Research Article About the Properties of a Modified Generalized Beverton-Holt Equation in Ecology Models

This paper is devoted to the study of a generalized modified version of the well-known Beverton-Holt equation in ecology. The proposed model describes the population evolution of some species in a certain habitat driven by six parametrical sequences, namely, the intrinsic growth rate (associated with the reproduction capability), the degree of sympathy of the species with the habitat (described by a so-called environment carrying capacity), a penalty term to deal with overpopulation levels, the harvesting (fishing or hunting) regulatory quota, or related to use of pesticides when fighting damaging plagues, and the independent consumption which basically quantifies predation. The independent consumption is considered as a part of a more general additive disturbance which also potentially includes another extra additive disturbance term which might be attributed to net migration from or to the habitat or modeling measuring errors. Both potential contributions are included for generalization purposes in the proposed modified generalized Beverton-Holt equation. The properties of stability and boundedness of the solution sequences, equilibrium points of the stationary model, and the existence of oscillatory solution sequences are investigated. A numerical example for a population of aphids is investigated with the theoretical tools developed in the paper.


Introduction
The use of mathematical models in ecology is nowadays of important research interest, 1-10 because such models lead to a more precise study of the dynamics and interactions of populations among them and with the habitat.This paper is devoted to the study of a modified version of the well-known Beverton-Holt equation in ecology which describes the population evolution of some species in a certain habitat.The associate dynamics evolves subject to an intrinsic growth rate associated with the reproduction capability and the degree of sympathy of the species with the habitat described by a so-called environment carrying capacity, 2-6, 11, 12 .The standard Beverton-Holt equation becomes the so-called generalized Beverton-Holt equation 12 , when extended by considering in the model two extra sequences, namely, the harvesting namely, fishing or hunting quota, or regulation through the use of pesticides and an additive disturbance sequence.Such a disturbance sequence consists of an independent consumption sequence, which describes a predation of the species though which it gains, develops, and grows plus a net migration sequence.The migration term plays a close role to that of the independent consumption in the population evolution and it is due to potential positive/negative migration towards/from the habitat under study.In general, the joint contribution of independent consumption plus positive or negative migration resp., immigration and emigration towards or from the habitat could also potentially include other extra effects of correction of contribution of population in the model.Some of those effects are, for instance, unexpected accidental mortality not fixed through properly adjusting the intrinsic growth rate, loss of population due to harvesting which has not been parameterized through the harvesting quota or measuring modeling errors in general.The study in this paper adds two more sequences to the generalized Beverton-Holt equation to conform a penalty term for eventual overpopulation at intermediate sampling points.This is the main novelty of the paper related to previous literature on the subject.The penalty term corrects the population evolution so that large amounts of population translate into the decrease of the number of individuals in the future.The motivation of the use of the penalty term is clear since it is well known that large amount of individuals in a habitat makes the overall population decrease by natural regulation, for instance, due to competition for food or difficulties for nesting.The properties of boundedness and stability of the solution, existence of equilibrium points of the limit stationary version of the equation, as well as the existence of oscillatory solutions are investigated.The following basic notation is being used through the manuscript.g x denotes the first derivative with respect to x of the differentiable real function g x .The symbols "∧" and "∨" stand for logic conjunction and disjunction, respectively.I n is the identity matrix of the nth order.
The population evolution is supposed to be governed by a modified Beverton-Holt equation, subject to initial conditions x −1 > 0, x 0 ≥ 0, which is parameterized by the following real sequences.
1 is the intrinsic growth rate which is associated to the population ability for reproduction.
is the environment carrying capacity which is associated to the sympathetic degree, more or less favorable, of the habitat towards the species.
is the external disturbance term which refers to joint effects of independent consumption plus migration contribution, namely, immigration i.e., positive net migration from outside towards the habitat , null, or emigration i.e., negative net migration from the habitat to outside including atypical mortality not included in the model by the growth rate: the disturbance.The additive disturbance contribution has the generic form d k d 0k d mk where d 0k is the independent consumption and d mk is the net migration contribution.The decomposition of d k d 0k d mk into the two separated parts d 0k and d mk is irrelevant for the modified generalized Beverton-Holt equation from an analysis point of view but it is quite relevant from a biological insight point of view, as pointed out in detail by one of the reviewers.The reason is that it allows the inclusion in the same evolution equation of two quite different effects, in nature, like predation and net migration.
are sequences which conform a penalty term x p k k e −α k x k to describe how the excess of population translates into a contributed tendency to its decrease at the next sampling time.The sequences are assumed to be subject to the constraints ∞ > p 2 ≥ p 1 ≥ 0, ∞ > α 2 ≥ α 1 ≥ 0, and α 1 0 ⇒ p 1 0.
A particular case of 1 is obtained with γ k 1 no harvesting activity , p k α k 0 no penalty correcting term for overpopulation control , d k 0 zero jointly net independent consumption plus migration resulting in the well-known standard Beverton-Holt equation, which is a particular version of the Hassell model for c k 1 and where a double sampling period is used related to 1.2 .In other words, only the recruitment and storage measurements are relevant to describe the population evolution since there is not necessary the evaluation at intermediate samples of a penalty term associated with overpopulation.Another particular case of 1.2 is obtained with p k α k 0 resulting in the so-called generalized Beverton-Holt equation, also described equivalently as a two-stage evolution equation, which includes discontinuities at sampling times due to harvesting and disturbance activities associated with impulses in the corresponding logistic continuous-time differential equation.Finally, note that the modified Beverton-Holt equation leads to the Ricker model which is also a limiting equation of the Hassell model and that of the Beverton-Holt one if c k 1 : in order to ensure that γ k < 1.Thus, the environment carrying capacity satisfies lim inf To investigate issues on the standard Beverton-Holt equation, it is useful to analyze the solution sequence of its inverse 11 .The mathematical properties of the modified generalized Beverton-Holt equation including boundedness properties, equilibrium points, and existence of oscillations are investigated.A numerical example is proposed for the study of the evolution of a population of aphids, a very resistive species of insects which causes considerable damage and reproduces according to several reproduction cycles per year 13, 14 .

Boundedness of the solutions
There is a wide variety of works concerning boundedness of the solutions, stability, investigation of equilibrium points, and existence of oscillatory solutions in discrete recursive sequences of several types see, e.g., 15-23 .Parallel problems are also relevant in continuous-time differential systems 16, 24-26 .To investigate those problems in the context of the modified Beverton-Holt equation 1.2 , define s k : x −1 k , k ∈ N, being the solution sequence of the inverse of the modified Beverton-Holt equation provided that x k / 0, k ∈ N. Thus, one gets directly from 1.2 where which can be rewritten as for the standard Beverton-Holt equation d k α k p k ≡ 0 implying ζ k ≡ 1 and γ k ≡ 1 and double sampling rate; see 1.3 , 2.2 -2.4 becomes in particular 11 The boundedness of the solutions for bounded initial conditions is a minimal requirement in order that a biological model is well posed.The subsequent result is concerned with boundedness and global Lyapunov's stability of the solution sequences of 1.2 and 2.3 -2.4 .
0 and {s k } ∞ 0 are positive bounded sequences.As a result, the modified Beverton-Holt equation 1.2 is globally Lyapunov's stable.
iii If d 1 ≥ 0, then {x k } ∞ 0 and {s k } ∞ 0 are nonnegative sequences, the first one being bounded from above.As a result, the modified Beverton-Holt equation 1.2 is globally Lyapunov's stable.Furthermore, since dζ x /dx 0 for α / 0 only at its relative maximum x p/α so that ζ p/α p/αe p max x∈R 0 ζ x since ζ 0 0 and lim x → ∞ ζ x 0 implying that the relative maximum is also the absolute maximum.As a result, for all k ∈ N and i has been proved. If 1 < ∞, for all k ∈ N and ii has been proved. If iii has been proved.Property iv follows directly from 1.2 .
For formulation coherency, the population cannot be negative at any sample.This idea motivates the following axiom and simple-related assertions which will be then useful for some mathematical proofs concerning the case of the eventual negativity of the joint disturbance contribution.
Proofs of Assertions 2.3-2.6.Assertion 2.4 is equivalent to prove that its contrapositive proposition x k ≤ 0 for some k ∈ N ⇒ d 2 ≥ 0 is true.Assume that x k ≤ 0 for some k ∈ N.Then, Axiom 2.2 implies that x k 0 so that ζ k 0 and then x k 1 d k ≥ 0 from 1.2 and Axiom 2.2 so that Assertion 2.3 is proved.Also,

2.8
Assertion 2.6 follows since for any k ∈ N, The extinction of the standard Beverton-Holt equation is investigated in 27 when the intrinsic growth rate is less then unity.Note that eventual extinction is also admitted by Axiom 2.2 for intrinsic growth rate exceeding unity and negative disturbance contribution.This situation includes as particular cases that of local emigration plus zero-independent consumption and that of negative-independent consumption with no net migration contribution.However, the axiom prevents against eventual negative levels of population which is not physically possible.
Remark 2.7.Note that Assertion 2.5 together with an intermediate result in the proof of Theorem 2.1 implies Now, Axiom 2.2 and Assertions 2.3-2.6 extend the results of Theorem 2.1 concerning positivity of the solutions and further stability results as follows.
i {x k } ∞ 0 and {s k } ∞ 0 are nonnegative sequences which are bounded and positive and bounded from below, respectively, independently of the disturbance contribution.As a result, the modified Beverton-Holt equation 1.2 is globally Lyapunov's stable.Furthermore, if d k > 0, for all k ∈ N, both sequences are positive and bounded. ii

2.11
Proof.i Note from 1.2 , Axiom 2.2, and Assertion 2.5 that, for d k < 0, 12 so that from 2.4 ii .As a result, Property i has been fully proved.
ii Recursive direct calculations from 2.4 yield taking i into account irrespective of the value of the disturbance contribution: is positive and bounded from Theorem 2.1.Thus, the first two constraints linked via logic conjunction follow directly from 2.14 since either However, from the equivalent contrapositive logic proposition to Assertion 2.6, iii It is proved using similar techniques as those used in the proof of ii by noting that |d k | is upper bounded with the bound of 2.12 for d k ≤ 0 and Remark 2.9.Note that the use of the contrapositive logic proposition to Assertion 2.6 is not feasible, as invoked in the last part of the proof of Theorem 2.8 ii , in order to prove some close result concerned with Theorem 2.8 iii .This follows since d k < 0 ⇒ ζ k > 0, for all k ∈ N from Assertion 2.5.

Equilibrium points
It is well known that both the standard Beveron-Holt equation, driven by two parameters, and the generalized one, driven by four parameters, have equilibrium points.In particular, the equilibrium point under nonextinction condition of the standard Beverton-Holt equation is the carrying capacity of the environment; see 2-6, 11-13, 27, 28 .The following result holds, which is concerned with the existence of equilibrium points in the limit equation 1.2 , that is, the stationary solutions lim k → ∞ x k x ≥ 0 of 1.2 as the parameters converge to those finite limits.ii If d < 0, then an equilibrium point x x on R 0 can exist only if the following two necessary conditions hold: If there exists such an equilibrium point, then at most two distinct equilibrium points satisfying 0 < x 1 ≤ x M ≤ x 2 can exist.
iii If d 0, then x 1 0 is an equilibrium point of 1.2 .Another equilibrium point might exist Proof.The equilibrium points of 1.2 , if any, are the real values of x ∈ R 0 such that 0 and has a relative maximum, which is also the absolute maximum in R 0 at x M p 1 /α which is g M p 1 /αe p 1 max g x : x ∈ R 0 .Also, g x is increasing on x ∈ 0, x M and decreasing on x ∈ x M , ∞ .Thus, the equilibrium points x x, if any, related to x b and x M satisfy the subsequent constraints.
a If d > 0, then f 0 −Kd < g 0 0. Thus, there is a unique equilibrium point x x i.e., f x g x on R 0 which is, in addition, positive which is subject either to x M > x ≥ x b , which requires α ∈ 0, p 1 /d as a necessary condition, or K|d| > g 0 0 and x a ≤ x b < 0. Thus, an equilibrium point x x i.e., f x g x on R 0 can exist only if the following two necessary conditions hold: since f 0 > g 0 and f x being increasing on 0, ∞ implies that solutions to f x g x might exist only if f 0 ≤ g M and the continuous function h x : g x − f x is monotonically strictly increasing i.e., h x g x − f x > 0 on some interval X M ⊂ 0, x M .
If there exists such an equilibrium point, then it can exist at most two distinct equilibrium points satisfying 0 c If d 0, then f 0 g 0 0 so that x 1 0 is an equilibrium point of 1.2 .Another equilibrium point might exist only if K γμKpx p e −αx p 1 − αx − 1 > 2 μ − 1 x in order that h x : g x − f x be monotonically strictly increasing on some interval X M ⊂ 0, x M .ii The case p α 0; Lower and upper bounds for the equilibrium points of the limit stationary equation 1.2 are investigated in the subsequent result.The use of those bounds is important when the exact equilibrium point cannot be calculated by imprecise knowledge of the model parameterization or computational difficulties.
i The equilibrium points x 1, 2 of the limit stationary modified Beverton-Holt equation 1.2 satisfy the following properties: provided that K > d μ − 1 so that x 2 0 if and only if d 0. If d < 0 and has a sufficiently small modulus, then 0 > −|d| x 22 ≤ x 2 ≤ x 12 < 0. Also,

3.4
ii Define the real functions ψ x γKμx/ K μ − 1 x and ζ x x p e −αx .Then, any equilibrium point x of the limit stationary Beverton-Holt equation 1.2 is locally asymptotically stable if the two constraints below jointly hold: Proof.Note from 3.1 that since ζ x ≤ p/αe p , for all x ∈ R 0 .From 3.1 , the equilibrium points are the nonnegative real solutions to h x 0, if any.The zeros of the convex parabola h 1 x are sufficiently small modulus, and x 11 < 0 if d < 0 of sufficiently large modulus if d ≥ 0 and x 12 < 0 if d < 0 of sufficiently small modulus or complex if d < 0 of sufficiently large modulus.The zeros of the convex parabola h 2 x are

3.9
In view of 3.6 , since h 1 x , h 2 x , and h x are convex parabolas, subject to 3.6 , the zeros x 1, 2 of h x , which are the equilibrium points of the limit stationary 1.2 , satisfy provided that d ≥ 0 and K > d μ − 1 so that x 2 0 if and only if d 0. If d < 0 of sufficiently small modulus, then 3.11 so that Property i has been proved.To prove Property ii , first note that the modified limit Beverton-Holt equation 1.2 may be written more compactly as where x p e −αx , 3.13 which implies

3.14
The equilibrium points x, if any, satisfy The linearized dynamics of 3.12 about the equilibrium points are 19 whose associated characteristic equation in the complex indeterminate λ is

3.21
Thus, from 3.18 , sufficient conditions for the equilibrium point x x 2 to be locally stable if p ≥ αd are from 3.19 to 3.21 :

3.22
Theorem 3.1 i -ii establish that there is a unique equilibrium point on R for d > 0 of the limit stationary equation 1.2 and that at most two equilibrium points might exist on R for d < 0 being of small absolute value.Those results are obtained by investigating the zeros of the upper-bounding and lower-bounding parabolas to h x .The subsequent result gives explicit conditions of the existence of two equilibrium points in such a situation provided that extra sufficiency-type conditions on the remaining limit parameters are fulfilled.
ii If d 0, then the limit equation 1.2 has a zero equilibrium point and another positive equilibrium points, and K > μ − 1 p/α / γ μ p/αe p − 1 .
Proof.i One gets from 3.6 for d < 0 that For x p/α > 0 where ζ x reaches its maximum value p/αe p , and p/α p 1 < 0 for x p/α ∈ R 0 and h : R 0 → R is a real continuous function, there exist x 1, 2 ∈ R with x 2 > x 1 such that h x i 0, i 1, 2 which are equilibrium points of the limit equation 1.2 and the proof is complete.
ii The proof is similar to that of i by noting that the particular condition |d| 0 < γμ p/αe p − 1 K − μ − 1 p/α / μ − 1 p αK p is guaranteed by the given constraint on K.
The following result proves that if there is only one equilibrium point d > 0 of the limit equation, then it is a globally stable attractor.Also, if there are two equilibrium points d ≤ 0 with sufficiently small |d| plus extra conditions on the parameters , then the smaller equilibrium point is locally unstable while the largest one is a globally stable attractor.
i If d > 0, then the unique positive equilibrium point is a global stable attractor.
ii If d ≤ 0 and of sufficiently small modulus, satisfying the constraints of Theorem 3.4,then there are two nonnegative distinct equilibrium points.The smallest one is locally unstable if p 0, and locally stable if p > 0 while the largest one is a global stable attractor.
Proof.i Consider the initial value problem x i x i ≥ 0 i 0, 1 .Thus, one has for d > 0

3.26
The equilibrium point is positive and unique from Theorem 3.1 i and within d, dK/ μ − 1 ⊂ d, μK/ μ − 1 d from Theorem 3.3 i .Assume that the equilibrium point is locally unstable.Then, the stability constraints 3.18 do not jointly hold.Since 3.26 holds and the equilibrium is locally unstable, a stable limit oscillatory solution has to exist subject to 3.26 .Then, there is θ ∈ N the oscillation period such that 0 The linearization about the limit oscillation limit cycle is defined by the dynamics z It is direct to see that A has an eigenvalue with modulus greater than unity if and only if 3.16 has a zero with modulus greater than unity since they coincide.Therefore, the limit cycle is unstable which leads to a contradiction to the fact that it should exist and be stable.Then, the unique equilibrium point is locally stable and there is no stable limit cycle.Since the equilibrium point is stable and unique, it is also a global stable attractor and Property i has been proved.
ii If d 0, then x 0 is an equilibrium point.For any small perturbation population at sampled time k, which has to be positive if p 0, x j > 0 for j ∈ J \ k for some finite integer J > k from inspection of the limit equation 1.2 .Then, the zero equilibrium is locally unstable.An alternative proof is the test of the stability conditions 3.18 for p x 0. The second condition |ψ 0 ζ 0 | ζ 0 ψ 0 ζ 0 ψ 0 γμ > 1 is violated so that x 0 is locally unstable.For x 0 and p > 0, both stability conditions are fulfilled as 1 0 < 1 so that the zero equilibrium point is locally stable.The largest equilibrium point is a positive global stable attractor from Property i .If d ≤ 0 and satisfies the conditions of Theorem 3.4, then there are two positive equilibrium points.The largest one is a stable global attractor proved under similar considerations as those used to prove Property i .

Existence of oscillatory solutions of the nonstationary-modified Beverton-Holt equation
The main motivation of the study of this section is that very commonly the evolution equation of populations in biology problems follows oscillatory seasonal patterns according to reproduction needs, food supply from the environment, or metabolic cycles of the species 13 .In Theorem 2.8, it has been proved that where which implies and it is implied by x k ∈ 0, ∞ ; for all k ∈ N .Furthermore, if d k > 0; for all k ∈ N then the above upper-bound is finite which implies and it is implied by x k ∈ 0, ∞ ; for all k ∈ N .It is of interest to investigate when the solution of the modified Beverton-Holt equation is oscillatory equivalent to the solution of its inverse to be oscillatory.It has to be pointed out that the generation of limit oscillatory solutions being independent of the initial conditions is very important in some engineering problems, like, for instance, all those associated with the synthesis of electronic oscillators see, e.g., 28 .To that end in the context of the modified Beverton-Holt equation 1.2 , define the discrete functions χ x : N 2 0 → R and χ x : N 0 × N 2 → R as follows: respectively.Thus, precise definitions of what is meant by oscillatory solution of 1.2 are given.
Definition 4.1.A particular solution of 1.2 is strongly oscillatory if for any k ∈ N 0 , there exist two finite natural numbers N i k ; i 1, 2, depending on k, such that It turns out that any oscillation of the Beverton-Holt equation implies, and is implied by, an oscillation of the solution of its inverse.Definitions 4.1 and 4.2 imply two alternate changes of sign at some three samples everywhere for the solution sequence strong oscillation or within some finite interval weak oscillation , respectively.For each interval k, k N 1 k , k N 1 k N 2 k , only one oscillation test is needed.If x k 0, then either the population is extinguished and remains at the zero equilibrium point or it is recovering positive values at the next samples generated by immigration so that the oscillation existence text can be performed later on formally.
, either the population is extinguished for all samples and no test for oscillation applies or some population is recovered from immigration positive disturbance at future samples and the inverse Beverton-Holt equation can be defined.

4.11
Then, the proof follows directly from Assertions 4.4-4.5 and direct computations since 4.11 is equivalent to the four equivalent propositions for the existence of strong oscillatory solution.
Remark 4.7.Note that the statement after the disjunction symbol in the third proposition in Theorem 4.6 for existence of strong oscillation may be expanded as follows: 12 since ζ x varies from zero to infinity, it is always possible to have strong oscillatory solutions by accomplishing with the inequalities with appropriate parameterizations of combinations of the harvesting quota and intrinsic growth rate at certain samples sufficiently far away from each former test for oscillation see also Assertion 4.4 .Similar considerations apply for the reverse inequalities guaranteeing also strong oscillatory solutions conditions for weak oscillatory solutions are similar but they only apply on some finite interval.

Numerical example
In this section, the evolution of a population of aphids is investigated under the modified generalized Beverton-Holt equation through a numerical tested example.A brief empirical description of those species follows.Aphids, or plant lice, are small, soft-bodied, pear-shaped insects which are commonly found on nearly all indoor and outdoor plants, as well as vegetables, field crops, and fruit trees.Most of them are about 1/10 inch long.They feed on plants by piercing them with syringe-like mouths pars and sucking the sap.Their diet is rich in carbohydrates and deficient in amino acids.Some of these amino acids cannot be synthesized by the insect but are supplied by the intracellular symbiont bacteria Buchnera aphidicola.Such a symbiont lives inside huge host cells bacteriocytes of which there are about 60-80 per individual and are transmitted to eggs through generations.Aphids have unusual and complex life cycles which allow them to build up huge levels of population in very short periods of time.Furthermore, they overwinter as fertilized eggs.Nymphs which hatch from these eggs become wingless females stem mothers which reproduce without mating holding new amounts of eggs in their bodies until hatching born alive youths.This pattern continues for as long as conditions are favorable.When the days get shorter in the fall and there are cooler temperatures, a generation appears which includes both males and females which are the production of fertilized eggs which overwinter 14 .As a short summary of the above description, it can be emphasized that the populations grow very fast in short periods of time during the same year and they also decrease fast to reach low survival thresholds but not up to extinction.
A common mathematical model for the population evolution of aphids is given in 13 as follows: where a k is the number of adult female aphids in the kth generation, p k fa k is the number of progeny in the kth generation, m is the fractional mortality of the young aphids, f is the number of progeny per female aphid and r is the ratio of female aphids to total adult aphids.The above parameters are all nonnegative by nature.The model might be generalized to extend the above parameters to be varying sequence.From standard stability results the model is globally asymptotically stable and also exponentially stable if and only if g : fr 1 − m < 1, globally stable with the population being constant if g 1 and unstable with the solution diverging at exponential rate if g > 1.It turns out that such a description does not fix properly the above empirical description from biology knowledge.Exhaustive simulation via the modified generalized Beverton-Holt equation shows that the nontrivial solutions are always bounded and positive as expected.They have equilibrium points which depend on parameters.For instance, for zero harvesting quota, zero disturbance contribution, and very small or zero powers p of the penalty term, the solutions reach an equilibrium which lies below that reached by the standard Beverton-Holt model, defined by the carrying capacity.However, there are several cycles of alternate fast increase/decrease of the populations during the transients.This model property is useful to describe the above biological behavior during one year of population evolution.If the parameters are readjusted, the solution changes accordingly while keeping a similar shape.For instance, the upper limit is smaller as the penalty term increases which also produces more abrupt cycles of increase/decrease of the population.
Figure 1 describes the population evolution for constant parameterizations μ 2, K 200, d 0, p 0.01, α 0.002, and γ 0.8.It is seen that the upper limit is smaller than the level of 200 corresponding to a standard Beverton-Hold equation, governed by μ and K, and also smaller than the level of 160, corresponding to a unmodified generalized Beverton-Holt equation, governed by μ, K, γ, and d 0. The reason is that a penalty term is introduced for the overpopulation of aphids.The penalty term translates into a certain level of population control since the population individuals compete in their habitat.The transient shows alternate cycles of increase/decrease of population levels.If α 0.001, with the remaining parameters being identical, then the upper limit increases as expected since both the standard and unmodified generalized Beverton-Holt equations are parameterized by a carrying capacity of 200.The population evolution is displayed in Figure 2 where the existence of the upper limit is obvious and a zoom for the first part of transient exhibiting the alternate cycles is given in Figure 3. Finally, Figure 4 shows the sample-to-sample ratio between the population evolution of the modified generalized Beverton-Holt equation to the standard one over a transient of 30 samples for zero harvesting quota.
On the other hand, Figures 5 and 6 display the population evolution under moderate disturbance contribution consisting of joint independent consumption plus net migration fixed to 1/10 of the initial population with harvesting quotas of 20% and 60%, respectively.Those quotas can be interpreted in the plagues context as the use of pesticides to fight the plague.In the first case, the population grows slightly for each cycle, but exhaustive simulation for more samples demonstrates that the population remains bounded to smaller values than the carrying capacity, as expected.In the second case, it is shown that the levels of population decrease at each successive life cycle.

Conclusions
This paper has been devoted to the study of stability and equilibrium point of a proposed modified generalized Beverton-Holt equation in ecology.Such an equation is governed, in general, by six parametrical sequences, namely, i the intrinsic growth rate and the environment carrying capacity which define the standard Beverton-Holt equation, ii the harvesting quota and the disturbance contribution which, in addition, to the two former ones parameterize the generalized Beverton-Holt equation, iii a penalty term for the eventual overpopulation which includes, in general, the product of a potential term and a decreasing exponential one of the previous population.
It has been proved that the proposed model is Lyapunov stable and possesses stable equilibrium points.It has also been proved that even under constant parameterizations the model can exhibit an oscillatory behavior to describe the alternative cycles of increasing/decreasing levels of population evolution.In this sense, it can be useful to correct the foreseen evolution through the standard Beverton-Holt equation which, under nonextinction conditions obtained from a carrying capacity greater than unity, leads to the population convergence to a finite set point limit defined by the environment carrying capacity.It can also be useful to introduce a correction of the behavior foreseen by its limiting equation, which is the Ricker model, which leads also to a limit set point being typically smaller than that associated with the standard Beveerton-Holt equation.The modified generalized Beverton-Holt equation is also shown to be useful to describe during its transient alternate cycles of the population levels for certain populations of insects since the solution is bounded and extinction-free under standard parameterization conditions so that it exhibits an oscillatory behavior within finite positive lower and upper bounds.In this context and in order to show its usefulness the proposed model has been applied to study a population of aphids which exhibit alternate cycles of increase and decrease of population and which state in latent levels of populations against very adverse conditions of their habitat.

Theorem 3 . 1 .
The following properties hold.iIf d > 0, then it exists a unique equilibrium point x x on R 0 which is, in addition, positive being subject either to x M > x ≥ x b , which requires α ∈ 0, p 1 /d as a necessary condition, or x M ≤ x < x b , which requires α ≥ p 1 /d.

Remark 3 . 2 .
Particular cases of equilibrium points of interest are the following.i The case p α d 0. Thus, the constraint 3.1 becomes f x μ − 1 x 2 Kx g x γμKx which yields an equilibrium point at x K γμ−1 / μ−1 , provided that 1 > γ > μ −1 .If γ 1, one obtains the equilibrium point of the standard Beverton-Holt equation x K.