Yingchao Hao and Cuiping Li

(School of Mathematical Sciences, Beihang University, Beijing 100083, China)

Abstract: Fractional linear maps have played a key role in mathematical biology, population dynamics, and other research areas. In this paper, a special kind of Ricatti map is studied in detail in order to determine the asymptotical behaviors of fixed points and periodic solutions. Making use of composition operation of maps and the methods of dynamical systems and qualitative theory, fixed points or periodic orbits are expressed precisely, average value of periodic solution is estimated concretely, and several different bounds are obtained for periodic solutions of the Beverton-Holt map when both intrinsic growth rate and carrying capacity change periodically. In addition, some sufficient conditions are given about the attenuation of periodic solution of the non-autonomous Beverton-Holt equation. Compared with present works in literature, our results about bounds of periodic solutions are more precise, and our proofs about the attenuation of periodic solution are more concise.

Keywords: Beverton-Holt equation; Cushing-Henson conjecture; attenuation; p-cycle

0 Introduction

Fractional linear maps have been studied extensively[1-3]. A special form of Ricatti map, so-called Beverton-Holt map

is deeply concerned. This map originated from an investigation of the reaction of species to periodically changing living environment, whereμ>0 denotes intrinsic growth rate andK>0 expresses carrying capacity of the environment.

The Beverton-Holt equation models density dependent growth which shows compensation as opposed to over-compensation. This equation takes the following form:

(1)

Obviously,x=0 is a fixed point of Eq.(1). It is globally asymptotical stable ifμ∈(0,1).x=Kis another fixed point, and is globally asymptotical stable forμ>1.

It is known that living environment surrounding the species influences the population dramatically. In Ref. [4], authors considered a situation where carrying capacityKfluctuates periodically with a minimal integer periodpas a result of seasonally varying environment. Then Eq. (1) becomes

(2)

whereKn+p=Kp,n=0,1,2,….

Let

then a (nonautonomous) difference system with periodpis obtained.

xn+1=fn(xn),fn+p(x)=fn(x),n=0,1,…

(3)

In Ref.[5], Cushing and Henson made two conjectures forp-periodic Beverton-Holt Eq. (2) whenμ>1 andp≥2:

Conjecture(Conj.)2Thisp-periodic solution is attenuant, that is

The correctness of Conj.2 means that a varying habitat harms the population. That is to say, the average population is less in a periodically oscillating habitat than it is in a constant habitat with the same average[6].

In Ref. [11], Haskell et al. studied the attenuation of Beverton-Holt equation whenμ=μnis alsop-periodic. They gave a condition onμnandKnto make the second conjecture true.

Refs.[12] and [13] proved the existence and globally asymptotically stability of periodic orbit of periodrfor periodic nonautonomous difference equations via the concept of skew-product dynamical systems.

In present work, we investigate the Beverton-Holt equation for the case of changingμnandKnperiodically with same periodp.Some sufficient conditions are obtained to guarantee the attenuation of periodic solution forp-periodic Beverton-Holt equation. Our proof about the attenuation is different from that in Ref. [11].

1 Average of Periodic Cycle

As mentioned above,it is already known that Conj. 1 and Conj. 2 are correct for constantμand periodically changingKn.However, the situation is drastically different ifμis alsop-periodic. That is to say, the intrinsic growth rate of the species also changes periodically with time.

Consider Eq. (2) withp-periodicμn,

(4)

whereμnandKnhave the same minimal periodp≥2 satisfyingμn>1,Kn>0.

Ref.[14] extends Conj.2 to Eq. (4). The following inequality is obtained:

ProofDefine

whereEisatisfies the linear difference equation below:

Ei=KiEi-1+(μi-1)μi-1…μ0Ki-1…K0

E0=μ0-1,i=0,1,…,p-1

So

where coefficientsriare defined as follows:

(5)

IfKi≠Ki+1for at least onei∈{0,1,…,p-1}, there is

(6)

whereF1(x1)=f0°fp-1°fp-2°…°f1(x1). By the symmetry, there is

where

Kp=K0,μp=μ0

and

(7)

Similar to Eq. (6), there is

(8)

By the same way,there is

(9)

Proposition2IfKi≠Ki+1for at least onei∈{0,1,…,p-1}, then for anyp-periodic solution of difference Eq. (4) withμn>1,Kn>0, there is

ProofThe proposition is correct because

The following theorem summarizes the above conclusions.

Theorem1For anyp-periodic solution of difference Eq. (4) withμn>1,Kn>0 andKi≠Ki+1for at least onei∈{0,1,…,p-1}, the following inequalities hold:

2 Results

In this section,an extension of Conj. 2 to Eq. (4) is discussed. The following example illustrates that Conj. 2 may not hold for somep-periodicμnandKn.

Example1Takingp=2,μ0=2,μ1=3,K0=3,K1=4 in Eq. (4), 2-periodic solution {15/4,10/3} andav(Kn)=85/24>av(Kn)=7/2 can be easily obtained.

Two theorems about the attenuation of periodic solution of Eq. (4) for periodically changingμiandKiare proved in this section. Our proofs are more direct and different from those in Ref. [11]. Firstly, a lemma is proved which will be used below.

Lemma1The following inequality is true forai>0,xi>0,i=1,2,…n,

ProofThis lemma can be proved easily by applying Jessen’s inequality to the convex functionh(x)=x-1defined on (0,+∞).

Next the conditions on Eq.(4) is explored to prove Conj. 2 is true.

where allriare given by Eq. (5). With Lemma 1, there is

(10)

Since

the following inequality is obtained:

(11)

In the same way,there is

i=2,…,p-1.

It is easy to check

Therefore

i=2,…,p-1

(12)

By equation

and Lemma 1 again,the following expression is obtained:

μp-1r0K0

(13)

Adding up inequalities from (10) to (13) yields

μp-1)riKi+…+(1+μp-1…μ1+μp-1…μ2+

…+μp-1)r0K0=Kp-1+Kp-2+…+K0+

That is

(14)

where

si=

(15)

Or

(16a)

(16b)

(μi-1…μ0+…+μi-1)+

(16c)

in whichi=1,2,…,p-2.

(1+μp-1…μ1+μp-1…μ2+…+μp-1)r0=

or

s0+s1+…+sp-1=0

(17)

The following lemma shows how the signs ofsichange withμ0,μ1,…,μp-1.

Lemma2Letμn>1,Kn>0 be bothp-periodic, andsibe defined by Eq.(15). Ifp-periodic sequence {μn} satisfiesμ0≤μ1≤…≤μp-1, then the following statements are true.

(S1)s0<0,sp-1>0；

(S2) Ifsi-1≥0, thensi>0 fori=2,…,p-1.

Proof(S1) can be proved easily. By Eq. (15), there is

Obviously,s0<0 andsp-1>0 because ofμ0=μminandμp-1=μmax.

(S2) is proved next. With Eq. (15) again,siis rewritten as follows:

(18)

Supposesi≤0 andsi-1≥0 for somei∈{2,3,…,p-1}(p>3).Then the following two inequalities are obtained:

(19)

where

(20a)

(μp-2-μi-1)+1

(20b)

(20c)

(20d)

The following inequality is obtained by Eq.(19):

(21)

On the other hand, there isC>A>0 andB>D>0 by assumptions onμi.This contradiction means thatsi>0 wheneversi-1≥0.So (S2) of lemma holds.

Now the main result of this paper is given.

ProofBy Lemma 2, it is known thats0<0,sp-1>0 andsi≥0 for some integerN(i=N,N+1,…,p-1).That is

s0<0,s1<0,…,sN≥0

sN+1>0,…,sp-1>0

By Eq. (17), there is

The conclusion is proved by using Theorem 2.

3 Conclusions