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

Abstract

We consider discrete models of the form 𝑥_𝑛+1=𝑥_𝑛𝑓(𝑥_𝑛−1)+ℎ_𝑛, where ℎ_𝑛 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.

1. Introduction

In mathematical ecology, difference equations of the form 𝑥_𝑛+1=𝑥_𝑛𝑓(𝑥_𝑛), 𝑛∈ℕ∶={0,1,…} are used to model single species with nonoverlapping generations [1, 2], where 𝑥_𝑛 denotes the number of sexually mature individuals at discrete time 𝑛, and 𝑓(𝑥_𝑛) 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 𝑥_𝑛+1=𝑥_𝑛𝑓𝑥_𝑛−𝑘, (1) 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 𝑥_𝑛+1=𝑥_𝑛𝑓𝑥_𝑛−1+ℎ_𝑛, (2) where {ℎ_𝑛} 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 ℎ_𝑛=0, then there is no long-term survival regardless of the initial density of the population.

This paper is organized as follows. In Section 2, we give some preliminary results concerning local stability, boundedness, and global stability of (2) when the stocking sequence is 1-periodic, that is, when ℎ_𝑛=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, ℎ_𝑗=ℎ>0 for all 𝑗=0,1,…,𝑝−1. Thus, (2) becomes 𝑥_𝑛+1=𝑥_𝑛𝑓𝑥_𝑛−1+ℎ. (3)

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 𝑦_𝑛+1−𝑓(𝑥)𝑦_𝑛−𝑥𝑓^′(𝑥)𝑦_𝑛−1=0. (4)

Define 𝑝∶=𝑓(𝑥) and 𝑞∶=−𝑥𝑓^′(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 𝜆_𝑗,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 𝜆_𝑗,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=−𝑥𝑓′(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 and ℎ>0. The positive equilibrium 𝑥_2,ℎ of (3) is locally asymptotically stable.

Proof. Since 𝑥_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 that 𝑥_𝑘≥ℎ for all 𝑘≥1. On the other hand, since 𝑥_𝑛+1=𝑥_𝑛−1𝑓𝑥_𝑛−1𝑓𝑥_𝑛−2+ℎ𝑓𝑥_𝑛−1+ℎ≤𝑥_𝑛−1𝑓𝑥_𝑛−1𝑏+ℎ𝑏+ℎ, (5) 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 {𝑥_𝑛} of (3) is called oscillatory about a curve 𝐻(𝑥,𝑦)=0 if the sequence {𝑢_𝑛=(𝑥_𝑛−1,𝑥_𝑛)} 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 {𝑥_𝑛} oscillates about 𝑥_2,ℎ, but it is not oscillatory about 𝑦=𝑥. So, {𝑥_𝑛} is either eventually increasing or eventually decreasing, which contradicts the assumption that 𝑥_𝑛 is oscillatory about 𝑥_2,ℎ. Conversely, suppose {(𝑥_𝑛−1,𝑥_𝑛)} oscillates about 𝑦=𝑥, but {𝑥_𝑛} does not oscillate about 𝑥_2,ℎ. First, we consider the case 𝑥_𝑛≤𝑥_2,ℎ for all 𝑛≥𝑛₀. If 𝑥_𝑚>𝑥_𝑚−1 for some 𝑚>𝑛₀, then 𝑓(𝑥_𝑚)<𝑓(𝑥_𝑚−1) and consequently 𝑥_𝑚+1=𝑥_𝑚𝑓𝑥_𝑚−1+ℎ>𝑥_𝑚𝑓𝑥_𝑚+ℎ>𝑥_𝑚. (6)
So, we can induce an eventually increasing sequence which contradicts our assumption. If 𝑥_𝑚≤𝑥_𝑚−1 for some 𝑚>𝑛₀, then 𝑥_𝑚+1≤𝑥_𝑚𝑓(𝑥_𝑚)+ℎ. Thus, either 𝑥_𝑚+1≤𝑥_𝑚, and the induction leads to a decreasing sequence that must converge which is not possible, or 𝑥_𝑚+1>𝑥_𝑚, and we go back to the first scenario. Finally, the case 𝑥_𝑛≥𝑥_2,ℎ for all 𝑛≥𝑛₀ can be handled similarly.

Next, we define the map 𝑇𝑥,𝑦=𝑦,𝑦𝑓(𝑥)+ℎ. (7)

The map 𝑇 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 𝑆∶=limsup𝑥_𝑛,𝐼∶=liminf𝑥_𝑛. (8)

From the equation 𝑥_𝑛+2=𝑥_𝑛𝑓(𝑥_𝑛)𝑓(𝑥_𝑛−1)+ℎ𝑓(𝑥_𝑛)+ℎ and using the fact that 𝑡𝑓(𝑡) is increasing, we obtain (𝑆)𝑓(𝐼)(𝐼)(𝐼)𝑓(𝑆)(𝑆)𝑆≤𝑆𝑓+ℎ𝑓+ℎ,𝐼≥𝐼𝑓+ℎ𝑓+ℎ. (9)

When ℎ>0, we have 𝑆≥𝐼>0. So, we can multiply the first inequality by 𝐼 and the second one by 𝑆 to obtain 𝑆𝑓(𝑆)+1≤𝐼𝑓(𝐼)+1. (10)

Since 𝑡(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 solution 𝑥_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, 𝑅_ℎ is an invariant of (3) if 𝑇(𝑥,𝑦)∈𝑅_ℎ for all (𝑥,𝑦)∈𝑅_ℎ. 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 constant 𝑐_ℎ≥ℎ such that 𝐺_ℎ(𝑡)=(𝑏𝑡+ℎ)𝑓(𝑡)≤𝑏𝑐_ℎ for all 𝑡≥0. Furthermore, 𝑐_ℎ can be taken as 𝑐_ℎ∶=(1/𝑏)sup_𝑡𝐺_ℎ(𝑡).

Proof. Use the fact that 𝑡𝑓(𝑡) is bounded and 𝑓(𝑡) is decreasing with 𝑓(0)=𝑏 and lim_𝑡→∞𝑓(𝑡)=0 to obtain the result.

Next, define the curves Γ_𝑗,𝑗=0,1,2,3,4 to be the line segments that connect the points (0,0),(0,ℎ),(𝑐_ℎ,𝑏𝑐_ℎ+ℎ), (𝑏𝑐_ℎ+ℎ,𝑏𝑐_ℎ+ℎ), (𝑏𝑐_ℎ+ℎ,0), and (0,0), respectively. Now, define 𝑅_ℎ to be the region bounded by the curves of Γ_𝑗,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 𝑅_ℎ becomes a quadrilateral rather than a pentagon.

Theorem 5. The region 𝑅_ℎ as defined above gives a compact invariant for (3).

Proof. Consider the map 𝑇(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 𝑅_ℎ. 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 𝑇𝑐_ℎ,𝑏𝑐_ℎ+ℎ=𝑏𝑐_ℎ+ℎ,𝑏𝑐_ℎ+ℎ𝑓𝑐_ℎ,𝑇+ℎ𝑏𝑐_ℎ+ℎ,𝑏𝑐_ℎ=+ℎ𝑏𝑐_ℎ+ℎ,𝑏𝑐_ℎ+ℎ𝑓𝑏𝑐_ℎ+ℎ+ℎ. (11)
By the choice of 𝑐_ℎ given in Proposition 4, we have 𝑏𝑐_ℎ+ℎ𝑓𝑏𝑐_ℎ≤+ℎ+ℎ𝑏𝑐_ℎ+ℎ𝑓𝑐_ℎ+ℎ≤𝑏𝑐_ℎ+ℎ. (12)
Thus, 𝑇(Γ₂)⊂Γ₃. Next, 𝑇(Γ₃)⊂𝑅_ℎ and 𝑇(Γ₄)=(0,ℎ) are straightforward to observe. Finally, we show that 𝑇(Γ₁)⊂𝑅_ℎ. For 0≤𝑡≤𝑐_ℎ, we have 𝑇(𝑡,𝑏𝑡+ℎ)=𝑏𝑡+ℎ,(𝑏𝑡+ℎ)𝑓(𝑡)+ℎ; (13) however, ℎ≤𝑏𝑡+ℎ≤𝑏𝑐_ℎ+ℎ and (𝑏𝑡+ℎ)𝑓(𝑡)+ℎ≤𝑏𝑐_ℎ+ℎ by the choice of 𝑐_ℎ, which completes the proof. Figure 2 illustrates the region 𝑅_ℎ 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 region 𝑅ℎ 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 𝑥_𝑛+1=𝑥_𝑛𝑓𝑥_𝑛−1+ℎ_𝑛, (14) where ℎ_𝑛 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 ∑^𝑝−1_𝑗=0ℎ_𝑗>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 {𝑇_𝑗,𝑗=0,1,…,𝑝−1}, where 𝑇_𝑗(𝑥,𝑦)=(𝑦,𝑦𝑓(𝑥)+ℎ_𝑗). It is obvious that if we replace ℎ by ℎ_𝑗 in Theorem 5, then 𝑅_ℎ𝑗 forms a compact and invariant region for the individual map 𝑇_𝑗, 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 𝑅_ℎ𝑗 by Γ_𝑗,𝑖,𝑖=0,…,4, where Γ_𝑗,𝑖 corresponds to Γ_𝑖 in the autonomous case and that are associated with the individual map 𝑇_𝑗. Also, the constant 𝑐_ℎ in Proposition 4 will be replaced by 𝑐_ℎ𝑗, and this is associated with the individual map 𝑇_𝑗.

3.1. Existence of a Periodic Solution

We start by establishing a compact invariant region for (14). Define ℎ_𝑚∶=maxℎ₀,ℎ₁,…,ℎ_𝑝−1,𝑐_𝑚∶=max𝑐_ℎ𝑗∶𝑗=0,…,𝑝−1, (15) where 𝑐_ℎ𝑗 is as taken in Proposition 4; that is, 𝑐_ℎ𝑗=(1/𝑏)sup_𝑡𝐺_ℎ𝑗(𝑡); then use ℎ_𝑚 and 𝑐_𝑚 to define the region 𝑅_ℎ𝑚 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 {𝑇_𝑗}. Each of the following holds true.(i) One has 𝑅_ℎ𝑖⊆𝑅_ℎ𝑗 whenever ℎ_𝑖≤ℎ_𝑗.(ii)𝑅_ℎ𝑚 is a compact invariant for each individual map 𝑇_𝑗.(iii)𝑅_ℎ𝑚 is a compact invariant for the map 𝑇∶=𝑇_𝑝−1∘𝑇_𝑝−2∘⋯∘𝑇₀.

Proof. (i) When ℎ_𝑖≤ℎ_𝑗, we obtain 𝐺_ℎ𝑖(𝑡)≤𝐺_ℎ𝑗(𝑡) for all 𝑡≥0. Thus, 𝑐_ℎ𝑖≤𝑐_ℎ𝑗, and the result becomes obvious from Proposition 4 and the geometric structure of the regions 𝑅_ℎ𝑖 and 𝑅_ℎ𝑗. To prove (ii), let (𝑥,𝑦)∈𝑅_ℎ𝑚, we show that 𝑇_𝑗(𝑥,𝑦)∈𝑅_ℎ𝑚. Since 𝑇_𝑗𝑥,𝑦=𝑦,𝑦𝑓(𝑥)+ℎ_𝑗=𝑦,𝑦𝑓(𝑥)+ℎ_𝑚−0,ℎ_𝑚−ℎ_𝑗=𝑇_𝑚𝑥,𝑦−0,ℎ_𝑚−ℎ_𝑗, (16) then the first component of 𝑇_𝑗(𝑥,𝑦) is the same as the first component of 𝑇_𝑚(𝑥,𝑦) and the second component of 𝑇_𝑗(𝑥,𝑦) is lower than the second component of 𝑇_𝑚(𝑥,𝑦). Now, the fact that 𝑇_𝑚(𝑥,𝑦)∈𝑅_ℎ𝑚 and the geometric structure of 𝑅_ℎ𝑚 assures that 𝑇_𝑗(𝑥,𝑦)∈𝑅_ℎ𝑚. 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 equation 𝑥_𝑛+1=𝐹(𝑥_𝑛,𝑥_𝑛−1,…,𝑥_𝑛−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 an 𝑟-periodic solution of the equation 𝑥_𝑛+1=𝐹(𝑥_𝑛,𝑥_𝑛−1,…,𝑥_𝑛−𝑘)+ℎ_𝑛 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 𝐹_𝑖∶=𝐹+ℎ_𝑖,𝑖=0,1,…,𝑝−1; then for each 0≤𝑖≤𝑑−1, the maps {𝐹_{𝑘𝑑+𝑖},𝑘=0,1,…,𝑝/𝑑−1} must agree at the point 𝑋_𝑖∶=(𝑥_𝑖,𝑥_𝑖−1,…,𝑥_𝑖−𝑘), where the components 𝑥_𝑖−𝑘,𝑥_{𝑖+1−𝑘},…,𝑥_𝑖 are consecutive elements of the 𝑝-periodic solution. This implies ℎ_𝑖=ℎ_𝑑+𝑖=ℎ_2𝑑+𝑖=⋯=ℎ_{(𝑝/𝑑−1)𝑑+𝑖} (17) for all 𝑖=0,1,…,𝑑−1, which contradicts the minimality of the period of the 𝑝-periodic difference equation.

Theorem 7 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). Let 𝑀 be a nonempty, convex, and compact subset of ℝ^𝑛. If 𝑇∶𝑀→𝑀 is continuous, then 𝑇 has a fixed point in 𝑀.

Theorem 9. The 𝑝-periodic difference equation in (14) has a 𝑝-periodic solution.

Proof. Consider the map 𝑇∶=𝑇_𝑝−1∘𝑇_𝑝−2∘⋯∘𝑇₀, then using Lemma 6, we obtain 𝑇∶𝑅_ℎ𝑚→𝑅_ℎ𝑚. Furthermore, 𝑅_ℎ𝑚 is nonempty, compact, and obviously convex. So, by Lemma 8, 𝑇 has a fixed point in 𝑅_ℎ𝑚. 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 When 𝑝=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 {𝑥_2𝑛} and the one with odd indices {𝑥_2𝑛}. Thus, we have 𝑥_2𝑛+1=𝑥_2𝑛𝑓𝑥_2𝑛−1+ℎ₀,𝑥_2𝑛+2=𝑥_2𝑛+1𝑓𝑥_2𝑛+ℎ₁. (18)

Since the solutions are bounded, we define liminf𝑥_2𝑛+𝑖=𝐼_𝑖,limsup𝑥_2𝑛+𝑖=𝑆_𝑖,𝑖=0,1. (19)

Now, the second iterate of (18) gives us 𝑥_2𝑛+2=𝑥_2𝑛𝑓𝑥_2𝑛𝑓𝑥_2𝑛−1+ℎ₀𝑓𝑥_2𝑛+ℎ₁,𝑥_2𝑛+3=𝑥_2𝑛+1𝑓𝑥_2𝑛+1𝑓𝑥_2𝑛+ℎ₁𝑓𝑥_2𝑛+1+ℎ₀. (20)(21)

Use the fact that 𝑓(𝑡) is decreasing and 𝑡𝑓(𝑡) is increasing in (20) to obtain 𝑆₀≤𝑆₀𝑓𝑆₀𝑓𝐼₁+ℎ₀𝑓𝐼₀+ℎ₁,𝐼₀≥𝐼₀𝑓𝐼₀𝑓𝑆₁+ℎ₀𝑓𝑆₀+ℎ₁. (22)(23)

Also, (21) gives us 𝑆₁≤𝑆₁𝑓𝑆₁𝑓𝐼₀+ℎ₁𝑓𝐼₁+ℎ₀,𝐼₁≥𝐼₁𝑓𝐼₁𝑓𝑆₀+ℎ₁𝑓𝑆₁+ℎ₀. (24)(25)

Multiply inequality (22) by 𝐼₀ and inequality (23) by 𝑆₀ to obtain 𝑆₀𝐼₀𝑓𝐼₀𝑓𝑆₁+𝑆₀ℎ₀𝑓𝑆₀+ℎ₁≤𝐼₀𝑆₀𝑓𝑆₀𝑓𝐼₁+𝐼₀ℎ₀𝑓𝐼₀+ℎ₁. (26)

Since 𝐼₀(ℎ₀𝑓(𝐼₀)+ℎ₁)≤𝑆₀(ℎ₀𝑓(𝑆₀)+ℎ₁), we obtain 𝑓𝐼₀𝑓𝑆₁≤𝑓𝑆₀𝑓𝐼₁. (27)

Also, multiply inequality (24) by 𝐼₁ and inequality (25) by 𝑆₁ to obtain 𝑆₁𝐼₁𝑓𝐼₁𝑓𝑆₀+𝑆₁ℎ₁𝑓𝑆₁+ℎ₀≤𝐼₁𝑆₁𝑓𝑆₁𝑓𝐼₀+𝐼₁ℎ₁𝑓𝐼₁+ℎ₀. (28)

Since 𝐼₁(ℎ₁𝑓(𝐼₁)+ℎ₀)≤𝑆₁(ℎ₁𝑓(𝑆₁)+ℎ₀), we obtain 𝑓𝐼₁𝑓𝑆₀≤𝑓𝐼₀𝑓𝑆₁. (29)

Using inequalities (27) and (29), we obtain the following result.

Lemma 10. Consider 𝐼₀,𝐼₁,𝑆₀,𝑆₁ as defined in (19); then 𝑓(𝐼₀)𝑓(𝑆₁)=𝑓(𝐼₁)𝑓(𝑆₀).

Next, we give the following result.

Theorem 11. For 𝑝=2, the 2-periodic solution of (14) is a global attractor.

Proof. Use the result of Lemma 10 in inequality (26) to obtain 𝑆₀ℎ₀𝑓𝑆₀+ℎ₁≤𝐼₀ℎ₀𝑓𝐼₀+ℎ₁. (30)
Since 𝑔(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 𝑆₁ℎ₁𝑓𝑆₁+ℎ₀≤𝐼₁ℎ₁𝑓𝐼₁+ℎ₀, (31) and consequently 𝑆₁=𝐼₁. Hence, 𝐼_𝑖=𝑆_𝑖,𝑖=0,1, and the proof is complete.

Remark 12. Observe that the approach of this section proves not only the global attractivity of the 𝑝-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 function 𝑓 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(𝑥_𝑛−𝑚)=𝜇𝐾/(𝐾+(𝜇−1)𝑥_𝑛−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 𝑦_𝑛+1=𝑏𝑦_𝑛/(1+𝑦_𝑛−1), which takes the following form after forcing stocking: 𝑦_𝑛+1=𝑏𝑦_𝑛1+𝑦_𝑛−1+ℎ_𝑛. (32)

When ℎ_𝑛=0, (32) has the positive equilibrium 𝑥_2,0=𝑏−1 which is globally asymptotically stable. When ℎ_𝑛=ℎ>0,𝑥_2,ℎ=(1/2)(𝑏+ℎ−1)+(1/2)(𝑏+ℎ−1)²+ℎ inherits the global asymptotic stability of 𝑥_2,0 as shown in [21]. Now, consider {ℎ_𝑛} 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 𝑦−ℎ₀1+𝑦=𝑏𝑥,𝑥−ℎ₁(1+𝑥)=𝑏𝑦. (33)

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 choosing ℎ₀=𝑏+1/(𝑏−1) and ℎ₁=𝑏. In this case, the 2-periodic solution {𝑥,𝑦} is given by 𝑏𝑥=𝑏−1+^3/2√,𝑏−11𝑦=𝑏++𝑏−1𝑏(𝑏−1). (34)

Next, use the function 𝑓(𝑡)=𝑏/(1+𝑡) in Section 3.2 and follow the same steps to find liminf𝑦_2𝑛=𝐼₀=𝑆₀=limsup𝑦_2𝑛,liminf𝑦_2𝑛+1=𝐼₁=𝑆₁=limsup𝑦_2𝑛+1. (35) 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 stable 2-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 𝑥_𝑠𝑣1∶=2𝑥+𝑦=1212𝑏−1++𝑏𝑏−1√^3/2+𝑏−1𝑏(𝑏−1). (36)

On the other hand, when we take the constant stocking ℎ=(1/2)(ℎ₀+ℎ₁)=𝑏+1/2(𝑏−1), we obtain the globally attracting equilibrium 𝑥=4𝑏²−6𝑏+3+16𝑏⁴−32𝑏³+28𝑏²−12𝑏+14().𝑏−1 (37) Figure 4 shows that 𝑥_𝑎𝑣>𝑥.

Figure 4 

This graph shows the average of the attracting 2-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 equation 𝑥_𝑛+1=𝑥_𝑛𝑓(𝑥_𝑛−1)+ℎ_𝑛, 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.

Acknowledgments

The author thanks one of the anonymous referees for his suggestions and remarks. This work is partially supported by SQU Internal Grant IG/SCI/DOMS/11/14.