Global Stability and Oscillation of a Discrete Annual Plants Model

and Applied Analysis 3 Combining 1.1 and 1.2 , we see that the Watkinson model of the annual plants is given by the difference equation N n 1 λN n 1 aN n γ λmN n , n 0, 1, 2, . . . . 1.3 In this model the density-independent mortality is not included and the growth of the population occurs only during the vegetative phase of the life cycle. Watkinson in 18 assumed that density-independent mortality during the seed phase of the life cycle can easily be incorporated by multiplying λ by the probability that a seed will survive from the time of its formation to germination and establishment. Also in this model it is clear that the past history of the population is ignored, that is, the growth of the population is governed by a principle of causality, that is, the future state of population is independent of the past and is determined solely by the present. In fact in a single species population there is a time delay because of the time it takes a female animal or a plant to mature before it can begin to reproduce. A more realistic model must include some of the past history of population. Accordingly Kocić and Ladas 19 considered the model N n 1 λN n 1 aN n − 1 γ mN n − 1 , n 0, 1, 2, . . . , 1.4 and proved that ifN −1 ≥ 0,N 0 > 0 and λ ∈ 1,∞ , a ∈ 0,∞ , γ ∈ 0, 1 , m ∈ 0,∞ , 1.5 then limn→∞N n N, whereN is the unique fixed point of 1.4 . Note that the assumption γ ≤ 1 is different from the assumption γ > 1 that has been proposed byWatkinson 18 , which reflects the fact that an increasing density leads to a less efficient use of the resources with a given area in terms of total dry matter population. In 11 the authors considered the general equation with two delays of the form N n 1 λN n 1 aN n − k γ λmN n − l , n 0, 1, 2, . . . , 1.6 where λ ∈ 1,∞ , a, γ,m ∈ 0,∞ , l, k ∈ {0, 1, 2, 3, . . .}. 1.7 The authors in 11, Theorem 6.3.1 proved that if Nγa ( 1 aN )γ−1 ( 1 aN )γ λmN mN l k / 1, 1.8 4 Abstract and Applied Analysis then every solution of 1.6 oscillates about N if and only if every solution of the linearized delay difference equation y n 1 − y n Nγa ( 1 aN )γ−1 λ y n − k mNy n − l 0, 1.9 oscillates. In 11, Open Problem 6.3.1 the authors mentioned that the global asymptotic stability of the fixed-point N of 1.6 has not been investigated yet. Our aims in this paper is to consider this open problem, when k l, and establish some sufficient conditions for the global stability of the positive fixed point of the delay difference equation N n 1 λN n 1 aN n − k γ λmN n − k , n 0, 1, 2, . . . , 1.10 whereN n in 1.10 represents the number of mature population in the nth and the function F N n − k : λ 1 aN n − k γ λmN n − k , 1.11 represents the number of mature population that were produced in the n − k th cycle and survived to maturity in the nth cycle. We also establish an explicit sufficient condition for oscillation of all solutions of 1.10 about the fixed point. We note that whenm 0, and k 0 1.10 reduces to N n 1 λN n 1 aN n γ , n 0, 1, 2, . . . . 1.12 This equation has been proposed by Hassell 1 to describe the growth of the population of insects. On the other hand, when γ 1 and m 0, 1.10 becomes the Pielou equation 20 N n 1 λN n 1 aN n − k , n 0, 1, 2, . . . . 1.13 By the biological interpretation, we assume that the initial condition of 1.10 is given by N −k ,N −k 1 ,N −k 2 , . . . ,N 1 ∈ 0,∞ , N 0 > 0. 1.14 By a solution of 1.10 , we mean a sequence N n which is defined for n ≥ −k and satisfies 1.10 for n ≥ 0 and by a positive solution, we mean that the terms of the sequence {N n }n 1 are all positive. Then, it is easy to see that the initial value problem 1.10 and 1.14 has a unique positive solution N n . In the sequel, we will only consider positive solutions of 1.10 . We say that N is a fixed of 1.10 if N (( 1 aN )γ λmN ) λN, 1.15 Abstract and Applied Analysis 5 that is, the constant sequence {N n }n −k withN n N for all n ≥ −k is a solution of 1.10 . In the following, we prove that 1.10 has a unique positive fixed point. Let g N : 1 aN γ λmN − λ. 1.16 Then g 0 −λ < 0 and g ∞ ∞, so that there exists N > 0 such that g N 0. Also g ′ N γa 1 aN γ−1 λm > 0, ∀N > 0. 1.17 It follows that g N 0 has exactly one solution and so 1.10 has a unique positive fixed point which is denoted byN and obtained from the solution of ( 1 aN )γ λmN − λ 0. 1.18and Applied Analysis 5 that is, the constant sequence {N n }n −k withN n N for all n ≥ −k is a solution of 1.10 . In the following, we prove that 1.10 has a unique positive fixed point. Let g N : 1 aN γ λmN − λ. 1.16 Then g 0 −λ < 0 and g ∞ ∞, so that there exists N > 0 such that g N 0. Also g ′ N γa 1 aN γ−1 λm > 0, ∀N > 0. 1.17 It follows that g N 0 has exactly one solution and so 1.10 has a unique positive fixed point which is denoted byN and obtained from the solution of ( 1 aN )γ λmN − λ 0. 1.18 The stability of equilibria is one of the most important issues in the studies of population dynamics. The fixed-pointN of 1.10 is locally stable if the solution of the population model N n approaches N as time increases for all N 0 in some neighborhood of N. The fixed point N of 1.10 is globally stable if for all positive initial values the solution of the model approachesN as time increases. A model is locally or globally stable if its positive fixed point is locally or globally stable. The fixed-point N is globally asymptotically stable if its locally and globally stable. A solution {N n } of 1.10 is said to be oscillatory aboutN ifN n −N is oscillatory, where the sequenceN n −N is said to be oscillatory ifN n −N is not eventually positive or eventually negative. For the delay equations, for completeness, we present some global stability conditions of the zero solution of the delay difference equation x n 1 − x n A n x n − k 0, 1.19 that we will use in the proof of the main global stability results. Györi and Pituk 21 , proved that if lim n→∞ sup n−1 ∑ i n−k A i < 1, 1.20 then the zero solution is globally stable. Erbe et al. 22 improved 1.20 and proved that if lim n→∞ sup n ∑ i n−k A i < 3 2 1 2 k 1 , 1.21 then the zero solution is globally stable. Also Kovácsvölgy 23 proved that if lim n→∞ sup n−1 ∑ i n−2k A i < 7 4 , 1.22 6 Abstract and Applied Analysis then every zero solution of 1.19 is globally stable, whereas the result by Yu and Cheng 24 gives an improvement over 1.22 to lim n→∞ sup n ∑ i n−2k A i < 2. 1.23 The paper is organized as follows: in Section 2, we establish some sufficient conditions for global stability of N. The results give a partial answer to the open problem posed by Kocić and Ladas 11, Open problem 6.3.1 and improve the results that has been established by Kocić and Ladas 19 in the sense that the restrictive condition γ ∈ 0, 1 is not required. In Section 3, we will use an approach different from the method used in 11 and establish an explicit sufficient condition for oscillation of the positive solutions of the delay equation 1.10 about N. Some illustrative examples and simulations are presented throughout the paper to demonstrate the validity and applicability of the results. 2. Global Stability Results In this section, we establish some sufficient conditions for local and global stability of the positive fixed-point N. First, we establish a sufficient condition for local stability of 1.10 . The linearized equation associated with 1.10 at N is given by y n 1 − y n Nγa ( 1 aN )γ−1 λmN ( 1 aN )γ λmN y n − k 0. 2.1 Applying the local stability result of Levin and May 25 on 2.1 , we have the following result. Theorem 2.1. Assume that 1.14 holds and γ, λ > 1. If L ( a, γ, λ,m, k,N ) Nγa ( 1 aN )γ−1 λmN ( 1 aN )γ λmN < 2 cos πk 2k 1 , 2.2 then the fixed-point N of 1.10 is locally asymptotically stable. To prove the main global stability results for 1.10 , we need to find some upper and lower bounds for positive solutions of 1.10 which oscillate about N. Theorem 2.2. Assume that 1.14 holds and γ, λ > 1. LetN n be a positive solution of 1.10 which oscillates about N. Then there exists n1 > 0 such that for all n ≥ n1, one has Y1 : λN (( 1 aNλk )γ mNλk 1 )k ≤ N n ≤ Nλ k : Y2, 2.3 Abstract and Applied Analysis 7 Proof. First we will show the upper bound in 2.3 . The sequence N n is oscillatory about the positive periodic solution N in the sense that there exists a sequence of positive integers {nl} for l 1, 2, . . . such that k ≤ n1 < n2 < · · · < nl < · · · with liml→∞nl ∞,N nl < N and N nl 1 ≥ N. We assume that some of the terms N j with nl < j ≤ nl 1 are greater thanN and some are less thanN. Our strategy is to show that the upper bound holds in each interval nl, nl 1 . For each l 1, 2, . . ., let ζl be the integer in the interval nl, nl 1 such that N ζl 1 max { N ( j ) : nl < j ≤ nl 1 } . 2.4 Then N ζl 1 ≥ N ζl which implies that ΔN ζl ≥ 0. To show the upper bound on 2.3 , it suffices to show that N ζl ≤ Nλ Y2. 2.5and Applied Analysis 7 Proof. First we will show the upper bound in 2.3 . The sequence N n is oscillatory about the positive periodic solution N in the sense that there exists a sequence of positive integers {nl} for l 1, 2, . . . such that k ≤ n1 < n2 < · · · < nl < · · · with liml→∞nl ∞,N nl < N and N nl 1 ≥ N. We assume that some of the terms N j with nl < j ≤ nl 1 are greater thanN and some are less thanN. Our strategy is to show that the upper bound holds in each interval nl, nl 1 . For each l 1, 2, . . ., let ζl be the integer in the interval nl, nl 1 such that N ζl 1 max { N ( j ) : nl < j ≤ nl 1 } . 2.4 Then N ζl 1 ≥ N ζl which implies that ΔN ζl ≥ 0. To show the upper bound on 2.3 , it suffices to show that N ζl ≤ Nλ Y2. 2.5 We assume that N ζl > N, otherwise there is nothing to prove. Now, since ΔN ζl ≥ 0, it follows from 1.10 that 1 ≤ N ζl 1 N ζl λ 1 aN ζl − k γ λmN ζl − k , 2.6 and hence λ 1 aN n − k γ λmN n − k > 1 λ ( 1 aN )γ λmN . 2.7 This implies that N ζl − k < N. Now, since N ζl > N and N ζl − k < N, there exists an integer ζl in the interval ζl − k, ζl , such that N ζl ≤ N and N j > N for j ζl 1, . . . , ζl. From 1.10 , we see that 


Introduction
Most populations live in seasonal environments and, because of this, have annual rhythms of reproduction and death.In addition, measurements are often made annually because interest is centered on population changes from year to year rather than on the obvious and predictable changes that occur seasonally.Continuous differential equations are not well suited to these kinds of processes and data.Thus, practical ecologists have long employed discrete-time difference equations for studying the dynamics of resource and pest populations.In particular one can consider the difference equation N n 1 f N n , n 0, 1, 2, . . ., 1.1 as a measure of the population growth, where N n 1 is the size of the population at time n 1, N n is the size of the population at time n, and the function f N is the densitydependent growth rate from generation to generation and in general it is a nonlinear function of N. The skills in modelling a specific population's growth lie in determining the appropriate form of f N to reflect the known observations or the facts of the species under consideration.Density dependent is a dependence of per capita population growth rate on present and/or past population densities.Hassell 1 proposed that the models of the population dynamics in a limited environment are based on the following two fundamentals: 1 population have the potential to increase exponentially; 2 there is a density-dependent feedback that progressively reduces the actual rate of increase.
In fact, in population dynamics, there is a tendency for that variable N n to increase from one generation to the next when it is small, and decrease when it is large.In 2 Cull showed that for population dynamics models the nonlinear function f N often has the following properties: f 0 0 and there is a unique positive fixed-point N such that f N N, f N > N for 0 < N < N, and f N < N for N < N, and such that if f N has a maximum N M in 0, N then f N decreases monotonically as N increases beyond N N M such that f N > 0, see Figure 1 .
In recent decades the dynamics of discrete models in different areas have been extensively investigated by many authors.For contributions, we refer the reader to 2-17 and the references cited therein.
For population models of plants, Watkinson 18 assumed that the function f N represents the number of seeds produced per parent plant which survived to flowering in the next generation and reproduce seasonality and have effectively nonoverlapping generations, even if a seed bank is present 18 .Using these assumptions Watkinson derived some different forms of the function f N for seven different cases.For the annual plants Watkinson 18 assumed that the density-dependent function is given by where λ is the growth rate and m represents the reciprocal of the asymptotic value of N when the initial plant density tends to infinity and it is called the degree of self-thinning.
The parameter a has the dimension of the area and 1/a can be considered as the density of plants at which mutual interference between individuals becomes appreciable and γ is the density-dependent parameter where the biological significance is rather unclear.In 18 the author proposed that γ > 1, which reflects the fact that an increasing density leads to a less-efficient use of the resources with a given area in terms of total dry matter population.
Combining 1.1 and 1.2 , we see that the Watkinson model of the annual plants is given by the difference equation In this model the density-independent mortality is not included and the growth of the population occurs only during the vegetative phase of the life cycle.
Watkinson in 18 assumed that density-independent mortality during the seed phase of the life cycle can easily be incorporated by multiplying λ by the probability that a seed will survive from the time of its formation to germination and establishment.Also in this model it is clear that the past history of the population is ignored, that is, the growth of the population is governed by a principle of causality, that is, the future state of population is independent of the past and is determined solely by the present.In fact in a single species population there is a time delay because of the time it takes a female animal or a plant to mature before it can begin to reproduce.A more realistic model must include some of the past history of population.Accordingly Kocić and Ladas 19 considered the model where N is the unique fixed point of 1.4 .Note that the assumption γ ≤ 1 is different from the assumption γ > 1 that has been proposed by Watkinson 18 , which reflects the fact that an increasing density leads to a less efficient use of the resources with a given area in terms of total dry matter population.
In 11 the authors considered the general equation with two delays of the form The authors in 11, Theorem 6.
represents the number of mature population that were produced in the n − k th cycle and survived to maturity in the nth cycle.We also establish an explicit sufficient condition for oscillation of all solutions of 1.10 about the fixed point.We note that when m 0, and k 0 1.10 reduces to N n 1 λN n 1 aN n γ , n 0, 1, 2, . . . .

1.12
This equation has been proposed by Hassell 1 to describe the growth of the population of insects.On the other hand, when γ 1 and m 0, 1.10 becomes the Pielou equation 20 By the biological interpretation, we assume that the initial condition of 1.10 is given by By a solution of 1.10 , we mean a sequence N n which is defined for n ≥ −k and satisfies 1.10 for n ≥ 0 and by a positive solution, we mean that the terms of the sequence {N n } ∞ n 1 are all positive.Then, it is easy to see that the initial value problem 1.10 and 1.14 has a unique positive solution N n .In the sequel, we will only consider positive solutions of 1.10 .We say that N is a fixed of 1.10 if 10 .In the following, we prove that 1.10 has a unique positive fixed point.Let g N : 1 aN γ λmN − λ. 1.16 Then g 0 −λ < 0 and g ∞ ∞, so that there exists N > 0 such that g N 0. Also 1.17 It follows that g N 0 has exactly one solution and so 1.10 has a unique positive fixed point which is denoted by N and obtained from the solution of The stability of equilibria is one of the most important issues in the studies of population dynamics.The fixed-point N of 1.10 is locally stable if the solution of the population model N n approaches N as time increases for all N 0 in some neighborhood of N. The fixed point N of 1.10 is globally stable if for all positive initial values the solution of the model approaches N as time increases.A model is locally or globally stable if its positive fixed point is locally or globally stable.The fixed-point N is globally asymptotically stable if its locally and globally stable.A solution {N n } of 1.10 is said to be oscillatory about N if N n −N is oscillatory, where the sequence N n −N is said to be oscillatory if N n −N is not eventually positive or eventually negative.
For the delay equations, for completeness, we present some global stability conditions of the zero solution of the delay difference equation that we will use in the proof of the main global stability results.Gy öri and Pituk 21 , proved that if then the zero solution is globally stable.

Global Stability Results
In this section, we establish some sufficient conditions for local and global stability of the positive fixed-point N. First, we establish a sufficient condition for local stability of 1.10 .The linearized equation associated with 1.10 at N is given by Applying the local stability result of Levin and May 25 on 2.1 , we have the following result.
then the fixed-point N of 1.10 is locally asymptotically stable.
To prove the main global stability results for 1.10 , we need to find some upper and lower bounds for positive solutions of 1.10 which oscillate about N. Theorem 2.2.Assume that 1.14 holds and γ, λ > 1.Let N n be a positive solution of 1.10 which oscillates about N. Then there exists n 1 > 0 such that for all n ≥ n 1 , one has Proof.First we will show the upper bound in 2.3 .The sequence N n is oscillatory about the positive periodic solution N in the sense that there exists a sequence of positive integers We assume that some of the terms N j with n l < j ≤ n l 1 are greater than N and some are less than N. Our strategy is to show that the upper bound holds in each interval n l , n l 1 .For each l 1, 2, . .., let ζ l be the integer in the interval n l , n l 1 such that To show the upper bound on 2.3 , it suffices to show that We assume that N ζ l > N, otherwise there is nothing to prove.Now, since ΔN ζ l ≥ 0, it follows from 1.10 that From 1.10 , we see that Multiplying this inequality from ζ l to ζ l − 1, we have and so which immediately gives 2.5 .Hence, there exists an n 1 > 0 such that N n ≤ Y 2 for all n ≥ n 1 .Now, we show the lower bound in 2.3 for n ≥ n 1 k.For this, let μ l be the integer in the interval n l , n l 1 such that N μ l 1 min N j : n l < j ≤ n l 1 .

2.12
Then N μ l 1 ≤ N μ l which implies that ΔN μ l ≤ 0. We assume that N μ l < N, otherwise there is nothing to prove.Then, it suffices to show that Since ΔN μ l ≤ 0, we have from 1.10 that

2.15
This leads to N μ l − k > N. Now, since N μ l < N and N μ l − k > N, then there exists a μ l ∈ μ l − k, μ l such that N μ l ≥ N and N j < N for j μ l 1, . . ., μ l .From 1.10 and 2.3 , we have

2.16
Multiplying the last inequality from μ l to μ l − 1, we have and this implies that which immediately leads to 2.13 .The proof is complete.

Abstract and Applied Analysis 9
One of the techniques used in the proof of the global stability of the zero solution of the nonlinear equation is the application of what is called a linear method see 15, 16 .To apply this method, we have to prove that the solution is bounded and the solution of 1.10 , say z n , is a solution of the corresponding linear equation.This can be done by using the main value theorem, which implies where ζ n lies between zero and z n − k .Therefore, we obtain Applying the global stability results presented in Section 1, we can obtain some sufficient conditions for global stability provided that the solution is bounded.With this idea and using the fact that the solutions are bounded, we are now ready to state and prove the main global stability results for 1.10 .

2.23
Proof.First, we prove that every positive solution N n which does not oscillate about N satisfies 2.23 .Assume that N n > N for n sufficiently large the proof when N n < N is similar and will be omitted since uh u > for u / 0 see below .Let N n Ne z n .

2.24
To prove that 2.23 holds it suffices to prove that lim n → ∞ z n 0. From 1.10 and 2.24 , we see that z n > 0 and satisfies where

2.26
Note that h 0 0 and h u > 0 for u > 0. It follows from 2.25 that Hence, z n is decreasing and there exists a nonnegative real number α ≥ 0 such that lim n → ∞ z n α.If α > 0, then there exists a positive integer n 2 > n 1 such that α/2 ≤ z n − k ≤ 3α/2 for n > n 2 .This implies from 2.25 that where η min α/2≤u≤3α/2 h u > 0. Thus summing up the last inequality from n 2 to n − 1, we obtain

2.29
This contradicts the fact that z n is positive.Then α 0, that is, lim n → ∞ z n 0. Thus 2. 23

2.33
From the last inequality and the assumption 2.22 , we see that

2.34
Then by the results of Gy öri and Pituk then every positive solution of N n of 1.10 satisfies 2.23 .
Also by employing the result due to Kovácsv ölgy 23 , we have the following result.
then every positive solution of N n of 1.10 satisfies 2.23 .
The result by Yu and Cheng 24 when applied gives us the following result.Corollary 2.7.Assume that 1.14 holds, λ, γ > 1.
then every positive solution of N n of 1.10 satisfies 2.23 .
We illustrate the main results with the following examples.

2.39
Then the condition 2.22 of Theorem 2.3 is satisfied and then the fixed-point N 1 is globally stable.For illustration, we plotted the iterations in the phase plane and the time series x N n , y N n − 1 , x, y N, N , where we found that the solutions oscillate and converge to the fixed-point x, y N, N

Oscillation Results
In this section, we establish an explicit sufficient condition for oscillation of 1.10 about the positive fixed-point N. In 1. 13 Pielou assumed that there is a delay k in the response of the growth rate per individual to density changes.Pielou showed that the tendency to oscillate is a property of the populations themselves and is independent of any extrinsic factors.That is, population size oscillates even though the environment remains constant according to Pielou; oscillation can be set up in a population if its growth rate is governed by a density-dependent mechanism and if there is a delay in the response of the growth rate to density changes.In this section, we consider 1.10 and also prove that under some conditions on the parameters the solutions oscillate about the fixed-point N even though the environment remains constant.Oscillatory behavior of the solutions is very significant which implies the prevalence of the mature plants around the positive fixed point.We, first prove the following theorem which proves that oscillation of 1.10 about the positive fixed point is equivalent to oscillation of a linear difference equation about zero.Note that when k ≥ 1 then the condition 1.8 is already satisfied.Theorem 3.1.Assume that 1.14 holds.Furthermore, assume that there exists ε > 0, such that every solution of oscillates.Then every positive solution of 1.10 oscillates about N.
Proof.Without loss of generality we assume that 1.10 has a solution N n exceeding N and define z n as in 2.24 .From the transformation 2.24 it is clear that N n oscillates about N if and only if z n oscillates about zero and transforms 1.10 to Note that f 0 0, and uf u > 0, for u / 0, lim u → 0 f u u 1.

3.4
From 3.4 it follows that for any given arbitrarily small ε > 0, there exists δ > 0 such that for all 0 < u < δ, we have f u ≥ 1 − ε u similarly, for all −δ < u < 0, we have f u ≤ 1 − ε u .In view of Theorem 2.2, since z n → 0, thus for sufficiently large n we can use this estimate in 3.2 , to conclude that z n is a positive solution of the differential inequality Then, by Lemma 1 in 26 the delay difference equation 3.1 also has an eventually positive solution, which contradicts the assumption that every solution of 3.1 oscillates.Thus every positive solution of 1.10 oscillates about N, which completes the proof.
To establish the condition for oscillation of all positive solution of 1.10 about the positive fixed point, we need the following result which is extracted from 27 .To illustrate the main result of Theorem 3.3, we consider the following example.3.9 Here a 1, m 0.02, k 1, γ 2, and λ 1 1 2 / 1 − 0.02 4.0816.In this case the positive fixed-point N 1 and the condition 3.8 is satisfied.Then by Theorem 3.3, every positive solutions oscillates about the positive fixed-point N 1, see Figure 5 where we plotted the iterations and the time series in a focus type.Remark 3.5. 1 We note that the condition 1.8 that has been proposed in 11, Theorem 6.3.1 is not required in the proof of the oscillation results in Theorem 3.3.
N ζ l − k < N. Now, since N ζ l > N and N ζ l − k < N, there exists an integer ζ l in the interval ζ l − k, ζ l , such that N ζ l ≤ N and N j > N for j ζ l 1, . . ., ζ l .
Erbe et al. 22 improved 1.20 and proved that if lim The results give a partial answer to the open problem posed by Kocić and Ladas 11, Open problem 6.3.1 and improve the results that has been established by Kocić andLadas 19in the sense that the restrictive condition γ ∈ 0, 1 is not required.In Section 3, we will use an approach different from the method used in 11 and establish an explicit sufficient condition for oscillation of the positive solutions of the delay equation 1.10 about N. Some illustrative examples and simulations are presented throughout the paper to demonstrate the validity and applicability of the results.
holds, and any positive solution of 1.10 which does not oscillate about N satisfies 2.23 .To complete the global attractivity results we prove that every oscillatory about N satisfies 2.23 .From the transformation 2.24 it is clear that N n oscillates about N if and only if z n oscillates about zero.So to complete the proof, we have to demonstrate that lim n → ∞ z n 0, where z n is a solution of 2.25 .Equation2.25 is the same as lies between N and N n − k .In Theorem 2.2, we proved that the oscillatory solutions of 1.10 are bounded and the upper bound of solution is given by Y 2 .Using this, since the values of A are increasing in η, we have max 21 , we deduce that the zero solution of 2.31 is globally stable that is, lim n → ∞ z n 0, and hence lim n → ∞ N n N. The proof is complete.Remark 2.4.From Theorem 2.3, it is clear that the global attractivity of N is equivalent to the global attractivity of zero solution of the linear difference equation 2.31 .By employing the results by Gy öri and Pituk 21 , we showed that if 2.22 holds then 2.23 is satisfied.Now, we apply different result which improves the condition 2.22 based on the improvement of the global attractivity condition that has been given Gy öri and Pituk 21 for the difference equation 2.31 .Applying the result by Erbe et al. 22 we have the following result.
Here, we take a 0.0521, m 0.04, γ 1.5, λ 30, and N 22.347.One can easily check that the condition 2.22 of Theorem 2.3 is not satisfied and then the fixed-point N 22.347 is not globally stable.For illustration, we plotted the iterations and the time series where we found that the solutions oscillate and does not converge to the fixed-point N 22.347, see Figure4.Remark 2.12.We note that the results in 19 cannot be applied on 2.44 , since γ 1.5 > 1.Note also that the condition of local stability for 2.44 which reads