point, a better approach to reaching discrete time models is to derive discrete models as approximation of continuous time processes. There are at least two different easily applicable ways to ‘discretize’

I Supported by the NSFC and NCET-08-0755 (MF,JMZ) and NSERC, CFI and ERA of Canada (HPZ).

∗

Corresponding author. E-mail addresses: [email protected] (J. Zhang), [email protected] (M. Fan), [email protected] (H. Zhu).

0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.03.048

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J. Zhang et al. / Mathematical and Computer Modelling 52 (2010) 515–521

the continuous time overlapping generalization model. The first method is based on direct discretization of the derivative in the ordinary differential equations (see [2] for more details) while the second derivation employs differential equations with piece-wise arguments (see [6] for more details). For example, consider the following continuous time single population model with multiple delays [7] x˙ (t ) = x(t )[a(t ) − g (t , x(t − τ1 (t )), . . . , x(t − τn (t )))],

t ∈ R.

(1.1)

Based on the above-mentioned discretization methods, one can easily derive the following discrete equivalents of (1.1), respectively

1x(t ) = x(t )[a(t ) − g (t , x(t − τ1 (t )), . . . , x(t − τn (t )))],

t ∈ Z,

(1.2)

see [8], and x(t + 1) = x(t ) exp {a(t ) − g (t , x(t − τ1 (t )), . . . , x(t − τn (t )))} ,

t ∈ Z.

(1.3)

In the past decades, theoretical evidences to date indicate that many population and community patterns represent intricate interactions between biology and variation in the physical environment (see [9] and other papers in the same issue). As a consequence, a mathematical model must be nonautonomous when the environmental fluctuation is taken into account. These guidelines address that we can take advantage of the properties of those varying parameters. For example, the parameters are periodic for seasonal reasons. It is well known that temporal fluctuations in the physical environment are a major driver of population fluctuations. Therefore, it is reasonable to seek conditions under which the resulting periodic nonautonomous system would have a periodic solution. With the help of the calculus on time scales and a continuation theorem in coincidence degree, and based on a dynamic equation on time scales, would hope to explore the existence of periodic solution of (1.1) and (1.3) in a unified way following a similar argument in [10,11]. Unfortunately, the approach in [10,11] does not work due to the difficulties in dealing with the priori estimates. The principle aim of this paper is to explore the existence of periodic solutions of a more general single species model on time scales, which incorporate both (1.1) and (1.3) as special cases. In Section 2, for the reader’s convenience, we will present some basic results from the calculus on time scales, then study the existence of periodic solutions in Section 3. We will also apply the main results of the paper to some more concrete single population models to end the paper. This study will reveal that the existence of periodic solutions of ordinary differential equations and their discrete equivalents discretized by the first method can be explored in a unified way and also provide a framework for studying such existence problem. In addition, the main technique in the paper can be applied to deal with other dynamic equations on time scales. 2. Basic results of calculus on time scales In this section, we present some basic definitions and preliminary results from the calculus on time scales [4] so that the paper is self-contained. Throughout this paper, the symbol T denotes a time scale, i.e., an arbitrary nonempty closed subset of the real numbers R. Throughout, the time scale T is assumed to be ω-periodic, i.e., t ∈ T implies t + ω ∈ T. In particular, the time scale T under consideration is unbounded above and below. Some examples of such time scales are

[ [2k, 2k + 1],

[[

1



R,

Z,

Ta,b

[ = [k(a + b), k(a + b) + a] with ω = a + b, a > 0, b > 0.

k∈Z

k+

k∈Z n∈N

n

,

k∈Z

Definition 2.1. We define the forward jump operator σ : T → T, the backward jump operator ρ : T → T, and the graininess µ : T → R+ = [0, ∞) for t ∈ T by

σ (t ) := inf{s ∈ T : s > t }, ρ(t ) := sup{s ∈ T : s < t }, µ(t ) = σ (t ) − t respectively. If σ (t ) = t, t is called right-dense (otherwise: right-scattered), and if ρ(t ) = t, then t is called left-dense (otherwise: left-scattered). Definition 2.2. Assume f : T → R is a function and let t ∈ T. Then we define f 1 (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 1 (t )[σ (t ) − s] ≤ ε|σ (t ) − s| for all s ∈ U . In this case, f 1 (t ) is called the delta (or Hilger) derivative of f at t. Moreover, f is said to be delta or Hilger differentiable on T if f 1 (t ) exists for all t ∈ T. A function F : T → R is called an antiderivative of f : T → R provided F 1 (t ) = f (t ) for all

J. Zhang et al. / Mathematical and Computer Modelling 52 (2010) 515–521

517

t ∈ T. Then we define s

Z

f (t )1t = F (s) − F (r ) for s, r ∈ T. r

Theorem 2.3. Assume f : T → R is a function and let t ∈ T. Then we have the following: (i) If f is differentiable at t, then f is continuous at t. (iv) If f is differentiable at t, then f (σ (t )) = f (t ) + µ(t )f 1 (t ). Definition 2.4. A function f : T → R is said to be rd-continuous if it is continuous at all right-dense points in T and its left-sided limits exists (finite) at all left-dense points in T. The set of rd-continuous functions f : T → R will be denoted by Crd (T). Lemma 2.5. Every rd-continuous function has an antiderivative. Definition 2.6. We say that a function p : T → R is regressive provided 1 + µ(t )p(t ) 6= 0 for all t ∈ T holds. The set of all regressive and rd-continuous functions p : T → R will be denote in this paper by R = R(T) = R(T, R). Definition 2.7. If p ∈ R, then the exponential function is defined as ep (t , s) = exp

t

Z

ξµ(τ ) (p(τ ))1τ



( with ξh (z ) =

s

Log(1 + hz )

z

h

,

if h 6= 0, if h = 0,

where t , s ∈ T and Log is the principal logarithm. Theorem 2.8. If p ∈ R and t , s, r ∈ T, then ep (t , t ) ≡ 1, ep (t , s) =

1 ep (s, t )

= e p (s, t ),

ep (t , s)ep (s, r ) = ep (t , r ),

[ep (·, s)]1 = pep (·, s). 3. Existence of periodic solutions In this paper, we assume that the graininess µ is also ω-periodic. To facilitate the discussion below, we now introduce some notations to be used throughout this paper. Let

κ = min{[0, ∞) ∩ T}, Iω = [κ, κ + ω] ∩ T, T + = [κ, ∞) ∩ R, Z Z κ+ω 1 1 g = g (s)1s = g (s)1s, ω Iω ω κ where g ∈ Crd (T) is an ω-periodic real function, i.e., g (t + ω) = g (t ) for all t ∈ T. Next, let us recall the continuation theorem in coincidence degree theory, borrowing notations and terminology from [12], which will come into play later on. 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 Im L is closed in Z . If L is a Fredholm mapping of index zero and there exist continuous projections P : X → X and Q : Z → Z such that Im P = Ker L, Im L = Ker Q = Im (I − Q ), then it follows that L|Dom L ∩ Ker P : (I − P )X → Im L is invertible. We denote the inverse of that map by KP . If Ω is an open bounded subset of X , the mapping N will be called L-compact on Ω if QN (Ω ) is bounded and KP (I − Q )N : Ω → X is compact. Since Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L. Lemma 3.1 (Continuation Theorem). Let L be a Fredholm mapping of index zero and N be L-compact on Ω . Suppose (a) For each λ ∈ (0, 1), every solution z of Lz = λNz is such that z 6∈ ∂ Ω ; (b) QNz = 6 0 for each z ∈ ∂ Ω ∩ Ker L and the Brouwer degree deg{JQN , Ω ∩ Ker L, 0} 6= 0. Then the operator equation Lz = Nz has at least one solution lying in Dom L ∩ Ω . Consider the following scalar dynamic equation on time scales x1 (t ) = x(t )[a(t ) − g (t , x(t − τ1 (t )), . . . , x(t − τn (t )))],

t ∈ T,

(3.1)

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J. Zhang et al. / Mathematical and Computer Modelling 52 (2010) 515–521

where:

(H1 ) a : T → (0, ∞) and g : T × (R+ )n → R+ are both rd-continuous and ω-periodic in t ∈ T, where ω > 0 is called the period of (3.1). Moreover, g (t , ·) is continuous on (R+ )n for fixed t ∈ T. (H2 ) τi : T → R+ and τi (t + ω) = τi (t ) such that t − τi (t ) ∈ T for 1 ≤ i ≤ n. It is obvious that if T = R and T = Z, then (3.1) reduces to (1.1) and (1.3), respectively. In order to explore the existence of positive periodic solutions of (3.1), first we should embed our problem in the frame of coincidence degree theory. Define L ω = u ∈ C (T, R+ ) : u(t + ω) = u(t )



kuk = max |u(t )| for u ∈ L ω .

for all t ∈ T ,



t ∈Iω

It is not difficult to show that (L ω , k·k) is a Banach space. Let L0ω = {u ∈ L ω : u = 0} ,

Lcω = u ∈ L ω : u(t ) ≡ h ∈ R+ for t ∈ T .





Then it is easy to show that L0ω and Lcω are both closed linear subspaces of L ω , L ω = L0ω ⊕ Lcω , and dim Lcω = 1. To achieve the priori estimation in the case of the dynamic equations (3.1) on a time scales T, the following several lemmas are introduced: Lemma 3.2. Assume that g is rd-continuous and ω-periodic function on T, then we have t +ω

Z

g (s)1s =

κ+ω

Z κ

t

g (s)1s,

t ∈ T.

Proof. Let t = κ + nω + r , 0 ≤ r < ω, it is not difficult to show that t +ω

Z

g (s)1s = t

κ+nω+r +ω

Z

κ+nω+r Z κ+ω

g (s)1s +

= κ

Z

g (s)1s =

κ+ω

= κ

Z

Z

κ+r +ω

κ+r κ+r +ω

g (s)1s

g (s)1s −

κ+ω

κ+r

Z κ

g (s)1s

g (s)1s.

Hence, we obtain the desired result.



Lemma 3.3. If x(t ) is a nonnegative ω-periodic solution of (3.1), then min x(t ) ≥ δkxk, t ∈Iω

where δ = e a (κ + ω, ω).

Proof. Since x(σ (t )) = x(t ) + µ(t )x1 (t ), we have x1 (t ) = a(t )(x(σ (t )) − µ(t )x1 (t )) − x(t )g [t , x(t − τ1 (t )), . . . , x(t − τn (t ))]. This equation is equivalent to the following system

(1 + a(t )µ(t ))x1 (t ) − a(t )x(σ (t )) = −x(t )g [t , x(t − τ1 (t )), . . . , x(t − τn (t ))]. It follows from 1 + a(t )µ(t ) 6= 0 that x1 (t ) −

a(t ) 1 + a(t )µ(t )

x(σ (t )) = −

x(t ) 1 + a(t )µ(t )

g [t , x(t − τ1 (t )), . . . , x(t − τn (t ))].

Furthermore, x1 (t ) + ax(σ (t )) = −

x( t ) 1 + a(t )µ(t )

g [t , x(t − τ1 (t )), . . . , x(t − τn (t ))].

By multiplying e a (t , κ) on both sides of the above equation, it is easy to show that

(e a (t , κ)x(t ))1 = −

e a (t , κ)x(t ) 1 + a(t )µ(t )

g [t , x(t − τ1 (t )), . . . , x(t − τn (t ))].

Integrating from t to t + ω, we obtain x(t )e a (t , κ)(e a (t + ω, t ) − 1) = −

t +ω

Z t

e a (s, κ)x(s) 1 + a(s)µ(s)

g (s, x(s − τ1 (s)), . . . , x(s − τn (s)))1s,

J. Zhang et al. / Mathematical and Computer Modelling 52 (2010) 515–521

519

and hence, x(t ) =

e a (s, t )

t +ω

Z

x(s)

1 − e a (κ + ω, κ) 1 + a(s)µ(s)

t

κ+ω

Z =

e a ( s, t )

g (s, x(s − τ1 (s)), . . . , x(s − τn (s)))1s

x(s)

1 − e a (κ + ω, κ) 1 + a(s)µ(s)

κ

g (s, x(s − τ1 (s)), . . . , x(s − τn (s)))1s.

(3.2)

Let G(t , s) =

e a (s, t ) 1 − e a (κ + ω, κ)

,

for t ≤ s ≤ t + ω.

Then one can show that A1 :=

e a (κ + ω, κ) 1 − e a (κ + ω, κ)

=

e a (t + ω, t ) 1 − e a (κ + ω, κ)

≤ G(t , s) ≤

1 1 − e a (κ + ω, κ)

=: A2 .

Using (3.2), we can get κ+ω

Z kxk ≤ A2

x( s) 1 + a(s)µ(s)

κ

g (s, x(s − τ1 (s)), . . . , x(t − τn (s)))1s

and min x(t ) ≥ A1

κ+ω

Z

t ∈Iω

x(s) 1 + a(s)µ(s)

κ

g (s, x(s − τ1 (s)), . . . , x(t − τn (s)))1s.

From the above argument, it is not difficult to show that A1

min x(t ) ≥

A2

t ∈Iw

kxk = δkxk.

The proof is complete.



Now we consider the existence of positive periodic solution of (3.1). Theorem 3.4. Assume that ( H1 ) and ( H2 ) hold. If

(H3 ) there exist constants M2 > M1 > 0 such that if ui ≥ M2 for all 1 ≤ i ≤ n, then g (t , u1 , u2 , . . . , un ) > a(t ),

t ∈ Iω

and if 0 < ui ≤ M1 for all 1 ≤ i ≤ n, then g (t , u1 , u2 , . . . , un ) < a(t ),

t ∈ Iω .

Then (3.1) has at least one positive ω-periodic solution. Proof. Let X = Z = L ω and define Nx = x(t )[a(t ) − g (t , x(t , τ1 (t )), . . . , x(t − τn (t )))], Lx = x1 ,

Px = Qx = x.

Then Ker L = Lc , Im L = L0ω , and dim Ker L = 1 = codim Im L. Since L0ω is closed in L ω , it follows that L is a Fredholm mapping of index zero. It is not difficult to show that P and Q are continuous projections such that Im P = Ker L and Im L = Ker Q = Im (I − Q ). Furthermore, the generalized inverse (to L) KP : Im L → Ker P ∩ Dom L exists and is given by ω

KP (x) = xˆ − xˆ ,

where xˆ (t ) =

t

Z κ

x(s)1s.

Thus QNx =

1

ω

κ+ω

Z κ

x(s)[a(s) − g (s, x(s − τ1 (s)), . . . , x(s − τn (s)))]1s

and KP (I − Q )Nx =

Z κ+ω Z t 1 (Nx)(s)1s − (Nx)(s)1s1t ω κ κ κ   Z κ+ω 1 − t −κ − (t − κ)1t Nx. ω κ

Z

t

520

J. Zhang et al. / Mathematical and Computer Modelling 52 (2010) 515–521

Obviously, QN and KP (I − Q )N are continuous. Since X is a Banach space, using the Arzelà–Ascoli theorem, it is easy to show that KP (I − Q )N (Ω ) is compact for any open bounded set Ω ⊂ X . Moreover, QN (Ω ) is bounded. Thus, N is L-compact on Ω with any open bounded set Ω ⊂ X . Now we are in the position to identify an appropriate open, bounded subset Ω for the application of the continuation theorem Lemma 3.1. For the operator equation Lx = λNx, λ ∈ (0, 1), we have x1 (t ) = λx(t )[a(t ) − g (t , x(t − τ1 (t )), . . . , x(t − τn (t )))].

(3.3)

Assume that x ∈ X is an arbitrary solution of equation (3.3) for a certain λ ∈ (0, 1). Following a similar strategy in Lemma 3.3, one can reach the following conclusion min x(t ) ≥ kxke λa (κ + ω, ω). t ∈Iω

Therefore, we have min x(t ) ≥ δkxk.

(3.4)

t ∈Iω

Integrating both sides of (3.3) over the interval [κ, κ + ω], we obtain κ+ω

Z κ

x(t )[a(t ) − g (t , x(t − τ1 (t )), . . . , x(t − τn (t )))]1t = 0.

(3.5)

Next, we will show that kx(t )k < δ2 . If it is not true, that is, kxk ≥ δ2 , by (3.4), we have mint ∈T x(t ) = mint ∈Iω x(t ) ≥ δkxk ≥ M2 . From (H3 ), it is easy to show that M

M

g (t , x(t − τ1 (t )), . . . , x(t − τn (t ))) > a(t ),

for t ∈ T.

This contradicts the equality (3.5). So we have kx(t )k < mint ∈Iω x(t ) > δ M1 . Now we define



Ω := x ∈ X : δ M1 < x(t ) <

M2

δ2

M2

δ

. Similarly, it follows from (3.4) and (3.5) and (H3 ) that



, t ∈ Iω .

It is clear that Ω satisfies the requirement (a) in Lemma 3.1. If x ∈ ∂ Ω ∩ Ker L, then x is constant with x = δ M1 or x = δ2 , and we have M

QNx =

1

ω

κ+ω

Z κ

h

i

x(s) a(s) − g (s, x(s − τ1 (s)), . . . , x(s − τn (s))) 1s 6= 0.

Moreover, note that J = I since Im Q = Ker L. In order to compute the Brouwer degree, let us consider the homotopy H (ν, x) = ν

  1

2

δ M1 +

M2



δ



− x + (1 − ν)QNx,

ν ∈ [0, 1].

For any x ∈ ∂ Ω ∩ Ker L, ν ∈ [0, 1], we have H (ν, x) 6= 0. By the homotopic invariance of topological degree, we have deg{JQN , Ω ∩ Ker L, 0} = deg{QNx, Ω ∩ Ker L, 0}

  = deg

1

2

δ M1 +

M2

δ



 − x, Ω ∩ Ker L, 0

6= 0, where deg(·, ·, ·) is the Brouwer degree. Now we have proved that Ω satisfies all requirements in Lemma 3.1. Thus Lx = Nx has at least one solution in Dom L ∩ Ω , that is, (3.1) has at least one positive ω-periodic solution in Dom L ∩ Ω . The proof is complete.  By a similar argument as above, we can prove the following result. Theorem 3.5. Assume that ( H1 ) and ( H2 ) hold. If

(H3 ) there exist constants M4 > M3 > 0 such that if ui ≥ M4 for all 1 ≤ i ≤ n, then g (t , u1 , u2 , . . . , un ) < a(t ),

for t ∈ Iω

and if 0 < ui ≤ M3 for all 1 ≤ i ≤ n, then g (t , u1 , u2 , . . . , un ) > a(t ),

for t ∈ Iω .

Then the equation (3.1) has at least one positive ω-periodic solution.

J. Zhang et al. / Mathematical and Computer Modelling 52 (2010) 515–521

521

In order to illustrate some features of our maim theorems, we explore the existence of positive periodic solutions of some more concrete single population models, which have been extensively studied in the literature. Consider the following single population models



x1 (t ) = a(t )x(t ) 1 −

" 1

x (t ) = x(t ) a(t ) −

x(t − τ (t ))



K (t )

n X

,

(3.6)

# ai (t )x(t − τi (t )) ,

(3.7)

i =1

" 1

x (t ) = a(t )x(t ) 1 −

n Y x(t − τi (t ))

K (t )

i=1

" 1

x (t ) = a(t )x(t ) 1 −

" 1

x (t ) = x(t ) a(t ) −

# ,

n X

ai (t )x(t − τi (t ))

i=1

1 + ci (t )x(t − τi (t ))



x(t ) K (t )

θ #

(3.8)

# ,

,

(3.9)

(3.10)

where a, ai , ci , K : T → R are rd-continuous and ω-periodic functions such that a(t ) > 0, ai (t ) ≥ 0, ci (t ) ≥ 0, K (t ) > 0, θ > 0, τi : T → R+ and τi (t + ω) = τi (t ) satisfy t − τi (t ) ∈ T for 1 ≤ i ≤ n. When T = R or T = Z, (3.6)–(3.10) reduce to the well-known continuous or discrete time nonautonomous Logistic equation with single delay [2], and with multi-delays [8,7], delayed multiplicative Logistic equation [13,14], Michaelis–Menton type single species growth model with several deviating arguments [15,16], and nonautonomous Gilpin–Ayala single species model [17,18], which have been explored extensively in the literatures. By Theorem 3.4, we can conclude that Theorem 3.6. Each of (3.6)–(3.10) has at least one positive ω-periodic solution. References [1] S.E. Kingsland, Modeling nature: Episodes in the History of Population Ecology, 2nd edition, University of Chicago Press, Chicago, IL. [2] P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Princeton University Press, Princeton and Oxford, 2003. [3] F.B. Christiansen, T.M. Fenchel, Theories of Population in Biological Communities, in: Lecture Notes in Ecological Studies, vol 20, Springer-Verlag, Berlin, 1977. [4] M. Bohner, A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001. [5] S. Hilger, Analysis on measure chains — a unified approach to continuous and discrete calculus, Results Math. 18 (1990) 18–56. [6] M. Fan, K. Wang, Periodic solutions of a discrete time nonautonomous ratio-dependent predator–prey system, Math. Comput. Model. 35 (2002) 951–961. [7] Y.K. Li, Existence and global attractivity of positive periodic solution for a class of delay differential equations (in Chinese), Sci. China Ser. A. 28 (1998) 108–118. [8] D.Q. Jiang, R.P. Agarwal, Existence of positive periodic solutions for a class of difference equations with several deviating arguments, Comput. Math. Appl. 45 (2003) 1303–1309. [9] P. Chesson, Understanding the role of environmental variation in population and community dynamics, Theor. Popul. Biol. 64 (2003) 253–254. [10] M. Bohner, M. Fan, J.M. Zhang, Existence of periodic solutions in predator–prey and competition dynamic systems, Nonlinear Anal. Real World. Appl. 7 (2006) 1193–1204. [11] M. Bohner, M. Fan, J.M. Zhang, Periodicity of scalar dynamic equations and applications to population models, J. Math. Anal. Appl. 330 (2007) 1–9. [12] R.E. Gaines, J.L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, in: Lecture Notes in Mathematics, vol. 568, Springer-Verlag, Berlin, Heidelberg, NY, 1977. [13] K. Gopalsamy, S.B. Lalli, Oscillatory and asymptotic behaviour of a multiplicative delay logistic equation, Dynam. Stab. Syst. 7 (1992) 35–42. [14] B.G. Zhang, K. Gopalsamy, Global attractivity in the delay logistic equation, J. Math. Anal. Appl. 150 (1990) 274–283. [15] Y. Kuang, Global stability for a class of nonlinear nonautonomous delay logistic equations, Nonlinear Anal. TMA 17 (1991) 627–634. [16] I. Kubiaczyk, S.H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Math. Comput. Model. 35 (2002) 295–301. [17] M. Fan, D. Ye, P.J.Y. Wong, R.P. Agarwal, Periodicity in a class of nonautonomous scalar equations with deviating arguments and applications to population models, Dynam. Syst. 19 (2004) 279–301. [18] M.E. Gilpin, F.J. Ayala, Global models of growth and competition, Proc. Natl. Acad. Sci. USA 70 (1973) 3590–3593.