Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 5963594, 15 pages
https://doi.org/10.1155/2017/5963594

Research Article

Global Asymptotic Stability for Discrete Single Species Population Models

A. Bilgin and M. R. S. Kulenović

Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA

Correspondence should be addressed to M. R. S. Kulenović; mkulenovic@mail.uri.edu

Received 28 March 2017; Accepted 16 May 2017; Published 13 June 2017

Academic Editor: Rigoberto Medina

Copyright © 2017 A. Bilgin and M. R. S. Kulenović. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We present some basic discrete models in populations dynamics of single species with several age classes. Starting with the basic Beverton-Holt model that describes the change of single species we discuss its basic properties such as a convergence of all solutions to the equilibrium, oscillation of solutions about the equilibrium solutions, Allee’s effect, and Jillson’s effect. We consider the effect of the constant and periodic immigration and emigration on the global properties of Beverton-Holt model. We also consider the effect of the periodic environment on the global properties of Beverton-Holt model.

1. Introduction

The following difference equation is known as Beverton-Holt model:

where is a rate of change (growth or decay) and is the size of population at th generation. It was introduced by Beverton and Holt in 1957 and depicts density dependent recruitment of a population with limited resources in which resources are not shared equally. It assumes that the per capita number of offspring is inversely proportional to a linearly increasing function of the number of adults.

Let be the size of a population in generation .

Suppose that is the net reproductive rate, that is, the number of offspring that each individual leaves before dying if there is no limitation in resources. Then the Beverton-Holt model is given by

where is carrying capacity. This leads to the equation

which by the substitution reduces to (1), where .

The model is well studied and understood and exhibits the following properties.

Theorem 1. Equation (1) has the following properties:

(1)Equation (1) has two equilibrium points and when .(2)All solutions of (1) are monotonic (increasing or decreasing sequences).(3)If , then the equilibrium point is the global attractor (i.e., ).(4)If , then the equilibrium point is the global attractor (i.e., , when ).(5)Both equilibrium points are globally asymptotically stable; that is, they are global attractors with the property that small changes of initial condition result in small changes of the corresponding solution .(6)Equation (1) can be solved explicitly and has the following solution:

which implies all the above-mentioned properties of (1).

See, for example, [1–6].

The following difference equation is known as Beverton-Holt model with constant immigration or stocking

where is a rate of change (growth or decay), is a constant immigration, and is the size of population at th generation. The simple substitution reduces (5) to the so-called Riccati’s equation

which is well studied and understood (see [7, 8]) and exhibits the following properties.

Theorem 2. (1) Equation (5) has one positive equilibrium point .
(2) All solutions of (5) are monotonic (increasing or decreasing sequences) and bounded.
(3) The equilibrium point is a global attractor and is globally asymptotically stable.
(4) Furthermore, (5) can be solved explicitly and has the following solution:

where and and

The biological implications of this model are that the constant immigration eliminates the possibility of zero equilibrium and so all solutions get attracted to the unique positive equilibrium solution.

The Beverton-Holt model with emigration or harvesting leads to the equation

where is a rate of change (growth or decay) and is a constant emigration. The solution of (9) is given by (7), where should be replaced by .

Equation (9) has quite different dynamics than (5), since it can have , , or equilibrium solutions, depending on the values of the expression

The following results hold for (9).

Theorem 3. Equation (9) exhibits one of the following three dynamic scenarios:

(1)If then every solution of (9) satisfies . In addition, every solution is increasing.(2)If then (9) has a single equilibrium which is nonhyperbolic and every solution of (9) satisfies if and if . In addition, every solution is decreasing.(3)If then (9) has two positive equilibrium solutions , where is a repeller and is locally asymptotically stable. Every solution of (9) satisfies if and if . In addition, every solution which starts in is increasing, while the other nonconstant solutions are decreasing.

The biological implications of this model are that the constant emigration or harvesting introduces the possibility of the threshold such that if the initial population is below that threshold the population goes to extinction.

See Elaydi [2], Kulenović and Ladas [7], Kulenović and Merino [8], and Thieme [6].

The following difference equation is known as Beverton-Holt model with periodic immigration or stocking:

where is a rate of change (growth or decay), is a periodic immigration or stocking, and is the size of population at th generation. The substitution reduces (11) to the so-called Riccati’s equation with periodic coefficients

which is, very recently, studied and understood and exhibits the following properties; see [9, 10].

Theorem 4. Equation (11) has the following properties:

(1)Equation (11) has the unique nonnegative periodic solution , whose period equals the period of .(2)The periodic solution is the global attractor of all solutions of (11).(3)There is a procedure for finding the explicit solution of (11). In particular, there are explicit formulas for the cases when is periodic sequence with periods .

The biological implications of model (11) are that the periodic immigration imposes its periodicity on the solutions of the model and so all solutions get attracted to the unique periodic solution whose period equals the period of immigration.

Case of periodic emigration is quite different as this emigration may introduce the periodic threshold which would imply the extinction scenario if the initial population is below that threshold.

See Grove et al. [9, 10].

The following difference equation is known as the Beverton-Holt model with periodic environment:

where is a rate of change (growth or decay), is a periodic sequence of period modeling periodicity of environment (periodic supply of food, energy, etc.), and is the size of population at th generation.

Assuming and rewriting (13) as

the substitution reduces (13) to the linear nonautonomous equation

where . The solution of (15) is given as

and it is well studied and understood and exhibits the following properties.

Theorem 5. Equation (15) has the following properties:

(1)Equation (15) has the unique nonnegative periodic solution , with period equal to .(2)The periodic solution is the global attractor of all solutions of (15).(3)The periodic environment is deleterious in the sense that the size of population in periodic environment is smaller than the average of sizes in constant environments. We say that in this case the periodic solution is an attenuant cycle. Mathematically, this means that

where is the equilibrium of (1) when .

Theorem 5 gives an example of so-called Jillson’s effect that refers to any change in global behavior caused by a periodic fluctuation of the environment; see [11, 12].

See [9, 11, 13–17] for some mathematical models with periodic coefficients.

The following difference equation, known as sigmoid Beverton-Holt model, is mathematically the simplest Beverton-Holt type model that exhibits Allee’s effect:

where is a rate of change (growth or decay) and is the size of population at th generation. The model is well studied and understood and exhibits the following properties.

Theorem 6. (1) Equation (1) has either one equilibrium point when or two equilibrium points and , when , or three equilibrium points and when .
(2) All solutions of (1) are monotonic (increasing or decreasing sequences) and bounded.
(3) If , then the equilibrium point is the global attractor.
(4) If , then the equilibrium point is the attractor with the basin of attraction   and is nonhyperbolic with the basin of attraction .
(5) If , then there are two attractors: with the basin of attraction and with the basin of attraction .

The biological implications of this model are that it exhibits so-called Allee’s effect (the social dysfunction and failure to mate successfully when population density falls below a certain threshold) in the sense that if the initial size is smaller than the population goes to extinction. See [18].

In this paper we extend Theorems 1–6 to the case of several generation model with special emphasis on three-generation model. We prove general results about asymptotic stability, both local and global, which cover all kinds of transition or response functions such as linear (also known as Holling type I functions) [19], Beverton-Holt (also known as Holling type II functions or Holling hyperbolic functions), sigmoid Beverton-Holt (also known as Holling type III functions or sigmoid functions), and exponential functions. In order to do so, we introduce some tools in Section 2 which contains some global attractivity results for monotone systems and some difference inequalities results which lead to precise global attractivity results for nonautonomous asymptotically autonomous difference equations. In Sections 3 and 4 we obtain fairly general results for local and global asymptotic stability of th generations model that extends all results in this section. In the special case of three-generation model we find the precise basins of attraction of all locally stable equilibrium solutions and locally stable period-two solutions.

2. Preliminaries

In this part we present basic tools which we use to extend the results in Section 1 to more general models that includes several age groups or generations.

2.1. Global Attractivity Results for Monotone Systems

Let be a partial order on with nonnegative cone . For the order interval is the set of all such that . We say if and and if . A map on a subset of is order preserving if whenever , strictly order preserving if whenever , and strongly order preserving if whenever .

Let be a map with a fixed point and let be an invariant subset of that contains . We say that is stable (asymptotically stable) relative to if is a stable (asymptotically stable) fixed point of the restriction of to .

The next result in [20] is stated for order preserving maps on . See [21] for a more general version valid in ordered Banach spaces. See [22–24] for related results.

Theorem 7. For a nonempty set and , a partial order on , let be an order preserving map, and let be such that and . If and , then is an invariant set and

(i)there exists a fixed point of in ,(ii)if is strongly order preserving, then there exists a fixed point in which is stable relative to ,(iii)if there is only one fixed point in , then it is a global attractor in and therefore asymptotically stable relative to .

The following result in [20] is a direct consequence of the trichotomy result of Dancer and Hess in [21].

Corollary 8. If the nonnegative cone of is a generalized quadrant in , and if has no fixed points in other than and , then the interior of is either a subset of the basin of attraction of or a subset of the basin of attraction of .

Consider the general th order difference equation

where is continuous function and suppose that and are two equilibrium solutions of (19). By introducing new variables

we rewrite (19) as the system

whose corresponding map has the form

The map is nondecreasing map with respect to the ordering in defined as

where , , in the sense that

for every . Set , . The interval is an invariant set for the map , that is, . Consequently, by Corollary 8, the interior of the interval is a part of the basin of attraction of one of two fixed points .

The reasoning given in the above discussion leads to the following result for general difference equation (19).

Theorem 9. Consider (19), where is continuous, nondecreasing in all variables and bounded function with the lower and upper bound and , respectively. If (19) has two equilibrium points , such that is unstable and is asymptotically stable, then the equilibrium is globally asymptotically stable within its basin of attraction which contains and the equilibrium is globally asymptotically stable within its basin of attraction which contains .

2.2. Difference Inequalities

In this section we give some basic results on difference inequalities which we will use later to extend some of our results for autonomous equation to the case of asymptotically autonomous difference equations. See [25, 26].

Theorem 10. Let and be a nondecreasing function in and for any fixed . Suppose that, for , the inequalities

hold. Then

implies that

Proof. Suppose that (27) is not true. Then, there exists a smallest such that

By using (25) and (26) and the monotone character of , it follows from

that

which is a contradiction.

Applying Theorem 10 twice, we obtain the following result.

Corollary 11. Suppose that and are two functions defined on and nondecreasing with respect to and . Let

Then

where and are the solutions of the difference equations

An immediate extension of Theorem 10 is the following result.

Theorem 12. Let , be sequences satisfying

where is nondecreasing with respect to its argument. Then,

implies

Theorem 13. Consider the difference equation

where is nondecreasing function. Assume that

and let

be the limiting difference equation. Assume that there exists such that every solution of difference equation

converges to a constant solution   for every . If

then every solution of the difference equation (37) satisfies

Proof. In view of (39) for every there exists such that

which implies

Now, assume that and consider two equations of the form (40), where and . By Corollary 11 we have that

where satisfies

and satisfies

In view of the assumptions

which completes the proof.

Example 14. The difference equation

where , , and , has a solution

which is convergent if and only if is convergent. In this case and the limiting equation is

Similarly, can be divergent and yet with the limiting equation (51). This shows that if the limiting equation is nonhyperbolic the dynamics of original equation can be very diverse.

Example 15. The difference equation

where , , , and , can be transformed into

where , . Solving (53) and going back to we obtain

that is convergent if and only if and are convergent, where , .
In this case and the limiting equation is

This shows that if the limiting equation is nonhyperbolic the dynamics of original equation can be very diverse and not well described by dynamics of limiting equation.

Example 16. The difference equation (52), where , , , and , has simple behavior in the hyperbolic case, that is, when . Indeed Theorem 13 implies that in this case the global dynamics of (52) are the same as the global dynamics of the limiting equation (1), described in Theorem 1. Thus the following result holds:

Similarly an application of Theorem 13 gives the following result.

Example 17. Consider the difference equation

where (38) holds. Then the following result holds:

where are the positive equilibrium solutions of the corresponding limiting equation

In this case difference equation exhibits Allee’s effect.

Theorem 18. Consider the difference equation

where , are nondecreasing functions and

and the limiting difference equation

Assume that there exists such that every solution of difference equation

converges to a constant solution   for every and . If

where is an equilibrium solution of the limiting difference equation (62), then every solution of the difference equation (60) satisfies

Proof. In view of (61) for all there exist and such that

Let . Then implies

Now, assume that and consider two equations of the form (62). In view of Corollary 11 we have that

where , satisfy

In view of the assumption (64) we have that

which by (68) implies (65).

Theorem 18 has an immediate extension to the th order difference equation of the form

Theorem 19. Consider the difference equation (71), where , , are nondecreasing functions and

and the limiting difference equation

Assume that there exists such that every solution of difference equation

converges to a constant solution for every . If

where is an equilibrium solution of the limiting difference equation (73), then every solution of the difference equation (71) satisfies (65).

3. Single Species Two-Generation Models

We start with an example of cooperative system which is feasible mathematical model in population dynamics that illustrates Theorems 7, 9, and 10 and Corollary 8. This system can be considered as cooperative Leslie two-generation population model, where each generation helps growth of the other.

Example 20. Consider the cooperative system

where , . The equilibrium solutions satisfy equation

which implies that system (76) has always the zero equilibrium and if it has positive equilibrium solutions then it is necessarily , , in which case there is the unique equilibrium solution given as

when

The Jacobian matrix of the map associated with system (76) is

Thus the Jacobian matrix of the map at the zero equilibrium is

and at the positive equilibrium is

The local stability of system (76) is described by the following result.

Claim 1. Consider system (76).

(1)The positive equilibrium of system (76) is locally asymptotically stable when (79) holds.(2)The zero equilibrium of system (76) is locally asymptotically stable if ; it is a saddle point if ; it is a nonhyperbolic equilibrium if .

Proof. (1) After simplification the characteristic equation of becomes

In view of Theorem   in [7] or Theorem   in [8] is locally asymptotically stable if

holds. The inequality is equivalent to (79) and the inequality is equivalent to , which, in view of , , is always satisfied.
(2) The characteristic equation of becomes

In view of Theorem   in [8] is locally asymptotically stable if and it is a saddle point if . Finally, if then is a nonhyperbolic equilibrium point () of stable type () if , of unstable type if , and of resonance type if .

By using Theorems 7, 9, 10, and 18 and Corollary 8 we can formulate the following result which describes the global dynamics of system (76).

Theorem 21. Consider system (76).

(1)If then and if and , if .(2)If then and if and , if .(3)The positive equilibrium of system (76) is globally asymptotically stable when (79) holds.(4)The zero equilibrium of system (76) is globally asymptotically stable when , , and

hold.

Proof. (1) If then the first equation of system (76) implies , which shows that is an increasing sequence, and because there is no positive equilibrium in this case we have that . In view of Theorem 18   is converging to the asymptotic solution of the limiting equation

which completes the proof in this case.
(2) The proof in this case is similar to the proof of case (1) and is omitted.
(3) Assume that (79) holds. In view of Claim 1   is a saddle point and is locally asymptotically stable. By using Corollary 8 we conclude that the interior of ordered interval is attracted to . Furthermore, any solution of system (76) different from which starts on coordinate axis in one step enters the interior of ordered interval and so converges to . Every solution of system (76) satisfies

which, in view of Theorem 10, means that

where , satisfy

Since, , , we obtain that

for some and . In view of (iii) of Theorem 7 every solution which starts in the interior of ordered interval is attracted to . Since system (76) is strictly cooperative we conclude that the whole ordered interval is attracted to . If a solution starts at in the complement of one can find the points and such that . Since is monotone map we have that for , which implies that because .
(4) In view of condition (86), system (76) has only the zero equilibrium. By using (91) we have that the map associated with system (76) has an invariant rectangle , with the unique equilibrium point , and by Theorem 7 every solution which starts in this rectangle must converge to . The fact that the rectangle is also attractive completes the proof.

The following difference equation is known as density dependent Leslie matrix model with two age classes, juveniles and adults:

where the parameters , , , and are positive real numbers and the initial conditions and are nonnegative real numbers. Here is the size of population at th generation. This model was considered first by Kulenović and Yakubu [27] in 2004 and later by Franke and Yakubu [15–17], where the extensions of this model to the periodic environment were considered.

Theorem 22. The density dependent Leslie matrix model (92) exhibits the following properties:

(1)Equation (92) has always the zero equilibrium and when has the unique positive equilibrium .(2)Assume . Then the zero equilibrium of (92) is globally asymptotically stable.(3)Assume . Then the unique positive equilibrium of (92) is globally asymptotically stable.(4)Assume that holds. All solutions of (92) satisfy

where are the real roots of characteristic equation at the equilibrium .(5)Equation (92) has both, the oscillatory and nonoscillatory solutions. The oscillatory solutions have the semicycles of length one, with the possible exception of the first semicycle.

An application of Theorems 18 and 22 yields the following.

Example 23. Consider the difference equation

where the parameters , , , and are positive real sequences and numbers and the initial conditions and are nonnegative real numbers. Assume

If either or , then the global dynamics of nonautonomous difference equation (94) are the same as the global dynamics of the limiting equation (92) and are described by Theorem 22.

Remark 24. Similarly, as for the Beverton-Holt equation the difficult case is when the limiting equation (92) is nonhyperbolic. In this case the dynamics of nonautonomous equation can be quite different than the dynamics of the limiting equation.

4. Local and Global Dynamics of Several Generation Models

We consider the following difference equation as a generalization of density dependent Leslie matrix model with two age classes:

where the parameters , , , and satisfy the following conditions:

for all . Equation (96) is called density dependent Leslie matrix model with age classes. See [28] for some global asymptotic stability results for such model.

Examples of functions which satisfy condition (97) are

The first three functions in (98) are also called Holling functions of types I, II, and III (see [1, 6, 19]) and are widely used in modeling. Condition (97) implies that , , and that is always an equilibrium of (96). Furthermore, if there exists a positive equilibrium , then .

First we state and prove the local stability result for (96) which is sharp.

Theorem 25. Consider (96) subject to condition (97) and assume that the functions , , are differentiable at the equilibrium of (96). Then the equilibrium of (96) is one of the following:

(a)Locally asymptotically stable if (b)Nonhyperbolic and locally stable if (c)Unstable if .

Proof. This result is the consequence of Theorem in Janowski and Kulenović [29] applied to the linearization

of (96) at the equilibrium .

The global result is simple to state and apply.

Theorem 26. Consider (96) subject to condition (97). The zero equilibrium of (96) is globally asymptotically stable if

Proof. The result is an immediate consequence of Corollary in [29] applied to the following linearization of (96):

In this case

and Corollary in [29] implies the global asymptotic stability of the zero equilibrium.

Assume that (96) has a positive equilibrium . Then (100) is not satisfied; that is, .

The global result requires an additional condition which is well known.

Theorem 27. Consider (96) subject to condition (97) and

where are constants, for all . The equilibrium of (96) is globally asymptotically stable if

is satisfied.

Proof. If is differentiable on then condition (103) follows from the condition

The result is an immediate consequence of Corollary in [29] applied to the following linearization of (96):

which, by substitution , becomes the linearized equation

where

Now we have

which in view of Corollary in [29] proves global asymptotic stability of .

By using the monotone convergence results in [7, 8, 30] we obtain the following powerful global asymptotic stability result for (96).

Theorem 28. Consider (96) subject to the condition

where is nondecreasing for every . If there exists a constant such that

then if (96) has the unique positive equilibrium , it is globally asymptotically stable.

Proof. Set

Then . Furthermore, if there exists a constant such that , , then in view of (111) we would have

This shows that the interval is an invariant interval for the function , which is nondecreasing in all its arguments. In view of the theorem in [7, 30], the fact that (96) has the unique positive equilibrium implies that this equilibrium is globally asymptotically stable.

An application of Theorems 25–27 gives the following result for Leslie or Beverton-Holt model with generations.

Theorem 29. The Beverton-Holt model with generations

has the following properties:

(a)If , then the zero equilibrium is globally asymptotically stable.(b)If , then the zero equilibrium is nonhyperbolic and locally stable.(c)If , then the positive equilibrium is globally asymptotically stable.

Proof. Now (a) follows from Theorems 25 and 26 and the fact that

In the case of (c), the global asymptotic stability of the positive equilibrium follows from Theorem 28 with and as and .

Theorem 30. The sigmoid Beverton-Holt model with generations

where the initial conditions are nonnegative numbers, has the following properties:

(a)If , then the zero equilibrium is globally asymptotically stable.(b)If , then the zero equilibrium is globally asymptotically stable within its basin of attraction which contains . The positive equilibrium is locally nonhyperbolic and is global attractor within its basin of attraction which contains .(c)If , then the zero equilibrium is globally asymptotically stable within its basin of attraction which contains . The larger positive equilibrium is globally asymptotically stable within its basin of attraction which contains .

Proof. The linearized equation of (116) at an equilibrium point is

and the characteristic equation of (117) is

Theorem 25 implies the local stability of the zero and the positive equilibrium points. Observe that the equilibrium equation of (116) can have at most two positive solutions.
When (116) has only the zero equilibrium and can be linearized as

Since for every we have

which by Theorem   in [29] or Theorem 27 implies that the zero equilibrium is globally asymptotically stable.
The positive equilibrium satisfies

which either has one positive solution when or has two positive solutions when .
In the case when , the characteristic equation (118) takes the form

with as a solution, which shows that is stable and nonhyperbolic equilibrium.
In the case when , we have two positive equilibrium solutions which satisfy (121) and so . In view of Theorem   in [29] is locally asymptotically stable if and only if

which by (121) implies

Similarly one can show that the equilibrium is unstable if and only if . Consequently, is locally asymptotically stable and is unstable.
Every solution of (116) satisfies , . Using this it follows that the interval , where , is invariant set for monotone map , which contains the unique fixed point . In view of Theorem 7 every orbit of converges to , which means that as .
Assume that . Then (116) has two positive equilibrium solutions , where is unstable and is asymptotically stable. In a similar way to the above the interior of the ordered interval is a subset of the basin of attraction , and the union of the interiors of the ordered intervals and is a subset of the basin of attraction .

An application of Theorem 19 and the global attractivity result proved in [27] for constant coefficient case gives the following global attractivity result for asymptotically constant -stage Beverton-Holt model:

Example 31. Consider the difference equation (125) where (72) holds. Then the following result holds:

Example 32. Consider the difference equation

where (72) holds. Then the following result holds:

where is the smaller and is the bigger positive equilibrium.

It should be noticed that Examples 31 and 32 do not cover a nonhyperbolic case, which is the case when and , respectively.

Finally, in the case of the nonautonomous difference equation (71), where the sequences , , , , and satisfy (97), the global attractivity and global stability results in [29] give some robust global stability results.

Theorem 33. Consider (71) subject to (97). If are bounded sequences for and

(i)

 for some constant , then the zero equilibrium is globally asymptotically stable,(ii)

 then the zero equilibrium is stable.

In the special case , based on the results in [20], we obtain more precise description of the basins of attraction of the equilibrium points as well as the period-two solutions. See also [31–33]. The general theory of competitive and cooperative systems is given in [3, 4, 22, 23, 34–36]. The Leslie model with 2 generations which exhibits Allee’s effect may possess up to three minimal period-two solutions which in certain cases may have substantial basins of attraction as it was shown in [37]. We summarize the results in [37] as follows.

Theorem 34. Consider the difference equation

where , . Then

(a)if then the zero equilibrium of (131) is globally asymptotically stable,(b)if , then the zero equilibrium of (131) is globally asymptotically stable within its basin of attraction which is the region below the global stable manifold , in the Northeast ordering. The positive equilibrium is locally nonhyperbolic and is global attractor within its basin of attraction, which is the region above the global stable manifold in the Northeast ordering,(c)if , then (131) has three equilibrium points and zero, one, two, or three minimal period-two solutions. If there is no minimal period-two solutions and the equilibrium point is a saddle point, then the zero equilibrium is globally asymptotically stable within its basin of attraction which is the region below the global stable manifold , in the Northeast ordering. The larger positive equilibrium is the global attractor within its basin of attraction, which is the region above the global stable manifold in the Northeast ordering,(d)If and (131) has one minimal period-two solution which is a saddle point then the boundary between the basins of attraction of and is the union of the stable manifolds of and the stable manifolds of the period-two points and , , ,(e)If and (131) has two minimal period-two solutions , , where one of them is a saddle point and the other one is nonhyperbolic solution of the stable type, then the boundary between the basins of attraction of and is the union of the stable manifolds of and the period-two points and , , . Furthermore, the basin of the attraction of the nonhyperbolic period-two solution is the region between the stable manifolds of two period-two solutions , ,(f)If and (131) has three minimal period-two solutions , , such that , where , , are saddle points and is locally asymptotically stable, then the boundary of the basins of attraction of is the stable manifold of , the boundary of the basins of attraction of is the stable manifold of , and the basin of the attraction of the period-two solution is the region between the stable manifolds of two period-two solutions , .

See Figure 1 for visual interpretation of cases (d) and (f). See Figure 2 for visual interpretation of case (e).

(a)

(b)

Figure 1: (a) Visual illustration of Theorem 34, case (d). (b) Visual illustration of Theorem 34, case (f).

(a)

(b)

Figure 2: (a) Visual illustration of Theorem 34, case (e), where larger period-two solution is a saddle point. (b) Visual illustration of Theorem 34, case (e), where smaller period-two solution is a saddle point.

Proof. The linearized equation of (131) at an equilibrium point is

and the characteristic equation of (132) is

The linearized equation (133) at the zero equilibrium is , which shows that the zero equilibrium is locally asymptotically stable. The linearized equation (133) at the equilibrium is , with the eigenvalues , , which shows that is the nonhyperbolic equilibrium of stable type. The necessary and sufficient condition for the equilibrium to be nonhyperbolic is . The necessary and sufficient condition for (131) to have three equilibrium points , , is . When the direct calculation shows that is a saddle point and is locally asymptotically stable.
As a consequence of the results in [38] every solution of (131) has two eventually monotone subsequences and , which in view of the boundedness of solutions of (131) implies that all solutions of (131) converge either to the equilibrium or to period-two solutions.
When (131) has only the zero equilibrium and can be linearized as

Since for every we have

which by Theorem   in [29] or Theorem 27 implies that the zero equilibrium is globally asymptotically stable.
The cases and are explained in great details in [37].

Remark 35. The algebraic conditions on the coefficients , for having , , or period-two solutions for model (116) are given in [37]. We have also shown in [37] that another feasible model

where and the initial conditions , are nonnegative, has similar dynamic scenarios as well as the combination of Beverton-Holt and sigmoid Beverton-Holt model:

where and the initial conditions , are nonnegative. In both cases the conditions for existence of period-two solutions are rather complicated.
The biological implications of models (116), (136), and (137) are that for some values of parameters of these models period-two behavior emerges with substantial basin of attraction.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

L. J. S. Allen, An Introduction to Mathematical Biology, Prentice Hall, 2006.

S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005.

M. P. Hassell, H. N. Comins, and R. M. May, “Spatial structure and chaos in insect population dynamics,” Nature, vol. 353, no. 6341, pp. 255–258, 1991.

R. M. May, “Stability and Complexity in Model Ecosystems,” Princeton University Press Princeton, Princeton, NJ, USA, 2001.

H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, vol. 15, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.

H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, USA, 2003.

M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002.

M. R. Kulenović and O. Merino, Discrete dynamical systems and difference equations with Mathematica, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2002.

J. E. Franke and J. F. Selgrade, “Attractors for discrete periodic dynamical systems,” Journal of Mathematical Analysis and Applications, vol. 286, no. 1, pp. 64–79, 2003.

E. A. Grove, Y. Kostrov, G. Ladas, and S. W. Schultz, “Riccati difference equations with real period-2 coefficients,” Communications on Applied Nonlinear Analysis, vol. 14, no. 2, pp. 33–56, 2007.

J. M. Cushing, “Periodically forced nonlinear systems of difference equations,” Journal of Difference Equations and Applications, vol. 3, no. 5-6, pp. 547–561, 1998.

D. A. Jillson, “Insect populations respond to fluctuating environments,” Nature, vol. 288, no. 5792, pp. 699-700, 1980.

J. M. Cushing and S. M. Henson, “Global dynamics of some periodically forced, monotone difference equations,” Journal of Difference Equations and Applications, vol. 7, no. 6, pp. 859–872, 2001.

S. M. Henson and J. M. Cushing, “The effect of periodic habitat fluctuations on a nonlinear insect population model,” Journal of Mathematical Biology, vol. 36, no. 2, pp. 201–226, 1997.

J. E. Franke and A.-A. Yakubu, “Attenuant cycles in periodically forced discrete-time age-structured population models,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 69–86, 2006.

J. E. Franke and A.-A. Yakubu, “Globally attracting attenuant versus resonant cycles in periodic compensatory Leslie models,” Mathematical Biosciences, vol. 204, no. 1, pp. 1–20, 2006.

J. E. Franke and A.-A. Yakubu, “Signature function for predicting resonant and attenuant population 2-cycles,” Bulletin of Mathematical Biology, vol. 68, no. 8, pp. 2069–2104, 2006.

J. M. Cushing, “The Allee effect in age—structured population dynamics,” in Mathematical ecology, T. G. Hallam, L. J. Gross, and S. A. Levin, Eds., pp. 479–505, World Scientific Publishing Co., Singapore, 1988.

C. S. Holling, “The functional response of predators to prey density and its role in mimicry and population regulation,” Memoirs of the Entomological Society of Canada, vol. 97, supplement 45, pp. 5–60, 1965.

M. R. Kulenović and O. Merino, “Invariant manifolds for competitive discrete systems in the plane,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 20, no. 8, pp. 2471–2486, 2010.

E. N. Dancer and P. Hess, “Stability of fixed points for order-preserving discrete-time dynamical systems,” Journal fur die Reine und Angewandte Mathematik, vol. 1991, no. 419, pp. 125–140, 1991.

M. W. Hirsch and H. Smith, Monotone Dynamical Systems, Handbook of Differential Equations, Ordinary Differential Equations (second volume), Elsevier, Amsterdam, the Netherlands, 2005.

M. W. Hirsch and H. Smith, “Monotone maps: a review,” Journal of Difference Equations and Applications, vol. 11, no. 4-5, pp. 379–398, 2005.

M. R. S. Kulenović and O. Merino, “A global attractivity result for maps with invariant boxes,” Discrete and Continuous Dynamical Systems - Series B, vol. 6, no. 1, pp. 97–110, 2006.

R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY. USA, 2nd edition, 2000.

V. Lakshmikantham and D. Trigiante, Theory of difference equations: numerical methods and applications, vol. 251 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, Inc., New York, NY, USA, 2nd edition, 2002.

M. R. Kulenović and A.-A. Yakubu, “Compensatory versus overcompensatory dynamics in density-dependent Leslie models,” Journal of Difference Equations and Applications, vol. 10, no. 13-15, pp. 1251–1265, 2004.

E. A. Grove, C. M. Kent, G. Ladas, S. Valicenti, and R. Levins, “Global stability in some population models,” in Communications in difference equations (Poznan, 1998), pp. 149–176, Gordon and Breach, Amsterdam, 2000.

E. J. Janowski and M. R. Kulenović, “Attractivity and global stability for linearizable difference equations,” Computers & Mathematics with Applications. An International Journal, vol. 57, no. 9, pp. 1592–1607, 2009.

M. R. Kulenović and O. Merino, “Competitive-exclusion versus competitive-coexistence for systems in the plane,” Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences, vol. 6, no. 5, pp. 1141–1156, 2006.

H. L. Smith, “Planar competitive and cooperative difference equations,” Journal of Difference Equations and Applications, vol. 3, no. 5-6, pp. 335–357, 1998.

H. L. Smith, “Invariant curves for mappings,” SIAM Journal on Mathematical Analysis, vol. 17, no. 5, pp. 1053–1067, 1986.

H. L. Smith, “Periodic competitive differential equations and the discrete dynamics of competitive maps,” Journal of Differential Equations, vol. 64, no. 2, pp. 165–194, 1986.

M. P. Hassell and H. N. Comins, “Discrete time models for two-species competition,” Theoretical Population Biology, vol. 9, no. 2, pp. 202–221, 1976.

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, vol. 247 of Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Harlow, UK, 1991.

R. M. May and W. J. Leonard, “Nonlinear aspects of competition between three species,” SIAM Journal on Applied Mathematics, vol. 29, no. 2, pp. 243–253, 1975.

A. Bilgin, M. R. Kulenović, and E. Pilav, “Basins of attraction of period-two solutions of monotone difference equations,” Advances in Difference Equations, vol. 2016, no. 74, 25 pages, 2016.

A. M. Amleh, E. Camouzis, and G. Ladas, “On the Dynamics of a Rational Difference Equation, Part I,” Journal of Difference Equations and Applications, vol. 3, pp. 1–35, 2008.