This paper is concerned with the global behavior of higher-order difference equation of the form yn+1=(ynexp(β(1-2∑i=0kaiyn-i)))/(1-yn+ynexp(β(1-2∑i=0kaiyn-i))), n=0,1,2…, y-k,y-k+1,…,y0∈(0,1). Under some certain assumptions, it is proved that the positive equilibrium is globally asymptotical stable.

1. Introduction and Preliminaries

Nonlinear difference equations of order greater than one are of paramount importance in applications where the (n+1)th generation of the system depends on the previous k generations. Such equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering, and economics [1–8]. The global character of difference equations is a most important topic and there have been many recent investigations and interest in the topic [1, 5–7, 9–14]. In particular, many researchers have paid attention to the global attractivity and convergence of the kth-order recursive sequence [2, 10, 14–19] and several approaches have been developed for finding the global character of difference equations; see [2, 6, 7, 9–15, 17, 19–21]. Moreover, we refer to [3, 4, 6, 7, 16] and the references therein for the oscillation and nonoscillation of difference equations. However, a large number of the literatures concerned with the rational difference equations and it is not enough to understand the global dynamics of a general difference equations, particularly irrational difference equation.

In this paper we study the global behavior of higher-order difference equation of some genotype selection model:yn+1=ynexp(β(1-2∑i=0kaiyn-i))1-yn+ynexp(β(1-2∑i=0kaiyn-i)),n=1,2,…,
where β∈(0,∞), k∈{0,1,2,…},a0,a1,…,ak∈[0,1], ∑i=0kai=1, and initial conditions are y-k,y-k+1,…,y0∈(0,1). When a0=a1=⋯=ak-1=0,ak=1, (1.1) reduces toyn+1=ynexp(β(1-2yn-k))1-yn+ynexp(β(1-2yn-k)),n=1,2,…,
which was introduced by [6] as an example of a map generation by a simple mode for frequency dependent natural selection. The local stability of positive equilibrium y¯=1/2 of (1.2) was investigated by [6].

We note that the appearance of yn-i(i=0,1,…,k) in the selection coefficient reflects the fact that the environment at the present time depends upon the activity of the population at some time in the past and that this in turn depends upon the gene frequency at that time. The points 0, 1/2, and 1 are the only equilibrium solutions of (1.1). One can easily see that yn∈[0,1] for all n=1,2,…. If yN=0 for some N∈ℕ, then yn=0 for all n≥N and if yN=1 for some N∈ℕ, then yn=1 for all n≥N, So in the following, we will restrict our attention to the difference equation:yn+1=ynexp(β(1-2∑i=0kaiyn-i))1-yn+ynexp(β(1-2∑i=0kaiyn-i)),y-k,y-k+1,…,y0∈(0,1).
By introducing the substitution yn=xn1+xn,
then (1.3) becomesxn+1=xnexp(β(2∑i=0kai1+xn-i-1)),n=1,2,…,
where β∈(0,∞), k∈{0,1,2,…}, ∑i=0kai=1, and x-k,x-k+1,…,x0∈(0,∞) are arbitrary initial conditions.

In the sequel we will consider (1.5). It is clear that (1.5) has unique positive equilibrium point x¯=1.

In the following, we give some results which will be useful in our investigation of the behavior of solutions of (1.5), and the proof of lemmas can be found in [6].

Definition 1.1.

The equilibrium point x¯ of the equation xn+1=F(xn,xn-1,…,xn-k),n=0,1,…, is the point that satisfies the condition x¯=F(x¯,x¯,…,x¯).

Definition 1.2.

(a) A sequence {xn} is said to be oscillate about zero or simply oscillate if the terms xn are neither eventually all positive nor eventually all negative. Otherwise the sequence is called nonoscillatory. A sequence {xn} is called strictly oscillatory if for every n0≥0, there exist n1,n2≥n0 such that xn1xn2<0.

(b) A sequence {xn} is said to be oscillate about x¯ if the sequence {xn-x¯} oscillates. The sequence is called strictly oscillatory about x¯ if the sequence {xn-x¯} is strictly oscillatory.

Definition 1.3.

Let y¯ be an equilibrium point of equation yn+1=f(yn,yn-1,…,yn-k); then the equilibrium point y¯ is called

locally stable if for every ɛ>0 there exists δ>0 such that for all yk,y-k+1,…,y0∈I with |y-k-y¯|+|y-k+1-y¯|+⋯+|y0-y¯|<δ, we have |yn-y¯|<ɛ, for all n>-1;

locally asymptotically stable if it is locally stable and if there exists γ>0 such that for all y-k,y-k+1,…,y0∈I with |y-k-y¯|+|y-k+1-y¯|+⋯+|y0-y¯|<γ, we have limn→∞yn=y¯;

a global attractor if for all y-k,y-k+1,…,y0∈I, we have limn→∞yn=y¯.

globally asymptotically stable if y¯ is locally stable and y¯ is a global attractor.

Lemma 1.4.

Assume that β is a positive real number and k is a nonnegative integer; then the following statements are true.

If k=0, then every solution of (1.2) oscillates about y¯=1/2 if and only if β>2.

If k≥1, then every solution of (1.2) oscillates about y¯=1/2 if and only if
β>2kk(k+1)k+1.

Lemma 1.5.

The linear difference equation
xn+k+∑i=1kpixn+k-i=0,n=0,1,2,…,
where p1,p2,…,pm∈ℝ, is asymptotically stable provided that
∑i=1k|pi|<1.

Lemma 1.6.

The linear difference equation
xn+1-xn+∑i=1mpixn-ki=0,n=1,2,…,
where p1,…,pm∈(0,∞) and k1,k2,…,km are positive integers, is asymptotically stable provided that
∑i=1mkipi<1.

Lemma 1.7.

Consider the difference equation
xn+1=xnf(xn,xn-k1,…,xn-kr);
one assumes that the function f satisfies the following hypotheses.

f∈C[[0,∞)×[0,∞)r,(0,∞)], and g∈C[[0,∞)r+1,(0,∞)], where
g(u0,u1,…,ur)=u0f(u0,u1,…,ur)foru0∈(0,∞),u1,u2,…,ur∈[0,∞),g(0,u1,…,ur)=limuo→0+g(u0,u1,…,ur).

f(u0,u1,…,ur) is nonincreasing in u1,…,ur.

The equation
f(x,x,…,x)=1
has a unique positive solution x¯.

Either the function f(u0,u1,…,ur) does not depend on u0 or for every x>0 andu≥0[f(x,u,…,u)-f(x¯,u,…,u)](x-x¯)≤0
with
[f(x,u,…,u)-f(x¯,u,…,u)](x-x¯)<0forx≠x¯.

The function f(u0,u1,…,ur) does not depend on u0 or that for every x,y∈(0,∞) withx≠x¯[xf(x,y,…,y)-x¯f(x¯,y,…,y)]>0.

Furthermore assume that the function F(x) is given by
F(x)=x¯f(x¯,x,…,x)k+1
and has no periodic orbits of prime period 2; then x¯ is a global attractor of all positive solutions of (1.11).Lemma 1.8.

Let F∈C[[0,∞),(0,∞)] be a nonincreasing function, and let x¯ denote the (unique) fixed point of F. Then the following statements are equivalent:

x¯ is only fixed point of F2 in (0,∞);

F2(x)>x for 0<x<x¯.

2. Main Results

In this section, we will investigate the asymptotic stability and global behavior of the positive equilibrium point of (1.5).

Theorem 2.1.

Assume that β is a positive real number and k is a nonnegative integer; then the following statements are true.

If k=0, then every solution of (1.5) oscillates about x¯=1 if and only if β>2.

If k≥1,0<β<4k(k+1),a0=0, then x¯=1 is an asymptotically stable solution of (1.5).

If k≥1,0<β≤2,1/2<a0≤1, then x¯=1 is an asymptotically stable solution of (1.5).

Proof.

When k=0, then a0=1,a1=a2=⋯=ak=0, (1.5) becomes (1.2), and case (a) follows from Lemma 1.4(a).

When k≠0, the linearized equation of (1.5) about equilibrium point x¯=1 is
zn+1-(1-βa02)zn+∑i=1kβai2zn-i=0.
If a0=0, by applying the Lemma 1.6, (2.1) is asymptotically stable provided that
∑i=1kiβai2<1,
so for
0<β<4k(k+1),
and x¯=1 is an asymptotically stable solution of (1.5). The proof of (b) is completed.

If 1/2<a0≤1, by applying Lemma 1.5, (2.1) is asymptotically stable provided that
|(1-βa02)|+∑i=1k|βai2|<1,
so for 0<β<2, and x¯=1 is an asymptotically stable solution of (1.5). The proof of (c) is completed.

Theorem 2.2.

Assume that β is a positive real number and k is a positive integer. Then the equilibrium x¯=1 of (1.5) is globally asymptotically stable, when one of the following three cases holds:

1≤k≤7,0<β<1/2(k+1),a0=0;

k≥8,0<β<4/k(k+1),a0=0;

k≥1,0<β<2c0/(k+1),1/2<a0<1, where c0 is a constant with 1/2<c0<1.

Proof.

If one of the three conditions (a), (b), (c) is satisfied, by applying Theorem 2.1, x¯=1 is an asymptotically stable solution of (1.5). To complete the proof it remains to show that x¯ is a global attractor of all positive solutions of (1.5). We will employ Lemma 1.7; set
f(u0,u1,…,uk)=exp[β(2∑i=0kai1+ui-1)];
clearly equation xn+1=xnf(xn,xn-1,…,xn-k) satisfies the hypotheses (H1)–(H4) of Lemma 1.7. Set
h(x,y)=xf(x,y,…,y)=xexp(β(1-y-2a01+y+2a01+x)).
If 0≤a0≤1,x≠x¯,0<β<2, then
∂h∂x=exp(β(1-y-2a01+y+2a01+x))(1-2βa0x(1+x)2)=exp(β(1-y-2a01+y+2a01+x))(x2+(2-2βa0)x+1(1+x)2)>0;
hence, the function f(u0,u1,…,ur) does not depend on u0 or that for every x,y∈(0,∞) with x≠x¯,
[xf(x,y,…,y)-x¯f(x¯,y,…,y)]>0.
Furthermore assume that the function F(x) is given by
F(x)=x¯f(x¯,x,…,x)k+1=exp((k+1)β(2(1-a0)1+x+a0-1)).
Set
Φ(x)=x-F(F(x));
then
Φ'(x)=1-F′(F(x))F′(x)=1-[exp((k+1)β(2(1-a0)1+x+a0-1))(k+1)β2(1-a0)(1+x)2]|F(x)×exp((k+1)β(2(1-a0)1+x+a0-1))(k+1)β2(1-a0)(1+x)2.
When 0<β(k+1)(1-a0)<c0,0<x<1, then 1<F(x)<exp((k+1)β(1-a0))<ec0. Hence,
Φ′(x)=1-[exp((k+1)β(2(1-a0)1+x+a0-1))(k+1)β2(1-a0)(1+x)2]|F(x)×(exp((k+1)β(2(1-a0)1+x+a0-1))(k+1)β2(1-a0)(1+x)2≥1-[exp((k+1)β(2(1-a0)1+x+a0-1))(k+1)β2(1-a0)(1+x)2]|F(x)=1×exp((k+1)β(2(1-a0)1+x+a0-1))(k+1)β2(1-a0)(1+x)2|x=0≥1-(k+1)2β2(1-a0)2ec0≥1-c02ec0=0,
where c0 is a constant with 1-c02ec0=0, and clearly 1/2<c0<1. So Φ(x) is an increasing function, hence
Φ(x)=x-F2(x)<Φ(1)=0,
and so, F2(x)>x for 0<x<1, by Lemma 1.8, we know that x¯ is only fixed point of F2 in (0,∞), and by Lemma 1.7, x¯ is a global attractor of all positive solutions of (1.5). The proof is complete.

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