Global stability of discrete dynamical systems via exponent analysis : applications to harvesting population models

We present a novel approach to study the local and global stability of families of one-dimensional discrete dynamical systems, which is especially suitable for difference equations obtained as a convex combination of two topologically conjugated maps. This type of equations arise when considering the effect of harvest timing on the stability of populations.

Corresponding author.Email: jperan@ind.uned.es In other situations, the solution is much harder.Here, we consider a problem proposed by Cid, Liz and Hilker in [8,Conjecture 3.5].They conjectured that if equation (1.1) has a locally asymptotically stable (L.A.S.) equilibrium, then the difference equation x t+1 = (1 − θ)F 0 (x t ) + θF 1 (x t ), t = 0, 1, 2, . . ., (1.3) also has a locally asymptotically stable equilibrium for each θ ∈ [0, 1], provided that f is a compensatory population map [7].In this paper, we show that this conjecture is true for a broad family of population maps.Indeed, for all maps in that family, we prove that the equilibrium of (1.3) is not only L.A.S. but globally asymptotically stable (G.A.S.).In other words, we provide sufficient conditions for (1.3) to inherit the global asymptotic behavior of (1.1) independently of the value of θ ∈ [0, 1].Equation (1.3) arises when the effect of harvest timing on population dynamics is considered.Together with many other factors, harvest time conditions the persistence of exploited populations, especially for seasonally reproducing species [6,19,28,31], which on the other hand are particularly suitable to be modeled by discrete difference equations [20].A key question in management programmes is to ensure the sustainability of the tapped resources, thus the issue is generating an increasing interest.However, most previous studies have focused on population size and few have addressed population stability.A model proposed in [32] and based on constant effort harvesting-also known as proportional harvesting-allows for the consideration of any intervention moment during the period between two consecutive breeding seasons, a period that from now on we will call the harvesting season for the sake of simplicity.For this model, two topologically conjugated systems are obtained when the removal of individuals takes place at the beginning or at the end of the harvesting season-namely difference equations (1.1) and (1.2).For these two conjugated systems, harvesting with a certain effort-namely the value of r-can create an asymptotically stable positive equilibrium.When individuals are removed at an intermediate moment during the harvesting season, the dynamics of the population follow a convex combination of these limit cases-namely (1.3).In this framework, Conjecture 3.5 in [8] has a clear meaning with important practical consequences: delaying harvest could not destabilize populations with compensatory dynamics.
Previous works have addressed the problem considered here.Cid et al. proved in [8] that the local stability of the positive equilibrium is not affected by the time of intervention for populations governed by the Ricker model [30].They also obtained a sharp global stability result for the quadratic map [25] and the Beverton-Holt model [5].Global stability is always desirable as it allows to predict the fate of populations with independence of their initial size.Yet, proving it is in general a difficult task, this being reflected in the fact that many different schemes have been used in the literature for this purpose.In [14,15], the authors showed that harvest time does not affect the global stability in the Ricker case by using well-known tools, namely results independently proved by Allwright [2] and Singer [34] for unimodal maps with negative Schwarzian derivative and a sufficient condition for global stability in [35,Corollary 9.9].
Little is known about the effect of the moment of intervention on the stability of populations governed by equations different from the Ricker model, the Beverton-Holt model or the quadratic map (although see [14,Proposition 2], where it was proved that the moment of intervention does not affect the stability when the harvesting effort is high enough).To reduce this gap, we introduce an innovative approach that is especially useful to prove the global stability of a broad family of population models, namely those encompassed in the so called generalized α-Ricker model [24].Among others, the Bellows, the Maynard Smith-Slatkin and the discretized version of the Richards models are covered by our analysis [4,26,29].Interestingly, these three models can be seen, respectively, as generalizations of the already studied Ricker, Beverton-Holt and quadratic maps where the term related to the density dependence includes a new exponent parameter α.In the proposed new method, the focus is on α: under certain conditions, we provide sharp results of both local and global stability of the positive equilibrium of the system depending on the value of α.In particular, these results can be considered as the proof, for a wide range of population models, of [8,Conjecture 3.5].It is important to stress that this does not prove the aforementioned conjecture in general, which is impossible since it is false [14], but supports its validity when restricted to meaningful population maps used in population dynamics.
The proposed new method can be applied whenever the per capita production function g has a strictly negative derivative.The domain (0, ρ) of g can be bounded or unbounded.All bounded cases can be easily reduced to the case ρ = 1.The range (g(ρ), g(0)) can also be bounded or unbounded, provided that 0 ≤ g(ρ) < 1 < g(0) ≤ +∞.
The applications that we present in this paper focus on the cases g(0) < +∞ and g(ρ) = 0.In particular, our examples deal with the following models: • The Bellows model, which includes the Ricker model as a particular case (Subsection 4.1).
• The discretization of the Richards model, which includes the quadratic model as a particular case (Subsection 4.2).
• The Maynard Smith-Slatkin model, which includes the Beverton-Holt model as a particular case (Subsection 4.3).
• The Thieme model, which includes the Hassell model as a particular case (Subsection 4.4).
The paper is organized as follows.Section 2 describes the harvesting population model that motivates our study and lists the families of per capita production functions that we will consider in Section 4. Section 3 states and proves the main results.Section 4 is divided in several subsections, each of them consisting in an example of the applicability of the main results.Finally, Section 5 focuses on the "L.A.S. implies G.A.S." and the "stability implies G.A.S" properties.

Per capita production functions
First-order difference equations are commonly used to describe the population dynamics of species reproducing in a short period of the year.Usually, these equations take the general form where x t corresponds to the population size at generation t and map g to the per capita production function, which naturally has to be assumed as non-negative.In addition, g is frequently assumed to be strictly decreasing, because of the negative effect of the intraspecific competition in the population size, and when that condition holds the population is said compensatory [7,20].Theoretical ecologists have developed several concrete families of per capita production functions.These families depend on one or several parameters, which are essential to fit the functions to the experimental data.
Our results cover some of the most relevant families of compensatory population maps, which, as it was pointed out in [24], can be described in a unified way using the map where α, κ ∈ R ++ and p ∈ R \ {−∞}, with R ++ denoting the set of positive real numbers and R := [−∞, +∞] the extended real line.
The following models are obtained for different values of the parameters: [M1] For p = 1 and α = 1, the Beverton-Holt model [5], in which g(x) = κ 1+x .
Models [M1-M3] are compensatory.Nevertheless, [M2-M3] are always overcompensatory [7,9] (map xg(x) is unimodal) and can have very rich and complicate dynamics, whereas [M1] is never overcompensatory (the map xg(x) is increasing) and has pretty simple dynamics: all solutions monotonically tend to the same equilibrium which, consequently, is G.A.S. Map (2.2) also includes models that are overcompensatory or not depending on the values of the parameters: [M4] For p = 1, the Maynard Smith-Slatkin model [26], in which g(x) = κ 1+x α .

Models [M7-M8] generalize [M2-M3
] by including a new exponent parameter α, which determines the severity of the density dependence and makes the models more flexible to describe datasets [4].This is the announced exponent parameter playing a central role in our study.
Before presenting the harvesting model where these population production functions will be plugged in, it is convenient to make some remarks.First, we point out that the domain of g is bounded for models [M2] and [M7], whereas it is unbounded for the rest of models.When the domain of g is bounded, there is a restriction in the parameters involved in the map for which (2.1) is well-defined.On the other hand, a suitable rescaling allows to obtain other frequently used expressions of these eight models depending on an extra parameter, e.g.g(x) = κ(1 − mx) for the quadratic model or g(x) = κe −mx for the Ricker model.This extra parameter is irrelevant for the dynamics of (2.1).

Modelling harvest timing
Assume that a population described by (2.1) is harvested at the beginning of the harvesting season t and a fraction γ ∈ [0, 1) of the population is removed.Then, it is well established that the population dynamics are given by (2.3) When individuals are removed at the end of the harvesting season, the population dynamics follow ). (2.4) The above situations represent the two limit cases of our problem.To model the dynamics of populations harvested at any time during the harvesting season, we consider the framework introduced by Seno in [32].Let θ ∈ [0, 1] represent a fixed time of intervention during the harvesting season, in such a way that θ = 0 corresponds to removing individuals at the beginning of the season and θ = 1 at the end.Assume that the reproductive success at the end of the season depends on the amount of energy accumulated during it.Given that the per capita production function depends on x t before θ and on (1 − γ)x t afterwards, Seno assumed that the population production is proportional to the time period before/after harvesting.This leads to the convex combination of (2.3) and (2.4) given by In particular, substituting θ = 0 in (2.5) yields (2.3), and (2.4) is obtained for θ = 1.The two maps derived from (2.5) for θ = 0 and θ = 1 are topologically conjugated.Thus, if the equilibrium for θ = 0 is G.A.S., then the equilibrium for θ = 1 is also G.A.S., and vice versa.From a practical point of view, this implies that for these two limit cases we can predict the long-run behavior of the system with independence of the initial condition.In view of this, it is natural to study to what extent the same is true if individuals are removed at any intermediate moment during the harvesting season.
Substituting map (2.2) into (2.5),we obtain an intricate model depending on up to five parameters for which establishing general local or global stability results is a tricky task.For that purpose, we develop a general method in the following section.

Exponent analysis method
Consider the difference equation Notice that the domain of h can be the open bounded interval (0, 1) or the open unbounded interval (0, +∞), covering all the models described in the previous section.In addition, the image of h can be bounded or unbounded, although the applications presented in this paper are restricted to the bounded case.
For ρ = 1, it is not obvious that the difference equation x t+1 = x t g s (x t ) is well-defined, i.e. xg s (x) ∈ (0, ρ) for x ∈ (0, ρ).Next, we study when the difference equation x t+1 = x t g s (x t ) is well-defined and has a unique equilibrium.We establish some notation first.Being the function a diffeomorphism from (0, ρ) to (ν s b, µb), where we denote by j s its inverse diffeomorphism, i.e., the function j s : (ν s b, µb) → (0, ρ) satisfying for all z ∈ (ν s b, µb).Obviously, when ρ = +∞, one has where ν s is given by (3.1).Then, the map xg s (x) has a unique fixed point in (0, ρ) if and only if s > s * .Moreover, this fixed point is (j s (1)) and that ν s depends continuously on s.Since g s maps (0, ρ) onto (ν s b, µb) and νb < 1 < µb holds, we have that the equation g s (x) = 1, for x ∈ (0, ρ), has solution if and only if s > s * .We have already stressed that ν s = ν, for ρ = +∞.Hence, we have s * = 0 for ρ = +∞.
In the conditions of Lemma 3.1, for each s ∈ (0, 1] we define the function Now, we study under which conditions the difference equation x t+1 = x t g s (x t ) is well- defined.
Proof.We consider separately the cases ρ = +∞ and ρ = 1.The case ρ = +∞ is trivial.For ρ = 1, we have The latter inequality always holds if z ≤ 1 µb , because g s ((0, 1)) = (ν s b, µb).Hence, Since ρ = 1, we have that τ s (z) > 0 for z ∈ 1 µb , 1 and which finishes the proof of the first affirmation.For the second one, notice that α s decreases as we increase s, because j s decreases with s.Therefore, Now, in the conditions of Lemma 3.1, for each s ∈ (s * , 1], we write and define the function Proof.A direct application of L'Hôpital's rule shows that σ s is a continuous function: On the other hand, to see that σ s takes values on R ++ note that z → ln(j s (z)) is a decreasing function and that j s is a diffeomorphism, so j s (1) < 0. Finally, for ρ = 1, one has The function σ s , given in (3.8), is related to the fixed points of the map f s • f s with f s (x) = xg s (x), as we will see next.Assuming α < α s , for the map f s • f s to be well-defined, and rearranging for α in the fixed points equation we have (see Lemma 3.1) In other words, the difference equation x t+1 = x t g s (x t ) has a nontrivial period-2 orbit if and only if there exists z ∈ (1/b s , b s ) \ {1} and α < α s such that σ s (z) = α.Consequently, considering σ s for the study of the global stability of the equilibrium of x t+1 = x t g s (x t ) is natural since, by the main theorem in [10], the absence of nontrivial period-2 orbits for x t+1 = x t g s (x t ) is equivalent to the global asymptotic stability of this equilibrium.More specifically, we will use the following result: 1) and apply the Sharkovsky Forcing Theorem [33] to see that , then x ∞ would not be stable for the map f (2) , since x j = f (2nj) (x 0 ) would be a decreasing sequence, for all x 0 ∈ (a 1 , x ∞ ).Applying the same argument for the interval (x ∞ , a 2 ), we conclude that ( In particular, replacing x with f (m) (x), one has ( Therefore, the subsequence of f (n) (x) n formed by the terms smaller (respectively, greater) than the x ∞ is increasing (respectively, decreasing).Then, lim n→∞ f (n) (x) = x ∞ , for all x ∈ I.The converse is obvious.
Moreover, we point out that the following theorem (which is the main result of this paper) can be applied under very general conditions.In particular, it holds when the per capita production function has unbounded range.
In what follows, ρ, µ, ν, b and c will be considered as constants, while s and α will be mostly seen as parameters.
By the symmetry of σ s and applying an analogous argument as the one presented in (3.9)-(3.12)we obtain that To prove (ii), in view of (i) above, (3.9)-(3.12)and Lemma 3.4, just consider the following four scenarios: Therefore, the equilibrium x ∞ is unstable.
• In any other case, the equation x t+1 = f s (x t ) has nonconstant periodic solutions.Therefore, the equilibrium x ∞ is not G.A.S.
(B).We start by verifying that the function .
Since condition (H 1 ) holds and j s is a decreasing diffeomorphism, we have that the function z → ∂(ln j s (z))/∂s is non-decreasing in (1/b s , b s ) for each s ∈ (s * , 1].Thus, ) is well-defined for s = 1, by Lemma 3.2, we know that (3.13) is well-defined for s ∈ (s * , 1), and, if its equilibrium is G.A.S. for s = 1, (A)-(ii) and the fact that Therefore, (3.13) is well-defined and its equilibrium is G.A.S. for all s ∈ (s * , 1]. (C).Following the same reasoning as in the previous case but using (H 2 ) instead of (H 1 ), it is easy to see that the function s → σ s (z) is decreasing for each z ∈ (1/b s , b s ).As a consequence, if the equilibrium of (3.13) is L.A.S. for s = 1, the application of (A)-(i) yields α ≤ σ 1 (1) < σ s (1), for all s ∈ (s * , 1], and (3.13) is well-defined and its equilibrium is L.A.S. for all s ∈ (s * , 1].Remark 3.7.Note that σ s • exp is an even function, which makes it more suitable for graphical representations than σ s itself.Theorem 3.6 reduces the study of the local or global stability to the study of the relative position of the graph of σ s with respecto to α. Figure 3.1 illustrates this.For a fixed s, the relative position of min z∈(1,b s ) σ s (z), σ s (1) and α determines the local and global stability of the equilibrium of (3.13).Suppose that the graph of σ s corresponds to the black curve in Figure 3.1-A.From (i) and (ii) in Theorem 3.6, we obtain that the equilibrium of (3.13) is unstable for α > σ s (1), L.A.S. but not G.A.S. for min z∈(1,b s ) σ s (z) < α < σ s (1), and G.A.S. for α < min z∈(1,b s ) σ s (z). Figure 3.1-B illustrates the special case when the function σ s attains a strict global minimum at z = 1.In such a situation, the range of values of α for which the equilibrium is L.A.S., thanks to (i) in Theorem 3.6, is contained in the range of values of α for which it is G.A.S., thanks to (ii) in Theorem 3.6.Hence, in this case, Theorem 3.6 completely characterizes the stability of the equilibrium of (3.13): it is G.A.S. for α ≤ σ s (1) and unstable for α > σ s (1).
it is L.A.S. but not G.A.S., and for α < min z∈(1,b s ) σ s (z) it is G.A.S. B: Since σ s attains at z = 1 a strict global minimum, the equilibrium of (3.13) is G.A.S. for α ≤ σ s (1).C: The assumption that σ 1 attains a strict global minimum at z = 1 and condition (H 1 ) are sufficient to guarantee that the graphs of the family of functions {σ s } 0<s≤1 are above the graph of σ 1 and, consequently, the equilibrium of (3.13) is G.A.S. for each s ∈ (0, 1] and α ≤ σ 1 (1).
Figure 3.1-C deals with the last part of Theorem 3.6.Assume that σ 1 (1) is a global minimum of σ 1 (z) and that condition (H 1 ) holds.Then, all the graphs of the family of functions {σ s } 0<s≤1 are above the graph of σ 1 (z) and, therefore, the equilibrium of (3.13) is G.A.S. for each α ≤ σ 1 (1) and 0 < s ≤ 1.
Apart from condition (H 1 ), Theorem 3.6-(B) assumes that (3.13) is well-defined and that its equilibrium is G.A.S. for s = 1.But we have already mentioned that guaranteeing the G.A.S. of an equilibrium is a difficult task.Nevertheless, when the logarithmically scaled diffeomorphism φ s (u) := ln (j s (e u )) is C 3 , we can derive a sufficient condition for σ s (1) to be the strict global minimum of σ s (z).Lemma 3.8.If φ s (u) := ln (j s (e u )) is three times continuously differentiable with φ s (u) < 0 for all u ∈ (− ln b s , ln b s ), then σ s (z) attains at z = 1 its strict global minimum value.
Proof.It is routine to check that for j = 0, 1, 2, and that

Application to some population models
The next result characterizes the elements of the family of per capita production functions (2.2) for which condition (H 1 ) in Theorem 3.6 holds.(1 + qx) 1/q is a decreasing diffeomorphism.Moreover, h satisfies (H 1 ) for each s ∈ (0, 1) if and only if p ≥ −1.
Finally, the result is straightforward for p = 0 since h(x) = e −x and h (x) h (sx) = e −(1−s)x .
The following subsections deal with the study of the harvesting model (2.5) for the per capita production functions in Subsection 2.1.We use a similar procedure for all of them, based on the following five steps: 1. First, we rewrite the difference equation that we want to study, which will depend on certain original parameters, as (3.13) with parameters b, c, s, α, ν, µ and ρ.
2. We check that h satisfies condition (H 1 ), thanks to Lemma 4.1.
3. If necessary, we check that (3.13) is well-defined for s = 1.Next, we invoke Lemma 3.8 to guarantee that the rewritten difference equation, with s = 1, has an equilibrium which is G.A.S.
4. Then, we use statement (B) in Theorem 3.6 to conclude the global stability result for s ∈ (s * , 1].
5. Finally, we interpret the result in terms of the original parameters.
In this example, it is natural to assume that (4.2) is well-defined for γ = 0, i.e., that the population model without harvesting makes sense.As mentioned when we presented this per capita production function in Subsection 2.1, equation (4.2) is well-defined for γ = 0 if and only if ακ < (1 + α) Clearly, the function h(x) is a decreasing diffeomorphism from (0, 1) to (0, 1) and, by Lemma 4.1, satisfies condition (H 1 ).
We aim to obtain a global stability result for (3.13) with s = 1, which is equivalent to (4.2) with θ = 1.Note that (3.13) is well-defined for s = 1 because αb In order to use Theorem 3.6, we need to impose But, for the selected values of the parameters, this is always true because where we have used that ) is well-defined and has a unique equilibrium.If, in addition, θ = 1, then the equilibrium of (4.2) is unstable for κ To our knowledge, Proposition 4.3 gives the first global stability result for the discretization of the Richards model even in the case without harvesting.Notice that the results in [22] cannot be used in this case since ρ = +∞.In the harvesting framework, Proposition 4.3 includes [8, Proposition 3.6] as a particular result, where the quadratic model was considered.
In [8], following [1, Appendix S1] and [23, Theorem 1], it was stated that the equilibrium of (4. No result is known about global dynamics of (4.3), in the general case.However, this model can be easily handled thanks to Theorem 3.6 and Lemma 4.1.
It is interesting to note that considering the exponent parameter α in the quadratic model, i.e., studying the discretization of the Richards model, unveils the complete parallelism between the Maynard Smith-Slatkin model and the quadratic model with respect to stability results.

Hassell and Thieme models
As already mentioned, topologically conjugated production functions give rise to equivalent dynamical behaviors.However, when a convex combination of the type of (2.5) is applied to two topologically conjugated production functions, the transformed systems could exhibit different dynamical behaviors.
When applying Theorem 3.6, while working in the case s = 1, we can replace our production function by a topologically conjugated one, for which calculations are simpler.This replacement is no longer valid when checking condition (H 1 ).
In this subsection, we put into practice the previous approach to study the two models still left: Thieme's and Hassell's models.Since Thieme's model has Hassell's model as a particular case, we only consider the former.Besides, without loss of generality, we assume the per capita production function of the Thieme model to be given by Now, the change of variables y t = x 1/β t shows that the dynamics of the difference equation are identical of those of the equation , whose per capita production function, g(x) = κ 1/β 1 + x αβ , belongs to the Maynard Smith-Slatkin family of maps.This provides a straightforward way to characterize the global stability of the Thieme model.The previous result improves the global stability condition presented in [35] with a simpler proof than the one used in [22], which relies in calculating the sign of a certain Schwarzian derivative.
The Seno model (2.5) for the Thieme production function is Again, in order to apply the results in Section 3, we set b = κ(1 − γ) > 1, c = κ(1 − γ)θ and h(x) = 1 (1+x) β , which is a decreasing diffeomorphism from (0, +∞) to (0, 1) satisfying condition (H 1 ), thanks to Lemma 4.1.And we get the following new result about the Thieme model under harvesting.Altogether, we have shown that [8,Conjecture 3.5] holds when restricted to the per capita production functions [M1-M8].Indeed, we have shown that a stronger result holds since we are able to guarantee that the equilibrium is G.A.S. for θ ∈ (0, 1).Furthermore, using part C of Theorem 3.6 we obtain the following general local stability result in the spirit of [8,Conjecture 3.5].

On the "L.A.S. implies G.A.S." property
Global stability is a desirable property of systems as it allows to predict the fate of orbits with independence of the initial condition.For many well-known discrete population models, the local asymptotic stability of the equilibrium implies its global asymptotic stability; e.g., see [3,11,13,16,21,22,27].However, determining whether this popular statement in population dynamics is true when delayed harvesting is exerted on populations governed by any of the well-known models [M3-M8] is an open problem.Although the property "L.A.S. implies G.A.S." has been cited many times, we have not been able to find a definition of it in the literature, so we provide the following here.Definition 5.1.Let F 0 be a set of maps f : U f ⊂ R + → U f , each of them having a unique fixed point x f ∈ U f .Given F ⊂ F 0 , we say that the family of discrete equations with f ∈ F (or F itself) satisfies the "L.A.S. implies G.A.S." property when Analogously, we say that F satisfies the "stability implies G.A.S." property when Note that both properties are inherited by subsets.It should also be stressed that a family F trivially satisfies both properties when x f is G.A.S. for all f ∈ F.Moreover, these properties lack interests when F is a singleton, F = { f }.We will avoid affirming that such families satisfy these properties in that case.
A direct application of our main theorem leads to the following results about the "stability implies G.A.S." and "L.A.S. implies G.A.S." properties.
Corollary 5.2.Let F ⊂ F 0 .The family F satisfies the "stability implies G.A.S" property if and only if the map σ s (z) attains its strict global minimum at z = 1, for each function in F.
Proof.Suppose that there exists a function in F for which the corresponding map σ s satisfies σ s (z 0 ) ≤ σ s (1) for some z 0 = 1 in dom σ s .On the one hand, if σ s (z 0 ) < σ s (1), by considering α ∈ (σ s (z 0 ), σ s (1)), we find a case for which the equilibrium x ∞ is L.A.S., but not G.A.S. (apply part (A) in Theorem 3.6).On the other hand, if σ s (z) ≥ σ s (1) for all z ∈ dom σ s , we consider α = σ s (1).In this case (see (3.14)), we would have for all x ∈ (0, ρ), and thus the equilibrium x ∞ is stable.However, this equilibrium is not G.A.S., because α = σ s (z 0 ) (apply part (A) in Theorem 3.6 again).In both cases, the above family of discrete equations does not satisfy the "stability implies G.A.S." property.
Conversely, assume that there is a function in F and an admissible value of α > 0 for which the corresponding equilibrium x ∞ is stable, but not G.A.S.By part (A) in Theorem 3.6, α ≤ σ s (1) and there exists z 0 ∈ dom σ s with α = σ s (z 0 ).In this case, the map σ s (z) does not attain its strict global minimum at z = 1.
We consider now F 1 to be the set of those functions f ∈ F 0 for which s = 1 and h satisfies condition (H 1 ) and F 2 to be the set of those functions f ∈ F 0 for which α ≤ σ 1 (1) and h satisfies condition (H 1 ).Corollary 5.3.Let F ⊂ F 0 be such that F ∩ F 1 satisfies the "L.A.S. implies G.A.S." property.Then, x f is G.A.S. for every function f ∈ F ∩ F 2 .
The above statement can be reformulated as follows.
Proof.Apply the part (B) in Theorem 3.6 and Corollary 5.2.
Finally, consider the set F 3 ⊂ F 0 made up of those functions for which the logarithmically scaled diffeomorphism φ s (u) := ln (j s (e u )) is C 3 with negative third derivative.
The above statement can be reformulated as follows.

L.A.S. does not imply G.A.S. for Seno's model
In Section 4, it was proved that for models [M1-M8] the property "L.A.S implies G.A.S." is true for θ∈ {0, 1}.In view of this, it would be natural to conjecture the validity of the property for (2.5) with any intervention moment θ ∈ [0, 1].Nevertheless, the conjecture would be false.
To prove this, under the conditions of Theorem 3.6, it is enough to find population parameters for which min z∈(1,b) σ s (z) < σ s (1).Surprisingly, the counterexample can be found using one of the models for which we have seen that harvest time is not destabilizing, namely the Maynard Smith-Slatkin model.Consider If we fix α = 5.9, the aforementioned equation is equivalent to (4.3) for κ = 46, γ = 0.775 and θ = 0.8853.From the biological point of view, the latter corresponds to a certain population that is harvested at a given moment during the harvesting season.Let us study the effect that changing the moment of intervention would have on the stability of the equilibrium size of this population.If harvesting was exerted at the beginning or at the end of the season (i.e., θ = 0 or θ = 1), the equilibrium would be unstable (the derivative of the production function at that point is approximately −4.33).The above discussion shows that harvest timing can be stabilizing by itself in this case, since we have seen that the equilibrium is L.A.S. for θ ≈ 0.8853.Numerical simulations reveal that this happens not only for this intervention moment but for all those ranging from 0.8546 to 0.9368, approximately.However, the asymptotic stability of the equilibrium is only local for all these harvest times, given that the inequality min z∈(1,b s ) σ s (z) ≤ α < σ s (1) holds for all of them.
The local stability of the equilibrium implies that nearby orbits are attracted towards it, being the convergence speed determined by the absolute value of the derivative of the production function at that point.In the case considered above, harvest time is not only stabilizing but can also turn the equilibrium into superstable by reducing that derivative to zero.This happens for two intervention moments, namely θ ≈ 0.8853 and θ ≈ 0.918668 (cf., Figure 5.1-B).On the other hand, we have seen that for θ ≈ 0.8853 the equilibrium is not G.A.S., and thus some nonzero orbits escape from its attraction.It would be possible that this only happened for few initial conditions, but it is not the case and the equilibrium coexists with another positive attractor, namely an attracting 12-cycle (cf., Figure 5.1-C).
This example shows that the stabilization of a population through a delay in the time of intervention is a sensitive issue: we could achieve the local stability of the equilibrium, even with the fastest possible convergence of nearby orbits and, at the same time, induce bistability in the global population dynamics.Such a situation is in general undesirable, since small perturbations could lead to sudden sharp changes in the population size.Finally, this example provides a family of first order difference equations arising in population dynamics where "L.A.S. does not imply G.A.S.".We note that for higher order difference equations motivated by population dynamics, it was showed recently in [18] that "L.A.S. does not imply G.A.S." for Clark's equation with a nonlinearity with negative Schwarzian derivative if the order of the equation is at least four.3) for κ = 46, α = 5.9, γ = 0.775 for varying harvest time in the range for which the equilibrium is locally asymptotically stable.Red dots correspond to the initial condition x 0 = 6, and blue dots correspond to x 0 = 6.6.The vertical dashed lines represent the intervention moments for which the equilibrium is superstable (namely θ ≈ 0.8853 and θ ≈ 0.918668).