# Dynamics of Planar Systems That Model Stage-Structured Populations

Authors: N. Lazaryan; H. Sedaghat
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

We study a general discrete planar system for modeling stage-structured
populations. Our results include conditions for the global convergence of
orbits to zero (extinction) when the parameters (vital rates) are time and density dependent. When the parameters are
periodic we obtain weaker conditions for extinction. We also study a rational
special case of the system for Beverton-Holt type interactions and show that
the persistence equilibrium (in the positive quadrant) may be globally
attracting even in the presence of interstage competition. However, we
determine that with a sufficiently high level of competition, the persistence equilibrium becomes unstable (a saddle point) and the system exhibits period two
oscillations.


---

## Body

## 1. Introduction

Stage-structured models of single-species populations with lowest dimension in discrete time are expressed as planar systems of difference equations. For a general expression of these models, consider system(1a)
A
t
+
1
=
s
1
t
σ
1
c
11
t
J
t
,
c
12
t
A
t
J
t
+
s
2
t
σ
2
c
21
t
J
t
,
c
22
t
A
t
A
t
,
(1b)
J
t
+
1
=
b
t
ϕ
c
1
t
J
t
,
c
2
t
A
t
A
tfrom [1] in which J
(
t
) and A
(
t
) are numbers (or densities) of juveniles and adults, respectively, remaining after t (juvenile) periods. The vital rates s
i and b (survival and inherent low density fertility) as well as the competition coefficients c
i and c
i
j in (1a) and (1b) may be density dependent; that is, they may depend on J and A and also explicitly on time; that is, the system may be nonautonomous. Early examples of matrix models used in species populations dynamics can be found in [2–5] and their comprehensive treatment is provided in [6].Under certain constraints on the various functions, including periodic vital rates and competition coefficients having the same common periodp, sufficient conditions for global convergence to zero (extinction) as well as the existence of periodic orbits for (1a) and (1b) are established in [1]. If μ is the mean fertility rate (the mean value of b
(
t
) above), then it is also shown that orbits of period p appear when μ exceeds a critical value μ
c, while global convergence to 0 or extinction occurs when μ
<
μ
c. On the other hand, conditions under which the species survives (i.e.permanence) were studied in [7, 8].In this paper, we study the following abstraction of the matrix model (1a) and (1b): (2a)
x
n
+
1
=
σ
1
,
n
x
n
,
y
n
y
n
+
σ
2
,
n
x
n
,
y
n
x
n
,
(2b)
y
n
+
1
=
ϕ
n
x
n
,
y
n
x
n
,where for each time period n
≥
0 the functions σ
1
,
n
,
σ
2
,
n
,
ϕ
n
:
[
0
,
∞
)
2
→
[
0
,
∞
) are bounded on the compact sets in [
0
,
∞
)
2. This feature allows for (
0,0
) to be a fixed point of the system and it is true if, for example, σ
1
,
n, σ
2
,
n, ϕ
n are continuous functions for every n. Under biological constraints on the parameters, we may think of x
n
=
A
(
n
) and y
n
=
J
(
n
) as in (1a) and (1b).System (2a) and (2b) includes typical stage-structured models in the literature. For instance, the tadpole-adult model for the green tree frogHyla cinerea population that is proposed in [9] may be expressed as(3a)
x
n
=
y
n
a
+
k
1
y
n
+
x
n
c
+
k
2
x
n
,
(3b)
y
n
=
b
n
x
n
.This is a system of type (2a) and (2b) with Beverton-Holt type functions σ
1 and σ
2. Competition in (3a) and (3b) occurs separately among juveniles and adults but not between the two classes, as they feed on separate resources; thus σ
1 and σ
2 do not depend on both juvenile and adult numbers and ϕ is independent of both numbers. Two cases are analyzed in [9]: (i) continuous breeding with constant b
n
=
b so that (3a) and (3b) is autonomous and (ii) seasonal breeding where b
n is periodic. In addition to considering extinction and survival in the autonomous case, it is shown that seasonal breeding may be deleterious (relative to continuous breeding) for populations with high birth rates, but it can be beneficial with low birth rates.Another system of type (2a) and (2b) is the autonomous stage-structured model with harvesting that is discussed in [10, 11], which may be written as(4a)
x
n
+
1
=
1
-
h
j
s
j
y
n
+
1
-
h
a
s
a
x
n
,
(4b)
y
n
+
1
=
x
n
f
1
-
h
a
x
n
.The numbersh
j
,
h
a
∈
[
0,1
] denote the harvest rates of juveniles and adults, respectively. The stock-recruitment function f
:
[
0
,
∞
)
→
[
0
,
∞
) may be compensatory (e.g., Beverton-Holt [12]) or overcompensatory (e.g., Ricker [13]). Compensatory recruitment is used in populations where recruitment increases with increase in densities before reaching an asymptote, while in overcompensatory models recruitment declines as density increases (see [11, 14]). A thorough analysis of the dynamics of (4a) and (4b) with the Ricker function appears in [10]. The results in [10, 11] clarify many issues with regard to the effects of harvesting in stage-structured models such as global convergence to 0 and the existence of a stable survival equilibrium as well as the so-calledhydra effect for different harvesting scenarios and with different recruitment functions; this refers to the counter-intuitive situation where an increase in the harvest or mortality rate results in a corresponding increase in the total population; for example, see [15–17].Also studied in [10] is the occurrence of periodic and nonperiodic attractors and chaotic behavior for certain parameter ranges.Next, the model in [18] studies the harvesting and predation of sex- and age-structured populations. Although the added stage for two sexes results in a three-dimensional model, the existence of an attracting, invariant planar manifold reduces the study of the asymptotics of the system to that of the planar system:(5a)
x
n
+
1
=
p
s
Y
y
n
+
s
x
n
,
(5b)
y
n
+
1
=
x
n
f
y
n
+
x
n
p
,where the density-dependent per capita reproductive rate f may be Beverton-Holt or Ricker similarly to f in (4b). Here x
n is the number of females and y
n is the number of young members in the population (the male population is a fixed proportion of the females).We also mention the adult-juvenile model(6a)
x
n
+
1
=
s
1
y
n
,
(6b)
y
n
+
1
=
x
n
f
x
n
,
y
nin which all adults are removed through harvesting, predation, migration, or just dying after one period, as in the case of semelparous species, that is, an organism that reproduces only once before death. In [19] conditions for the global attractivity of the positive fixed point and the occurrence of two cycles for (6a) and (6b) are obtained. A significant difference between (5a), (5b), (6a), and (6b) and systems (3a), (3b), (4a), and (4b) is the fact that y
n
+
1 in (5b) or in (6b) may depend on both x
n and y
n.We study the qualitative properties of the orbits of (2a) and (2b) such as uniform boundedness and global convergence to 0 under minimal restrictions on time-dependent parameters. Biological constraints may be readily imposed to obtain special cases relevant to population models.We also investigate convergence to zero with periodic parameters (extinction in a periodic environment). In particular, we show that convergence to zero occurs even if the mean value ofσ
2
,
n exceeds 1, a situation that cannot occur if σ
2
,
n is constant in n; see Remark 16 below.In the final section we study the dynamics of a rational special case of (2a) and (2b). Sufficient conditions for the global asymptotic stability of a fixed point in the positive quadrant [
0
,
∞
)
2 as well as conditions for the occurrence of orbits of prime period two are obtained. In particular, we establish that a sufficiently high level of interspecies competition tends to destabilize the survival fixed point and result in periodic oscillations.Discrete population models generally have been studied by numerous authors; see, for example, [20–32] and the references therein.


## 2. Uniform Boundedness of Orbits

Conditions under which the orbits of (2a) and (2b) are bounded are not transparent. In this section we obtain general results about the uniform boundedness of orbits of (2a) and (2b) in the positive quadrant 0
,
∞
2. We begin with a simple, yet useful lemma.Lemma 1.
Letα
>
0, let 0
<
β
<
1, and let x
0
≥
0. If for all n
≥
0
(7)
x
n
+
1
≤
α
+
β
x
nthen for every ɛ
>
0 and all sufficiently large values of n
(8)
x
n
≤
α
1
-
β
+
ɛ
.Proof.
Letu
0
=
x
0 and note that every solution of the linear, first-order equation u
n
+
1
=
α
+
β
u
n converges to its fixed point α
/
(
1
-
β
). Furthermore, (9)
x
1
≤
α
+
β
x
0
=
α
+
β
u
0
=
u
1
,
x
2
≤
α
+
β
x
1
≤
α
+
β
u
1
=
u
2and, by induction, x
n
≤
u
n. Since u
n
→
α
/
(
1
-
β
) for every ɛ
>
0 and all sufficiently large n,(10)
x
n
≤
u
n
≤
α
1
-
β
+
ɛ
.Theorem 2.
Letσ
1
,
n, σ
2
,
n, ϕ
n be bounded on the compact sets in [
0
,
∞
)
2 for each n
=
0,1
,
2
,
… and suppose that for some r
,
M
>
0
(11)
sup
u
,
v
∈
0
,
r
2
⁡
σ
2
,
n
u
,
v
≤
M
∀
n
≥
0
;that is, the sequence of functions {
σ
2
,
n
} is uniformly bounded on the square [
0
,
r
]
2. If there are numbers M
0
,
M
1
>
0 and σ
¯
∈
(
0,1
) such that uniformly for all n
(12)
u
ϕ
n
u
,
v
≤
M
0
i
f
u
,
v
∈
0
,
∞
2
,
(13)
σ
1
,
n
u
,
v
≤
M
1
i
f
u
,
v
∈
0
,
∞
×
0
,
M
0
,
(14)
σ
2
,
n
u
,
v
≤
σ
¯
i
f
u
,
v
∈
r
,
∞
×
0
,
M
0
,then all orbits of (2a) and (2b) are uniformly bounded and for all sufficiently large values of n satisfy(15)
0
≤
x
n
≤
M
0
M
1
+
r
M
+
σ
¯
1
-
σ
¯
,
y
n
≤
M
0
.Proof.
By (2b) and (12) y
n
≤
M
0 for n
≥
1 so by (2a) and (13)(16)
0
≤
x
n
+
1
≤
M
0
M
1
+
σ
2
,
n
u
,
v
x
n
.
By (11) and (14)(17)
0
≤
x
n
+
1
≤
M
0
M
1
+
max
⁡
σ
¯
x
n
,
M
r
≤
σ
¯
x
n
+
M
0
M
1
+
r
M
.
Next, applying Lemma1 with ɛ
=
σ
¯
/
(
1
-
σ
¯
), we obtain for all (large) n
(18)
0
≤
x
n
≤
M
0
M
1
+
r
M
1
-
σ
¯
+
ɛ
=
M
0
M
1
+
r
M
+
σ
¯
1
-
σ
¯as stated.Corollary 3.
For functionsσ
1
,
n, σ
2
,
n, ϕ
n defined on [
0
,
∞
)
2 for n
=
0,1
,
2
,
… assume that there are numbers M
0
,
M
1
>
0 and σ
¯
∈
(
0,1
) such that for all (
u
,
v
)
∈
[
0
,
∞
)
2 and all n
(19)
u
ϕ
n
u
,
v
≤
M
0
,
σ
1
,
n
u
,
v
≤
M
1
,
σ
2
,
n
u
,
v
≤
σ
¯
.Then all orbits of (2a) and (2b) are uniformly bounded and for all sufficiently large values of n
(20)
0
≤
x
n
≤
M
0
M
1
+
σ
¯
1
-
σ
¯
,
y
n
≤
M
0
.Theorem2 is more general than the preceding corollary. For instance, Corollary 3 does not apply to system (21)
x
n
+
1
=
a
x
n
+
b
y
n
2
1
+
c
x
n
,
y
n
+
1
=
α
x
n
1
+
β
x
n
+
γ
y
n
.However, ifa
∈
(
0,1
), b
,
α
,
β
>
0, and c
,
γ
≥
0, then all orbits of this system with initial values in [
0
,
∞
)
2 are uniformly bounded by Theorem 2.


## 3. Global Attractivity of the Origin

In this section we obtain general sufficient conditions for the convergence of all orbits of the system to(
0,0
). For population models these yield conditions that imply the extinction of species.


### 3.1. General Results

We start with the following lemma; see [33] for the proof and some background on this result.Lemma 4.
Letα
∈
(
0,1
) and assume that the functions f
n
:
0
,
∞
k
+
1
→
0
,
∞ satisfy the inequality(22)
f
n
u
0
,
…
,
u
k
≤
α
max
⁡
u
0
,
…
,
u
kfor all (
u
0
,
…
,
u
k
)
∈
[
0
,
∞
) and all n
≥
0. Then for every solution {
x
n
} of the difference equation(23)
x
n
+
1
=
f
n
x
n
,
x
n
-
1
,
…
,
x
n
-
kthe following is true:(24)
x
n
≤
α
n
/
k
+
1
max
⁡
x
0
,
x
-
1
,
…
,
x
-
k
.Note that (22) implies that x
n
=
0 is a constant solution of (23) and furthermore (24) implies that this solution is globally exponentially stable.Throughout this section we assume thatσ
i
,
n, ϕ
n are all bounded functions for i
=
1,2 and every n
=
0,1
,
2
,
…. Then the following are well-defined sequences of real numbers:(25)
σ
¯
i
,
n
=
sup
u
,
v
≥
0
⁡
σ
i
,
n
u
,
v
,
ϕ
¯
n
=
sup
u
,
v
≥
0
⁡
ϕ
n
u
,
v
.Theorem 5.
If the inequality(26)
limsup
n
→
∞
⁡
σ
¯
1
,
n
ϕ
¯
n
-
1
+
σ
¯
2
,
n
<
1holds, then l
i
m
n
→
∞
x
n
=
0 for every orbit {
(
x
n
,
y
n
)
} of the planar system (2a) and (2b) in the positive quadrant 0
,
∞
2. If also either the sequence {
ϕ
¯
n
} is bounded or the inequality(27)
liminf
n
→
∞
⁡
σ
¯
1
,
n
>
0
,holds, then every orbit of (2a) and (2b) converges to (
0,0
).Proof.
By (26) there is δ
∈
(
0,1
) such that σ
¯
1
,
n
ϕ
¯
n
-
1
+
σ
¯
2
,
n
≤
δ for all (large) n. From (2a)(28)
y
n
≤
ϕ
¯
n
-
1
x
n
-
1so for all (large) n (2b) yields(29)
x
n
+
1
≤
ϕ
¯
n
-
1
σ
¯
1
,
n
x
n
-
1
+
σ
¯
2
,
n
x
n
≤
σ
¯
1
,
n
ϕ
¯
n
-
1
+
σ
¯
2
,
n
max
⁡
x
n
,
x
n
-
1
≤
δ
max
⁡
x
n
,
x
n
-
1
.
Lemma4 now implies that l
i
m
n
→
∞
x
n
=
0. Furthermore either by hypothesis there is a positive number μ such that ϕ
¯
n
≤
μ or by (27) there is a positive number ρ such that σ
¯
1
,
n
≥
ρ for all (large) n so that(30)
ϕ
¯
n
-
1
≤
δ
-
σ
¯
2
,
n
σ
¯
1
,
n
≤
δ
ρfor all sufficiently large values of n
. Now, if M
=
μ or M
=
δ
/
ρ as the case may be, then from (2b) in the planar system we see that(31)
lim
n
→
∞
⁡
y
n
≤
lim
n
→
∞
⁡
ϕ
¯
n
-
1
x
n
-
1
≤
M
lim
n
→
∞
⁡
x
n
-
1
=
0and the proof is complete.Remark 6.
(
1
) Theorem 5 is valid even if the separate sequences {
σ
1
,
n
} or {
ϕ
¯
n
} are not bounded by 1 as long as for all n large enough σ
¯
1
,
n
ϕ
¯
n
-
1
≤
δ
-
σ
¯
2
,
n.
(
2
) If (26) is satisfied but {
ϕ
¯
n
} is unbounded and {
σ
¯
1
,
n
} does not satisfy (27) then y
n may not converge to 0; see the example following Corollary 18 below.We consider an application of Theorem5 to “noisy” autonomous system next. Let ɛ
n, ɛ
i
,
n, i
=
1,2, be bounded sequences of real numbers and let(32)
ɛ
¯
=
sup
n
≥
1
⁡
ɛ
n
,
ɛ
¯
i
=
sup
n
≥
1
⁡
ɛ
i
,
n
,
i
=
1,2
.Additionally, letσ
1
,
σ
2
,
ϕ
:
[
0
,
∞
)
2
→
[
0
,
∞
) be bounded functions and denote their supremums over [
0
,
∞
)
2 by σ
¯
1, σ
¯
2, ϕ
¯, respectively. If in (2a) and (2b) we have(33)
ϕ
n
x
n
,
y
n
=
ϕ
x
n
,
y
n
+
ɛ
n
,
σ
i
,
n
x
n
,
y
n
=
σ
i
x
n
,
y
n
+
ɛ
i
,
n
,
i
=
1,2then we refer to (2a) and (2b) as an autonomous system with low-amplitude disturbances or fluctuations in the rates σ
1, σ
2, ϕ, assuming that all three of these are positive functions and for all u
,
v
≥
0
(34)
ɛ
¯
≤
ϕ
u
,
v
,
ɛ
¯
i
≤
σ
i
u
,
v
,
i
=
1,2
.These inequalities ensure that the functionsϕ
n and σ
i
,
n are positive, as required for (2a) and (2b).Corollary 7.
Suppose that (2a) and (2b) is an autonomous system with low-amplitude disturbances or fluctuations in the above sense. If(35)
σ
¯
1
+
ɛ
¯
1
ϕ
¯
+
ɛ
¯
+
σ
¯
2
+
ɛ
¯
2
<
1then the origin is the unique, globally asymptotically stable fixed point of (2a) and (2b) relative to the positive quadrant [
0
,
∞
).Note that (35) holds for nontrivial sequences ɛ
n, ɛ
i
,
n of real numbers if σ
¯
1
ϕ
¯
+
σ
¯
2
<
1.Remark 8.
Since in the above discussion the sequencesϵ
n, ϵ
i
,
n,  i
=
1,2, are arbitrary bounded sequences, they can also be sequences of random variables that are drawn from distributions with finite support. For example, ϵ
n, ϵ
i
,
n can be drawn from uniform distribution on some interval [
0
,
θ
] so long as(36)
σ
¯
1
+
θ
ϕ
¯
+
θ
+
σ
¯
2
+
θ
<
1
.
Corollary7 will hold, implying that the origin is globally attracting even in the presence of noise.In the autonomous case where the three parameter functionsσ
1
,
n, σ
2
,
n, ϕ
n do not depend on n at all, we have the following planar system: (37a)
x
n
+
1
=
σ
1
x
n
,
y
n
y
n
+
σ
2
x
n
,
y
n
x
n
,
(37b)
y
n
+
1
=
ϕ
x
n
,
y
n
x
n
.If in Corollary7 we set ɛ
¯
i
,
ɛ
¯
=
0 in (35) then we obtain the following result for the above autonomous system.Corollary 9.
Assume thatσ
1
,
σ
2
,
ϕ
:
[
0
,
∞
)
2
→
[
0
,
∞
) are bounded functions and the following inequality holds:(38)
σ
¯
1
ϕ
¯
+
σ
¯
2
<
1
;then the origin is the unique, globally asymptotically stable fixed point of (37a) and (37b) relative to the positive quadrant [
0
,
∞
)
2.Inequality (38) may be explicitly related to the local asymptotic stability of the origin for (37a) and (37b) when the functions σ
1, σ
2, ϕ are smooth. Consider the associated mapping(39)
F
u
,
v
=
u
σ
u
,
v
+
v
σ
1
u
,
v
,
u
ϕ
u
,
vwhose linearization at (
0,0
) has eigenvalues(40)
λ
±
=
σ
2
0,0
±
σ
2
0,0
2
+
4
σ
1
0,0
ϕ
0,0
2
.These are real and a routine calculation shows thatλ
±
<
1 if(41)
σ
1
0,0
ϕ
0,0
+
σ
2
0,0
<
1
.Under suitable differentiability hypotheses, this inequality is implied by (38) and is equivalent to it if the suprema of σ
2 and σ
1
ϕ occur at (
0,0
).


### 3.2. Folding the System

In the next and later sections it will be convenient to fold system (2a) and (2b) to a second-order equation; see [34] for more details on folding. System (2a) and (2b) in general folds as follows: substitute for y
n
+
1 from (2b) into (2a) to obtain(42)
x
n
+
2
=
σ
1
,
n
+
1
x
n
+
1
,
ϕ
n
x
n
,
h
n
x
n
,
x
n
+
1
x
n
ϕ
n
x
n
,
h
n
x
n
,
x
n
+
1
x
n
+
σ
2
,
n
+
1
x
n
+
1
,
ϕ
n
x
n
,
h
n
x
n
,
x
n
+
1
x
n
x
n
+
1
,where(43)
h
n
x
n
,
x
n
+
1
=
y
nis derived by solving (2a) for y
n. Although an explicit formula for h
n is not feasible in general, it is readily obtained in typical cases; for instance, suppose that σ
2
,
n
(
u
,
v
)
=
σ
2
,
n
(
u
) and σ
1
,
n
(
u
,
v
)
=
σ
1
,
n
(
u
) are both independent of (or constant in) v for all n; note that systems (3a), (3b), (4a), (4b), (5a), (5b), (6a), and (6b) are all of this type. In this case it is clear that(44)
y
n
=
h
n
x
n
,
x
n
+
1
=
x
n
+
1
-
σ
2
,
n
x
n
x
n
σ
1
,
n
x
nand furthermore (42) reduces to(45)
x
n
+
2
=
σ
1
,
n
+
1
x
n
+
1
ϕ
n
x
n
,
x
n
+
1
-
σ
2
,
n
x
n
x
n
σ
1
,
n
x
n
x
n
+
σ
2
,
n
+
1
x
n
+
1
x
n
+
1
.The pair of first-order equations (44) and (45) represents folding of (2a) and (2b). Note that with positive parameter functions, each pair x
0
,
y
0
≥
0 generates an orbit {
(
x
n
,
y
n
)
} of (2a) and (2b) that is in [
0
,
∞
)
2 for all n. So we have x
n
+
1
,
x
n
≥
0 and also by (43) h
n
(
x
n
,
x
n
+
1
)
≥
0 so ϕ
n
(
x
n
,
h
n
(
x
n
,
x
n
+
1
)
) is well defined for every such orbit of (2a) and (2b).Remark 10.
An even simpler reduction than the above is possible ifϕ
n
(
u
,
v
)
=
ϕ
n
(
u
) is independent of (or constant in) v. In this case,(46)
x
n
+
2
=
σ
1
,
n
+
1
x
n
+
1
,
ϕ
n
x
n
x
n
ϕ
n
x
n
x
n
+
σ
2
,
n
+
1
x
n
+
1
,
ϕ
n
x
n
x
n
x
n
+
1and it is not necessary to solve (2a) for y
n implicitly (i.e., the system folds without inversions). Special cases of this type include systems (3a), (3b), (4a), and (4b).


### 3.3. Global Convergence to Zero with Periodic Parameters

The results in this section show that global convergence to zero may occur even if (26) does not hold; see Remark 16 below. Recall from the proof of Theorem 5 that(47)
x
n
+
1
≤
σ
1
,
n
ϕ
¯
n
-
1
x
n
-
1
+
σ
¯
2
,
n
x
n
.The right-hand side of the above inequality is a linear expression. Consider the linear difference equation(48)
u
n
+
1
=
a
n
u
n
+
b
n
u
n
-
1
,
a
n
+
p
1
=
a
n
,
b
n
+
p
2
=
b
n
,where the coefficients a
n and b
n are nonnegative and their periods p
1 and p
2 are positive integers with least common multiple p
=
lcm
⁡
(
p
1
,
p
2
); we say that the linear difference equation (48) is periodic with period p. In this study we assume that(49)
a
n
,
b
n
≥
0
,
n
=
0,1
,
2
,
…
.By Lemma4 every solution of (48) converges to zero if a
n
+
b
n
<
1 for all n. However, it is known that convergence to zero may occur even when a
n
+
b
n exceeds 1 (for infinitely many n in the periodic case). We use the approach in [35] to examine the consequences of this issue when the planar system has periodic parameters. The following result is an immediate consequence of Theorem 13 in [35].Lemma 11.
Assume thatα
j, β
j for j
=
1,2
,
…
,
p are obtained by iteration from (48) from the real initial values:(50)
α
0
=
0
,
α
1
=
1
;
β
0
=
1
,
β
1
=
0
.
Suppose that the quadratic polynomial(51)
α
p
r
2
+
β
p
-
α
p
+
1
r
-
β
p
+
1
=
0is proper, that is, not 0
=
0, and suppose that it has a real root r
1
≠
0. If the recurrence(52)
r
n
+
1
=
a
n
+
b
n
r
ngenerates nonzero real numbers r
2
,
…
,
r
p then {
r
n
}
n
=
1
∞ is periodic with preiod p and yields a triangular system of first-order equations that is equivalent to (48) as follows:(53)
t
n
+
1
=
-
b
n
r
n
t
n
,
t
1
=
u
1
-
r
1
u
0
,
(54)
u
n
+
1
=
r
n
+
1
u
n
+
t
n
+
1
.System (53) and (54) is also known as asemiconjugate factorization of (48); see [36] for an introduction to this concept. The sequence {
r
n
} that is generated by (52) is said to be(unitary) eigensequence of (48). Eigenvalues are essentially constant eigensequences for if p
=
1 in Lemma 11 then (51) reduces to (55)
α
1
r
2
+
β
1
-
α
2
r
-
β
2
=
0
,
r
2
-
a
1
r
-
b
1
=
0and the latter equation is the standard characteristic equation of (48) with constant coefficients; see [35] for more details on the semiconjugate factorization of linear difference equations.Each of (53) and (54) readily yields a solution by iteration as follows:(56)
t
n
=
t
1
-
1
n
-
1
b
1
b
2
⋯
b
n
-
1
r
1
r
2
⋯
r
n
-
1
,
(57)
u
n
=
r
n
r
n
-
1
⋯
r
2
u
1
+
r
n
r
n
-
1
⋯
r
3
t
2
+
⋯
r
n
t
n
-
1
+
t
n
=
r
n
r
n
-
1
⋯
r
2
r
1
u
0
+
∑
i
=
1
n
-
1
r
n
r
n
-
1
⋯
r
i
+
1
t
i
+
t
n
.Lemma 12.
Suppose that the numbersα
n and β
n are defined as in Lemma 11, though we do not assume that (48) is periodic here. Then(a)
β
n
=
0 for all n
≥
2 if and only if b
1
=
0;
(b)
if (49) holds then for all n
≥
2
(58)
α
n
≥
a
1
a
2
⋯
a
n
-
1
,
β
n
≥
b
1
a
2
⋯
a
n
-
1
,
(59)
α
2
n
-
1
≥
b
2
b
4
⋯
b
2
n
-
2
,
β
2
n
≥
b
1
b
3
⋯
b
2
n
-
1
.Proof.
(a) Letb
1
=
0. Then β
2
=
b
1
=
0 and since β
1
=
0 by definition it follows that β
3
=
0. Induction completes the proof that β
n
=
0 if n
≥
2. The converse is obvious since b
1
=
β
2.
(b) Sinceα
2
=
a
1 and β
2
=
b
1 the stated inequalities hold for n
=
2. If (58) is true for some k
≥
2 then (60)
α
k
+
1
=
a
k
α
k
+
b
k
α
k
-
1
≥
a
k
α
k
≥
a
1
a
2
⋯
a
k
-
1
a
k
,
β
k
+
1
=
a
k
β
k
+
b
k
β
k
-
1
≥
a
k
β
k
≥
b
1
a
2
⋯
a
k
-
1
a
k
.
Now, the proof is completed by induction. The proof of (59) is similar since(61)
α
3
=
a
2
α
2
+
b
2
α
1
≥
b
2
,
β
4
=
a
3
β
3
+
b
3
β
2
≥
b
3
b
1and if (59) holds for some k
≥
2 then (62)
α
2
k
+
1
≥
b
2
k
α
2
k
-
1
≥
b
2
b
4
⋯
b
2
k
-
2
b
2
k
,
β
2
k
+
2
≥
b
2
k
+
1
β
2
k
≥
b
1
b
3
⋯
b
2
k
-
1
b
2
k
+
1which establishes the induction step.Lemma 13.
Assume that (49) holds with a
i
>
0 for i
=
1
,
…
,
p and (48) is periodic with period p
≥
2. Then one has the following.
(a) Equation (48) has a positive (hence unitary) eigensequence {
r
n
} of period p.
(b) Ifb
i
>
0 for i
=
1
,
…
,
p then(63)
r
1
r
2
⋯
r
p
=
1
2
α
p
+
1
+
β
p
+
α
p
+
1
-
β
p
2
+
4
α
p
β
p
+
1
.
Hence,r
1
r
2
⋯
r
p
<
1 if(64)
α
p
β
p
+
1
<
1
-
α
p
+
1
1
-
β
p
.
(c) Ifb
i
<
1 for i
=
1
,
…
,
p then r
1
r
2
⋯
r
p
>
b
1
b
2
⋯
b
p.Proof.
(a) Lemma12 shows that α
i
>
0 for i
=
2
,
…
,
p
+
1. Now, either (i) b
1
>
0 or (ii) b
1
=
0. In case (i), the root r
+ of the quadratic polynomial (51) is positive since by Lemma 12  β
p
+
1
>
0 and thus(65)
r
+
=
α
p
+
1
-
β
p
+
α
p
+
1
-
β
p
2
+
4
α
p
β
p
+
1
2
α
p
>
α
p
+
1
-
β
p
+
α
p
+
1
-
β
p
2
α
p
≥
0
.
Ifr
1
=
r
+ then from (52) r
i
=
a
i
-
1
+
b
i
-
1
/
r
i
-
1
≥
a
i
-
1
>
0 for i
=
2
,
…
,
p
+
1. Thus, by Lemma 11, (48) has a unitary (in fact positive) eigensequence of period p. If b
1
=
0 then by Lemma 12  β
p
=
β
p
+
1
=
0 and (51) reduces to(66)
α
p
r
2
-
α
p
+
1
r
=
0which has a root r
+
=
α
p
+
1
/
α
p
>
0. As in the previous case it follows that (48) has a positive eigensequence of period p.
(b) To establish (63), let r
1
=
r
+ and note that (51) can be written as(67)
r
1
=
α
p
+
1
r
1
+
β
p
+
1
α
p
r
1
+
β
p
.
Since{
r
n
} has period p, r
p
+
1
=
r
1 so from (52) and the definition of the numbers α
n and β
n it follows that (68)
a
p
+
b
p
r
p
=
r
p
+
1
=
α
p
+
1
r
1
+
β
p
+
1
α
p
r
1
+
β
p
=
a
p
α
p
+
b
p
α
p
-
1
r
1
+
a
p
β
p
+
b
p
β
p
-
1
α
p
r
1
+
β
p
=
a
p
α
p
r
1
+
β
p
+
b
p
α
p
-
1
r
1
+
β
p
-
1
α
p
r
1
+
β
p
=
a
p
+
b
p
α
p
r
1
+
β
p
/
α
p
-
1
r
1
+
β
p
-
1
.
Sinceb
p
≠
0 it follows that(69)
r
p
=
α
p
r
1
+
β
p
α
p
-
1
r
1
+
β
p
-
1
.
We claim that ifb
i
≠
0 for i
=
1
,
…
,
p then(70)
r
p
-
j
=
α
p
-
j
r
1
+
β
p
-
j
α
p
-
j
-
1
r
1
+
β
p
-
j
-
1
,
j
=
0,1
,
…
,
p
-
2
.
This claim is easily seen to be true by induction; we showed that it is true forj
=
0 and if (70) holds for some j then by (52) (71)
a
p
-
j
-
1
+
b
p
-
j
-
1
r
p
-
j
-
1
=
r
p
-
j
=
a
p
-
j
-
1
α
p
-
j
-
1
+
b
p
-
j
-
1
α
p
-
j
-
2
r
1
+
a
p
-
j
-
1
β
p
-
j
-
1
+
b
p
-
j
-
1
β
p
-
j
-
2
α
p
-
j
-
1
r
1
+
β
p
-
j
-
1
=
a
p
-
j
-
1
α
p
-
j
-
1
r
1
+
β
p
-
j
-
1
+
b
p
-
j
-
1
α
p
-
j
-
2
r
1
+
β
p
-
j
-
2
α
p
-
j
-
1
r
1
+
β
p
-
j
-
1
=
a
p
-
j
-
1
+
b
p
-
j
-
1
α
p
-
j
-
2
r
1
+
β
p
-
j
-
2
α
p
-
j
-
1
r
1
+
β
p
-
j
-
1from which it follows that(72)
r
p
-
j
-
1
=
α
p
-
j
-
1
r
1
+
β
p
-
j
-
1
α
p
-
j
-
2
r
1
+
β
p
-
j
-
2and the induction argument is complete. Now, using (70), we obtain(73)
r
p
r
p
-
1
⋯
r
2
r
1
=
α
p
r
1
+
β
p
α
p
-
1
r
1
+
β
p
-
1
α
p
-
1
r
1
+
β
p
-
1
α
p
-
2
r
1
+
β
p
-
2
⋯
α
2
r
1
+
β
2
α
1
r
1
+
β
1
r
1
=
α
p
r
1
+
β
p
.
Given thatr
1
=
r
+ (73) implies that (74)
r
1
r
2
⋯
r
p
=
α
p
α
p
+
1
-
β
p
+
α
p
+
1
-
β
p
2
+
4
α
p
β
p
+
1
2
α
p
+
β
p
=
1
2
α
p
+
1
+
β
p
+
α
p
+
1
-
β
p
2
+
4
α
p
β
p
+
1and (63) is obtained. Hence, r
1
r
2
⋯
r
p
<
1 if(75)
α
p
+
1
+
β
p
+
α
p
+
1
-
β
p
2
+
4
α
p
β
p
+
1
<
2
.
Upon rearranging terms and squaring,(76)
α
p
+
1
-
β
p
2
+
4
α
p
β
p
+
1
<
4
-
4
α
p
+
1
+
β
p
+
α
p
+
1
+
β
p
2which reduces to (64) after straightforward algebraic manipulations.
(c) First, assume thatp is odd. Then by (59)(77)
α
p
β
p
+
1
=
b
2
b
4
⋯
b
p
-
1
b
1
b
3
⋯
b
p
=
b
1
b
2
⋯
b
pso from (63)(78)
r
1
r
2
⋯
r
p
>
α
p
β
p
+
1
=
b
1
b
2
⋯
b
p
.
Ifb
i
<
1 for i
=
1
,
…
,
p then b
1
b
2
⋯
b
p
<
1 so b
1
b
2
⋯
b
p
>
b
1
b
2
⋯
b
p as required. Now let p be even. Then from (63) and (59)(79)
r
1
r
2
⋯
r
p
>
α
p
+
1
+
β
p
2
≥
b
2
b
4
⋯
b
p
+
b
1
b
3
⋯
b
p
-
1
2
.
Ifb
i
<
1 for i
=
1
,
…
,
p then b
2
b
4
⋯
b
p
≥
b
1
b
2
⋯
b
p and b
1
b
3
⋯
b
p
-
1
≥
b
1
b
2
⋯
b
p and the proof is complete.Some of the numbersa
i may exceed 1 in Lemma 13 without affecting the conclusions of the lemma. Additionally, not all the conditions in Lemma 13 are necessary. For instance, if b
1
=
0 then Lemma 13(c) holds trivially. Additionally, by Lemma 12(a), β
n
=
0 for n
≥
2 so the following equality must hold instead of (63):(80)
r
1
r
2
⋯
r
p
=
α
p
+
1
.This is in fact true becauser
1
=
r
+
=
α
p
+
1
/
α
p so repeating the argument in the proof of Lemma 13(b) yields r
p
-
j
=
α
p
-
j
/
α
p
-
j
-
1 for j
=
0,1
,
…
,
p
-
2. Hence(81)
r
p
r
p
-
1
⋯
r
2
r
1
=
α
p
α
p
-
1
α
p
-
1
α
p
-
2
⋯
α
2
α
1
α
p
+
1
α
p
=
α
p
+
1as claimed. These observations establish the following version of Lemma 13.Lemma 14.
Leta
i
>
0 and let b
i
≥
0 for i
=
1
,
…
,
p with b
1
=
0. Then the linear equation (48) has a positive (hence unitary) eigensequence {
r
n
} of period p given by(82)
r
1
=
α
p
+
1
α
p
,
r
j
=
α
j
α
j
-
1
,
j
=
2
,
…
,
pand 0
=
b
1
b
2
⋯
b
p
<
r
1
r
2
⋯
r
p
<
1 if α
p
+
1
<
1.In Lemma14 some of the numbers a
i or b
i may exceed 1 without affecting the conclusions of the lemma.Theorem 15.
Assume that (27) holds and the sequences and {
σ
¯
1
,
n
ϕ
¯
n
-
1
} and {
σ
¯
2
,
n
} have period p with σ
¯
2
,
i
>
0 and σ
¯
1
,
i
ϕ
¯
i
-
1
≥
0 for i
=
1
,
…
,
p. Additionally let the numbers α
n
,
β
n be as previously defined with a
n
=
σ
¯
2
,
n and b
n
=
σ
¯
1
,
n
ϕ
¯
n
-
1. All nonnegative orbits of the planar system converge to (
0,0
) if either one of the following holds:(a)
0
<
σ
¯
1
,
i
ϕ
¯
i
-
1
<
1 and (64) holds.
(b)
σ
¯
1,1
ϕ
¯
0
=
0 and α
p
+
1
<
1.Proof.
Let{
u
n
} be a solution of the linear equation (48) with a
n
=
σ
¯
2
,
n, b
n
=
σ
¯
1
,
n
ϕ
¯
n
-
1, u
0
=
x
0, and u
1
=
x
1. Then by (47) (83)
x
2
≤
σ
¯
1,1
ϕ
¯
0
x
0
+
σ
¯
2,1
x
1
=
σ
¯
1,1
ϕ
¯
0
u
0
+
σ
¯
2,1
u
1
=
u
2
,
x
3
≤
σ
¯
1,2
ϕ
¯
2
x
2
+
σ
¯
2,2
x
2
≤
σ
¯
1,2
ϕ
¯
1
u
1
+
σ
¯
2,2
u
2
=
u
3
.
By induction it follows thatx
n
≤
u
n. If (64) holds then, by Lemma 13, l
i
m
n
→
∞
u
n
=
0 so {
x
n
} converges to 0. Furthermore, l
i
m
n
→
∞
y
n
=
0 as in the proof of Theorem 5 and the proof is complete.Remark 16.
(
1
) Condition (64) involves the numbers α
j, β
j rather than the coefficients of (48) directly. In the case of period p
=
2 the role of a
i and b
i is more apparent. Inequality (64) in this case is (84)
α
2
β
3
<
1
-
α
3
1
-
β
2
,
a
1
a
2
b
1
<
1
-
b
2
-
a
1
a
2
1
-
b
1and simple manipulations reduce the last inequality to(85)
a
1
a
2
<
1
-
b
1
1
-
b
2
.
(
2
) Inequality (85) holds even if a
1
>
1 or a
2
>
1 thus showing how global convergence to (
0,0
) my occur when (26) does not hold. Furthermore, it is possible that (85) holds together with(86)
a
1
+
a
2
2
>
1
.
Note that (85) holds even with arbitrarily large mean value in (86) if say a
1
→
0 as a
2
→
∞. In population models this implies that if (85) holds with a
n
=
σ
¯
2
,
n and b
n
=
σ
¯
1
,
n
ϕ
¯
n
-
1 then extinction may still occur after restocking the adult population to raise the mean value of the composite parameter σ
¯
2
,
n above 1 by a wide margin.
(
3
) In Theorem 15 the individual sequences σ
¯
1
,
n, ϕ
¯
n need not be periodic or even bounded. Therefore, the theorem applies to (2a) and (2b) even if the system itself is not periodic as long as the combination σ
¯
1
,
n
ϕ
¯
n
-
1 of parameters is periodic along with σ
¯
2
,
n.


## 3.1. General Results

We start with the following lemma; see [33] for the proof and some background on this result.Lemma 4.
Letα
∈
(
0,1
) and assume that the functions f
n
:
0
,
∞
k
+
1
→
0
,
∞ satisfy the inequality(22)
f
n
u
0
,
…
,
u
k
≤
α
max
⁡
u
0
,
…
,
u
kfor all (
u
0
,
…
,
u
k
)
∈
[
0
,
∞
) and all n
≥
0. Then for every solution {
x
n
} of the difference equation(23)
x
n
+
1
=
f
n
x
n
,
x
n
-
1
,
…
,
x
n
-
kthe following is true:(24)
x
n
≤
α
n
/
k
+
1
max
⁡
x
0
,
x
-
1
,
…
,
x
-
k
.Note that (22) implies that x
n
=
0 is a constant solution of (23) and furthermore (24) implies that this solution is globally exponentially stable.Throughout this section we assume thatσ
i
,
n, ϕ
n are all bounded functions for i
=
1,2 and every n
=
0,1
,
2
,
…. Then the following are well-defined sequences of real numbers:(25)
σ
¯
i
,
n
=
sup
u
,
v
≥
0
⁡
σ
i
,
n
u
,
v
,
ϕ
¯
n
=
sup
u
,
v
≥
0
⁡
ϕ
n
u
,
v
.Theorem 5.
If the inequality(26)
limsup
n
→
∞
⁡
σ
¯
1
,
n
ϕ
¯
n
-
1
+
σ
¯
2
,
n
<
1holds, then l
i
m
n
→
∞
x
n
=
0 for every orbit {
(
x
n
,
y
n
)
} of the planar system (2a) and (2b) in the positive quadrant 0
,
∞
2. If also either the sequence {
ϕ
¯
n
} is bounded or the inequality(27)
liminf
n
→
∞
⁡
σ
¯
1
,
n
>
0
,holds, then every orbit of (2a) and (2b) converges to (
0,0
).Proof.
By (26) there is δ
∈
(
0,1
) such that σ
¯
1
,
n
ϕ
¯
n
-
1
+
σ
¯
2
,
n
≤
δ for all (large) n. From (2a)(28)
y
n
≤
ϕ
¯
n
-
1
x
n
-
1so for all (large) n (2b) yields(29)
x
n
+
1
≤
ϕ
¯
n
-
1
σ
¯
1
,
n
x
n
-
1
+
σ
¯
2
,
n
x
n
≤
σ
¯
1
,
n
ϕ
¯
n
-
1
+
σ
¯
2
,
n
max
⁡
x
n
,
x
n
-
1
≤
δ
max
⁡
x
n
,
x
n
-
1
.
Lemma4 now implies that l
i
m
n
→
∞
x
n
=
0. Furthermore either by hypothesis there is a positive number μ such that ϕ
¯
n
≤
μ or by (27) there is a positive number ρ such that σ
¯
1
,
n
≥
ρ for all (large) n so that(30)
ϕ
¯
n
-
1
≤
δ
-
σ
¯
2
,
n
σ
¯
1
,
n
≤
δ
ρfor all sufficiently large values of n
. Now, if M
=
μ or M
=
δ
/
ρ as the case may be, then from (2b) in the planar system we see that(31)
lim
n
→
∞
⁡
y
n
≤
lim
n
→
∞
⁡
ϕ
¯
n
-
1
x
n
-
1
≤
M
lim
n
→
∞
⁡
x
n
-
1
=
0and the proof is complete.Remark 6.
(
1
) Theorem 5 is valid even if the separate sequences {
σ
1
,
n
} or {
ϕ
¯
n
} are not bounded by 1 as long as for all n large enough σ
¯
1
,
n
ϕ
¯
n
-
1
≤
δ
-
σ
¯
2
,
n.
(
2
) If (26) is satisfied but {
ϕ
¯
n
} is unbounded and {
σ
¯
1
,
n
} does not satisfy (27) then y
n may not converge to 0; see the example following Corollary 18 below.We consider an application of Theorem5 to “noisy” autonomous system next. Let ɛ
n, ɛ
i
,
n, i
=
1,2, be bounded sequences of real numbers and let(32)
ɛ
¯
=
sup
n
≥
1
⁡
ɛ
n
,
ɛ
¯
i
=
sup
n
≥
1
⁡
ɛ
i
,
n
,
i
=
1,2
.Additionally, letσ
1
,
σ
2
,
ϕ
:
[
0
,
∞
)
2
→
[
0
,
∞
) be bounded functions and denote their supremums over [
0
,
∞
)
2 by σ
¯
1, σ
¯
2, ϕ
¯, respectively. If in (2a) and (2b) we have(33)
ϕ
n
x
n
,
y
n
=
ϕ
x
n
,
y
n
+
ɛ
n
,
σ
i
,
n
x
n
,
y
n
=
σ
i
x
n
,
y
n
+
ɛ
i
,
n
,
i
=
1,2then we refer to (2a) and (2b) as an autonomous system with low-amplitude disturbances or fluctuations in the rates σ
1, σ
2, ϕ, assuming that all three of these are positive functions and for all u
,
v
≥
0
(34)
ɛ
¯
≤
ϕ
u
,
v
,
ɛ
¯
i
≤
σ
i
u
,
v
,
i
=
1,2
.These inequalities ensure that the functionsϕ
n and σ
i
,
n are positive, as required for (2a) and (2b).Corollary 7.
Suppose that (2a) and (2b) is an autonomous system with low-amplitude disturbances or fluctuations in the above sense. If(35)
σ
¯
1
+
ɛ
¯
1
ϕ
¯
+
ɛ
¯
+
σ
¯
2
+
ɛ
¯
2
<
1then the origin is the unique, globally asymptotically stable fixed point of (2a) and (2b) relative to the positive quadrant [
0
,
∞
).Note that (35) holds for nontrivial sequences ɛ
n, ɛ
i
,
n of real numbers if σ
¯
1
ϕ
¯
+
σ
¯
2
<
1.Remark 8.
Since in the above discussion the sequencesϵ
n, ϵ
i
,
n,  i
=
1,2, are arbitrary bounded sequences, they can also be sequences of random variables that are drawn from distributions with finite support. For example, ϵ
n, ϵ
i
,
n can be drawn from uniform distribution on some interval [
0
,
θ
] so long as(36)
σ
¯
1
+
θ
ϕ
¯
+
θ
+
σ
¯
2
+
θ
<
1
.
Corollary7 will hold, implying that the origin is globally attracting even in the presence of noise.In the autonomous case where the three parameter functionsσ
1
,
n, σ
2
,
n, ϕ
n do not depend on n at all, we have the following planar system: (37a)
x
n
+
1
=
σ
1
x
n
,
y
n
y
n
+
σ
2
x
n
,
y
n
x
n
,
(37b)
y
n
+
1
=
ϕ
x
n
,
y
n
x
n
.If in Corollary7 we set ɛ
¯
i
,
ɛ
¯
=
0 in (35) then we obtain the following result for the above autonomous system.Corollary 9.
Assume thatσ
1
,
σ
2
,
ϕ
:
[
0
,
∞
)
2
→
[
0
,
∞
) are bounded functions and the following inequality holds:(38)
σ
¯
1
ϕ
¯
+
σ
¯
2
<
1
;then the origin is the unique, globally asymptotically stable fixed point of (37a) and (37b) relative to the positive quadrant [
0
,
∞
)
2.Inequality (38) may be explicitly related to the local asymptotic stability of the origin for (37a) and (37b) when the functions σ
1, σ
2, ϕ are smooth. Consider the associated mapping(39)
F
u
,
v
=
u
σ
u
,
v
+
v
σ
1
u
,
v
,
u
ϕ
u
,
vwhose linearization at (
0,0
) has eigenvalues(40)
λ
±
=
σ
2
0,0
±
σ
2
0,0
2
+
4
σ
1
0,0
ϕ
0,0
2
.These are real and a routine calculation shows thatλ
±
<
1 if(41)
σ
1
0,0
ϕ
0,0
+
σ
2
0,0
<
1
.Under suitable differentiability hypotheses, this inequality is implied by (38) and is equivalent to it if the suprema of σ
2 and σ
1
ϕ occur at (
0,0
).


## 3.2. Folding the System

In the next and later sections it will be convenient to fold system (2a) and (2b) to a second-order equation; see [34] for more details on folding. System (2a) and (2b) in general folds as follows: substitute for y
n
+
1 from (2b) into (2a) to obtain(42)
x
n
+
2
=
σ
1
,
n
+
1
x
n
+
1
,
ϕ
n
x
n
,
h
n
x
n
,
x
n
+
1
x
n
ϕ
n
x
n
,
h
n
x
n
,
x
n
+
1
x
n
+
σ
2
,
n
+
1
x
n
+
1
,
ϕ
n
x
n
,
h
n
x
n
,
x
n
+
1
x
n
x
n
+
1
,where(43)
h
n
x
n
,
x
n
+
1
=
y
nis derived by solving (2a) for y
n. Although an explicit formula for h
n is not feasible in general, it is readily obtained in typical cases; for instance, suppose that σ
2
,
n
(
u
,
v
)
=
σ
2
,
n
(
u
) and σ
1
,
n
(
u
,
v
)
=
σ
1
,
n
(
u
) are both independent of (or constant in) v for all n; note that systems (3a), (3b), (4a), (4b), (5a), (5b), (6a), and (6b) are all of this type. In this case it is clear that(44)
y
n
=
h
n
x
n
,
x
n
+
1
=
x
n
+
1
-
σ
2
,
n
x
n
x
n
σ
1
,
n
x
nand furthermore (42) reduces to(45)
x
n
+
2
=
σ
1
,
n
+
1
x
n
+
1
ϕ
n
x
n
,
x
n
+
1
-
σ
2
,
n
x
n
x
n
σ
1
,
n
x
n
x
n
+
σ
2
,
n
+
1
x
n
+
1
x
n
+
1
.The pair of first-order equations (44) and (45) represents folding of (2a) and (2b). Note that with positive parameter functions, each pair x
0
,
y
0
≥
0 generates an orbit {
(
x
n
,
y
n
)
} of (2a) and (2b) that is in [
0
,
∞
)
2 for all n. So we have x
n
+
1
,
x
n
≥
0 and also by (43) h
n
(
x
n
,
x
n
+
1
)
≥
0 so ϕ
n
(
x
n
,
h
n
(
x
n
,
x
n
+
1
)
) is well defined for every such orbit of (2a) and (2b).Remark 10.
An even simpler reduction than the above is possible ifϕ
n
(
u
,
v
)
=
ϕ
n
(
u
) is independent of (or constant in) v. In this case,(46)
x
n
+
2
=
σ
1
,
n
+
1
x
n
+
1
,
ϕ
n
x
n
x
n
ϕ
n
x
n
x
n
+
σ
2
,
n
+
1
x
n
+
1
,
ϕ
n
x
n
x
n
x
n
+
1and it is not necessary to solve (2a) for y
n implicitly (i.e., the system folds without inversions). Special cases of this type include systems (3a), (3b), (4a), and (4b).


## 3.3. Global Convergence to Zero with Periodic Parameters

The results in this section show that global convergence to zero may occur even if (26) does not hold; see Remark 16 below. Recall from the proof of Theorem 5 that(47)
x
n
+
1
≤
σ
1
,
n
ϕ
¯
n
-
1
x
n
-
1
+
σ
¯
2
,
n
x
n
.The right-hand side of the above inequality is a linear expression. Consider the linear difference equation(48)
u
n
+
1
=
a
n
u
n
+
b
n
u
n
-
1
,
a
n
+
p
1
=
a
n
,
b
n
+
p
2
=
b
n
,where the coefficients a
n and b
n are nonnegative and their periods p
1 and p
2 are positive integers with least common multiple p
=
lcm
⁡
(
p
1
,
p
2
); we say that the linear difference equation (48) is periodic with period p. In this study we assume that(49)
a
n
,
b
n
≥
0
,
n
=
0,1
,
2
,
…
.By Lemma4 every solution of (48) converges to zero if a
n
+
b
n
<
1 for all n. However, it is known that convergence to zero may occur even when a
n
+
b
n exceeds 1 (for infinitely many n in the periodic case). We use the approach in [35] to examine the consequences of this issue when the planar system has periodic parameters. The following result is an immediate consequence of Theorem 13 in [35].Lemma 11.
Assume thatα
j, β
j for j
=
1,2
,
…
,
p are obtained by iteration from (48) from the real initial values:(50)
α
0
=
0
,
α
1
=
1
;
β
0
=
1
,
β
1
=
0
.
Suppose that the quadratic polynomial(51)
α
p
r
2
+
β
p
-
α
p
+
1
r
-
β
p
+
1
=
0is proper, that is, not 0
=
0, and suppose that it has a real root r
1
≠
0. If the recurrence(52)
r
n
+
1
=
a
n
+
b
n
r
ngenerates nonzero real numbers r
2
,
…
,
r
p then {
r
n
}
n
=
1
∞ is periodic with preiod p and yields a triangular system of first-order equations that is equivalent to (48) as follows:(53)
t
n
+
1
=
-
b
n
r
n
t
n
,
t
1
=
u
1
-
r
1
u
0
,
(54)
u
n
+
1
=
r
n
+
1
u
n
+
t
n
+
1
.System (53) and (54) is also known as asemiconjugate factorization of (48); see [36] for an introduction to this concept. The sequence {
r
n
} that is generated by (52) is said to be(unitary) eigensequence of (48). Eigenvalues are essentially constant eigensequences for if p
=
1 in Lemma 11 then (51) reduces to (55)
α
1
r
2
+
β
1
-
α
2
r
-
β
2
=
0
,
r
2
-
a
1
r
-
b
1
=
0and the latter equation is the standard characteristic equation of (48) with constant coefficients; see [35] for more details on the semiconjugate factorization of linear difference equations.Each of (53) and (54) readily yields a solution by iteration as follows:(56)
t
n
=
t
1
-
1
n
-
1
b
1
b
2
⋯
b
n
-
1
r
1
r
2
⋯
r
n
-
1
,
(57)
u
n
=
r
n
r
n
-
1
⋯
r
2
u
1
+
r
n
r
n
-
1
⋯
r
3
t
2
+
⋯
r
n
t
n
-
1
+
t
n
=
r
n
r
n
-
1
⋯
r
2
r
1
u
0
+
∑
i
=
1
n
-
1
r
n
r
n
-
1
⋯
r
i
+
1
t
i
+
t
n
.Lemma 12.
Suppose that the numbersα
n and β
n are defined as in Lemma 11, though we do not assume that (48) is periodic here. Then(a)
β
n
=
0 for all n
≥
2 if and only if b
1
=
0;
(b)
if (49) holds then for all n
≥
2
(58)
α
n
≥
a
1
a
2
⋯
a
n
-
1
,
β
n
≥
b
1
a
2
⋯
a
n
-
1
,
(59)
α
2
n
-
1
≥
b
2
b
4
⋯
b
2
n
-
2
,
β
2
n
≥
b
1
b
3
⋯
b
2
n
-
1
.Proof.
(a) Letb
1
=
0. Then β
2
=
b
1
=
0 and since β
1
=
0 by definition it follows that β
3
=
0. Induction completes the proof that β
n
=
0 if n
≥
2. The converse is obvious since b
1
=
β
2.
(b) Sinceα
2
=
a
1 and β
2
=
b
1 the stated inequalities hold for n
=
2. If (58) is true for some k
≥
2 then (60)
α
k
+
1
=
a
k
α
k
+
b
k
α
k
-
1
≥
a
k
α
k
≥
a
1
a
2
⋯
a
k
-
1
a
k
,
β
k
+
1
=
a
k
β
k
+
b
k
β
k
-
1
≥
a
k
β
k
≥
b
1
a
2
⋯
a
k
-
1
a
k
.
Now, the proof is completed by induction. The proof of (59) is similar since(61)
α
3
=
a
2
α
2
+
b
2
α
1
≥
b
2
,
β
4
=
a
3
β
3
+
b
3
β
2
≥
b
3
b
1and if (59) holds for some k
≥
2 then (62)
α
2
k
+
1
≥
b
2
k
α
2
k
-
1
≥
b
2
b
4
⋯
b
2
k
-
2
b
2
k
,
β
2
k
+
2
≥
b
2
k
+
1
β
2
k
≥
b
1
b
3
⋯
b
2
k
-
1
b
2
k
+
1which establishes the induction step.Lemma 13.
Assume that (49) holds with a
i
>
0 for i
=
1
,
…
,
p and (48) is periodic with period p
≥
2. Then one has the following.
(a) Equation (48) has a positive (hence unitary) eigensequence {
r
n
} of period p.
(b) Ifb
i
>
0 for i
=
1
,
…
,
p then(63)
r
1
r
2
⋯
r
p
=
1
2
α
p
+
1
+
β
p
+
α
p
+
1
-
β
p
2
+
4
α
p
β
p
+
1
.
Hence,r
1
r
2
⋯
r
p
<
1 if(64)
α
p
β
p
+
1
<
1
-
α
p
+
1
1
-
β
p
.
(c) Ifb
i
<
1 for i
=
1
,
…
,
p then r
1
r
2
⋯
r
p
>
b
1
b
2
⋯
b
p.Proof.
(a) Lemma12 shows that α
i
>
0 for i
=
2
,
…
,
p
+
1. Now, either (i) b
1
>
0 or (ii) b
1
=
0. In case (i), the root r
+ of the quadratic polynomial (51) is positive since by Lemma 12  β
p
+
1
>
0 and thus(65)
r
+
=
α
p
+
1
-
β
p
+
α
p
+
1
-
β
p
2
+
4
α
p
β
p
+
1
2
α
p
>
α
p
+
1
-
β
p
+
α
p
+
1
-
β
p
2
α
p
≥
0
.
Ifr
1
=
r
+ then from (52) r
i
=
a
i
-
1
+
b
i
-
1
/
r
i
-
1
≥
a
i
-
1
>
0 for i
=
2
,
…
,
p
+
1. Thus, by Lemma 11, (48) has a unitary (in fact positive) eigensequence of period p. If b
1
=
0 then by Lemma 12  β
p
=
β
p
+
1
=
0 and (51) reduces to(66)
α
p
r
2
-
α
p
+
1
r
=
0which has a root r
+
=
α
p
+
1
/
α
p
>
0. As in the previous case it follows that (48) has a positive eigensequence of period p.
(b) To establish (63), let r
1
=
r
+ and note that (51) can be written as(67)
r
1
=
α
p
+
1
r
1
+
β
p
+
1
α
p
r
1
+
β
p
.
Since{
r
n
} has period p, r
p
+
1
=
r
1 so from (52) and the definition of the numbers α
n and β
n it follows that (68)
a
p
+
b
p
r
p
=
r
p
+
1
=
α
p
+
1
r
1
+
β
p
+
1
α
p
r
1
+
β
p
=
a
p
α
p
+
b
p
α
p
-
1
r
1
+
a
p
β
p
+
b
p
β
p
-
1
α
p
r
1
+
β
p
=
a
p
α
p
r
1
+
β
p
+
b
p
α
p
-
1
r
1
+
β
p
-
1
α
p
r
1
+
β
p
=
a
p
+
b
p
α
p
r
1
+
β
p
/
α
p
-
1
r
1
+
β
p
-
1
.
Sinceb
p
≠
0 it follows that(69)
r
p
=
α
p
r
1
+
β
p
α
p
-
1
r
1
+
β
p
-
1
.
We claim that ifb
i
≠
0 for i
=
1
,
…
,
p then(70)
r
p
-
j
=
α
p
-
j
r
1
+
β
p
-
j
α
p
-
j
-
1
r
1
+
β
p
-
j
-
1
,
j
=
0,1
,
…
,
p
-
2
.
This claim is easily seen to be true by induction; we showed that it is true forj
=
0 and if (70) holds for some j then by (52) (71)
a
p
-
j
-
1
+
b
p
-
j
-
1
r
p
-
j
-
1
=
r
p
-
j
=
a
p
-
j
-
1
α
p
-
j
-
1
+
b
p
-
j
-
1
α
p
-
j
-
2
r
1
+
a
p
-
j
-
1
β
p
-
j
-
1
+
b
p
-
j
-
1
β
p
-
j
-
2
α
p
-
j
-
1
r
1
+
β
p
-
j
-
1
=
a
p
-
j
-
1
α
p
-
j
-
1
r
1
+
β
p
-
j
-
1
+
b
p
-
j
-
1
α
p
-
j
-
2
r
1
+
β
p
-
j
-
2
α
p
-
j
-
1
r
1
+
β
p
-
j
-
1
=
a
p
-
j
-
1
+
b
p
-
j
-
1
α
p
-
j
-
2
r
1
+
β
p
-
j
-
2
α
p
-
j
-
1
r
1
+
β
p
-
j
-
1from which it follows that(72)
r
p
-
j
-
1
=
α
p
-
j
-
1
r
1
+
β
p
-
j
-
1
α
p
-
j
-
2
r
1
+
β
p
-
j
-
2and the induction argument is complete. Now, using (70), we obtain(73)
r
p
r
p
-
1
⋯
r
2
r
1
=
α
p
r
1
+
β
p
α
p
-
1
r
1
+
β
p
-
1
α
p
-
1
r
1
+
β
p
-
1
α
p
-
2
r
1
+
β
p
-
2
⋯
α
2
r
1
+
β
2
α
1
r
1
+
β
1
r
1
=
α
p
r
1
+
β
p
.
Given thatr
1
=
r
+ (73) implies that (74)
r
1
r
2
⋯
r
p
=
α
p
α
p
+
1
-
β
p
+
α
p
+
1
-
β
p
2
+
4
α
p
β
p
+
1
2
α
p
+
β
p
=
1
2
α
p
+
1
+
β
p
+
α
p
+
1
-
β
p
2
+
4
α
p
β
p
+
1and (63) is obtained. Hence, r
1
r
2
⋯
r
p
<
1 if(75)
α
p
+
1
+
β
p
+
α
p
+
1
-
β
p
2
+
4
α
p
β
p
+
1
<
2
.
Upon rearranging terms and squaring,(76)
α
p
+
1
-
β
p
2
+
4
α
p
β
p
+
1
<
4
-
4
α
p
+
1
+
β
p
+
α
p
+
1
+
β
p
2which reduces to (64) after straightforward algebraic manipulations.
(c) First, assume thatp is odd. Then by (59)(77)
α
p
β
p
+
1
=
b
2
b
4
⋯
b
p
-
1
b
1
b
3
⋯
b
p
=
b
1
b
2
⋯
b
pso from (63)(78)
r
1
r
2
⋯
r
p
>
α
p
β
p
+
1
=
b
1
b
2
⋯
b
p
.
Ifb
i
<
1 for i
=
1
,
…
,
p then b
1
b
2
⋯
b
p
<
1 so b
1
b
2
⋯
b
p
>
b
1
b
2
⋯
b
p as required. Now let p be even. Then from (63) and (59)(79)
r
1
r
2
⋯
r
p
>
α
p
+
1
+
β
p
2
≥
b
2
b
4
⋯
b
p
+
b
1
b
3
⋯
b
p
-
1
2
.
Ifb
i
<
1 for i
=
1
,
…
,
p then b
2
b
4
⋯
b
p
≥
b
1
b
2
⋯
b
p and b
1
b
3
⋯
b
p
-
1
≥
b
1
b
2
⋯
b
p and the proof is complete.Some of the numbersa
i may exceed 1 in Lemma 13 without affecting the conclusions of the lemma. Additionally, not all the conditions in Lemma 13 are necessary. For instance, if b
1
=
0 then Lemma 13(c) holds trivially. Additionally, by Lemma 12(a), β
n
=
0 for n
≥
2 so the following equality must hold instead of (63):(80)
r
1
r
2
⋯
r
p
=
α
p
+
1
.This is in fact true becauser
1
=
r
+
=
α
p
+
1
/
α
p so repeating the argument in the proof of Lemma 13(b) yields r
p
-
j
=
α
p
-
j
/
α
p
-
j
-
1 for j
=
0,1
,
…
,
p
-
2. Hence(81)
r
p
r
p
-
1
⋯
r
2
r
1
=
α
p
α
p
-
1
α
p
-
1
α
p
-
2
⋯
α
2
α
1
α
p
+
1
α
p
=
α
p
+
1as claimed. These observations establish the following version of Lemma 13.Lemma 14.
Leta
i
>
0 and let b
i
≥
0 for i
=
1
,
…
,
p with b
1
=
0. Then the linear equation (48) has a positive (hence unitary) eigensequence {
r
n
} of period p given by(82)
r
1
=
α
p
+
1
α
p
,
r
j
=
α
j
α
j
-
1
,
j
=
2
,
…
,
pand 0
=
b
1
b
2
⋯
b
p
<
r
1
r
2
⋯
r
p
<
1 if α
p
+
1
<
1.In Lemma14 some of the numbers a
i or b
i may exceed 1 without affecting the conclusions of the lemma.Theorem 15.
Assume that (27) holds and the sequences and {
σ
¯
1
,
n
ϕ
¯
n
-
1
} and {
σ
¯
2
,
n
} have period p with σ
¯
2
,
i
>
0 and σ
¯
1
,
i
ϕ
¯
i
-
1
≥
0 for i
=
1
,
…
,
p. Additionally let the numbers α
n
,
β
n be as previously defined with a
n
=
σ
¯
2
,
n and b
n
=
σ
¯
1
,
n
ϕ
¯
n
-
1. All nonnegative orbits of the planar system converge to (
0,0
) if either one of the following holds:(a)
0
<
σ
¯
1
,
i
ϕ
¯
i
-
1
<
1 and (64) holds.
(b)
σ
¯
1,1
ϕ
¯
0
=
0 and α
p
+
1
<
1.Proof.
Let{
u
n
} be a solution of the linear equation (48) with a
n
=
σ
¯
2
,
n, b
n
=
σ
¯
1
,
n
ϕ
¯
n
-
1, u
0
=
x
0, and u
1
=
x
1. Then by (47) (83)
x
2
≤
σ
¯
1,1
ϕ
¯
0
x
0
+
σ
¯
2,1
x
1
=
σ
¯
1,1
ϕ
¯
0
u
0
+
σ
¯
2,1
u
1
=
u
2
,
x
3
≤
σ
¯
1,2
ϕ
¯
2
x
2
+
σ
¯
2,2
x
2
≤
σ
¯
1,2
ϕ
¯
1
u
1
+
σ
¯
2,2
u
2
=
u
3
.
By induction it follows thatx
n
≤
u
n. If (64) holds then, by Lemma 13, l
i
m
n
→
∞
u
n
=
0 so {
x
n
} converges to 0. Furthermore, l
i
m
n
→
∞
y
n
=
0 as in the proof of Theorem 5 and the proof is complete.Remark 16.
(
1
) Condition (64) involves the numbers α
j, β
j rather than the coefficients of (48) directly. In the case of period p
=
2 the role of a
i and b
i is more apparent. Inequality (64) in this case is (84)
α
2
β
3
<
1
-
α
3
1
-
β
2
,
a
1
a
2
b
1
<
1
-
b
2
-
a
1
a
2
1
-
b
1and simple manipulations reduce the last inequality to(85)
a
1
a
2
<
1
-
b
1
1
-
b
2
.
(
2
) Inequality (85) holds even if a
1
>
1 or a
2
>
1 thus showing how global convergence to (
0,0
) my occur when (26) does not hold. Furthermore, it is possible that (85) holds together with(86)
a
1
+
a
2
2
>
1
.
Note that (85) holds even with arbitrarily large mean value in (86) if say a
1
→
0 as a
2
→
∞. In population models this implies that if (85) holds with a
n
=
σ
¯
2
,
n and b
n
=
σ
¯
1
,
n
ϕ
¯
n
-
1 then extinction may still occur after restocking the adult population to raise the mean value of the composite parameter σ
¯
2
,
n above 1 by a wide margin.
(
3
) In Theorem 15 the individual sequences σ
¯
1
,
n, ϕ
¯
n need not be periodic or even bounded. Therefore, the theorem applies to (2a) and (2b) even if the system itself is not periodic as long as the combination σ
¯
1
,
n
ϕ
¯
n
-
1 of parameters is periodic along with σ
¯
2
,
n.


## 4. Dynamics of a Beverton-Holt Type Rational System

In this section we apply some of the preceding results and obtain some new ones to study boundedness, extinction, and modes of survival in some rational special cases of (2a) and (2b). In population models these types of systems include the Beverton-Holt type interactions. Specifically, we consider the following nonautonomous system and some of its special cases:(87a)
x
n
+
1
=
α
1
,
n
y
n
1
+
β
1
,
n
x
n
+
γ
1
,
n
y
n
+
α
2
,
n
x
n
1
+
β
2
,
n
x
n
+
γ
2
,
n
y
n
,
(87b)
y
n
+
1
=
b
n
x
n
1
+
c
1
,
n
x
n
+
c
2
,
n
y
n
,where we assume that for all n
≥
0 and i
=
1,2
(88)
α
1
,
n
>
0
,
b
n
,
α
2
,
n
,
β
i
,
n
,
γ
i
,
n
,
c
i
,
n
≥
0
b
n
>
0
for  infinitely  many
n
.For example, if we think ofα
i as the natural survival rates then the population model (3a) and (3b) is a special case of (87a) and (87b). If we allow α
i to include additional factors such as harvesting rates then (87a) and (87b) is an extension of the model in [11] (with a Beverton-Holt recruitment function) in the sense that the competition coefficients β
i
,
n, γ
i
,
n, and c
i
,
n may be nonzero as well as time-dependent.


### 4.1. Uniform Boundedness and Extinction

We now examine boundedness and global convergence to 0 (extinction) in (87a) and (87b). The next result is in part a consequence of Corollary 3.Corollary 17.
Assume that (88) holds.
(a) Let the sequence{
α
1
,
n
} be bounded and l
i
m
s
u
p
n
→
∞
α
2
,
n
<
1. If there is M
0
>
0 such that b
n
≤
M
0
c
1
,
n for all n larger than a given positive integer then all orbits of (87a) and (87b) are uniformly bounded.
(b) Let the sequence{
b
n
} be bounded and suppose that there is M
>
0 such that(89)
α
1
,
n
≤
M
γ
1
,
n
,
α
2
,
n
≤
M
β
2
,
nfor all n larger than a given positive integer. Then all orbits of (87a) and (87b) are uniformly bounded.Proof.
(a) By hypothesis, for all (large)n,(90)
b
n
x
n
1
+
c
1
,
n
x
n
+
c
2
,
n
y
n
≤
M
0
c
1
,
n
x
n
1
+
c
1
,
n
x
n
+
c
2
,
n
y
n
<
M
0
.
Next, let(91)
σ
1
,
n
u
,
v
=
α
1
,
n
1
+
β
1
,
n
u
+
γ
1
,
n
v
,
σ
2
,
n
u
,
v
=
α
2
,
n
1
+
β
2
,
n
u
+
γ
2
,
n
v
.
By hypothesis, there isM
1
>
0 and δ
∈
(
0,1
) such that for all u
,
v
≥
0 and all sufficiently large values of n
(92)
σ
1
,
n
u
,
v
≤
α
1
,
n
≤
M
1
,
σ
2
,
n
u
,
v
≤
α
2
,
n
≤
δ
.
Now an application of Corollary3 completes the proof of (a).
(b) By (89) for all large n it follows that(93)
α
1
,
n
y
n
1
+
β
1
,
n
x
n
+
γ
1
,
n
y
n
≤
M
γ
1
,
n
y
n
1
+
β
1
,
n
x
n
+
γ
1
,
n
y
n
<
Mand, likewise,(94)
α
2
,
n
x
n
1
+
β
2
,
n
x
n
+
γ
2
,
n
y
n
≤
M
β
2
,
n
x
n
1
+
β
2
,
n
x
n
+
γ
2
,
n
y
n
<
Mfor all large n. Therefore, x
n
≤
2
M. Next, if {
b
n
} is bounded then y
n
≤
2
M
b
n is also bounded and the proof is complete.The next result follows readily from Theorem5.Corollary 18.
The origin(
0,0
) attracts every orbit of (87a) and (87b) in 0
,
∞
2 if(95)
limsup
n
→
∞
⁡
α
1
,
n
b
n
-
1
+
α
2
,
n
<
1and either b
n is bounded or l
i
m
i
n
f
n
→
∞
α
1
,
n
>
0.The above corollary is false when (95) holds if b
n is unbounded and thus α
1
,
n has a subsequence that converges to 0.Example 19.
Consider system(96)
x
n
+
1
=
α
-
n
y
n
+
s
x
n
,
y
n
+
1
=
β
α
n
x
n
1
+
c
x
n
,where α
>
1, β
>
0, 0
≤
s
<
1, c
≥
0, σ
1
,
n
=
α
-
n, and b
n
=
β
α
n. Then (95) is satisfied, so l
i
m
n
→
∞
x
n
=
0. But y
n does not approach 0 for large enough α; this may be inferred from Lemma 4 which shows that x
n converges to 0 at an exponential rate δ
n
/
2 where δ
=
s
+
β
/
α
∈
(
0,1
). Thus(97)
y
n
=
1
α
-
n
x
n
+
1
-
s
x
n
=
α
n
x
n
+
1
-
s
x
nwill not converge to 0 if α is sufficiently large.Corollary17 takes a simpler form for the autonomous special case of (87a) and (87b); namely,(98a)
x
n
+
1
=
α
1
y
n
1
+
β
1
x
n
+
γ
1
y
n
+
α
2
x
n
1
+
β
2
x
n
+
γ
2
y
n
,
(98b)
y
n
+
1
=
b
x
n
1
+
c
1
x
n
+
c
2
y
nwith constant parameters (99)
α
1
,
b
>
0
,
α
2
,
β
i
,
γ
i
,
c
i
≥
0
.The following result is applicable to (3a) and (3b) as well as special cases of (4a), (4b), (5a), and (5b) with rational f.Corollary 20.
Assume that (99) holds. All orbits of (98a) and (98b) in 0
,
∞
2 are uniformly bounded if either one of the following conditions holds:(a)
α
2
<
1 and c
1
>
0.
(b)
γ
1
,
β
2
>
0.It is noteworthy that if in part (a) abovec
1
=
0 then (98a) and (98b) may have unbounded solutions as in, for example, system (100)
x
n
+
1
=
α
1
y
n
,
y
n
+
1
=
b
x
n
1
+
c
2
y
n
,where α
2
=
c
1
=
0 and the remaining parameters are positive. This system folds to the second-order rational equation(101)
x
n
+
2
=
α
1
2
b
x
n
α
1
+
c
2
x
n
+
1which is known to have unbounded solutions if α
1
b
>
1; see [37].Corollary18 likewise simplifies in the autonomous case.Corollary 21.
Assume that (99) holds with α
1
b
+
α
2
<
1. Then the origin (
0,0
) is the globally asymptotically stable fixed point of (98a) and (98b) relative to 0
,
∞
2.


### 4.2. Persistence and the Role of Competition

We now explore the effects of competition in the autonomous system (98a) and (98b). There are 6 different competition coefficients and to reduce the number of different cases we focus on the special case below where β
i
,
γ
i
=
0:(102)
x
n
+
1
=
α
1
y
n
+
α
2
x
n
,
(103)
y
n
+
1
=
b
x
n
1
+
c
1
x
n
+
c
2
y
n
.Ifα
i define the natural survival rates s
i, then this system is complementary to (3a), (3b), (4a), and (4b) in the sense that in both of those systems c
2
=
0.By the last two corollaries, all orbits of the rational system (102) and (103) in [
0
,
∞
)
2 are uniformly bounded if c
1
>
0 and α
2
<
1 and they converge to the origin if α
1
b
+
α
2
<
1. We now examine this rational system in more detail using its folding, namely, the second-order rational equation(104)
x
n
+
2
=
a
x
n
+
1
+
σ
x
n
1
+
A
x
n
+
1
+
B
x
n
,where(105)
a
=
α
2
,
σ
=
α
1
b
,
A
=
c
2
α
1
,
B
=
1
α
1
α
1
c
1
-
α
2
c
2
.See (45); y-component is given by (44) or calculated directly using (102) as(106)
y
n
=
1
α
1
x
n
+
1
-
α
2
x
n
.With initial valuesx
0 and x
1
=
α
1
y
0
+
α
2
x
0 derived from (
x
0
,
y
0
)
∈
0
,
∞
2, x-component of the orbits {
(
x
n
,
y
n
)
} of the system is obtained by iterating (104). The equation in (106) is passive in the sense that after x-component of the orbit is generated by the core equation (104) y-component is derived from (106) without any further iterations. This observation also establishes the nontrivial fact that solutions of (104) that correspond to the orbits of the system in 0
,
∞
2 are nonnegative and well-defined even for B
<
0.Ifα
1
b
+
α
2
<
1, that is, σ
<
1
-
a, then zero is the only fixed point of (104). Corollary 21 establishes that, in this case, zero is globally asymptotically stable relative to [
0
,
∞
). On the other hand, when α
1
b
+
α
2
>
1, that is, σ
>
1
-
a, then 0 is no longer a stable fixed point of (104). By routine calculations, one can show that zero is a saddle point when 1
-
a
<
σ
<
1
+
a and if σ
>
1
+
a then zero is a repeller.In addition, whenσ
>
1
-
a and a
=
α
2
<
1, system (102) and (103) also has a fixed point in (
0
,
∞
)
2 given by(107)
x
¯
=
σ
-
1
-
a
1
-
a
A
+
B
=
α
1
α
1
b
+
α
2
-
1
1
-
α
2
α
1
c
1
+
1
-
α
2
c
2
,
y
¯
=
1
-
α
2
α
1
x
¯
.We note thatx
¯ is also a positive fixed point of folding (104). Under certain conditions, x
¯ attracts all solutions of (104) with positive initial values, and it is thus a survival equilibrium. We state the following result from literature; see [38].Lemma 22.
LetI be an open interval of real numbers and suppose that f
∈
C
(
I
m
,
R
) is nondecreasing in each coordinate. Let x
¯
∈
I be a fixed point of the difference equation(108)
x
n
+
1
=
f
x
n
,
x
n
-
1
,
…
,
x
n
-
m
+
1and assume that the function h
(
t
)
=
f
(
t
,
…
,
t
) satisfies the conditions(109)
h
t
>
t
i
f
t
<
x
¯
,
h
t
<
t
i
f
t
>
x
¯
,
t
∈
I
.Then I is an invariant interval of (108) and x
¯ attracts all solutions with initial values in I
.Theorem 23.
Leta
<
1
<
a
+
σ; that is, α
2
<
1
<
α
1
b
+
α
2. If the function(110)
f
u
,
v
=
a
u
+
σ
v
A
u
+
B
v
+
1is nondecreasing in both arguments, then the fixed point x
¯ attracts all solutions of (104) with initial values in (
0
,
∞
).Proof.
If we let(111)
h
t
=
a
t
+
σ
t
1
+
A
+
B
tthen the fixed point x
¯ is the solution of h
(
t
)
=
t. For t
>
0, we may write h
(
t
)
=
ϕ
(
t
)
t where(112)
ϕ
t
=
a
+
σ
1
+
A
+
B
t
with
ϕ
x
¯
=
h
x
¯
x
¯
=
1
.
Now,(113)
ϕ
′
t
=
-
σ
A
+
B
1
+
A
+
B
t
2
<
0for all t
>
0, so ϕ
(
t
) is strictly decreasing for all t
>
0. Therefore, (114)
t
<
x
¯
implies  that
h
t
=
ϕ
t
t
>
ϕ
x
¯
t
=
t
,
t
>
x
¯
implies  that
h
t
=
ϕ
t
t
<
ϕ
x
¯
t
=
t
.
The rest of the proof follows from Lemma22.Note that(115)
f
u
=
a
-
A
σ
v
A
u
+
B
v
+
1
2
,
f
v
=
σ
A
u
+
1
A
u
+
B
v
+
1
2
>
0
.Ifα
1
b
+
α
2
>
1 and c
2
=
0 then A
=
0, so f
u
,
f
v
>
0. Therefore by Theorem 23  x
¯ is globally asymptotically stable. However, if c
2
>
0, then f
u may not be positive, so the results of Theorem 23 may not apply to this case. The next result shows that orbits of the system may converge to x
¯ if c
2
>
0 but not too large.Theorem 24.
Letc
1
>
0 and let a
<
1
<
a
+
σ; that is, α
2
<
1
<
α
1
b
+
α
2. Then there exists c
>
0 such that for c
2
∈
[
0
,
c
] the fixed point x
¯ of (104) is globally asymptotically stable relative to (
0
,
∞
).Proof.
Since(116)
f
u
=
a
-
A
σ
v
A
u
+
B
v
+
1
2
=
a
A
u
+
B
v
2
+
2
A
a
u
+
a
+
2
a
B
-
A
σ
v
A
u
+
B
v
+
1
2to ensure that f
u
≥
0 it suffices for 2
a
B
-
A
σ
≥
0; that is,(117)
2
α
2
α
1
c
1
-
α
2
c
2
-
c
2
α
1
b
≥
0which is equivalent to(118)
c
2
≤
2
α
1
α
2
c
1
α
1
b
+
2
α
2
2
≐
cand the proof is complete.Ifc
2 is sufficiently large then f
u is not positive on (
0
,
∞
). Furthermore, x
¯ also becomes unstable for large enough c
2, which we establish next by examining the linearization of (104) around x
¯.The characteristic equation associated with the linearization of (104) at x
¯ is given by(119)
λ
2
-
p
λ
-
q
=
0
,where(120)
p
=
f
u
x
¯
,
x
¯
=
a
-
1
-
a
A
x
¯
1
+
A
+
B
x
¯
,
q
=
f
v
x
¯
,
x
¯
=
σ
-
1
-
a
B
x
¯
1
+
A
+
B
x
¯
.The roots of (119) are given by(121)
λ
1
=
p
-
p
2
+
4
q
2
,
λ
2
=
p
+
p
2
+
4
q
2
.Sincef
v
(
u
,
v
)
>
0 for all u
,
v
∈
(
0
,
∞
) it follows that q
>
0 and both roots are real with λ
1
<
0 and λ
2
>
0. Furthermore, λ
2
<
1 if(122)
p
+
p
2
+
4
q
2
<
1
that  is
q
<
1
-
pwhich is equivalent to(123)
2
1
-
a
A
+
B
x
¯
>
σ
-
1
-
a
.This inequality holds, sincex
¯
>
0 under our assumptions on the parameters. Therefore, λ
2
<
1. On the other hand, λ
1
>
-
1 if and only if(124)
p
-
p
2
+
4
q
2
>
-
1
that  is
p
+
1
>
qwhich is equivalent to(125)
2
A
a
+
B
x
¯
>
σ
-
1
+
a
.Note that when(
1
-
a
)
<
σ
<
(
1
+
a
) this is trivially the case since x
¯
>
0 under our assumptions on the parameters. Thus, x
¯ is locally asymptotically stable if σ
<
1
+
a.Next,λ
1
<
-
1 if σ
>
1
+
a and(126)
2
A
a
+
B
x
¯
<
σ
-
1
+
a
.We summarize the above results in the following lemma.Lemma 25.
Leta
<
1
<
a
+
σ; that is, α
2
<
1
<
α
1
b
+
α
2. Then the fixed point x
¯ of (104) is(a)
locally asymptotically stable if and only if (125) holds. In particular, this is true if(127)
1
-
a
<
σ
<
1
+
a
,
t
h
a
t
i
s
1
-
α
2
<
α
1
b
<
1
+
α
2
.
(b)
It is a saddle point if and only if (126) holds with σ
>
1
+
a; that is α
1
b
>
1
+
α
2.Inequality (126) implies a range for c
2 that we now determine. Let(128)
k
=
σ
-
1
+
a
σ
-
1
-
a
<
1
.Thenk
∈
(
0,1
) if σ
>
1
+
a,(129)
2
A
a
+
B
x
¯
<
σ
-
1
+
a
⟹
2
A
a
+
B
A
+
B
<
σ
-
1
+
a
σ
-
1
-
a
1
-
a
=
1
-
a
k
.Since(130)
2
A
a
+
B
=
2
α
1
c
2
α
2
+
c
1
α
1
-
c
2
α
2
=
2
c
1
,
A
+
B
=
1
α
1
c
1
α
1
+
1
-
α
2
c
2(129) is equivalent to(131)
2
c
1
α
1
c
1
α
1
+
1
-
α
2
c
2
<
1
-
a
k
=
1
-
α
2
k
.From the above inequality we obtain(132)
c
2
>
α
1
c
1
2
-
1
-
α
2
k
1
-
α
2
2
k
≐
c
¯
.Thus ifc
2
>
c
¯ then x
¯ is a saddle point and in particular the fixed point (
x
¯
,
y
¯
) is unstable. These observations lead to the following which may be compared with Theorem 24.Corollary 26.
Assume that (99) holds for system (102) and (103) and α
2
<
1
<
α
1
b
+
α
2. Then the fixed point (
x
¯
,
y
¯
) is unstable if c
2
>
c
¯.Our final result establishes that whenc
2
>
0 is sufficiently large system (102) and (103) can have a prime period two orbit which occurs as x
¯ becomes unstable. Existence of periodic orbits is established via the folding in (104).The difference equation in (104) has a positive prime period two solution if there exist real numbers m
,
M
>
0 and m
≠
M such that(133)
m
=
f
M
,
m
,
M
=
f
m
,
M
;that is,(134)
m
=
a
M
+
σ
m
A
M
+
B
m
+
1
,
M
=
a
m
+
σ
M
A
m
+
B
M
+
1from which we get(135)
m
-
a
M
A
M
+
B
m
+
1
=
σ
m
,
M
-
a
m
A
m
+
B
M
+
1
=
σ
M
;that is,(136)
A
m
M
+
B
m
2
+
m
-
A
a
M
2
-
a
B
M
m
-
a
M
=
σ
m
,
(137)
A
m
M
+
B
M
2
+
M
-
A
a
m
2
-
a
B
M
m
-
a
m
=
σ
M
.Taking the difference of the right and left sides of (136) and (137) yields (138)
B
m
2
-
M
2
+
m
-
M
-
A
a
M
2
-
m
2
-
M
-
m
=
σ
m
-
M
,
B
+
A
a
m
-
M
m
+
M
=
σ
-
1
+
a
m
-
M
.Whenm
≠
M, we get(139)
B
+
A
a
m
+
M
=
σ
-
1
+
aand since the left side of the last equation is positive this implies that σ
-
(
1
+
a
)
>
0. Stated differently, if σ
-
(
1
+
a
)
<
0, then (104) cannot have a positive prime period two solution.Similarly, taking the sum of the right and left sides of (136) and (137) yields(140)
2
A
m
M
+
B
m
2
+
M
2
+
m
+
M
-
A
a
m
2
+
M
2
-
2
a
B
M
m
-
a
m
+
M
=
σ
m
+
M
.Adding and subtracting2
(
B
-
A
a
) to and from the left hand side of the last expression in (140) yields(141)
2
A
-
a
B
-
B
+
A
a
M
m
+
B
-
A
a
m
+
M
2
=
σ
-
1
-
a
m
+
M
;that is,(142)
2
1
+
a
A
-
B
M
m
=
σ
-
1
-
a
m
+
M
-
B
-
A
a
m
+
M
2
=
m
+
M
σ
-
1
-
a
-
B
-
A
a
m
+
M
=
m
+
M
σ
-
1
-
a
-
B
-
A
a
σ
-
1
+
a
B
+
A
a
=
m
+
M
A
a
+
B
B
+
A
a
σ
-
1
-
a
-
B
-
A
a
σ
-
1
+
a
.Simplifying the right hand side, it follows that(143)
1
+
a
A
-
B
M
m
=
σ
-
1
+
a
A
a
+
B
2
A
a
σ
-
1
+
a
B
.Now, since we are assuming thatσ
-
(
1
+
a
)
>
0, then σ
-
1
>
0, so the right side of (143) is positive, which implies that A
-
B
>
0. Stated differently, if A
<
B, then (104) has no positive prime period two solution.From (143) we get(144)
M
m
=
σ
-
1
+
a
A
a
σ
-
1
+
a
B
c
1
+
a
A
-
B
A
a
+
B
2
≔
Qand let m
+
M
=
P, from which we obtain that M
=
P
-
m and m
=
P
-
M. This means that(145)
m
P
-
m
=
Q
,
M
P
-
M
=
Q
;that is, m and M are the roots of the quadratic(146)
S
t
=
t
2
-
P
t
+
Q
,where P
,
Q
>
0 and(147)
t
±
=
P
±
P
2
-
4
Q
2
.To ensure thatm and M are real, the roots of S
(
t
) must be real, which is the case if and only if P
2
-
4
Q
>
0; that is,(148)
σ
-
1
+
a
σ
-
1
+
a
-
4
A
a
σ
-
1
+
a
B
1
+
a
A
-
B
>
0
.We summarize the above results as follows.Theorem 27.
The second-order difference equation in (104) has a positive prime period two solution if and only if all of the following conditions are satisfied:(a)
σ
-
(
1
+
a
)
>
0.
(b)
A
-
B
>
0.
(c)
(
σ
-
(
1
+
a
)
)
σ
-
(
1
+
a
)
-
4
(
A
a
(
σ
-
1
)
+
a
B
)
/
(
1
+
a
)
(
A
-
B
)
>
0.The next result shows that a solution of period two appears whenx
¯ loses its stability.Corollary 28.
The second-order difference equation in (104) has a positive prime period two solution if and only if x
¯ is a saddle point.Proof.
Supposex
¯ is a saddle point. Then, by Lemma 25(b), 2
(
A
a
+
B
)
x
¯
<
σ
-
(
1
+
a
) from which we infer that σ
-
(
1
+
a
)
>
0.
Now2
(
A
a
+
B
)
x
¯
<
σ
-
(
1
+
a
) implies that(149)
2
A
a
+
B
1
-
a
A
+
B
σ
-
1
-
a
<
σ
-
1
+
awhich is true if and only if(150)
2
A
a
+
B
σ
-
1
-
a
<
1
-
a
A
+
B
σ
-
1
+
a
.
Adding and subtracting(
1
+
a
)
(
A
-
B
)
(
σ
-
(
1
+
a
)
) to and from the right hand side of the last expression in (150) yields (151)
1
+
a
A
-
B
σ
-
1
+
a
+
σ
-
1
+
a
1
-
a
A
+
B
-
1
+
a
A
-
B
=
1
+
a
A
-
B
σ
-
1
+
a
+
σ
-
1
+
a
2
B
-
2
A
a
.
Therefore,(152)
2
A
a
+
B
σ
-
1
-
a
+
2
A
a
-
B
σ
-
1
+
a
=
4
A
a
σ
-
1
+
a
B
<
1
+
a
A
-
B
σ
-
1
+
a
;that is,(153)
1
+
a
A
-
B
σ
-
1
+
a
-
4
A
a
σ
-
1
+
a
B
>
0from which we infer that A
-
B
>
0 and the roots of S
(
t
) are guaranteed to be real and positive. This satisfies all the conditions of Theorem 27 which completes the proof.Corollary 29.
Assume that (99) holds and furthermore α
2
<
1
<
α
1
b
+
α
2 and c
2
>
c
¯. Then system (102) and (103) has a cycle of period two in (
0
,
∞
)
2.Figure1 shows two orbits of system (102) and (103) from initial points (
x
0
,
y
0
)
=
(
2.3,1
) and (
x
0
,
y
0
)
=
(
0.0001,0.0001
). Although both orbits converge to the period two cycle, a shadow of the stable manifold of the fixed point is also seen in the initial segments of the two orbits. If the initial points start exactly on the stable manifold of x
¯ then the solutions converge to x
¯.Figure 1
Orbits illustrating period two oscillations and the saddle point.We studied the dynamics of a general planar system that includes many common stage-structured population models that evolve in discrete time. Our hypotheses regarding system (2a) and (2b) and its parameters are more general than what is typically assumed in population models with the aim of gaining a broader understanding of the mathematical properties of the system. The study in this paper is rigorous but incomplete and many issues remain. Generalizing the results in Section 4 to a level closer to that in Section 3 leads to a more comprehensive treatment of planar or two-stage, discrete population models. Among other things, this involves a consideration of systems involving the Ricker function where it is necessary to add the possibility of complex behavior. A resolution of these and related issues is left to future studies of system (2a) and (2b).


## 4.1. Uniform Boundedness and Extinction

We now examine boundedness and global convergence to 0 (extinction) in (87a) and (87b). The next result is in part a consequence of Corollary 3.Corollary 17.
Assume that (88) holds.
(a) Let the sequence{
α
1
,
n
} be bounded and l
i
m
s
u
p
n
→
∞
α
2
,
n
<
1. If there is M
0
>
0 such that b
n
≤
M
0
c
1
,
n for all n larger than a given positive integer then all orbits of (87a) and (87b) are uniformly bounded.
(b) Let the sequence{
b
n
} be bounded and suppose that there is M
>
0 such that(89)
α
1
,
n
≤
M
γ
1
,
n
,
α
2
,
n
≤
M
β
2
,
nfor all n larger than a given positive integer. Then all orbits of (87a) and (87b) are uniformly bounded.Proof.
(a) By hypothesis, for all (large)n,(90)
b
n
x
n
1
+
c
1
,
n
x
n
+
c
2
,
n
y
n
≤
M
0
c
1
,
n
x
n
1
+
c
1
,
n
x
n
+
c
2
,
n
y
n
<
M
0
.
Next, let(91)
σ
1
,
n
u
,
v
=
α
1
,
n
1
+
β
1
,
n
u
+
γ
1
,
n
v
,
σ
2
,
n
u
,
v
=
α
2
,
n
1
+
β
2
,
n
u
+
γ
2
,
n
v
.
By hypothesis, there isM
1
>
0 and δ
∈
(
0,1
) such that for all u
,
v
≥
0 and all sufficiently large values of n
(92)
σ
1
,
n
u
,
v
≤
α
1
,
n
≤
M
1
,
σ
2
,
n
u
,
v
≤
α
2
,
n
≤
δ
.
Now an application of Corollary3 completes the proof of (a).
(b) By (89) for all large n it follows that(93)
α
1
,
n
y
n
1
+
β
1
,
n
x
n
+
γ
1
,
n
y
n
≤
M
γ
1
,
n
y
n
1
+
β
1
,
n
x
n
+
γ
1
,
n
y
n
<
Mand, likewise,(94)
α
2
,
n
x
n
1
+
β
2
,
n
x
n
+
γ
2
,
n
y
n
≤
M
β
2
,
n
x
n
1
+
β
2
,
n
x
n
+
γ
2
,
n
y
n
<
Mfor all large n. Therefore, x
n
≤
2
M. Next, if {
b
n
} is bounded then y
n
≤
2
M
b
n is also bounded and the proof is complete.The next result follows readily from Theorem5.Corollary 18.
The origin(
0,0
) attracts every orbit of (87a) and (87b) in 0
,
∞
2 if(95)
limsup
n
→
∞
⁡
α
1
,
n
b
n
-
1
+
α
2
,
n
<
1and either b
n is bounded or l
i
m
i
n
f
n
→
∞
α
1
,
n
>
0.The above corollary is false when (95) holds if b
n is unbounded and thus α
1
,
n has a subsequence that converges to 0.Example 19.
Consider system(96)
x
n
+
1
=
α
-
n
y
n
+
s
x
n
,
y
n
+
1
=
β
α
n
x
n
1
+
c
x
n
,where α
>
1, β
>
0, 0
≤
s
<
1, c
≥
0, σ
1
,
n
=
α
-
n, and b
n
=
β
α
n. Then (95) is satisfied, so l
i
m
n
→
∞
x
n
=
0. But y
n does not approach 0 for large enough α; this may be inferred from Lemma 4 which shows that x
n converges to 0 at an exponential rate δ
n
/
2 where δ
=
s
+
β
/
α
∈
(
0,1
). Thus(97)
y
n
=
1
α
-
n
x
n
+
1
-
s
x
n
=
α
n
x
n
+
1
-
s
x
nwill not converge to 0 if α is sufficiently large.Corollary17 takes a simpler form for the autonomous special case of (87a) and (87b); namely,(98a)
x
n
+
1
=
α
1
y
n
1
+
β
1
x
n
+
γ
1
y
n
+
α
2
x
n
1
+
β
2
x
n
+
γ
2
y
n
,
(98b)
y
n
+
1
=
b
x
n
1
+
c
1
x
n
+
c
2
y
nwith constant parameters (99)
α
1
,
b
>
0
,
α
2
,
β
i
,
γ
i
,
c
i
≥
0
.The following result is applicable to (3a) and (3b) as well as special cases of (4a), (4b), (5a), and (5b) with rational f.Corollary 20.
Assume that (99) holds. All orbits of (98a) and (98b) in 0
,
∞
2 are uniformly bounded if either one of the following conditions holds:(a)
α
2
<
1 and c
1
>
0.
(b)
γ
1
,
β
2
>
0.It is noteworthy that if in part (a) abovec
1
=
0 then (98a) and (98b) may have unbounded solutions as in, for example, system (100)
x
n
+
1
=
α
1
y
n
,
y
n
+
1
=
b
x
n
1
+
c
2
y
n
,where α
2
=
c
1
=
0 and the remaining parameters are positive. This system folds to the second-order rational equation(101)
x
n
+
2
=
α
1
2
b
x
n
α
1
+
c
2
x
n
+
1which is known to have unbounded solutions if α
1
b
>
1; see [37].Corollary18 likewise simplifies in the autonomous case.Corollary 21.
Assume that (99) holds with α
1
b
+
α
2
<
1. Then the origin (
0,0
) is the globally asymptotically stable fixed point of (98a) and (98b) relative to 0
,
∞
2.


## 4.2. Persistence and the Role of Competition

We now explore the effects of competition in the autonomous system (98a) and (98b). There are 6 different competition coefficients and to reduce the number of different cases we focus on the special case below where β
i
,
γ
i
=
0:(102)
x
n
+
1
=
α
1
y
n
+
α
2
x
n
,
(103)
y
n
+
1
=
b
x
n
1
+
c
1
x
n
+
c
2
y
n
.Ifα
i define the natural survival rates s
i, then this system is complementary to (3a), (3b), (4a), and (4b) in the sense that in both of those systems c
2
=
0.By the last two corollaries, all orbits of the rational system (102) and (103) in [
0
,
∞
)
2 are uniformly bounded if c
1
>
0 and α
2
<
1 and they converge to the origin if α
1
b
+
α
2
<
1. We now examine this rational system in more detail using its folding, namely, the second-order rational equation(104)
x
n
+
2
=
a
x
n
+
1
+
σ
x
n
1
+
A
x
n
+
1
+
B
x
n
,where(105)
a
=
α
2
,
σ
=
α
1
b
,
A
=
c
2
α
1
,
B
=
1
α
1
α
1
c
1
-
α
2
c
2
.See (45); y-component is given by (44) or calculated directly using (102) as(106)
y
n
=
1
α
1
x
n
+
1
-
α
2
x
n
.With initial valuesx
0 and x
1
=
α
1
y
0
+
α
2
x
0 derived from (
x
0
,
y
0
)
∈
0
,
∞
2, x-component of the orbits {
(
x
n
,
y
n
)
} of the system is obtained by iterating (104). The equation in (106) is passive in the sense that after x-component of the orbit is generated by the core equation (104) y-component is derived from (106) without any further iterations. This observation also establishes the nontrivial fact that solutions of (104) that correspond to the orbits of the system in 0
,
∞
2 are nonnegative and well-defined even for B
<
0.Ifα
1
b
+
α
2
<
1, that is, σ
<
1
-
a, then zero is the only fixed point of (104). Corollary 21 establishes that, in this case, zero is globally asymptotically stable relative to [
0
,
∞
). On the other hand, when α
1
b
+
α
2
>
1, that is, σ
>
1
-
a, then 0 is no longer a stable fixed point of (104). By routine calculations, one can show that zero is a saddle point when 1
-
a
<
σ
<
1
+
a and if σ
>
1
+
a then zero is a repeller.In addition, whenσ
>
1
-
a and a
=
α
2
<
1, system (102) and (103) also has a fixed point in (
0
,
∞
)
2 given by(107)
x
¯
=
σ
-
1
-
a
1
-
a
A
+
B
=
α
1
α
1
b
+
α
2
-
1
1
-
α
2
α
1
c
1
+
1
-
α
2
c
2
,
y
¯
=
1
-
α
2
α
1
x
¯
.We note thatx
¯ is also a positive fixed point of folding (104). Under certain conditions, x
¯ attracts all solutions of (104) with positive initial values, and it is thus a survival equilibrium. We state the following result from literature; see [38].Lemma 22.
LetI be an open interval of real numbers and suppose that f
∈
C
(
I
m
,
R
) is nondecreasing in each coordinate. Let x
¯
∈
I be a fixed point of the difference equation(108)
x
n
+
1
=
f
x
n
,
x
n
-
1
,
…
,
x
n
-
m
+
1and assume that the function h
(
t
)
=
f
(
t
,
…
,
t
) satisfies the conditions(109)
h
t
>
t
i
f
t
<
x
¯
,
h
t
<
t
i
f
t
>
x
¯
,
t
∈
I
.Then I is an invariant interval of (108) and x
¯ attracts all solutions with initial values in I
.Theorem 23.
Leta
<
1
<
a
+
σ; that is, α
2
<
1
<
α
1
b
+
α
2. If the function(110)
f
u
,
v
=
a
u
+
σ
v
A
u
+
B
v
+
1is nondecreasing in both arguments, then the fixed point x
¯ attracts all solutions of (104) with initial values in (
0
,
∞
).Proof.
If we let(111)
h
t
=
a
t
+
σ
t
1
+
A
+
B
tthen the fixed point x
¯ is the solution of h
(
t
)
=
t. For t
>
0, we may write h
(
t
)
=
ϕ
(
t
)
t where(112)
ϕ
t
=
a
+
σ
1
+
A
+
B
t
with
ϕ
x
¯
=
h
x
¯
x
¯
=
1
.
Now,(113)
ϕ
′
t
=
-
σ
A
+
B
1
+
A
+
B
t
2
<
0for all t
>
0, so ϕ
(
t
) is strictly decreasing for all t
>
0. Therefore, (114)
t
<
x
¯
implies  that
h
t
=
ϕ
t
t
>
ϕ
x
¯
t
=
t
,
t
>
x
¯
implies  that
h
t
=
ϕ
t
t
<
ϕ
x
¯
t
=
t
.
The rest of the proof follows from Lemma22.Note that(115)
f
u
=
a
-
A
σ
v
A
u
+
B
v
+
1
2
,
f
v
=
σ
A
u
+
1
A
u
+
B
v
+
1
2
>
0
.Ifα
1
b
+
α
2
>
1 and c
2
=
0 then A
=
0, so f
u
,
f
v
>
0. Therefore by Theorem 23  x
¯ is globally asymptotically stable. However, if c
2
>
0, then f
u may not be positive, so the results of Theorem 23 may not apply to this case. The next result shows that orbits of the system may converge to x
¯ if c
2
>
0 but not too large.Theorem 24.
Letc
1
>
0 and let a
<
1
<
a
+
σ; that is, α
2
<
1
<
α
1
b
+
α
2. Then there exists c
>
0 such that for c
2
∈
[
0
,
c
] the fixed point x
¯ of (104) is globally asymptotically stable relative to (
0
,
∞
).Proof.
Since(116)
f
u
=
a
-
A
σ
v
A
u
+
B
v
+
1
2
=
a
A
u
+
B
v
2
+
2
A
a
u
+
a
+
2
a
B
-
A
σ
v
A
u
+
B
v
+
1
2to ensure that f
u
≥
0 it suffices for 2
a
B
-
A
σ
≥
0; that is,(117)
2
α
2
α
1
c
1
-
α
2
c
2
-
c
2
α
1
b
≥
0which is equivalent to(118)
c
2
≤
2
α
1
α
2
c
1
α
1
b
+
2
α
2
2
≐
cand the proof is complete.Ifc
2 is sufficiently large then f
u is not positive on (
0
,
∞
). Furthermore, x
¯ also becomes unstable for large enough c
2, which we establish next by examining the linearization of (104) around x
¯.The characteristic equation associated with the linearization of (104) at x
¯ is given by(119)
λ
2
-
p
λ
-
q
=
0
,where(120)
p
=
f
u
x
¯
,
x
¯
=
a
-
1
-
a
A
x
¯
1
+
A
+
B
x
¯
,
q
=
f
v
x
¯
,
x
¯
=
σ
-
1
-
a
B
x
¯
1
+
A
+
B
x
¯
.The roots of (119) are given by(121)
λ
1
=
p
-
p
2
+
4
q
2
,
λ
2
=
p
+
p
2
+
4
q
2
.Sincef
v
(
u
,
v
)
>
0 for all u
,
v
∈
(
0
,
∞
) it follows that q
>
0 and both roots are real with λ
1
<
0 and λ
2
>
0. Furthermore, λ
2
<
1 if(122)
p
+
p
2
+
4
q
2
<
1
that  is
q
<
1
-
pwhich is equivalent to(123)
2
1
-
a
A
+
B
x
¯
>
σ
-
1
-
a
.This inequality holds, sincex
¯
>
0 under our assumptions on the parameters. Therefore, λ
2
<
1. On the other hand, λ
1
>
-
1 if and only if(124)
p
-
p
2
+
4
q
2
>
-
1
that  is
p
+
1
>
qwhich is equivalent to(125)
2
A
a
+
B
x
¯
>
σ
-
1
+
a
.Note that when(
1
-
a
)
<
σ
<
(
1
+
a
) this is trivially the case since x
¯
>
0 under our assumptions on the parameters. Thus, x
¯ is locally asymptotically stable if σ
<
1
+
a.Next,λ
1
<
-
1 if σ
>
1
+
a and(126)
2
A
a
+
B
x
¯
<
σ
-
1
+
a
.We summarize the above results in the following lemma.Lemma 25.
Leta
<
1
<
a
+
σ; that is, α
2
<
1
<
α
1
b
+
α
2. Then the fixed point x
¯ of (104) is(a)
locally asymptotically stable if and only if (125) holds. In particular, this is true if(127)
1
-
a
<
σ
<
1
+
a
,
t
h
a
t
i
s
1
-
α
2
<
α
1
b
<
1
+
α
2
.
(b)
It is a saddle point if and only if (126) holds with σ
>
1
+
a; that is α
1
b
>
1
+
α
2.Inequality (126) implies a range for c
2 that we now determine. Let(128)
k
=
σ
-
1
+
a
σ
-
1
-
a
<
1
.Thenk
∈
(
0,1
) if σ
>
1
+
a,(129)
2
A
a
+
B
x
¯
<
σ
-
1
+
a
⟹
2
A
a
+
B
A
+
B
<
σ
-
1
+
a
σ
-
1
-
a
1
-
a
=
1
-
a
k
.Since(130)
2
A
a
+
B
=
2
α
1
c
2
α
2
+
c
1
α
1
-
c
2
α
2
=
2
c
1
,
A
+
B
=
1
α
1
c
1
α
1
+
1
-
α
2
c
2(129) is equivalent to(131)
2
c
1
α
1
c
1
α
1
+
1
-
α
2
c
2
<
1
-
a
k
=
1
-
α
2
k
.From the above inequality we obtain(132)
c
2
>
α
1
c
1
2
-
1
-
α
2
k
1
-
α
2
2
k
≐
c
¯
.Thus ifc
2
>
c
¯ then x
¯ is a saddle point and in particular the fixed point (
x
¯
,
y
¯
) is unstable. These observations lead to the following which may be compared with Theorem 24.Corollary 26.
Assume that (99) holds for system (102) and (103) and α
2
<
1
<
α
1
b
+
α
2. Then the fixed point (
x
¯
,
y
¯
) is unstable if c
2
>
c
¯.Our final result establishes that whenc
2
>
0 is sufficiently large system (102) and (103) can have a prime period two orbit which occurs as x
¯ becomes unstable. Existence of periodic orbits is established via the folding in (104).The difference equation in (104) has a positive prime period two solution if there exist real numbers m
,
M
>
0 and m
≠
M such that(133)
m
=
f
M
,
m
,
M
=
f
m
,
M
;that is,(134)
m
=
a
M
+
σ
m
A
M
+
B
m
+
1
,
M
=
a
m
+
σ
M
A
m
+
B
M
+
1from which we get(135)
m
-
a
M
A
M
+
B
m
+
1
=
σ
m
,
M
-
a
m
A
m
+
B
M
+
1
=
σ
M
;that is,(136)
A
m
M
+
B
m
2
+
m
-
A
a
M
2
-
a
B
M
m
-
a
M
=
σ
m
,
(137)
A
m
M
+
B
M
2
+
M
-
A
a
m
2
-
a
B
M
m
-
a
m
=
σ
M
.Taking the difference of the right and left sides of (136) and (137) yields (138)
B
m
2
-
M
2
+
m
-
M
-
A
a
M
2
-
m
2
-
M
-
m
=
σ
m
-
M
,
B
+
A
a
m
-
M
m
+
M
=
σ
-
1
+
a
m
-
M
.Whenm
≠
M, we get(139)
B
+
A
a
m
+
M
=
σ
-
1
+
aand since the left side of the last equation is positive this implies that σ
-
(
1
+
a
)
>
0. Stated differently, if σ
-
(
1
+
a
)
<
0, then (104) cannot have a positive prime period two solution.Similarly, taking the sum of the right and left sides of (136) and (137) yields(140)
2
A
m
M
+
B
m
2
+
M
2
+
m
+
M
-
A
a
m
2
+
M
2
-
2
a
B
M
m
-
a
m
+
M
=
σ
m
+
M
.Adding and subtracting2
(
B
-
A
a
) to and from the left hand side of the last expression in (140) yields(141)
2
A
-
a
B
-
B
+
A
a
M
m
+
B
-
A
a
m
+
M
2
=
σ
-
1
-
a
m
+
M
;that is,(142)
2
1
+
a
A
-
B
M
m
=
σ
-
1
-
a
m
+
M
-
B
-
A
a
m
+
M
2
=
m
+
M
σ
-
1
-
a
-
B
-
A
a
m
+
M
=
m
+
M
σ
-
1
-
a
-
B
-
A
a
σ
-
1
+
a
B
+
A
a
=
m
+
M
A
a
+
B
B
+
A
a
σ
-
1
-
a
-
B
-
A
a
σ
-
1
+
a
.Simplifying the right hand side, it follows that(143)
1
+
a
A
-
B
M
m
=
σ
-
1
+
a
A
a
+
B
2
A
a
σ
-
1
+
a
B
.Now, since we are assuming thatσ
-
(
1
+
a
)
>
0, then σ
-
1
>
0, so the right side of (143) is positive, which implies that A
-
B
>
0. Stated differently, if A
<
B, then (104) has no positive prime period two solution.From (143) we get(144)
M
m
=
σ
-
1
+
a
A
a
σ
-
1
+
a
B
c
1
+
a
A
-
B
A
a
+
B
2
≔
Qand let m
+
M
=
P, from which we obtain that M
=
P
-
m and m
=
P
-
M. This means that(145)
m
P
-
m
=
Q
,
M
P
-
M
=
Q
;that is, m and M are the roots of the quadratic(146)
S
t
=
t
2
-
P
t
+
Q
,where P
,
Q
>
0 and(147)
t
±
=
P
±
P
2
-
4
Q
2
.To ensure thatm and M are real, the roots of S
(
t
) must be real, which is the case if and only if P
2
-
4
Q
>
0; that is,(148)
σ
-
1
+
a
σ
-
1
+
a
-
4
A
a
σ
-
1
+
a
B
1
+
a
A
-
B
>
0
.We summarize the above results as follows.Theorem 27.
The second-order difference equation in (104) has a positive prime period two solution if and only if all of the following conditions are satisfied:(a)
σ
-
(
1
+
a
)
>
0.
(b)
A
-
B
>
0.
(c)
(
σ
-
(
1
+
a
)
)
σ
-
(
1
+
a
)
-
4
(
A
a
(
σ
-
1
)
+
a
B
)
/
(
1
+
a
)
(
A
-
B
)
>
0.The next result shows that a solution of period two appears whenx
¯ loses its stability.Corollary 28.
The second-order difference equation in (104) has a positive prime period two solution if and only if x
¯ is a saddle point.Proof.
Supposex
¯ is a saddle point. Then, by Lemma 25(b), 2
(
A
a
+
B
)
x
¯
<
σ
-
(
1
+
a
) from which we infer that σ
-
(
1
+
a
)
>
0.
Now2
(
A
a
+
B
)
x
¯
<
σ
-
(
1
+
a
) implies that(149)
2
A
a
+
B
1
-
a
A
+
B
σ
-
1
-
a
<
σ
-
1
+
awhich is true if and only if(150)
2
A
a
+
B
σ
-
1
-
a
<
1
-
a
A
+
B
σ
-
1
+
a
.
Adding and subtracting(
1
+
a
)
(
A
-
B
)
(
σ
-
(
1
+
a
)
) to and from the right hand side of the last expression in (150) yields (151)
1
+
a
A
-
B
σ
-
1
+
a
+
σ
-
1
+
a
1
-
a
A
+
B
-
1
+
a
A
-
B
=
1
+
a
A
-
B
σ
-
1
+
a
+
σ
-
1
+
a
2
B
-
2
A
a
.
Therefore,(152)
2
A
a
+
B
σ
-
1
-
a
+
2
A
a
-
B
σ
-
1
+
a
=
4
A
a
σ
-
1
+
a
B
<
1
+
a
A
-
B
σ
-
1
+
a
;that is,(153)
1
+
a
A
-
B
σ
-
1
+
a
-
4
A
a
σ
-
1
+
a
B
>
0from which we infer that A
-
B
>
0 and the roots of S
(
t
) are guaranteed to be real and positive. This satisfies all the conditions of Theorem 27 which completes the proof.Corollary 29.
Assume that (99) holds and furthermore α
2
<
1
<
α
1
b
+
α
2 and c
2
>
c
¯. Then system (102) and (103) has a cycle of period two in (
0
,
∞
)
2.Figure1 shows two orbits of system (102) and (103) from initial points (
x
0
,
y
0
)
=
(
2.3,1
) and (
x
0
,
y
0
)
=
(
0.0001,0.0001
). Although both orbits converge to the period two cycle, a shadow of the stable manifold of the fixed point is also seen in the initial segments of the two orbits. If the initial points start exactly on the stable manifold of x
¯ then the solutions converge to x
¯.Figure 1
Orbits illustrating period two oscillations and the saddle point.We studied the dynamics of a general planar system that includes many common stage-structured population models that evolve in discrete time. Our hypotheses regarding system (2a) and (2b) and its parameters are more general than what is typically assumed in population models with the aim of gaining a broader understanding of the mathematical properties of the system. The study in this paper is rigorous but incomplete and many issues remain. Generalizing the results in Section 4 to a level closer to that in Section 3 leads to a more comprehensive treatment of planar or two-stage, discrete population models. Among other things, this involves a consideration of systems involving the Ricker function where it is necessary to add the possibility of complex behavior. A resolution of these and related issues is left to future studies of system (2a) and (2b).



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