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 N n 1 λN n 1 aN n − k γ λmN n − k < λN n , 2.8 so that N n 1 < λN n . 2.9 Multiplying this inequality from ζl to ζl − 1, we have N ζl < N ( ζl ) λ ζl−ζl , 2.10 and so N ζl < Nλ, 2.11 8 Abstract and Applied Analysis which immediately gives 2.5 . Hence, there exists an n1 > 0 such that N n ≤ Y2 for all n ≥ n1. Now, we show the lower bound in 2.3 for n ≥ n1 k. For this, let μl be the integer in the interval nl, nl 1 such that N ( μl 1 ) min { N ( j ) : nl < j ≤ nl 1 } . 2.12 ThenN μl 1 ≤ N μl which implies thatΔN μl ≤ 0.We assume thatN μl < N, otherwise there is nothing to prove. Then, it suffices to show that N ( μl ) ≥ λ N (( 1 aNλk )γ mNλk 1 )k . 2.13 Since ΔN μl ≤ 0, we have from 1.10 that 1 ≥ N ( μl 1 ) N ( μl ) λ ( 1 aN ( μl − k ))γ λmNμl − k ) , 2.14 which implies that λ ( 1 aN ( μl − k ))γ λmNμl − k ) < 1 λ ( 1 aN )γ λmN . 2.15 This leads to N μl − k > N. Now, since N μl < N and N μl − k > N, then there exists a μl ∈ μl −k, μl such thatN μl ≥ N andN j < N for j μl 1, . . . , μl. From 1.10 and 2.3 , we have N n 1 λN n 1 aN n − k γ λmN n − k ≥ λN n ( 1 aNλk ))γ λmNλk . 2.16 Multiplying the last inequality from μl to μl − 1, we have N ( μl ) > N ( μl ) ⎛ ⎜⎝ λ ( 1 aNλk ))γ λmNλk ⎞ ⎟⎠ μl−μl , 2.17 and this implies that N ( μl ) > λN (( 1 aNλk )γ mNλk 1 )k , 2.18 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 Δz n h z n − k 0, 2.19and Applied Analysis 9 One of the techniques used in the proof of the global stability of the zero solution of the nonlinear equation Δz n h z n − k 0, 2.19 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 h z n − k h 0 z n − k h′ ζn , 2.20 where ζn lies between zero and z n − k . Therefore, we obtain Δz n h′ ζn z n − k 0. 2.21 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 . Theorem 2.3. Assume that 1.14 holds, λ, γ > 1 and N n is a positive solution of 1.10 . If G ( a, γ, λ,m, k,N ) γaNλ ( 1 aNλ )γ−1 mNλ 1 ( 1 aNλk )γ mNλk 1 < 1 k , 2.22 then lim n→∞ N n N. 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