# A Global Attractor in Some Discrete Contest Competition Models with Delay under the Effect of Periodic Stocking

Authors: Ziyad AlSharawi
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider discrete models of the formxn+1=xnf(xn−1)+hn, where hn is a nonnegative p-periodic sequence representing stocking in the population, and investigate their dynamics. Under certain conditions on the recruitment function f(x), we give a compact invariant region and use Brouwer fixed point theorem to prove the existence of a p-periodic solution. Also, we prove the global attractivity of the p-periodic solution when p=2. In particular, this study gives theoretical results attesting to the belief that stocking (whether it is constant or periodic) preserves the global attractivity of the periodic solution in contest competition models with short delay. Finally, as an illustrative example, we discuss Pielou's model with periodic stocking.


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## Body

## 1. Introduction

In mathematical ecology, difference equations of the formxn+1=xnf(xn), n∈ℕ:={0,1,…} are used to model single species with nonoverlapping generations [1, 2], where xn denotes the number of sexually mature individuals at discrete time n, and f(xn) is the density-dependent net growth rate of the population. The form of the function f(x) is chosen to reflect certain characteristics of the studied population such as intraspecific competition. For some background readings about models obtained by the various choices of f(x), we refer the reader to [1, 3, 4] in the discrete case. Also, we refer the reader to [5] and the references therein for the continuous case. Two classical types are known as the scrambled and contest competition models [4]. Our attention in this work is limited to the contest competition models where f(x) is assumed to be decreasing, xf(x) is increasing, and xf(x) is asymptotic to a certain level at high population densities. A prototype of such models is the Beverton-Holt model [6], which is obtained by considering f(x)=μKx/(K+(μ-1)x). Here, μ>1 is interpreted as the growth rate per generation, and K is the carrying capacity of the environment. In populations with substantial time needed to reach sexual maturity, certain delay effect must be included in the function f(x), which motivates us to consider difference equations of the form
(1)xn+1=xnf(xn-k),
where k is a fixed positive integer [7]. In general, it is widely known that long time delay has a destabilizing effect on the population's steady state, while short time delay can preserve stability [8–10]. However, when the delay is large, the dynamics of (1) is less tractable [11]. Furthermore, we are more interested here in the effect of stocking than the effect of delay, and therefore, we keep the time delay short to preserve stability in the absence of stocking. In particular, we fix the delay to be k=1.A substantial body of research has explored the effect of constant stocking on population models without delay [12–19]. In brief and general terms, it has been found that constant stocking can be used to suppress chaos, reverse the period doubling phenomena, lower the risk of extinction, and have a stabilizing effect on the population steady state. On the other hand, and to the best of our knowledge, little (if any) has been done to explore the effect of stocking (whether constant or periodic) on models with delay. So, our work here has a twofold objective which motivates us to consider difference equations of the form to study the effect of periodic stocking on contest competition models with delay and to complement the work of the author and his collaborators in [20], where the dynamics of (1) with k=1 was studied under the effect of constant yield harvesting. Recall that we have some accumulating restrictions on the function f(x) due to the nature of associating our equation with contest competition models. So, in an abstract mathematical form, our problem can be posed as follows. Consider the difference equation
(2)xn+1=xnf(xn-1)+hn,
where {hn} is a nonnegative p-periodic sequence representing stocking due to refuge, immigration, feeding, and so forth, and the function f(x) obeys the following conditions:(C1)
f(0)=b>1,(C2)
f∈C1([0,∞)) and f(x) is decreasing on [0,∞),(C3)
xf(x) is increasing and bounded. The condition in (C1) is a generic one in the absence of stocking, that is, if b≤1 and hn=0, then there is no long-term survival regardless of the initial density of the population.This paper is organized as follows. In Section2, we give some preliminary results concerning local stability, boundedness, and global stability of (2) when the stocking sequence is 1-periodic, that is, when hn=h>0 for all n∈ℕ. In Section 3, the period of the stocking sequence is taken to be larger than one. A compact invariant region has been established and a characterization of the periodic solutions is given. Also, the global asymptotic behavior of solutions has been investigated when p=2. As a particular case of (2), we discuss Pielou's equation with delay one in Section 4.


## 2. Preliminary Results: The Autonomous Case

In this section, we focus on the autonomous case, that is,hj=h>0 for all j=0,1,…,p-1. Thus, (2) becomes
(3)xn+1=xnf(xn-1)+h.Some results concerning (3) can be found in the literature [21]; however, for the sake of completeness and usage in the nonautonomous case, we give the following preliminary results.


### 2.1. Local Stability and Boundedness

Equation (3) has two equilibrium solutions at h=0, namely, 0 and f-1(1). For h>0, the origin slides downward to become negative, while the other equilibrium stays positive and slides upward. This fact becomes clear when we write t=tf(t)+h as 1-h/t=f(t). The left hand side is increasing in t while the right hand side is decreasing. Thus, we have only one positive equilibrium in the positive quadrant, which we denote in the sequel by x-2,h. Since x-2,h is positive and increasing in h, f(x-2,h)<1 for all h>0. The linearized equation associated with (3) at a fixed point x- is given by
(4)yn+1-f(x-)yn-x-f′(x-)yn-1=0.Definep:=f(x-) and q:=-x-f′(x-). For x-=x-2,h, we have 0<p<1 and q is nonnegative. The roots of λ2-pλ+q=0 determine the local stability of our equilibrium point. Since λj,h=(1/2)(p+(-1)jp2-4q),j=1,2,x-2,h starts as stable at h=0 and stays stable as long as q<1. Figure 1 clarifies the relationship between p,q and the magnitude of λj,h. We summarize these facts in the following proposition.Figure 1
This figure shows the magnitude of the characteristic roots of (4) depending on the location of the (p,q) values, where p=f(x-) and q=-x-f′(x-). Also, |λ1,h|>1 and |λ1,h|>1 when p,q>1;|λ1,h|<1 and |λ1,h|<1 when p<1 and q<1.Proposition 1.
Assume that conditions (C1) to (C3) are satisfied andh>0. The positive equilibrium x-2,h of (3) is locally asymptotically stable.Proof.
Sincex-2,h>x-2,0, we have p<1. Also, since F(t)=tf(t) is increasing, we obtain F′(t)=tf′(t)+f(t)>0 and consequently -q+p>0. Thus, we have q<p<1. Now, Figure 1 makes the rest of the proof clear.It is obvious thatxk≥h for all k≥1. On the other hand, since
(5)xn+1=xn-1f(xn-1)f(xn-2)+hf(xn-1)+h≤xn-1f(xn-1)b+hb+h,
the boundedness of y=tf(t) assures the boundedness of all solutions of (2).


### 2.2. Oscillations and Global Stability

A solution of (3) is called oscillatory if it is neither, eventually, less than nor larger than x-2,h [11]. Also, one can consider oscillations about a curve [20]. A solution {xn} of (3) is called oscillatory about a curve H(x,y)=0 if the sequence {un=(xn-1,xn)} does not eventually stay on one side of the curve. The latter definition can be more convenient in some cases; however, in (3), both are equivalent when we consider H(x,y)=y-x as we show in the following result.Proposition 2.
A solution of (3) is oscillatory if and only if it is oscillatory about the curve y=x.Proof.
Assume that{xn} oscillates about x-2,h, but it is not oscillatory about y=x. So, {xn} is either eventually increasing or eventually decreasing, which contradicts the assumption that xn is oscillatory about x-2,h. Conversely, suppose {(xn-1,xn)} oscillates about y=x, but {xn} does not oscillate about x-2,h. First, we consider the case xn≤x-2,h for all n≥n0. If xm>xm-1 for some m>n0, then f(xm)<f(xm-1) and consequently
(6)xm+1=xmf(xm-1)+h>xmf(xm)+h>xm.
So, we can induce an eventually increasing sequence which contradicts our assumption. Ifxm≤xm-1 for some m>n0, then xm+1≤xmf(xm)+h. Thus, either xm+1≤xm, and the induction leads to a decreasing sequence that must converge which is not possible, or xm+1>xm, and we go back to the first scenario. Finally, the case xn≥x-2,h for all n≥n0 can be handled similarly.Next, we define the map(7)T(x,y)=(y,yf(x)+h).The mapT portrays the solutions of (3) geometrically in the nonnegative quadrant, and therefore, it plays a prominent role in the sequel. Here, we used the nonnegative quadrant to denote the positive quadrant union of the axes on the boundary. By applying the map T on the regions above and below the curve y=x, one can observe that a nonequilibrium solution of (3) must be oscillatory. Also, using the map T, one can observe that stocking increases the frequency of oscillations in the following sense. The length of semicycles in the absence of stocking is longer than the length of semicycles in the existence of stocking, where a semicycle is used to denote the string of consecutive terms above or below the equilibrium.Since solutions of (3) are bounded, we define
(8)S:=limsupxn,I:=liminfxn.From the equationxn+2=xnf(xn)f(xn-1)+hf(xn)+h and using the fact that tf(t) is increasing, we obtain
(9)S≤Sf(S)f(I)+hf(I)+h,I≥If(I)f(S)+hf(S)+h.Whenh>0, we have S≥I>0. So, we can multiply the first inequality by I and the second one by S to obtain
(10)S(f(S)+1)≤I(f(I)+1).Sincet(f(t)+1) is increasing, we obtain I=S. This approach was used by Camouzis and Ladas in [22], and it was used by Nyerges in [21] to prove that x-2,h is globally attractive. This fact together with the local stability established in Proposition 1 shows the global asymptotic stability of x-2,h as we summarize in the following proposition.Proposition 3.
The equilibrium solutionx-2,h of (3) is globally asymptotically stable.Next, it is obvious that the positive quadrant forms an invariant for (3); however, since solutions are bounded, we are interested in a bounded invariant that can be developed to serve us in the periodic case. Notice that by invariance here we always mean forward invariance, that is, Rh is an invariant of (3) if T(x,y)∈Rh for all (x,y)∈Rh. To establish the existence of a bounded invariant region, we need to have in mind the following simple fact.Proposition 4.
There exists a finite constantch≥h such that Gh(t)=(bt+h)f(t)≤bch for all t≥0. Furthermore, ch can be taken as ch:=(1/b)suptGh(t).Proof.
Use the fact thattf(t) is bounded and f(t) is decreasing with f(0)=b and limt→∞f(t)=0 to obtain the result.Next, define the curvesΓj,j=0,1,2,3,4 to be the line segments that connect the points (0,0),(0,h),(ch,bch+h), (bch+h,bch+h), (bch+h,0), and (0,0), respectively. Now, define Rh to be the region bounded by the curves of Γj,j=0,…,4 including the boundary, then the following result gives a bounded invariant of (3). Here, it is worth mentioning that Γ0 shrinks to a point at h=0; however, our notation and arguments about the invariant region are still valid except that the boundary of Rh becomes a quadrilateral rather than a pentagon.Theorem 5.
The regionRh as defined above gives a compact invariant for (3).Proof.
Consider the mapT(x,y) as defined in (7). T is one-to-one on the positive quadrant. Thus, all we need is to test T on the boundary of Rh. It is straightforward computations to find that T(Γ0)⊆Γ1. Since horizontal line segments are mapped to vertical line segments under T, we test the end points of Γ2 to find
(11)T(ch,bch+h)=(bch+h,(bch+h)f(ch)+h),T(bch+h,bch+h)=(bch+h,(bch+h)f(bch+h)+h).
By the choice ofch given in Proposition 4, we have
(12)(bch+h)f(bch+h)+h≤(bch+h)f(ch)+h≤bch+h.
Thus,T(Γ2)⊂Γ3. Next, T(Γ3)⊂Rh and T(Γ4)=(0,h) are straightforward to observe. Finally, we show that T(Γ1)⊂Rh. For 0≤t≤ch, we have
(13)T(t,bt+h)=(bt+h,(bt+h)f(t)+h);
however, h≤bt+h≤bch+h and (bt+h)f(t)+h≤bch+h by the choice of ch, which completes the proof. Figure 2 illustrates the region Rh and its image under the map T when (bt+h)f(t) is increasing.Figure 2
The figure on the left shows the choice of the compact regionRh when y=(bt+h)f(t) is increasing, and the one on the right shows T(Rh) with blue boundary inside Rh.


## 2.1. Local Stability and Boundedness

Equation (3) has two equilibrium solutions at h=0, namely, 0 and f-1(1). For h>0, the origin slides downward to become negative, while the other equilibrium stays positive and slides upward. This fact becomes clear when we write t=tf(t)+h as 1-h/t=f(t). The left hand side is increasing in t while the right hand side is decreasing. Thus, we have only one positive equilibrium in the positive quadrant, which we denote in the sequel by x-2,h. Since x-2,h is positive and increasing in h, f(x-2,h)<1 for all h>0. The linearized equation associated with (3) at a fixed point x- is given by
(4)yn+1-f(x-)yn-x-f′(x-)yn-1=0.Definep:=f(x-) and q:=-x-f′(x-). For x-=x-2,h, we have 0<p<1 and q is nonnegative. The roots of λ2-pλ+q=0 determine the local stability of our equilibrium point. Since λj,h=(1/2)(p+(-1)jp2-4q),j=1,2,x-2,h starts as stable at h=0 and stays stable as long as q<1. Figure 1 clarifies the relationship between p,q and the magnitude of λj,h. We summarize these facts in the following proposition.Figure 1
This figure shows the magnitude of the characteristic roots of (4) depending on the location of the (p,q) values, where p=f(x-) and q=-x-f′(x-). Also, |λ1,h|>1 and |λ1,h|>1 when p,q>1;|λ1,h|<1 and |λ1,h|<1 when p<1 and q<1.Proposition 1.
Assume that conditions (C1) to (C3) are satisfied andh>0. The positive equilibrium x-2,h of (3) is locally asymptotically stable.Proof.
Sincex-2,h>x-2,0, we have p<1. Also, since F(t)=tf(t) is increasing, we obtain F′(t)=tf′(t)+f(t)>0 and consequently -q+p>0. Thus, we have q<p<1. Now, Figure 1 makes the rest of the proof clear.It is obvious thatxk≥h for all k≥1. On the other hand, since
(5)xn+1=xn-1f(xn-1)f(xn-2)+hf(xn-1)+h≤xn-1f(xn-1)b+hb+h,
the boundedness of y=tf(t) assures the boundedness of all solutions of (2).


## 2.2. Oscillations and Global Stability

A solution of (3) is called oscillatory if it is neither, eventually, less than nor larger than x-2,h [11]. Also, one can consider oscillations about a curve [20]. A solution {xn} of (3) is called oscillatory about a curve H(x,y)=0 if the sequence {un=(xn-1,xn)} does not eventually stay on one side of the curve. The latter definition can be more convenient in some cases; however, in (3), both are equivalent when we consider H(x,y)=y-x as we show in the following result.Proposition 2.
A solution of (3) is oscillatory if and only if it is oscillatory about the curve y=x.Proof.
Assume that{xn} oscillates about x-2,h, but it is not oscillatory about y=x. So, {xn} is either eventually increasing or eventually decreasing, which contradicts the assumption that xn is oscillatory about x-2,h. Conversely, suppose {(xn-1,xn)} oscillates about y=x, but {xn} does not oscillate about x-2,h. First, we consider the case xn≤x-2,h for all n≥n0. If xm>xm-1 for some m>n0, then f(xm)<f(xm-1) and consequently
(6)xm+1=xmf(xm-1)+h>xmf(xm)+h>xm.
So, we can induce an eventually increasing sequence which contradicts our assumption. Ifxm≤xm-1 for some m>n0, then xm+1≤xmf(xm)+h. Thus, either xm+1≤xm, and the induction leads to a decreasing sequence that must converge which is not possible, or xm+1>xm, and we go back to the first scenario. Finally, the case xn≥x-2,h for all n≥n0 can be handled similarly.Next, we define the map(7)T(x,y)=(y,yf(x)+h).The mapT portrays the solutions of (3) geometrically in the nonnegative quadrant, and therefore, it plays a prominent role in the sequel. Here, we used the nonnegative quadrant to denote the positive quadrant union of the axes on the boundary. By applying the map T on the regions above and below the curve y=x, one can observe that a nonequilibrium solution of (3) must be oscillatory. Also, using the map T, one can observe that stocking increases the frequency of oscillations in the following sense. The length of semicycles in the absence of stocking is longer than the length of semicycles in the existence of stocking, where a semicycle is used to denote the string of consecutive terms above or below the equilibrium.Since solutions of (3) are bounded, we define
(8)S:=limsupxn,I:=liminfxn.From the equationxn+2=xnf(xn)f(xn-1)+hf(xn)+h and using the fact that tf(t) is increasing, we obtain
(9)S≤Sf(S)f(I)+hf(I)+h,I≥If(I)f(S)+hf(S)+h.Whenh>0, we have S≥I>0. So, we can multiply the first inequality by I and the second one by S to obtain
(10)S(f(S)+1)≤I(f(I)+1).Sincet(f(t)+1) is increasing, we obtain I=S. This approach was used by Camouzis and Ladas in [22], and it was used by Nyerges in [21] to prove that x-2,h is globally attractive. This fact together with the local stability established in Proposition 1 shows the global asymptotic stability of x-2,h as we summarize in the following proposition.Proposition 3.
The equilibrium solutionx-2,h of (3) is globally asymptotically stable.Next, it is obvious that the positive quadrant forms an invariant for (3); however, since solutions are bounded, we are interested in a bounded invariant that can be developed to serve us in the periodic case. Notice that by invariance here we always mean forward invariance, that is, Rh is an invariant of (3) if T(x,y)∈Rh for all (x,y)∈Rh. To establish the existence of a bounded invariant region, we need to have in mind the following simple fact.Proposition 4.
There exists a finite constantch≥h such that Gh(t)=(bt+h)f(t)≤bch for all t≥0. Furthermore, ch can be taken as ch:=(1/b)suptGh(t).Proof.
Use the fact thattf(t) is bounded and f(t) is decreasing with f(0)=b and limt→∞f(t)=0 to obtain the result.Next, define the curvesΓj,j=0,1,2,3,4 to be the line segments that connect the points (0,0),(0,h),(ch,bch+h), (bch+h,bch+h), (bch+h,0), and (0,0), respectively. Now, define Rh to be the region bounded by the curves of Γj,j=0,…,4 including the boundary, then the following result gives a bounded invariant of (3). Here, it is worth mentioning that Γ0 shrinks to a point at h=0; however, our notation and arguments about the invariant region are still valid except that the boundary of Rh becomes a quadrilateral rather than a pentagon.Theorem 5.
The regionRh as defined above gives a compact invariant for (3).Proof.
Consider the mapT(x,y) as defined in (7). T is one-to-one on the positive quadrant. Thus, all we need is to test T on the boundary of Rh. It is straightforward computations to find that T(Γ0)⊆Γ1. Since horizontal line segments are mapped to vertical line segments under T, we test the end points of Γ2 to find
(11)T(ch,bch+h)=(bch+h,(bch+h)f(ch)+h),T(bch+h,bch+h)=(bch+h,(bch+h)f(bch+h)+h).
By the choice ofch given in Proposition 4, we have
(12)(bch+h)f(bch+h)+h≤(bch+h)f(ch)+h≤bch+h.
Thus,T(Γ2)⊂Γ3. Next, T(Γ3)⊂Rh and T(Γ4)=(0,h) are straightforward to observe. Finally, we show that T(Γ1)⊂Rh. For 0≤t≤ch, we have
(13)T(t,bt+h)=(bt+h,(bt+h)f(t)+h);
however, h≤bt+h≤bch+h and (bt+h)f(t)+h≤bch+h by the choice of ch, which completes the proof. Figure 2 illustrates the region Rh and its image under the map T when (bt+h)f(t) is increasing.Figure 2
The figure on the left shows the choice of the compact regionRh when y=(bt+h)f(t) is increasing, and the one on the right shows T(Rh) with blue boundary inside Rh.


## 3. Periodic Stocking

In this section, we force periodic stocking on (3) to obtain
(14)xn+1=xnf(xn-1)+hn,
where hn is a p-periodic sequence of stocking quotas, and p denotes the minimal period. Observe that some consecutive values of the stocking sequence can be zero; however, it is natural to assume that  ∑j=0p-1‍hj>0. As in the constant case, we associate (14) with a p-periodic sequence of two dimensional maps that we use in the sequel, namely {Tj,j=0,1,…,p-1}, where Tj(x,y)=(y,yf(x)+hj). It is obvious that if we replace h by hj in Theorem 5, then Rhj forms a compact and invariant region for the individual map Tj, which enables us to build a suitable machinery for establishing the existence of a periodic solution. It is convenient now to develop the notations of the previous section so it can suit the periodic case. We denote the line segments that form the boundary of Rhj by Γj,i,i=0,…,4, where Γj,i corresponds to Γi in the autonomous case and that are associated with the individual map Tj. Also, the constant ch in Proposition 4 will be replaced by chj, and this is associated with the individual map Tj.


### 3.1. Existence of a Periodic Solution

We start by establishing a compact invariant region for (14). Define
(15)hm:=max{h0,h1,…,hp-1},cm:=max{chj:j=0,…,p-1},
where chj is as taken in Proposition 4; that is, chj=(1/b)suptGhj(t); then use hm and cm to define the region Rhm as in the paragraph preceding Theorem 5. Now, we have the following result.Lemma 6.
Consider (14) together with the associated p-periodic sequence of maps {Tj}. Each of the following holds true.(i)
One hasRhi⊆Rhj whenever hi≤hj.(ii)
Rhm is a compact invariant for each individual map Tj.(iii)
Rhm is a compact invariant for the map T^:=Tp-1∘Tp-2∘⋯∘T0.Proof.
(i) Whenhi≤hj, we obtain Ghi(t)≤Ghj(t) for all t≥0. Thus, chi≤chj, and the result becomes obvious from Proposition 4 and the geometric structure of the regions Rhi and Rhj. To prove (ii), let (x,y)∈Rhm, we show that Tj(x,y)∈Rhm. Since
(16)Tj(x,y)=(y,yf(x)+hj)=(y,yf(x)+hm)-(0,hm-hj)=Tm(x,y)-(0,hm-hj),
then the first component of Tj(x,y) is the same as the first component of Tm(x,y) and the second component of Tj(x,y) is lower than the second component of Tm(x,y). Now, the fact that Tm(x,y)∈Rhm and the geometric structure of Rhm assures that Tj(x,y)∈Rhm. Finally, (iii) follows from (ii).Periodic stocking (or harvesting) has the effect of forcing population cycles to evolve and become multiples of the stocking/harvesting period as we show in the following result, which is more general than (14).Theorem 7.
Consider the general difference equationxn+1=F(xn,xn-1,…,xn-k) with p-periodic stocking (or harvesting). If a periodic solution exists, then the period is a multiple of p.Proof.
The proof is a contradiction; suppose that we have anr-periodic solution of the equation xn+1=F(xn,xn-1,…,xn-k)+hn for some r that is not a multiple of p. Then, the greatest common divisor between r and p (d:=gcd(r,p)) is not p. Define the maps Fi:=F+hi,i=0,1,…,p-1; then for each 0≤i≤d-1, the maps {Fkd+i,k=0,1,…,p/d-1} must agree at the point Xi:=(xi,xi-1,…,xi-k), where the components xi-k,xi+1-k,…,xi are consecutive elements of the p-periodic solution. This implies
(17)hi=hd+i=h2d+i=⋯=h(p/d-1)d+i
for all i=0,1,…,d-1, which contradicts the minimality of the period of the p-periodic difference equation.Theorem7 shows that (14) has no equilibrium solutions, and therefore, our previous notion of characterizing oscillatory solutions based on the oscillations about y=x is the valid one here. Thus, solutions of (14) are oscillatory about y=x because they cannot be monotonic. Although it is natural for fluctuations in the environment to create fluctuations in the population, we find it appropriate here to connect the loosely-defined term “fluctuation" with the mathematically well-defined term “oscillation.” Next, we use the Brouwer fixed theorem [23] (page 51) to prove the existence of a periodic solution of (14).Lemma 8 (Brouwer fixed-point theorem [23], page 51).
LetM be a nonempty, convex, and compact subset of ℝn. If T:M→M is continuous, then T has a fixed point in M.Theorem 9.
Thep-periodic difference equation in (14) has a p-periodic solution.Proof.
Consider the mapT^:=Tp-1∘Tp-2∘⋯∘T0, then using Lemma 6, we obtain T^:Rhm→Rhm. Furthermore, Rhm is nonempty, compact, and obviously convex. So, by Lemma 8, T^ has a fixed point in Rhm. This fixed point establishes a periodic solution of (14) with minimal period that divides p; however, Theorem 7 shows that the period must be p.


### 3.2. Global Attractivity of the Periodic Solution Whenp=2

Consider the periodicity of (14) to be p=2 and suppose h0+h1≠0. We partition the solutions of (14) into two subsequences, the one with even indices {x2n} and the one with odd indices {x2n}. Thus, we have
(18)x2n+1=x2nf(x2n-1)+h0,x2n+2=x2n+1f(x2n)+h1.Since the solutions are bounded, we define(19)liminf{x2n+i}=Ii,limsup{x2n+i}=Si,i=0,1.Now, the second iterate of (18) gives us
(20)x2n+2=x2nf(x2n)f(x2n-1)+h0f(x2n)+h1,(21)x2n+3=x2n+1f(x2n+1)f(x2n)+h1f(x2n+1)+h0.Use the fact thatf(t) is decreasing and tf(t) is increasing in (20) to obtain
(22)S0≤S0f(S0)f(I1)+h0f(I0)+h1,(23)I0≥I0f(I0)f(S1)+h0f(S0)+h1.Also, (21) gives us
(24)S1≤S1f(S1)f(I0)+h1f(I1)+h0,(25)I1≥I1f(I1)f(S0)+h1f(S1)+h0.Multiply inequality (22) by I0 and inequality (23) by S0 to obtain
(26)S0I0f(I0)f(S1)+S0(h0f(S0)+h1)≤I0S0f(S0)f(I1)+I0(h0f(I0)+h1).SinceI0(h0f(I0)+h1)≤S0(h0f(S0)+h1), we obtain
(27)f(I0)f(S1)≤f(S0)f(I1).Also, multiply inequality (24) by I1 and inequality (25) by S1 to obtain
(28)S1I1f(I1)f(S0)+S1(h1f(S1)+h0)≤I1S1f(S1)f(I0)+I1(h1f(I1)+h0).SinceI1(h1f(I1)+h0)≤S1(h1f(S1)+h0), we obtain
(29)f(I1)f(S0)≤f(I0)f(S1).Using inequalities (27) and (29), we obtain the following result.Lemma 10.
ConsiderI0,I1,S0,S1 as defined in (19); then f(I0)f(S1)=f(I1)f(S0).Next, we give the following result.Theorem 11.
Forp=2, the 2-periodic solution of (14) is a global attractor.Proof.
Use the result of Lemma10 in inequality (26) to obtain
(30)S0(h0f(S0)+h1)≤I0(h0f(I0)+h1).
Sinceg(t)=t(h0f(t)+h1) is increasing and I0≤S0, we must have I0=S0. Similarly, use the result of Lemma 10 in inequality (28) to obtain
(31)S1(h1f(S1)+h0)≤I1(h1f(I1)+h0),
and consequently S1=I1. Hence, Ii=Si,i=0,1, and the proof is complete.Remark 12.
Observe that the approach of this section proves not only the global attractivity of thep-periodic solution but also its existence; however, Theorem 7 is still significant here because it proves the minimality of the period. Also, establishing the compact invariant region in Lemma 6 deserves embracing regardless of the global attractivity of the periodic solution. Finally, proving the global attractivity for general p will be the topic of some future work.


## 3.1. Existence of a Periodic Solution

We start by establishing a compact invariant region for (14). Define
(15)hm:=max{h0,h1,…,hp-1},cm:=max{chj:j=0,…,p-1},
where chj is as taken in Proposition 4; that is, chj=(1/b)suptGhj(t); then use hm and cm to define the region Rhm as in the paragraph preceding Theorem 5. Now, we have the following result.Lemma 6.
Consider (14) together with the associated p-periodic sequence of maps {Tj}. Each of the following holds true.(i)
One hasRhi⊆Rhj whenever hi≤hj.(ii)
Rhm is a compact invariant for each individual map Tj.(iii)
Rhm is a compact invariant for the map T^:=Tp-1∘Tp-2∘⋯∘T0.Proof.
(i) Whenhi≤hj, we obtain Ghi(t)≤Ghj(t) for all t≥0. Thus, chi≤chj, and the result becomes obvious from Proposition 4 and the geometric structure of the regions Rhi and Rhj. To prove (ii), let (x,y)∈Rhm, we show that Tj(x,y)∈Rhm. Since
(16)Tj(x,y)=(y,yf(x)+hj)=(y,yf(x)+hm)-(0,hm-hj)=Tm(x,y)-(0,hm-hj),
then the first component of Tj(x,y) is the same as the first component of Tm(x,y) and the second component of Tj(x,y) is lower than the second component of Tm(x,y). Now, the fact that Tm(x,y)∈Rhm and the geometric structure of Rhm assures that Tj(x,y)∈Rhm. Finally, (iii) follows from (ii).Periodic stocking (or harvesting) has the effect of forcing population cycles to evolve and become multiples of the stocking/harvesting period as we show in the following result, which is more general than (14).Theorem 7.
Consider the general difference equationxn+1=F(xn,xn-1,…,xn-k) with p-periodic stocking (or harvesting). If a periodic solution exists, then the period is a multiple of p.Proof.
The proof is a contradiction; suppose that we have anr-periodic solution of the equation xn+1=F(xn,xn-1,…,xn-k)+hn for some r that is not a multiple of p. Then, the greatest common divisor between r and p (d:=gcd(r,p)) is not p. Define the maps Fi:=F+hi,i=0,1,…,p-1; then for each 0≤i≤d-1, the maps {Fkd+i,k=0,1,…,p/d-1} must agree at the point Xi:=(xi,xi-1,…,xi-k), where the components xi-k,xi+1-k,…,xi are consecutive elements of the p-periodic solution. This implies
(17)hi=hd+i=h2d+i=⋯=h(p/d-1)d+i
for all i=0,1,…,d-1, which contradicts the minimality of the period of the p-periodic difference equation.Theorem7 shows that (14) has no equilibrium solutions, and therefore, our previous notion of characterizing oscillatory solutions based on the oscillations about y=x is the valid one here. Thus, solutions of (14) are oscillatory about y=x because they cannot be monotonic. Although it is natural for fluctuations in the environment to create fluctuations in the population, we find it appropriate here to connect the loosely-defined term “fluctuation" with the mathematically well-defined term “oscillation.” Next, we use the Brouwer fixed theorem [23] (page 51) to prove the existence of a periodic solution of (14).Lemma 8 (Brouwer fixed-point theorem [23], page 51).
LetM be a nonempty, convex, and compact subset of ℝn. If T:M→M is continuous, then T has a fixed point in M.Theorem 9.
Thep-periodic difference equation in (14) has a p-periodic solution.Proof.
Consider the mapT^:=Tp-1∘Tp-2∘⋯∘T0, then using Lemma 6, we obtain T^:Rhm→Rhm. Furthermore, Rhm is nonempty, compact, and obviously convex. So, by Lemma 8, T^ has a fixed point in Rhm. This fixed point establishes a periodic solution of (14) with minimal period that divides p; however, Theorem 7 shows that the period must be p.


## 3.2. Global Attractivity of the Periodic Solution Whenp=2

Consider the periodicity of (14) to be p=2 and suppose h0+h1≠0. We partition the solutions of (14) into two subsequences, the one with even indices {x2n} and the one with odd indices {x2n}. Thus, we have
(18)x2n+1=x2nf(x2n-1)+h0,x2n+2=x2n+1f(x2n)+h1.Since the solutions are bounded, we define(19)liminf{x2n+i}=Ii,limsup{x2n+i}=Si,i=0,1.Now, the second iterate of (18) gives us
(20)x2n+2=x2nf(x2n)f(x2n-1)+h0f(x2n)+h1,(21)x2n+3=x2n+1f(x2n+1)f(x2n)+h1f(x2n+1)+h0.Use the fact thatf(t) is decreasing and tf(t) is increasing in (20) to obtain
(22)S0≤S0f(S0)f(I1)+h0f(I0)+h1,(23)I0≥I0f(I0)f(S1)+h0f(S0)+h1.Also, (21) gives us
(24)S1≤S1f(S1)f(I0)+h1f(I1)+h0,(25)I1≥I1f(I1)f(S0)+h1f(S1)+h0.Multiply inequality (22) by I0 and inequality (23) by S0 to obtain
(26)S0I0f(I0)f(S1)+S0(h0f(S0)+h1)≤I0S0f(S0)f(I1)+I0(h0f(I0)+h1).SinceI0(h0f(I0)+h1)≤S0(h0f(S0)+h1), we obtain
(27)f(I0)f(S1)≤f(S0)f(I1).Also, multiply inequality (24) by I1 and inequality (25) by S1 to obtain
(28)S1I1f(I1)f(S0)+S1(h1f(S1)+h0)≤I1S1f(S1)f(I0)+I1(h1f(I1)+h0).SinceI1(h1f(I1)+h0)≤S1(h1f(S1)+h0), we obtain
(29)f(I1)f(S0)≤f(I0)f(S1).Using inequalities (27) and (29), we obtain the following result.Lemma 10.
ConsiderI0,I1,S0,S1 as defined in (19); then f(I0)f(S1)=f(I1)f(S0).Next, we give the following result.Theorem 11.
Forp=2, the 2-periodic solution of (14) is a global attractor.Proof.
Use the result of Lemma10 in inequality (26) to obtain
(30)S0(h0f(S0)+h1)≤I0(h0f(I0)+h1).
Sinceg(t)=t(h0f(t)+h1) is increasing and I0≤S0, we must have I0=S0. Similarly, use the result of Lemma 10 in inequality (28) to obtain
(31)S1(h1f(S1)+h0)≤I1(h1f(I1)+h0),
and consequently S1=I1. Hence, Ii=Si,i=0,1, and the proof is complete.Remark 12.
Observe that the approach of this section proves not only the global attractivity of thep-periodic solution but also its existence; however, Theorem 7 is still significant here because it proves the minimality of the period. Also, establishing the compact invariant region in Lemma 6 deserves embracing regardless of the global attractivity of the periodic solution. Finally, proving the global attractivity for general p will be the topic of some future work.


## 4. Pielou's Equation with Stocking

As an illustrative example to our results, we consider the functionf in (14) to be f(t)=bt/(1+t). It is worth mentioning that in the absence of stocking, Pielou ([24], page 80) suggested taking f(xn-m)=μK/(K+(μ-1)xn-m) to account for certain fluctuating populations, which cannot be modeled by the Beverton-Holt equation. So, here we are dealing with the dimensionless Pielou's equation yn+1=byn/(1+yn-1), which takes the following form after forcing stocking:
(32)yn+1=byn1+yn-1+hn.Whenhn=0, (32) has the positive equilibrium x2,0=b-1 which is globally asymptotically stable. When hn=h>0,x2,h=(1/2)(b+h-1)+(1/2)(b+h-1)2+h inherits the global asymptotic stability of x2,0 as shown in [21]. Now, consider {hn} to be 2-periodic. To find the 2-periodic solution assured by Theorem 9, we substitute y-1=x and y0=y in (32) to obtain
(33)(y-h0)(1+y)=bx,(x-h1)(1+x)=by.Now, the solution is obvious graphically (it is the point of intersection between the two curves in the positive quadrant). However, the solution is not simple to write explicitly, and therefore, we proceed by choosingh0=b+1/(b-1) and h1=b. In this case, the 2-periodic solution {x-,y-} is given by
(34)x-=b-1+b3/2b-1,y-=b+1b-1+b(b-1).Next, use the functionf(t)=b/(1+t) in Section 3.2 and follow the same steps to find
(35)liminf{y2n}=I0=S0=limsup{y2n},liminf{y2n+1}=I1=S1=limsup{y2n+1}.
Thus, the odd iterates converge to a point, say u, while the even iterates converge to a point, say v. Substitute u and v in (18); then compare it with (33) to find that {u,v} is indeed {x-,y-}. Finally, Figure 3 shows the convergence to the 2-cycle for the specific values of the parameters.Figure 3
This graph shows the stable2-cycle for the 2-periodic equation in (32) when h0=b+1/(b-1) and h1=b, where b is fixed at 5/2.Another interesting notion that can be observed here is the resonance of the solutions of (32). The arithmetic average of the globally attracting 2-periodic solution is
(36)xsv:=12(x-+y-)=12(2b-1+1b-1+b3/2b-1+b(b-1)).On the other hand, when we take the constant stockingh=(1/2)(h0+h1)=b+1/2(b-1), we obtain the globally attracting equilibrium
(37)x-=4b2-6b+3+16b4-32b3+28b2-12b+14(b-1).
Figure 4 shows that xav>x-.Figure 4
This graph shows the average of the attracting2-cycle (blue color) in contrast with the equilibrium that results from constant stoking equals the average of h0 and h1, where h0=b+1/(b-1) and h1=b.


## 5. Conclusion and Discussion

In this paper, we investigated the dynamics of the periodic difference equationxn+1=xnf(xn-1)+hn, where f(x) is differentiable and decreasing on [0,∞), while xf(x) is increasing and bounded. This equation can be used as a discrete model to represent contest competition in species with periodic stocking. We found that periodic stocking forces the existence of a periodic solution that has the same period as the stocking period. In addition to the unbounded invariant given by the positive quadrant, we constructed a bounded invariant region, which we used to prove the existence of the periodic solution. Also, we proved that the periodic solution is globally attractive when the stocking period is 2. We conjecture that the periodic solution is globally attractive regardless of the stocking period. Although the steady state has evolved to become the periodic solution of the same period as the stocking period, our results show that periodic stocking preserves the global attractivity of the periodic solution.



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*Source: 101649-2013-10-27.xml*