Attenuance and Resonance in a Periodically Forced Sigmoid Beverton Holt Model

Transcription

1 International Journal of Difference Equations ISSN , Volume 7, Number 1, ) htt://camus.mst.edu/ijde Attenuance and Resonance in a Periodically Forced Sigmoid Beverton Holt Model Candace M. Kent Virginia Commonwealth University Deartment of Mathematics Richmond, VA 23284, USA cmkent@vcu.edu Vlajko L. Kocic and Yevgeniy Kostrov Xavier University of Louisiana Mathematics Deartment New Orleans, LA 70125, USA vkocic@xula.edu and ykostrov@xula.edu Abstract In this aer we continue to study the global behavior of the eriodically forced Sigmoid Beverton Holt model y n+1 = a nyn 1 + yn, n = 0, 1,..., where the initial condition y 0 is ositive and {a n } is a ositive sequence. First, we establish the conditions for the equivalence of the above equation and the equation of the form r x n x n+1 = Q n + r 1)x, n = 0, 1,..., n where > 0, r > 1, {Q} n is a sequence, and {x n } is a ositive sequence. Then we rove that if 0 < 1, the unique eriodic cycle { x n } which exists) is g-attenuant, that is 1 x i < 1 K i, where {K n } reresents the unique sequence of carrying caacities which will be defined in terms of. Also, we show that in the cases r > > 1 or > r > 1 Received February 17, 2012; Acceted February 19, 2012 Communicated by Gerasimos Ladas

2 36 C. M. Kent, V. L. Kocic, and Y. Kostrov then the eriodic cycle { x n } which exists) is g-attenuant with resect to carrying caacities {K n } and g-resonant with resect to Allee thresholds {T n } which will also be defined in terms of, that is, it satisfies 1 T i < 1 x i < 1 K i. AMS Subject Classifications: 39A23, 39A30. Keywords: Attenuance, resonance, eriodic difference equation, sigmoid Beverton Holt model. 1 Introduction In this aer we continue the study of the global asymtotic behavior of the eriodically forced Sigmoid Beverton Holt model initiated in [18]. More recisely, we will investigate the g-attenuance and g-resonance of eriodic solutions of eriodically forced Sigmoid Beverton Holt model with initial condition y n+1 = a nyn, n = 0, 1, ) 1 + yn y 0 > 0 and where {a n } is a ositive -eriodic sequence such that a n > 0, a n+ = a n, n = 0, 1,... and > ) The autonomous case of equation 1.1) y n+1 = ay n, n = 0, 1,..., 1.3) 1 + yn where a, > 0 has been introduced by Thomson [44] as a deensatory generalization of the Beverton Holt stock-recruitment relationshi used to develo a set of constraints designed to safeguard against overfishing. This model has been used in the study of fish oulation dynamics, articularly when overfishing is resent [16, 35 37, 43]. A very imortant feature of the Sigmoid Beverton Holt model is that, in the case > 1, it exhibits the so-called Allee effect. The Allee effect is originally attributed to W. C. Allee [1, 7, 42], who broadly defined it as a... ositive relationshi between any comonent of individual fitness and either numbers or density of consecifics. Practically seaking, the Allee effect causes, at low oulation densities, er caita birth rate declines. Under such a scenario, at low

3 Periodically Forced Sigmoid Beverton Holt Model 37 oulation densities, the oulation may slide into extinction. There are various scenarios in which the Allee effect aears in nature see [7, 8] and references cited therein) and can be attributed, for examle, to difficulties in finding mates when the oulation size is small, a higher mortality rate in juveniles when there are not enough adults to rotect them from redation, or uncontrollable harvesting, as in overfishing. On the other hand the Allee effect can be beneficial in some situations such as in controlling a oulation of fruit flies which are considered to be one of the worst insect ests in agriculture [8]. The techniques used to control them is the release of sterile males to create an Allee effect. Several discrete mathematical models exhibiting the Allee effect are known and studied in the literature [2, 16, 35 37, 43] and references cited therein). They all have the following features in common: i) the existence of three equilibrium oints: 0, T - Allee threshold, and K - carrying caacity of the environment 0 < T < K); ii) equilibria 0 and K are stable, while T is unstable; iii) if the oulation size dros below T, then the oulation slides into extinction, so it aroaches 0. Periodically forced oulation models exhibiting the Allee effect are relatively new in the literature [15, 18, 34, 45]. In [34] several eriodically forced discrete models exhibiting the Allee effect are studied, while a class of general unimodal mas with such roerties has been investigated in [15]. The effects of harvesting in some eriodically forced oulation models with Allee effect are studied in [45]. In the secial case when = 1, equation 1.1) reduces to the well-known eriodically forced Pielou logistic equation y n+1 = a ny n 1 + y n, n = 0, 1,... {a n } is a ositive -eriodic sequence) which is also equivalent to the eriodically forced Beverton Holt equation x n+1 = r n x n 1 + r n 1) xn K n, n = 0, 1, ) {K n } and {r n } are ositive -eriodic sequences, r n > 1). The dynamics and roerties of both equations have been studied in great detail in recent literature see for examle [3 6, 10 14, 17, 19 28, 32, 38, 41]). The concet of the attenuance and resonance of eriodic cycles was originally introduced by Cushing and Henson when they formulated the well-known conjecture for the eriodically forced Beverton Holt model 1.4) in the case r n = r > 1, n = 0, 1,... see [10]); namely we have Definition 1.1. A -eriodic cycle of a difference equation if such exists) { x n } is attenuant resonant) if its average value is less greater) than the average value of the carrying caacities {K n }, where {K n } is a q-eriodic sequence; that is, av x n ) < avk n ) av x n ) > avk n ))

4 38 C. M. Kent, V. L. Kocic, and Y. Kostrov where av x n ) = 1 1 k=0 x k and avk n ) = 1 q 1 K k. q In [22,23] the geometric mean was used as the average of the eriodic cycle and the carrying caacity instead of the arithmetic mean. That led to the concet of g-attenuance and g-resonance: Definition 1.2. A -eriodic cycle of a difference equation if such exists) { x n } is g- attenuant g-resonant) if its geometric mean is less greater) than the geometric mean of the carrying caacities {K n }, where {K n } is a q-eriodic sequence; that is, 1 / q 1 /q 1 / q 1 /q x i < K i x i > K i. Our motivation for using the geometric mean comes from oulation dynamics, where it is quite common ractice articularly when deriving oulation models by alying correlation and regression techniques [40]) to use a logarithm of the oulation density known as log oulation density ) instead of the oulation density itself. If {x n } denotes a oulation density then the corresonding log oulation density {X n } is defined as X n = lnx n ), n = 0, 1,.... Assume that a given oulation model has a -eriodic cycle { x n }. Then { Xn }, where k=0 X n = ln x n ), n = 0, 1,... is a corresonding -eriodic cycle for the log oulation density. So, we have av X n ) = 1 1 k=0 X k = 1 1 ln x k ) = ln k=0 1 x i, that is, the average of a -eriodic cycle of the log oulation density equals the logarithm of the geometric mean of a -eriodic cycle of the oulation density. Therefore, the g-attenuance or g-resonance can be interreted as the attenuance or resonance associated with the log oulation density, resectively. Next, we summarize some results from [18] that will be useful in the sequel: Lemma 1.3. Assume that {a n } is a ositive eriodic sequence with eriod and > 0. Let the sequence of functions {f n } be defined by f n x) = a nx, x > 0, n = 0, 1, ) 1 + x and let a crit = 1/ 1, rovided > 1. Then the following statements are true:

5 Periodically Forced Sigmoid Beverton Holt Model 39 i) If 0, 1), then function f n has the unique ositive fixed oint k n. ii) If > 1 and a n < a crit, then the function f n does not have ositive fixed oints and for all x > 0, f n x) < x. iii) If a n = a crit, then the function f n has the unique ositive fixed oint k n. iv) If a n > a crit, then the function f n has two ositive fixed oints t n and k n 0 < t n < a n 1)/ < k n ). In the case when > 1 and min{a 1,..., a } > a crit we have a n > a crit, n = 0, 1,..., so for every n = 0, 1,... f n has two ositive fixed oints t n and k n, resectively, such that 0 < t n < a n 1)/ < k n. Each of the sequences {t n } and {k n } is -eriodic. We call {k n } the sequence of carrying caacities, and {t n } the sequence of Allee thresholds. Similarly, in the case 0, 1), each of the functions f n has a unique fixed oint k n and the sequence {k n } reresents the sequence of carrying caacities. Theorem 1.4. Assume that {a n } is a ositive eriodic sequence with eriod and 1, ). Let min{a 1,..., a } > a crit. Then the following statements are true: i) Equation 1.1) has an invariant interval [α, β] where max{t 1,..., t } < α < min{k 1,..., k }, β max{a 1,..., a } and where {k n } and {t n } are -eriodic sequences of carrying caacities and Allee thresholds of 1.1), resectively. ii) In the interval [α, β], there exists a unique -eriodic solution {ȳ n } of equation 1.1) ȳ n [α, β] for n = 0, 1,...) which attracts all ositive solutions of equation 1.1) with initial conditions in [α, ). iii) In the interval [γ, ε] where 0 < γ < min{t 1,..., t }, max{t 1,..., t } < ε < min {a 1,..., a } 1)/ < min{k 1,..., k }. there exists a unique -eriodic solution {ỹ n } ỹ n [γ, ε]) of equation 1.1), which reels all ositive solutions with initial conditions in [γ, ε]. Theorem 1.5. Let {a n } be a ositive eriodic sequence with eriod and let 0, 1). Then equation 1.1) has a unique -eriodic ositive solution {ȳ n } which is a global attractor of all ositive solutions of 1.1).

6 40 C. M. Kent, V. L. Kocic, and Y. Kostrov 2 Preliminaries In this section we establish the condition for the equivalence of the Sigmoid Beverton Holt equation 1.1) and the equation of the form x n+1 = r x n Q n + r 1)x n 2.1) where > 0, r > 1, { } is a -eriodic sequence, and {x n } is a ositive sequence. Lemma 2.1. Consider the difference equation 1.1), where > 0, {a n } is a ositive -eriodic sequence, and the initial condition y 0 > 0. Assume that the equation where A = rr 1, 2.2) A = 1 has a solution r > 1. Let the sequence { } be defined by a i, 2.3) Q > 0 ) for n = 0 n = rr 1 Q n 1 a ) for n = 1, 2,.... i Then the sequence { } is -eriodic and the substitution y n = r 1 2.4) x n, n = 0, 1, ) transforms equation 1.1) into 2.1). In addition, if {y n } is a -eriodic solution of equation 1.1), then the sequence {x n }, defined by 2.5), is also a -eriodic solution of equation 2.1). Proof. First we will show that the sequence { } satisfies the equation: +1 = rr 1 1 a n. 2.6) Clearly, for n = 1, 2,... we have +1 = rr 1 1 ) n+1 Q n a i) = ) n rr 1 1 Q rr 1 1 n 1 a ) = i a n 1 rr 1) a n 1

7 Periodically Forced Sigmoid Beverton Holt Model 41 Next, we need to show that { } is, that is Clearly + =, n = 0, 1,.... rr 1 1 ) n+ ) ) n rr 1) rr = n+ 1 ) Q = a i ) rr 1 1 n+ 1 i=n a i ) n 1 a ) Q i = Since {a n } is, then n+ 1 i=n a i ). n+ 1 i=n and since r > 1 is a solution of the equation 1 a i = a i = A A = rr 1 1 we obtain rr 1 1 ) + = Q A n =. Furthermore, by substituting 2.5) into 1.1) we get r 1 x n+1 +1 = and then by alying equation 2.6) we obtain ) a n +1 r 1 1 x n x n+1 = ) x 1 + r 1) n = a n rr 1 1 a n which comletes the roof. r r 1) a n r 1 xn ) 1 + r 1 xn x n ) x n ) = ) r x n Q n + r 1)x n In the revious lemma the key factor for the transformation of equation 1.1) into 2.1) is that equation 2.2) has a solution r > 1. The next lemma establishes the conditions when such solution exists.

8 42 C. M. Kent, V. L. Kocic, and Y. Kostrov Lemma 2.2. Consider equation 2.2) where A, > 0. Let a crit = 1/ 1. Then the following statements are true: i) If 0 < < 1, then equation 2.2) has a unique solution r > 1. ii) If = 1 and A > 1, then equation 2.2) has a unique solution r = A > 1. iii) If > 1 and A < a crit, then equation 2.2) has no solutions r > 1. iv) If > 1 and A = a crit, then equation 2.2) has a unique solution r = > 1. v) If > 1 and A > a crit, then equation 2.2) has two solutions r 1, r 2 such that [ 1) 1 < r 1 < 1 + A] < r 2. Proof. i) Let u = r 1. Then r = 1 + u so equation 2.2) becomes or A = 1 + u )u 1 2.7) u + u 1 A = 0. Let φu) = u + u 1 A. Then φ0) = A < 0, φ ) = and, since < 1, φ u) = )u > 0. Therefore there exists a unique ū > 0 such that φū) = 0. So r = 1 + ū > 1 and the roof of art i) is comlete. ii) In the case = 1, equation 2.2) becomes A = r, and so it is trivial. iii-v) As in art i) we introduce u = r 1 so equation 2.2) becomes 2.7) which can be written as u Au = ) Consider the function ψu) = u Au Then ψ0) = 1, ψ ) = and ψ u) = u 1 A 1)u 2 = u 2 u A 1)). ) ) Since ψ A 1) A 1) < 0 on 0, and ψ > 0 on, the function ψ has a minimum at u A 1) =. Then ) ) 1 A 1) A 1) ψu ) = A + 1 = 1 A 1) 1.

9 Periodically Forced Sigmoid Beverton Holt Model 43 If A < 1/ 1, then ψu ) = 1 A 1) 1 > ) 1) 1 = 0. So ψu) ψu ) > 0 for u > 0 and equation 2.8) has no solutions. Therefore equation 2.2) also has no solutions. If A = 1/ 1, then ψu ) = 0 so the only solution of equation 2.8) is u = u = Therefore, equation 2.2) has a solution A 1) = 1. r = > 1. Finally, if A > 1 1 we find ψu ) < 0. Thus there are two solutions u 1 and u 2 such that 0 < u 1 < and so equation 2.2) has two solutions which comletes the roof. A 1) < u 2 [ ] A 1) 1 < r 1 < 1 + < r 2 The next lemma is the converse of Lemma 2.1. Lemma 2.3. Consider equation 2.1), where > 0, r > 1, { } is a ositive -eriodic sequence, and x 0 > 0. Then the substitution x n = r 1) 1/ y n, n = 0, 1, ) transforms 2.1) into 1.1), where {a n } is a ositive -eriodic sequence defined by a n = rr 1/ 1, n = 0, 1, ) +1 In addition, if {x n } is a -eriodic solution of equation 2.1), then the sequence {y n }, defined by 2.9), is also a -eriodic solution of equation 1.1).

10 44 C. M. Kent, V. L. Kocic, and Y. Kostrov Proof. The roof is trivial and will be omitted. Next we will focus on the existence of sequences of carrying caacities and Allee thresholds for equation 2.1). So, to find the carrying caacities and Allee thresholds we ut x n+1 = x n = x and solve for x : x = r x Q n + r 1)x. The following technical lemma will be useful in the sequel. Lemma 2.4. Consider the function ξ defined by Then the following statements are true: ξt) = 1 + r 1)t rt 1, t > ) i) If 0 < 1 < r, then t = 1 is the unique ositive zero of the function ξ. ii) If 1 < < r, then the function ξ has two ositive zeros t = 1 and t = t, where t 0, r 1)/r 1))) 0, 1). iii) If r = > 1, then t = 1 is the unique ositive zero of the function ξ. iv) If 1 < r <, then the function ξ has two ositive zeros t = 1 and t = t where t r 1)/r 1)), ) 1, ). v) If 1 < < r, then also if 1 < r <, then t < t > / 1 < 1; r 1 / 1 > 1. r 1 Proof. Clearly ξ1) = 0 so t = 1 is a ositive zero of the function ξ for all r > 1 and > 0. Next note ξ ) = and 1, > 1 ξ0+) = 1 r, = 1., 0 < < 1 The derivative of the function ξ is given by ξ t) = r 1)t 1 r 1)t 2. When 0 < 1, then ξ t) > 0 so t = 1 is the only ositive zero of the function ξ which comletes the roof of art i).

11 Periodically Forced Sigmoid Beverton Holt Model 45 On the other hand if > 1 we have r 1) < 0, 0 < t < r 1) ξ r 1) t) = = 0, t = r 1) r 1) > 0, t > r 1), and ξ 1) = r 1) r 1) = r. The following cases are ossible: Case 1. r > > 1.Since ξ 1) > 0, then there exists unique t 0, r 1)/r 1))) 0, 1) such that ξ t) = 0. Case 2. r = > 1. Then ξ 1) = ξ1) = 0 so t = 1 is the only solution of the equation 2.18). Case 3. > r > 1. Then ξ 1) < 0 so there exists t r 1)/r 1)), ) 1, ) such that ξ t) = 0 which comletes the roof of art iv). v) First observe that ξt) < 0 for t t, 1) and t 1, t), rovided r > > 1 and > r > 1, resectively. To comlete the roof it suffices to show ) / 1 ξ < ) r 1 Clearly where ξ Next, we find ) / 1 = 1 + r 1) r 1 ζ s) = ) 1 r r 1 = rr 1) 1 )/ r 1) 1)/ = rr 1) 1 )/ ζr) ζ)), ζs) = s s 2 s 1/ s 1) 1)/, s > 0. s r ) 1)/ 1 r 1 > 0, if 0 < s < = 0, if s = < 0, if s >. ) 1) 1)/ Since ζ decreases for s >, for r > > 1, we obtain ζr) < ζ) and 2.12) holds. Also, ζ increases for > r > 1, so we have ζr) < ζ) and again 2.12) is satisfied which comletes the roof.

12 46 C. M. Kent, V. L. Kocic, and Y. Kostrov The following lemma establishes the existence of carrying caacities and Allee thresholds of equation 2.1). Lemma 2.5. Consider equation 2.1), where > 0, r > 1, { } is a ositive -eriodic sequence, and x 0 > 0. Then the following statements are true: i) If 0 < 1, then equation 2.1) has a unique sequence of carrying caacities {K n } defined by K n =, n = 0, 1, ) and no Allee thresholds. ii) If r > > 1, then equation 2.1) has a unique sequence of carrying caacities {K n } and a unique sequence of Allee thresholds {T n } defined by K n =, T n = t, n = 0, 1,..., 2.14) where t is the unique solution of the equation in the interval 0, r 1)/r 1))). 1 + r 1)t rt 1 = ) iii) If r = > 1, then equation 2.1) has a unique sequence of carrying caacities {K n } defined by K n =, n = 0, 1,... and no Allee thresholds. iv) If > r > 1, then equation 2.1) has a unique sequence of carrying caacities {K n } and a unique sequence of Allee thresholds {T n } defined by K n = t, T n =, n = 0, 1,..., 2.16) where t is a unique solution of equation 2.15) in the interval r 1)/r 1)), ). Proof. Consider the equation x n+1 = rx n x n 1 + r 1) ) 1 x n ) 2.17) where > 0, r > 1 and { } is ositive sequence and x 0 > 0.

13 Periodically Forced Sigmoid Beverton Holt Model 47 To find carrying caacities and Allee thresholds we ut x n = x n+1 = x into equation 2.17) and solve for x to obtain x = ) 1 x rx ). x 1 + r 1) Clearly x = 0 is one of the solutions, so to find other solutions we have ) ) 1 x x 1 + r 1) r = 0. Let t = x/. Then the above equation becomes ξt) = 1 + r 1)t rt 1 = ) where the function ξ is defined by 2.11). Using Lemma 2.4 we have the following four cases: i) If 1, then t = 1 is the unique solution of equation 2.18) and hence {K n }, defined by K n =, n = 0, 1,..., is the unique sequence of carrying caacities of equation 2.17). ii) If r > > 1, then equation 2.18) has two ositive solutions t = 1 and t = t where 0 < t < r 1) r 1) < 1. Therefore equation 2.17) has carrying caacities {K n} and Allee thresholds {T n }, defined by K n =, T n = t, n = 0, 1,.... iii) If r = > 1. Then ξ 1) = ξ1) = 0 so t = 1 is the only solution of equation 2.18). Therefore equation 2.17) has only carrying caacities which coincide with Allee thresholds ) {K n }, where K n =, n = 0, 1,.... iv) > r > 1. Then ξ 1) < 0 so there exists t > r 1) r 1) > 1 such that ξ t) = 0 so equation 2.17) has carrying caacities {K n } and Allee thresholds are {T n }, defined by K n = t and T n =, n = 0, 1,..., resectively. Clearly, both carrying caacities {K n } and Allee thresholds {T n } are -eriodic sequences. The next theorem establishes sufficient conditions for the existence of -eriodic solutions of equation 2.1). Theorem 2.6. Consider equation 2.1), where > 0, r > 1, { } is a ositive - eriodic sequence and x 0 > 0. Then the following statements are true: i) If 0 < < 1, then there exists a unique -eriodic solution { x n } of equation 2.1) which attracts all ositive solutions.

14 48 C. M. Kent, V. L. Kocic, and Y. Kostrov ii) If > 1 and { Q min, Q 1,..., Q } 1 > Q 1 Q 2 Q 1)1/ 1, 2.19) rr 1) 1/ 1 then the equation 2.1) has two ositive -eriodic solutions { x n } and { x n }, x n < x n, for n = 0, 1,.... The solution { x n } is an attractor while { x n } is a reeller. Proof. i) It follows directly from Theorem 1.5 and Lemmas 2.1 and 2.2i). ii) Condition 2.19)) imlies that {a n } defined by 2.10) satisfies min{a 1,..., a } > a crit. Then from Theorem 1.4 it follows that equation 1.1) has two -eriodic solution {ȳ n } and {ỹ n } such that ỹ n < ȳ n. Finally, the existence of two ositive -eriodic solutions { x n } and { x n }, x n < x n, for n = 0, 1,... follows from Lemmas 2.1 and 2.2v) which comletes the roof. 3 G-attenuance and G-resonance In this section we examine the question of g-attenuance of eriodic cycles of equation 2.1). The following result is true: Theorem 3.1. Consider equation 2.1), where > 0, r > 1, { } is a ositive - eriodic sequence, and x 0 > 0. Let { x n } be a -eriodic solution of equation 2.1). Then the following statements are true: i) If 0 < 1, then the eriodic cycle { x n } is g-attenuant, that is 1 x i < 1 K i, where {K n } reresents the unique sequence of carrying caacities, defined by 2.13). ii) If r > > 1 or > r > 1, then the eriodic cycle { x n } is g-attenuant with resect to carrying caacities and g-resonant with resect to Alee thresholds, that is, it satisfies 1 T i < 1 x i < 1 K i, where {K n } reresents the unique sequence of carrying caacities and the sequence {T n } is the unique sequence of Allee thresholds, defined by 2.14) and 2.16), resectively.

15 Periodically Forced Sigmoid Beverton Holt Model 49 Proof. Consider the equation x n+1 = rx n x n 1 + r 1) ) 1 x n ). 3.1) Let { x n } be a solution of equation 3.1) rovided such solution exists. Then 1 r 1 x ) ) ) 1 1 x i i x i+1 = x 1 + r 1) i or equivalently 1 ) ) 1 ) ) 1 xi ) r 1) = r xi. 3.2) We introduce the function G by 1 Gx 0,..., x 1 ) = ln 1 + r 1) Then condition 3.2) is equivalent to Let xi ) ) 1 ln r 1) xi ln ). 3.3) G x 0,..., x 1 ) = ) F x 0,..., x 1 ) = 1 1 ln x i 3.5) Our goal is to find the extrema of F under the constraint G = 0. We will aly the Lagrange multiliers method. Let F + λg = 1 1 ln x i [ 1 + λ ln 1 + r 1) xi ) ) 1 ) ] xi ln r 1) ln. So, our first ste is to solve the following system with resect to x 0,..., x 1, λ : x i F + λg) = 0, i = 0,..., 1 G = 0.

16 50 C. M. Kent, V. L. Kocic, and Y. Kostrov The above system becomes x i F + λg) = 1 λ 1) x i 1 G = ln 1 + r 1) xi x i r 1) 1 + λ ) = 0, i = 0,..., 1 x 1 + r 1) i 3.6) ) ) 1 ) xi ln r 1) ln = 0 3.7) ) 1 and we observe that the substitution t = x i / reduces system 3.6)-3.7) to λr 1)t 1 + r 1)t = λ 1) 1 3.8) ln 1 + r 1)t ) ln r 1) ln t = ) So instead of solving system 3.6)-3.7) we have to find t and λ which satisfy 3.8)-3.9). From 3.8) we find λr 1)t = 1 + r 1)t ) λ 1) 1 ) and λ = On the other hand equation 3.9) can be written as: 1 + r 1)t 1) t r 1)). 3.10) ξt) = 1 + r 1)t rt 1 = 0, t > ) where the function ξ was defined by 2.11). We consider the following three cases: Case 1: 1 < r. From Lemma 2.4i) it follows t = 1 is the only ositive solution of the equation ξt) = 0. Then from 3.10) we find λ = r/ r))so the system 3.6)-3.7) has the unique solution x 0 = Q 0,..., x 1 = Q 1, λ = r/ r)). 3.12) Then Q 0,..., Q 1 ) is the unique critical oint of the function F + λg. Case 2: 1 < < r. From Lemma 2.4ii) the function ξ has two ositive zeros t = 1 and t = t, where t 0, r 1)/r 1))) 0, 1). Using 3.10) we find corresonding values for λ so the system 3.6)-3.7) has two solutions x 0 = Q 0,..., x 1 = Q 1, λ = x 0 = tq 0,..., x 1 = tq 1, λ = r r), 1 + r 1) t 1) t r 1)).

17 Periodically Forced Sigmoid Beverton Holt Model 51 Then the function F +λg has two critical oints Q 0,..., Q 1 ) and tq 0,..., tq 1 ), such that t 0, r 1)/r 1))) is the ositive zero of the function ξ. Case 3: r > > 1. Again, by alying Lemma 2.4iv) we find the function ξ has two ositive zeros t = 1 and t = t, where t r 1)/r 1)), ) 1, ). Similarly as in case 2, the system 3.6)-3.7) has two solutions r x 0 = Q 0,..., x 1 = Q 1, λ = r), x 0 = tq 0,..., x 1 = tq 1, λ = 1 + r 1) t 1) t r 1) ). Again, in this case the function F + λg has two critical oints Q 0,..., Q 1 ) and tq 0,..., tq 1 ), where t r 1)/r 1)), ) is the ositive zero of the function ξ. Next we will examine d 2 F + λg), at the critical oints Q 0,..., Q 1 ), and also at oints tq 0,..., tq 1 ) if r > > 1 and at tq 0,..., tq 1 ) when > r > 1 Consider 2 F + λg) x 2 i = λ 1) 1 x 2 i 2 F + λg) x i x j = 0, i j. + λr 1) Q 2 i [ 1) r 1) 1 + r 1) Furthermore we find that at the oint Q 0,..., Q 1 ) with λ = 2 F + λg) x 2 i = Q0,...,Q 1 ) So, for 1 < r and for r > > 1 while for > r > 1 r 1) 1 r) + Q 2 i r 1) r) r r 1) r) Q 2 i r 1) rq 2 i = Q 2 i r) + r 1) r 1) = + Q 2 i r) Q 2 i r r 1)2 = Q 2 i r r) 2 F + λg) x 2 i x i ) ] x i ) 2 r r) x i ) ) 2 [ ] 1) r 1) 1 + r 1)) 2 < 0, 3.13) xi = 2 F + λg) x 2 > ) i xi =

18 52 C. M. Kent, V. L. Kocic, and Y. Kostrov Next we will consider the case when x i = t and 1 < < r or x i = t ). In this case we have 1 + r 1) t λ = 1) t r 1)) and Since we get 2 F + λg) x 2 i 2 F + λg) x 2 i = xi = t = xi = t = λ 1) 1 t 2 Q 2 i + λr 1) Q 2 i λr 1) t λ 1) 1 = 1 + r 1) t λ 1) 1 t 2 Q 2 i λ 1) 1 t 2 Q 2 i [ 1) r 1) t ] t r 1) t ) 2. [ ] λ 1) 1 1) r 1) t + t 2 Q 2 i 1 + r 1) t [ ] 1 + r 1) t + 1) r 1) t 1 + r 1) t 1+r 1) t ) 1) 1 1 r 1) t = ) t 2 Q 2 i 1 + r 1) t = [ 1)1 + r 1) t ) 1) + r 1) t ] t 2 Q 2 i 1 + r 1) t ) 1 r 1) t ) 2 r 1) t = t 2 Q 2 i 1 + r 1) t ) 1 r 1) t ). In the case r > > 1, from Lemma 2.4v) we have t < 1)/r 1), so 1) r 1) t > 0 and we obtain 2 F + λg) x 2 i > ) xi = t Similarly we find 2 F + λg) x 2 i = xi = t 2 r 1) t t 2 Q 2 i 1 + r 1) t ) 1 r 1) t ) Then, in the case > r > 1, from Lemma 2.4v) we have t > 1)/r 1) so 1) r 1) t < 0 and we obtain 2 F + λg) x 2 < ) i xi = t

19 Periodically Forced Sigmoid Beverton Holt Model 53 To determine the character of critical oints we will examine d 2 F + λg) and critical oints. Since 2 F + λg)/ x i x j = 0, i, j = 0,..., 1, i j, we have d 2 F + λg) = 1 2 F + λg) dx 2 x 2 i. i Using 3.13) ) we obtain the following: d 2 F + λg) < 0, if 1 < r Q0,...,Q 1 < 0, if r > > 1 ) > 0, if > r > 1, d 2 F + λg) tq 0,..., tq 1 > 0, rovided r > > 1, ) d 2 F + λg) tq 0,..., tq 1 < 0, rovided > r > 1. ) Therefore, in the case 1 < r the function F attains a maximum subject to the constraint G = 0) at the oint Q 0,..., Q 1 ), so F x 0,..., x 1 ) F Q 0,..., Q 1 ). Let x i = x i, i = 0,..., 1, where { x n } is the solution of equation 2.1). Since { } can not be a solution of the equation 2.1), we have x 0,..., x 1 ) Q 0,..., Q 1 ) and the above inequality is strict: Finally, since {K n } = { } we obtain F x 0,..., x 1 ) < F Q 0,..., Q 1 ). 1 x n < 1 which comletes the roof of art i). Next we consider the case r > > 1. Since K n, d 2 F + λg) Q0,...,Q 1 ) < 0, and d2 F + λg) tq 0,..., tq 1 ) > 0, the function F attains the maximum and the minimum subject to the constraint G = 0) at the critical oints Q 0,..., Q 1 ) and tq 0,..., tq 1 ), resectively. Thus F tq 0,..., tq 1 ) F x 0,..., x 1 ) F Q 0,..., Q 1 ). Again, let x i = x i, i = 0,..., 1, where { x n } is the solution of equation 2.1). Since { } and { t } can not be solutions of the equation 2.1), we have x 0,..., x 1 )

20 54 C. M. Kent, V. L. Kocic, and Y. Kostrov Q 0,..., Q 1 ) and x 0,..., x 1 ) tq 0,..., tq 1 ) so the above inequalities are strict: F tq 0,..., tq 1 ) < F x 0,..., x 1 ) < F Q 0,..., Q 1 ). Finally, since {K n } = { } and {T n } = { t }, we obtain 1 T i < 1 x i < 1 K i. The roof in the remaining case > r > 1 is similar to the revious case so it is omitted. This comletes the roof of the theorem. Finally, we will address the question about g-attenuance and g-resonance of eriodic cycles of equation 1.1). First we will introduce the autonomous Sigmoid-Beverton Holt model z n+1 = Az n, n = 0, 1,..., 3.17) 1 + zn where z 0 > 0, {a n } is a ositive -eriodic sequence and A = i=1 a i / Equation 3.17) is associated with eriodically forced Sigmoid Beverton Holt model 1.1) in the way that its coefficient A reresents the geometric mean of one cycle of the sequence {a n }. Let K A and T A denote the carrying caacity and Allee threshold for equation 3.17), resectively. Theorem 3.2. Consider equation 1.1) and the associated autonomous equation 3.17), where > 0, {a n } is a ositive -eriodic sequence, A = a i /, and y 0, z 0 > 0. Let {ȳ n } be a -eriodic solution of equation 1.1). Then the following statements are true: i) If 0 < < 1, then the eriodic cycle {ȳ n } satisfies 1 / ȳ i < K A, where K A is the unique ositive carrying caacity of the associated autonomous equation 3.17). ii) If > 1 and A > a crit, then the eriodic cycle {ȳ n } satisfies T A < 1 ȳ i. < K A, where K A is the unique carrying caacity and T A is the unique Allee threshold of the associated autonomous equation 3.17).

21 Periodically Forced Sigmoid Beverton Holt Model 55 Proof. Since {ȳ n } is a -eriodic solution of 1.1), then, according to Lemmas 2.1 and 2.2, equation 2.1) has a -eriodic solution { x n } defined by x n = r 1) 1/ ȳ n, n = 0, 1,..., 3.18) where r is a ositive solution of 2.2) and { } is given by 2.4). First we consider the case i). Then 0 < < 1, and, by using Lemma 2.5 and Theorem 3.1 we have K n = and 1 / ȳ i = 1 / x i r 1/ 1 = r 1/ 1 < r 1/ K / i 1 Q / = r 1/. i x / i / 1 Let z = r 1/. Then r = 1 + z and since r is a ositive solution of the equation we obtain that z satisfies the equation A = rr 1 A = z 1 + z; 3.19) so z is a ositive equilibrium of the associated autonomous equation 3.17). Therefore and z = r 1/ = K A 1 / ȳ i < K A which comletes the roof of art i). Now, we consider the case ii). Since A > a crit, from Lemma [ 2.2 it follows that 1) equation 2.2) has two solutions r 1, r 2 such that 1 < r 1 < 1 + A] < r 2. So we will select r = r 2. Observe that A > a crit = 1/ 1 is equivalent to [ 1) 1 + A] > and we have r > > 1. Then by Lemma 2.5 and Theorem 3.1 we have K n = and T n = t n = 0, 1,..., where t is a unique solution of 1 + r 1)t rt 1 = ) such that 0 < t < r 1)/r 1)) < 1. Next we will show that z = r 1/ and z = tr 1/ are two ositive equilibria of the associated autonomous equation 3.17) and therefore K A = r 1/ > tr 1/ = T A.

22 56 C. M. Kent, V. L. Kocic, and Y. Kostrov Since we already showed in art i) that z = r 1/ satisfies equation 3.19) it remains to show that z = tr 1/ is also a solution of the same equation. Since t is a solution of equation 3.20) it follows t 1 = 1 r) t + r and we have A tr 1/ tr 1/ = A t 1 r 1) 1 )/ tr 1/ ) t = A r 1/ 1 r 1 + t ) 1 r) t = A r 1/ + r + t r 1 = A rr 1/ 1 = 0. Again, as in the case i) we use the fact that equation 2.1) has a -eriodic solution { x n } defined by 3.18) and by alying Theorem 3.1 we get 1 / 1 ȳ i = r 1/ 1 x / i 1 / 1 ȳ i = r 1/ Finally x / i 1 T A < / < r 1/ 1 / > r 1/ 1 1 / ȳ i < K A which comletes the roof of the theorem K / i 1 Q / = r 1/, i T / i 1 / = tr 1/. 4 Oen Problems In this section we formulate some oen roblems. Clearly we can not determine g-attenuance or g-resonance of a eriodic cycle of equation 2.1) in the case r = > 1, given the tool used in the roof of Theorem 3.1. So we formulate the following oen roblem Oen Problem 4.1. Consider equation 2.1), where r = > 1, { } is a ositive - eriodic sequence, and x 0 > 0. Let { x n } be a -eriodic solution of equation 2.1). Determine the g-attenuance or g-resonance of { x n } with resect to carrying caacities. Theorem 3.2 establishes bounds of the geometric mean of a eriodic cycle of equation 1.1) in terms of the carrying caacity K A and the Allee threshold T A of the associated autonomous equation 3.17). Clearly it does not reresent the result about g-attenuance or g-resonance of a eriodic cycle of equation 1.1). It is not clear how K A and T A are related to geometric means of carrying caacities and Allee thresholds of equation 1.1) so we formulate the following oen roblem.

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