Global Asymptotic Stability for Discrete Single Species Population Global Asymptotic Stability for Discrete Single Species Population Models Models

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.


Introduction
The following difference equation is known as Beverton-Holt model: where  > 0 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.
The model is well studied and understood and exhibits the following properties.
The following difference equation is known as Beverton-Holt model with constant immigration or stocking where  > 0 is a rate of change (growth or decay), ℎ > 0 is a constant immigration, and   is the size of population at th generation.The simple substitution   =   − ℎ reduces (5) to the so-called Riccati's equation which is well studied and understood (see [7,8]) and exhibits the following properties.
(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  ± = (1/2)(1 ± √ 1 − 4R) and R = /( + 1 + ℎ) 2 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  > 0 is a rate of change (growth or decay) and ℎ > 0 is a constant emigration.The solution of ( 9) is given by (7), where ℎ should be replaced by −ℎ.
(3) If  > 0 then (9)  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.
The following difference equation is known as Beverton-Holt model with periodic immigration or stocking: where  > 0 is a rate of change (growth or decay), ℎ  > 0 is a periodic immigration or stocking, and   is the size of population at th generation.The substitution   =   − ℎ  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].
(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 ℎ  is periodic sequence with periods 2, 3, 4.
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.
The following difference equation is known as the Beverton-Holt model with periodic environment: where  > 1 is a rate of change (growth or decay),   > 0 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   > 0 and rewriting (13) as the substitution   = 1/  reduces (13) to the linear nonautonomous equation where   = ( − 1)/  .The solution of ( 15) is given as and it is well studied and understood and exhibits the following properties.
(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   − 1 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].
The following difference equation, known as sigmoid Beverton-Holt model, is mathematically the simplest Beverton-Holt type model that exhibits Allee's effect: where  > 0 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.
(3) If  < 2, then the equilibrium point 0 is the global attractor.
(5) If  > 2, then there are two attractors: 0 with the basin of attraction B(0) = [0,  − ) 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  0 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 threegeneration 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.

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.

Global Attractivity Results for Monotone Systems.
Let ⪯ be a partial order on R  with nonnegative cone .For ,  ∈ R  the order interval ⟦, ⟧ is the set of all  such that  ⪯  ⪯ .We say  ≺  if  ⪯  and  ̸ =  and  ≪  if  −  ∈ int().A map  on a subset of R  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 R  .See [21] for a more general version valid in ordered Banach spaces.See [22][23][24] for related results.
Theorem 7.For a nonempty set  ⊂ R  and ⪯, a partial order on R  , 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 R  , and if  has no fixed points in ⟦ 1 ,  2 ⟧ other than  1 and  2 , then the interior of ⟦ 1 ,  2 ⟧ is either a subset of the basin of attraction of  1 or a subset of the basin of attraction of  2 .
Consider the general th order difference equation where  : R  → R is continuous function and suppose that 0 and  − > 0 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 R  defined as where for every u, k ∈ R  .Set 0 = (0, 0, . . ., 0), Consequently, by Corollary 8, the interior of the interval ⟦0, x − ⟧ is a part of the basin of attraction of one of two fixed points 0, x − .The reasoning given in the above discussion leads to the following result for general difference equation (19).Theorem 9. Consider (19), where  : R  → R 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 [,   ) +1 and the equilibrium   is globally asymptotically stable within its basin of attraction which contains

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  ∈  +  0 = { 0 ,  0 + 1, . ..} and (, , V) be a nondecreasing function in  and V for any fixed .Suppose that, for  ≥  0 , the inequalities hold.Then implies that Proof.Suppose that ( 27) is not true.Then, there exists a smallest  ∈  +  0 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  1 (, , V) and  2 (, , V) are two functions defined on  +  0 × R 2 and nondecreasing with respect to  and V. Let Then where   and   are the solutions of the difference equations An immediate extension of Theorem 10 is the following result.
Theorem 12. Let   ,  = 0, 1, 2, . .., be sequences satisfying where   is nondecreasing with respect to its argument.Then, Theorem 13.Consider the difference equation where  is nondecreasing function.Assume that and let be the limiting difference equation.Assume that there exists  0 > 0 such that every solution of difference equation converges to a constant solution   for every  ∈ (− 0 , + 0 ).If then every solution of the difference equation (37) satisfies Proof.In view of (39) for every  > 0 there exists  = () > 0 such that which implies Now, assume that  ≤  0 and consider two equations of the form (40), where  =  −  and  =  + .By Corollary 11 we have that where {ℓ  } satisfies and {  } satisfies In view of the assumptions which completes the proof.
Example 14.The difference equation where   ≥ 0,  = 0, 1, . .., and  0 ≥ 0, has a solution which is convergent if and only if ∑ ∞ =0   is convergent.In this case lim →∞   = 0 and the limiting equation is Similarly, ∑ ∞ =0   can be divergent and yet lim →∞   = 0 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  0 > 0,   > 0,  = 0, 1, . .., and lim →∞   = 1, can be transformed into where   = 1/  ,   = 1/  .Solving (53) and going back to   we obtain In this case lim →∞   = 0 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   > 0,  = 0, 1, . .., lim →∞   = , and  0 > 0, has simple behavior in the hyperbolic case, that is, when  ̸ = 1.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  1 ,  2 are nondecreasing functions and and the limiting difference equation Assume that there exists  0 > 0 such that every solution of difference equation converges to a constant solution  , for every  ∈ ( −  0 ,  +  0 ) and  ∈ ( −  0 ,  +  0 ).
where  is an equilibrium solution of the limiting difference equation ( 62 Let  = max( 1 ,  2 ).Then  ≥ () implies Now, assume that  ≤  0 and consider two equations of the form (62).In view of Corollary 11 we have that where {ℓ  }, {  } 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 and the limiting difference equation Assume that there exists  0 > 0 such that every solution of difference equation where  is an equilibrium solution of the limiting difference equation ( 73), then every solution of the difference equation ( 71) satisfies (65).

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 , , ,  > 0,  0 ,  0 ≥ 0. The equilibrium solutions (, ) satisfy equation which implies that system (76) has always the zero equilibrium  0 (0, 0) and if it has positive equilibrium solutions  + (, ) then it is necessarily  < 1,  < 1, 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  0 (0, 0) is and at the positive equilibrium  + (, ) is The local stability of system (76) is described by the following result.
(1) After simplification the characteristic equation of   ( + ) becomes In view of Theorem 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).
(4) The zero equilibrium  0 (0, 0) of system ( 76) is globally asymptotically stable when  < 1,  < 1, and Proof.(1) If  ≥ 1 then the first equation of system (76) implies  +1 >   ≥   , which shows that {  } ∞ =1 is an increasing sequence, and because there is no positive equilibrium in this case we have that lim →∞   = ∞.In view of Theorem 18 {  } ∞ =1 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  0 is a saddle point and  + is locally asymptotically stable.By using Corollary 8 we conclude that the interior of ordered interval ⟦ 0 ,  + ⟧ is attracted to  + .Furthermore, any solution of system (76) different from  0 which starts on coordinate axis in one step enters the interior of ordered interval ⟦ 0 ,  + ⟧ and so converges to  + .Every solution of system (76) satisfies which, in view of Theorem 10, means that where {  }, {V  } satisfy for some  0 > 0 and  ≥ ( 0 ).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 (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 ⟦ 0 , (  ,   )⟧, with the unique equilibrium point 0 , and by Theorem 7 every solution which starts in this rectangle must converge to  0 .The fact that the rectangle ⟦ 0 , (  ,   )⟧ 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  1 ,  2 ,  1 , and  2 are positive real numbers and the initial conditions  −1 and  0 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][16][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  1 +  2 > 1 has the unique positive equilibrium .
(4) Assume that  1 +  2 > 1 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  1 (),  2 (),  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.

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   ≥ 0,  = 0, . . ., , ∑  =0   > 0, and   () satisfy the following conditions: for all  = 0, 1, . . ., .Equation ( 96) is called density dependent Leslie matrix model with +1 age classes.See [28] for some global asymptotic stability results for such model.
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   ,  = 0, . . ., , are differentiable at the equilibrium  of (96).Then the equilibrium  of ( 96) is one of the following: Proof.This result is the consequence of Theorem 3 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  = 0 of ( 96) is globally asymptotically stable if Proof.The result is an immediate consequence of Corollary 1 in [29] applied to the following linearization of (96): In this case and Corollary 1 in [29] implies the global asymptotic stability the zero equilibrium.
Assume that (96) has a positive equilibrium  > 0. Then (100) is not satisfied; that is, ∑  =0   ≥ 1.The global result requires an additional condition which is well known.

Theorem 27. Consider (96) subject to condition (97) and
where   > 0 are constants, for all  = 0, . . ., .The equilibrium  of ( 96) is globally asymptotically stable if The result is an immediate consequence of Corollary 1 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 1 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).
where   () is nondecreasing for every .If there exists a constant  > 0 such that then if (96) has the unique positive equilibrium , it is globally asymptotically stable. Proof.Set Then ( 0 ,  1 , . . .,   ) ≤ ∑  =0     = .Furthermore, if there exists a constant  > 0 such that   ≥ ,  = 0, . . ., , then in view of (111) we would have This shows that the interval [, ] is an invariant interval for the function ( 0 ,  1 , . . .,   ), 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 ∑  =0   < 1, then the zero equilibrium is globally asymptotically stable.
then the zero equilibrium is nonhyperbolic and locally 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  = ∑  =0   and 0 <  ≤ .

Theorem 30. The sigmoid Beverton-Holt model with 𝑘 generations
where the initial conditions are nonnegative numbers, has the following properties: (a) If ∑  =0   < 2, then the zero equilibrium is globally asymptotically stable.
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 ∑  =0   < 2 (116) has only the zero equilibrium and can be linearized as which by Theorem 2 in [29] or Theorem 27 implies that the zero equilibrium is globally asymptotically stable.The positive equilibrium  satisfies which either has one positive solution  = 1 when ∑  =0   = 2 or has two positive solutions when ∑  =0   > 2. In the case when ∑  =0   = 2, the characteristic equation (118) takes the form with  = 1 as a solution, which shows that  = 1 is stable and nonhyperbolic equilibrium.
In the case when ∑  =0   > 2, we have two positive equilibrium solutions  ± which satisfy (121) and so  − < 1 <  + .In view of Theorem 3 in [29]  is locally asymptotically stable if and only if which by (121) implies (124) Similarly one can show that the equilibrium  > 1 is unstable if and only if  < 1.Consequently,  + is locally asymptotically stable and  − is unstable.Every solution of (116) satisfies   ≤ ∑  =0   = ,  =  + 1,  + 2, . ... Using this it follows that the interval ⟦x − , U⟧, where U = (, , . . ., ), is invariant set for monotone map , which contains the unique fixed point x − .In view of Theorem 7 every orbit of  converges to x − , which means that lim   =  − as  → ∞.
Assume that ∑  =0   > 2. 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 ⟦0, x − ⟧ is a subset of the basin of attraction B(0), and the union of the interiors of the ordered intervals ⟦x − , x + ⟧ and ⟦x + , U⟧ is a subset of the basin of attraction B(x + ).
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 32.Consider the difference equation where (72) holds.Then the following result holds: where  − is the smaller and  + is the bigger positive equilibrium.
In the special case  = 2, 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][32][33].The general theory of competitive and cooperative systems is given in [3,4,22,23,[34][35][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  0 ,  1 ≥ 0,  0 +    See Figure 1 for visual interpretation of cases (d) and (f).See Figure 2 for visual interpretation of case (e).
As a consequence of the results in [38] every solution {  } of (131) has two eventually monotone subsequences { 2 } and { 2+1 }, 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  0 +  1 < 2 (131) has only the zero equilibrium and can be linearized as  which by Theorem 2 in [29] or Theorem 27 implies that the zero equilibrium is globally asymptotically stable.The cases  0 + 1 = 2 and  0 + 1 > 2 are explained in great details in [37].
Remark 35.The algebraic conditions on the coefficients  0 ,  1 for having 1, 2, or 3 period-two solutions for model (116) are given in [37].We have also shown in [37] where ,  > 0 and the initial conditions  −1 ,  0 are nonnegative, has similar dynamic scenarios as well as the combination of Beverton-Holt and sigmoid Beverton-Holt model: ,  = 0, 1, . . ., where ,  > 0 and the initial conditions  −1 ,  0 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.

Figure 2 :
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.
1, and  2 are positive real sequences and numbers and the initial conditions  −1 and  0 are nonnegative real numbers.Assume