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

Authors: M. De La Sen
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

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.


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## Body

## 1. 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.Notation 1.1.
N and R are the sets of natural and real numbers,
respectively. (1.1)N0:=N∪{0};R+:={z∈R:z>0};R0+:=R+∪{0}={z∈R:z≥0}.k¯:={1,2,…,k};k¯0:=k∪{0}={0,1,2,…,k}.g′(x) denotes the first derivative with respect to x
of the differentiable real functiong(x).
The
symbols “⋀" and “∨" stand for logic conjunction and
disjunction, respectively.
In is the identity matrix of the nth
order.The
population evolution is supposed to be governed by a modified Beverton-Holt
equation,(1.2)xk+1=γkμkKkxkpke−αkxkxk−1Kk+(μk−1)xk−1+dk;k∈N subject
to initial conditions x−1>0,x0≥0,
which is parameterized by the following real sequences.γk:=1−bk,k∈N,
with bk∈[b¯1,b¯2] being the harvesting (fishing or
hunting) quota subject to 0<b¯1≤b¯2≤1.
Thus, γk∈[γ¯1,γ¯2],k∈N,
is the harvesting quota excess subject to 0<γ¯1:=1−b¯2≤γ¯2:=1−b¯1≤1.
If repopulation activity is incorporated to harvesting, then b¯1<0 when repopulation dominates
harvesting so that γ¯2>1.μk∈[μ¯1,μ¯2],k∈N,
with ∞>μ¯2≥μ¯1>1 is the intrinsic growth rate which is associated
to the population ability for reproduction.Kk∈[K¯1,K¯2],k∈N,
with ∞>K¯2≥K¯1>0, is the
environment carrying capacity which is associated to the sympathetic degree,
more or less favorable, of the habitat towards the species.dk∈[d¯1,d¯2],k∈N,
with ∞>d¯2≥d¯1>−∞, 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 dk=d0k+dmk where d0k is the independent consumption and dmk is the net migration contribution. The
decomposition of dk=d0k+dmk into the two separated parts dmk and dmk 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.pk∈[p¯1,p¯2],αk∈[α¯1,α¯2], with α¯k=0⇒p¯k=0,k∈N, are sequences which conform a penalty term xkpke−αkxk 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 withγk=1 (no harvesting activity), pk=αk=0 (no penalty correcting term for overpopulation
control), dk=0 (zero jointly net independent consumption plus
migration) resulting in the well-known standard Beverton-Holt equation, (1.3)xk+1=μkKkxkKk+(μk−1)xk,k∈N, which
is a particular version of the Hassell model xk+1=μkxk/(1+(μk−1)xk/Kk)ck,k∈N, for ck=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 pk=αk=0 resulting in the so-called generalized Beverton-Holt
equation, (1.4)xk+1=γkμkKkxkKk+(μk−1)xk+dk,k∈N, also
described equivalently as a two-stage evolution equation, (1.5)xk+1−=μkKkxkKk+(μk−1)xk,xk+1≡xk+1+=γkxk+1−+dk,k∈N, 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 ck=1): (1.6)xk+1=eμk(1−xk/Kk)xk,k∈N, if dk=0 and a harvesting
quota γk=((Kk+μk−1xk−1)/Kk(μk+xk−1))eμk,k∈N, is
applied provided that Kk>eμkμk−1/(1+μk−1/xk−1),k∈N, in order to ensure that γk<1.
Thus, the environment carrying capacity satisfies liminfxk−1→∞Kk>eμkμk−1 and Kk∈(eμkμk−1−f(xk−1),∞),k∈N.
The function f:R+→R0+ is defined by f(x)=μ2eμ/(x+μ) which is decreasing on R+ with limx→∞f(x)=0 with R0+:={z∈R:z≥0} and R+:={z∈R:z>0}=R0+∖{0}.
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].


## 2. 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 sk:=xk−1,k∈N, being the solution sequence of the inverse of
the modified Beverton-Holt equation provided that xk≠0,k∈N.
Thus, one gets directly from (1.2) (2.1)sk+1−1=γkμk−1Kk−1ζksk−1−1Kk−1+(μk−1−1)sk−1−1+dk=γkμk−1Kk−1ζkKk−1sk−1+μk−1−1+dk,k∈N, where ζk:=xkpe−αxk,k∈N. Then, (2.2)sk+1=Kk−1+(μk−1−1)xk−1γkμk−1Kk−1ζkxk−1+(Kk−1+(μk−1−1)xk−1)dk,k∈N(2.3)=Kk−1sk−1+μk−1−1γkμk−1Kk−1ζk+(Kk−1+sk−1+μk−1−1)dk,k∈N which
can be rewritten as (2.4)sk+1=Kk−1γkμk−1Kk−1ζksk−1+(μk−1−1)γkμk−1Kk−1ζk−Kk−1sk−1(Kk−1sk−1+μk−1−1)dkγkμk−1Kk−1ζk(γkμk−1Kk−1ζk+(Kk−1sk−1+μk−1−1)dk),k∈N for
the standard Beverton-Holt equation (dk=αk=pk≡0 implying ζk≡1 and γk≡1 and double sampling rate; see (1.3)), (2.2)–(2.4) becomes in particular [11] (2.5)sk+1=μk−1sk+(1−μk−1)Kk−1,k∈N.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).Theorem 2.1.
The following properties hold.
(i)
d¯1≤xk<∞,
for all k∈N.(ii)
Ifd¯1>0,
then {xk}0∞ and {sk}0∞ are positive bounded sequences. As a result,
the modified Beverton-Holt equation (1.2) is globally Lyapunov's
stable.(iii)
Ifd¯1≥0, then {xk}0∞ and {sk}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, {sk}0∞ is positive with limsupk→∞sk≤∞.(iv)
Ifdk≥0 and xk≥0, then 0≤xk+1<∞,
for all k∈N0.Proof.
(i)xk≥dk≥d¯1,
for all k∈N follows directly from (2.4). Also, ζk=1 if αk=pk=0.
Otherwise, αk≠0 so that (2.6)ζk≤max⁡k∈N(ζk:αkpk≠0)=max⁡k∈N(ζ(pkαk):αkpk≠0)=max⁡k∈N((pkαke)pk:αkpk≠0)≤(p¯2α¯1e)p¯2 since dζ(x)/dx=0 for α≠0 only at its relative maximum x=p/α so that ζ(p/α)=(p/αe)p=maxx∈R0+ζ(x) since ζ(0)=0 and limx→∞ζ(x)=0 implying that the relative maximum is
also the absolute maximum. As a result, ζk≤max(1,(p¯2/α¯1e)p¯2)<∞,for  allk∈N.
Then, from (1.2) (2.7)xk+1≤γ¯2μ¯2K¯2xk−1K¯1+(μ¯1−1)xk−1max⁡(1,(p¯2α¯1e)p¯2)+d¯2≤γ¯2μ¯2K¯2max⁡(1,(p¯2/α¯1e)p¯2)μ¯1−1+d¯2<∞, for
all k∈N and (i) has been proved.
Ifd¯1>0, then (i) implies 0<d¯1≤xk<∞ and 0<sk≤d¯1−1<∞,
for all k∈N and (ii) has been proved.
Ifd¯1≥0, then (i) implies 0≤xk<∞ and 0<sk≤∞, for all k∈N and, thus, liminfk→∞xk≥0,limsupk→∞xk<∞,liminfk→∞sk>0,
and limsupk→∞sk≤∞.
Property (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.Axiom 2.2.
xk≥0,  for  allk∈N.Assertion 2.3.
xk=0⇒dk≥0 for any k∈ℕ.Assertion 2.4.
d¯2<0⇒xk>0,for  allk∈N.Assertion 2.5.
dk<0⇒ζk>0 for any k∈N.Assertion 2.6.
ζk=0 only if xk=0 and dk≥0 for any k∈N.Proof.
Assertion2.4 is equivalent to prove that its contrapositive proposition xk≤0forsomek∈N⇒d¯2≥0 is true. Assume that xk≤0 for some k∈N. Then,
Axiom 2.2 implies that xk=0 so that ζk=0 and then xk+1=dk≥0 from (1.2) and Axiom 2.2 so that Assertion 2.3
is proved. Also, xk≤0⇒d¯2≥maxj∈N(dj)≥dk≥0⇔d¯2<0⇒xk>0,for  allk∈N what proves Assertion 2.4. Assertion 2.5 follows since Axiom 2.2⋀dk<0(2.8)ζk=(Kk−1+(μk−1−1)xk−1)|dk|γkμk−1Kk−1xk−1(xk+1≥0)≥(μ¯1−1)|dk|γ¯2μ¯2K¯2+K¯1|dk|γ¯2μ¯2K¯xk−1≥(μ¯1−1)|dk|γ¯2μ¯2K¯2>0foranyk∈N. Assertion
2.6 follows since for any k∈N,ζk=0⇒xk=0∨xk=∞.xk=∞ is impossible from Theorem 2.1(i) so
that for any k∈N,ζk=0⇒xk=0.
If xk=0,
then xk+1≥0 from Axiom 2.2 so that dk≥0 which is Assertion 2.3.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 Assertion2.5 together with an intermediate result in the proof
of Theorem 2.1 implies dk<0⇒∞>max(1,(p¯2/α¯1e)p¯2)≥ζk>0 for any k∈N, (2.9)dk≥0∞>max⁡(1,(p¯2α¯1e)p¯2)≥ζk≥0foranyk∈N. Then, ∞>ζk≥0, for all k∈N,
and ∞>limsupk→∞ζk≥liminfk→∞ζk≥0.
Also, if d¯2<0 or if dk<0 (although a nonnegative upper-bound be
assumed), for all k∈N, then ∞>limsupk→∞ζk≥liminfk→∞ζk>0.Now,
Axiom2.2 and Assertions 2.3–2.6 extend the results of Theorem 2.1 concerning
positivity of the solutions and further stability results as follows.Theorem 2.8.
The following properties hold under Axiom2.2.
(i)
{xk}0∞ and {sk}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 dk>0,for  allk∈N,
both sequences are positive and bounded.(ii)
Defineρk:=1/γkμk−1ζk,δk:=Kk−1sk−1+μk−1−1=(Kk−1+(μk−1−1)xk−1)/xk−1,ηk:=vk−ωk,vk:=(μk−1−1)/(ρk−1+δkdk), and ωk:=Kk−1sk−1δkdk/ρk−1(ρk−1+δkdk). If dk>0,for  allk∈N, then either (2.10)0<∏i=0k[ρi]<∞;∀k∈N(then0<lim⁡inf⁡k→∞∏i=0k[ρi]≤lim⁡sup⁡k→∞∏i=0k[ρi]<∞)∧0≤∑i=0k−1∏j=i+1k−1[ρj]ηi<∞(0≤∑i=0k−1∏j=i+1k−1[ρj]vi≤∞∧0≤∑i=0k−1∏j=i+1k−1[ρj]ωi≤∞),∀k∈N,orlimk→∞⁡∏i=0k[ρi]=−limk→∞∑i=0k−1∏j=i+1k−1[ρj]ηi=∞(lim⁡k→∞∑i=0k−1∏j=i+1k−1[ρj]ωi=∞).(iii)
Ifdk≤0,for  allk∈N, then either (2.11)0<∏i=0k[ρi]≤∞;∀k∈N∧0≤∑i=0k−1∏j=i+1k−1[ρj]ηi≤∞(0≤∑i=0k−1∏j=i+1k−1[ρj]vi≤∞∧0≤∑i=0k−1∏j=i+1k−1[ρj]ωi≤∞),∀k∈N,orlimk→∞⁡∏i=0k[ρi]=limk→∞∑i=0k−1∏j=i+1k−1[ρj]ηi=∞(limk→∞∑i=0k−1∏j=i+1k−1[ρj]ωi=∞).Proof.
(i) Note
from (1.2), Axiom 2.2, and Assertion 2.5
that, for dk<0, (2.12)ζk>0∧|dk|=−dk≤γk−1μk−1μk−1ζkKk−1+(μk−1−1)xk−1=γk−1μk−1μk−1ζksk−1Kk−1sk−1+(μk−1−1)>0 so
that from (2.4) (2.13)sk+1≥Kk−1γkμk−1Kk−1ζksk−1+(μk−1−1)γkμk−1Kk−1ζk+Kk−1sk−1(Kk−1sk−1+μk−1−1)|dk|γkμk−1Kk−1ζk(γkμk−1Kk−1ζk−(Kk−1sk−1+μk−1−1)|dk|)≥Kk−1γkμk−1Kk−1ζksk−1>00≤xk+1<∞ for
any k∈N0 if dk<0.
On the other hand, dk≥0⇒0≤xk+1<∞ and 0<sk+1≤∞ for any k∈N0 from Theorem 2.1(iv). Furthermore, dk>0⇒0<xk+1<∞ and 0<sk+1<∞ for any k∈N0 from Theorem 2.1(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: (2.14)0<sk=∏i=0k[ρi]s0+∑i=0k−1∏j=i+1k−1[ρj]ηi=∏i=0k[ρi]s0+∑i=0k−1∏j=i+1k−1[ρj](vi−ωi)≤∞,∀k∈N and,
furthermore, if dk>0;forallk∈N then (2.15)0<sk=∏i=0k[ρi]s0+∑i=0k−1∏j=i+1k−1[ρj]ηi=∏i=0k[ρi]s0+∑i=0k−1∏j=i+1k−1[ρj](vi−ωi)<∞,∀k∈N since {xk}0∞ is positive and bounded from Theorem 2.1. Thus, the first two constraints linked via logic conjunction follow directly from (2.4) follows directly from (2.14) since either ∏i=0k[ρi]<∞,for  allk∈N,
so that |∑i=0k−1∏j=i+1k−1[ρj]ηi|<∞,for  allk∈N, or limk→∞∏i=0k[ρi]=−limk→∞(∑i=0k−1∏j=i+1k−1[ρj]ηi)=∞,
or ∏i=0k[ρi]=−(∑i=0k−1∏j=i+1k−1[ρj]ηi)=∞ for
finite k∈N.
However, from the equivalent contrapositive logic proposition to Assertion 2.6, xk>0 (which holds from Theorem 2.1, since dk>0) ⇒ζk>0 which implies max(∏i=0k[ρi],|∑i=0k−1∏j=i+1k−1[ρj]ηi|)<∞ from (2.15) and then ∏i=0k[ρi]=−(∑i=0k−1∏j=i+1k−1[ρj]ηi)=∞ for finite k∈N is impossible. Property (ii) has been proved.
(iii)
It is proved using similar techniques as those used in the proof of (ii) by
noting that|dk| is upper bounded with the bound of (2.12) for dk≤0 and δk>0,ρk>0,ζk≥0,∞≥ηk≥0,for allk∈N.
The detailed proof is omitted.Remark 2.9.
Note that the use of the contrapositive logic proposition to Assertion2.6
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 dk<0⇒ζk>0,for  allk∈N from Assertion 2.5.


## 3. Equilibrium Points

Assume
thatlimk→∞Kk=K,limk→∞μk=μ, limk→∞γk=γ,limk→∞αk=γ,limk→∞pk=γ,limk→∞dk=d.
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, 12, 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 limk→∞xk=x¯≥0 of (1.2) as the parameters converge to those finite
limits.Theorem 3.1.
The following properties hold.
(i)
Ifd>0, then it exists a unique equilibrium point x=x¯ on R0+ which is, in addition, positive being subject
either to xM>x¯≥xb, which requires α∈[0,(p+1)/d) as a necessary condition, or xM≤x¯<xb, which requires α≥(p+1)/d.(ii)
Ifd<0, then an equilibrium point x=x¯ on R0+ can exist only if the following two necessary
conditions hold:(i)
|d|≤(1/K)((p+1)/αe)p+1;(ii)
K+(μ−1)|d|<γμK(p+1)xpe−αx−(γμKαxpe−αx+2(μ−1))x for x∈XM⊂[0,xM). If there exists such an equilibrium point, then at most two
distinct equilibrium points satisfying 0<x¯1≤xM≤x¯2 can exist.(iii)
Ifd=0, then x¯1=0 is an equilibrium point of (1.2). Another
equilibrium point might exist only if K[γμKpxpe−αx(p+1−αx)−1]>2(μ−1)x on some interval XM⊂[0,xM)Proof.
The
equilibrium points of (1.2), if any, are the (real) values of x∈R0+ such that (3.1)f(x):=(μ−1)x2+(K−d(μ−1))x−dK=g(x):=γμKζ(x)x. Note
that f:R→R is a convex parabola satisfying f(0)=−Kd whose zeros are (3.2)xa,b=(μ−1)d−K∓((K−(μ−1)d)2+4(μ−1)Kd)1/22(μ−1)={−Kμ−1<0d since ((K−(μ−1)d)2+4(μ−1)Kd)1/2=K+(μ−1)d. Also, g:R→R satisfies g(0)=0,g(∞)=limk→∞g(x)=0 and has a relative maximum, which is also the
absolute maximum in R0+) at xM=(p+1)/α which is gM=((p+1)/αe)p+1=max(g(x):x∈R0+). Also, g(x) is increasing on x∈[0,xM) and decreasing on x∈[xM,∞).
Thus, the equilibrium points x=x¯,
if any, related to xb and xM satisfy the subsequent constraints.
(a)
Ifd>0, then f(0)=−Kd<g(0)=0.
Thus, there is a unique equilibrium point x=x¯ (i.e., f(x¯)=g(x¯)) on R0+ which is, in addition, positive which is
subject either to xM>x¯≥xb, which requires α∈[0,(p+1)/d) as a necessary condition, or xM≤x¯<xb, which requires α≥(p+1)/d.(b)
Ifd<0, then f(0)=K|d|>g(0)=0 and xa≤xb<0.
Thus, an equilibrium point x=x¯ (i.e., f(x¯)=g(x¯)) on R0+ can exist only if the following two necessary
conditions hold:(i)
|d|≤(1/K)((p+1)/αe)p+1 implying f(0)≤gM since f(x) is increasing on [xb,∞) so that it is also increasing on R0+ since R0+⊂[xb,∞);(ii)
K+(μ−1)|d|<γμK(p+1)xpe−αx−(γμKαxpe−αx+2(μ−1))x for x∈XM⊂[0,xM) 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)≤gM 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 XM⊂[0,xM).
If there exists such an equilibrium point, then it can
exist at most two distinct equilibrium points satisfying0<x¯1≤xM≤x¯2.
(c)
Ifd=0, then f(0)=g(0)=0 so that x¯1 is an equilibrium point of (1.2). Another
equilibrium point might exist only if K[γμKpxpe−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 XM⊂[0,xM) on
some interval XM⊂[0,xM).Remark 3.2.
Particular cases of equilibrium points of interest are the following.
(i)
The casep=α=d=0.
Thus, the constraint (3.1) becomes f(x)=(μ−1)x2+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.(ii)
The casep=α=0;d>0.
Thus, x=γμKx/(K+(μ−1)x) holds for x=x¯1=K(γμ−1)/(μ−1) provided that 1>γ>μ−1, and which implies that limk→∞xk−=x¯1.
Also, x=γx+d holds for x=x¯2=d/(1−γ)>0 since d>0 and 1>γ>0,
which implies that limk→∞xk=limk→∞xk+=x¯2.
Then, x¯=x¯1=x¯2 is an equilibrium point of (1.2), provided
that 1>γ>μ−1,
implying that limk→∞xk=limk→∞xk+=limk→∞xk−=x¯ if x¯=K(γμ−1)/(μ−1)=d/(1−γ) if d=(1−γ)(γμ−1)K/(μ−1). 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.Theorem 3.3.
The following properties hold.
(i)
The
equilibrium pointsx¯1,2 of the limit stationary modified Beverton-Holt
equation (1.2) satisfy the following
properties: (3.3)0≤d=x22≤x¯2≤x12≤dKμ−1 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|=x22≤x¯2≤x12<0. Also, (3.4)−dKμ−1≥x11≤x¯1≤x21=−K(μ−1)<0ifd≥0,min⁡(x11,x12)≤x¯1≤max⁡(−K(μ−1),−|d|)<0,ifd<0.(ii)
Define the real functionsψ(x)=γKμx/(K+(μ−1)x) and ζ(x)=xpe−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: (3.5)1+ζ(x¯)ψ′(x¯)>0,|ψ(x¯)ζ′(x¯)|+ζ(x¯)ψ′(x¯)<1.Proof.
Note from (3.1) that (3.6)h2(x):=(μ−1)x2+(K−d(μ−1))x−dK≥h(x):=f(x)−g(x)=(μ−1)x2+(K−d(μ−1)−γμKζ(x))x−dK≥h1(x):=(μ−1)x2+(K−d(μ−1)−γμK(pαe)p)x−dK since ζ(x)≤(p/αe)p, for all x∈R0+. From
(3.1), the equilibrium points are the nonnegative real solutions to h(x)=0,
if any. The zeros of the convex parabola h1(x) are (3.7)x11=d(μ−1)+γμK(p/αe)p−K−((d(μ−1)+γμK(p/αe)p−K)2+4(μ−1)dK)1/22(μ−1)≤−dKμ−1≤0 if d≥0,x11>0  if d<0 of sufficiently small modulus, and x11<0 if d<0 of sufficiently large modulus (3.8)x12=d(μ−1)+γμK(p/αe)p−K+((d(μ−1)+γμK(p/αe)p−K)2+4(μ−1)dK)1/22(μ−1)≤dKμ−1 if d≥0 and x12<0 if d<0 of sufficiently small modulus or complex if d<0 of sufficiently large modulus.
The zeros of
the convex parabolah2(x) are (3.9)x21=d(μ−1)−K−((d(μ−1)+K)2)1/22(μ−1)=−K(μ−1)<0,x22=d(μ−1)−K+((d(μ−1)+K)2)1/22(μ−1)=d. In view of
(3.6), since h1(x),h2(x), and h(x) are convex parabolas, subject to (3.6), the
zeros x¯1,2 ofh(x), which are the equilibrium points of the limit stationary (1.2), satisfy (3.10)0≤d=x22≤x¯2≤x12=d(μ−1)+γμK(p/αe)p−K+((γμK(p/αe)p+d(μ−1)−K)2+4(μ−1)dK)1/22(μ−1)≤dKμ−1 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)0>−|d|=x22≤x¯2≤x12=γμK(p/αe)p−|d|(μ−1)−K+((γμK(p/αe)p−|d|(μ−1)−K)2−4(μ−1)|d|K)1/22(μ−1)<0,−dKμ−1≥x11≤x¯1≤x21=−K(μ−1)<0ifd≥0,min⁡(x11,x12)≤x¯1≤max⁡(−K(μ−1),−|d|)<0ifd<0, 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 (3.12)xk+1−d=ψ(xk−1)ζ(xk), where (3.13)ψ(x)=γKμxK+(μ−1)x,ζ(x)=xpe−αx, which implies (3.14)ψ′(x)=γK2μ(K+(μ−1)x)2;ζ′(x)=xp−1e−αx(p−αx). The
equilibrium points x¯,
if any, satisfy x¯−d=limk→∞xk−d=ψ(x¯)ζ(x¯).
The linearized dynamics of (3.12) about the equilibrium points are [19] (3.15)zk+1=ζ(x¯)ψ′(x¯)zk−1+ψ(x¯)ζ′(x¯)zk whose
associated characteristic equation in the complex indeterminate λ is (3.16)λ2−ψ(x¯)ζ′(x¯)λ−ζ(x¯)ψ′(x¯)=0 so
that the equilibrium point is locally asymptotically stable if both zeros of
(3.16) have modulus less than unity, namely, (3.17)1>−ζ(x¯)ψ′(x¯)>|ψ(x¯)ζ′(x¯)|−1 or, equivalently, (3.18)1+ζ(x¯)ψ′(x¯)>0;|ψ(x¯)ζ′(x¯)|+ζ(x¯)ψ′(x¯)<1. For d≥0,
since, x=x¯2≤x12,
it follows that ψ(x¯)≤γμx¯≤γμx¯12≤γμdK/(μ−1) and also that ψ(x¯)≤γKμ/(μ−1), for all x∈R0+ directly from (3.13) so that (3.19)ψ(x¯)≤min⁡(γKμ(μ−1),γμdKμ−1),dpe−αdK/(μ−1)≤ζ(x¯)≤max⁡(ζ(x):x∈R0+)=(pαe)p if d≥0 from the constraint. From that
constraint and 0≤d=x22≤x¯2≤x12 and (3.13)-(3.14), (3.20)γK2μ(K+dK(μ−1))2≤ψ′(x¯)≤γμmin⁡(1,K2(μ−1)2d2).ζ′(x) defined in (3.14) has a derivative (d/dx)ζ′(x)=xp−2e−αx[(p−1)p−(p+1)αx] which has a relative
extremum at x=(p−1)p/(p+1)α.
Thus, taking into account also (3.14) and 0≤d=x22≤x¯2≤x12 for d≥0,
one gets (3.21)dp−1e−αdK/(μ−1)(p−αdKμ−1)≤ζ′(x¯)≤min⁡((dKμ−1)(p−1)/2e−dα(p−αd),2pp+1((p−1)p(p+1)α)p−1e(p−1)p/(p+1)). 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)dpe−αdK/(μ−1)γK2μ(K+dK(μ−1))2>−1,|min⁡(γKμ(μ−1),γμdKμ−1)min⁡((dKμ−1)(p−1)/2e−dα(p−αd),2pp+1((p−1)p(p+1)α)p−1e(p−1)p/(p+1))|+γμ(pαe)pmin⁡(1,K2(μ−1)2d2)<1. 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.Theorem 3.4.
The following properties hold.
(i)
The limit equation (1.2) has two positive equilibrium points if d<0 and satisfies |d|<((γμ(p/αe)p−1)K−(μ−1)(p/α)/((μ−1)p+αK))p, provided that 1>γ>(((μ−1)p+Kα)/μKα)(αe/p)p,K>max(0,(μ−1)p/μα[(p/αe)p−1/μ]).(ii)
Ifd=0,
then the limit equation (1.2) has a zero equilibrium point and another positive
equilibrium points, and K>(μ−1)(p/α)/(γμ(p/αe)p−1).Proof.
(i) One gets from (3.6) for d<0 that (3.23)h(x)=(μ−1)x2+(K+(μ−1)|d|−γμKζ(x))x+K|d|. For x=p/α>0 where ζ(x) reaches its maximum value (p/αe)p,
and (3.24)h(pα)=((μ−1)pα+K)pα+((μ−1)pα+K)|d|−γμKep(pα)p+1<0, if 1>γ>(((μ−1)p+Kα)/μKα)(αe/p)p subject to (p/αe)p>((μ−1)p+Kα)/μKα=(μ−1)p/μKα+1/μ for  K>0⇔K>max(0,(μ−1)p/μα[(p/αe)p−1/μ]). Since h(0)=K|d|>0,h(∞)=limx→∞h(x)=∞,h(p/α)=((μ−1)(p/α)+K)(p/α)+((μ−1)(p/α)+K)|d|−(γμK/ep)(p/α)p+1<0 for x=p/α∈R0+ and h:R0+→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.Theorem 3.5.
The following properties hold.
(i)
Ifd>0, then the unique positive equilibrium point is
a global stable attractor.(ii)
Ifd≤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 problemxi=x^i≥0(i=0,1). Thus,
one has for d>0(3.25)0<d≤xk+1=μKe−αxkxk−1K+(μ−1)xk−1+d≤x^k+1≤μKe−αx^kμ−1+d≤lim⁡sup⁡k→∞x^k≤μKμ−1+d<∞. Thus, (3.26)0<d≤lim⁡inf⁡k→∞xk≤lim⁡sup⁡k→∞xk≤μKμ−1+d<∞. 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<d≤limk→∞xk+k1=x¯k+k1≤μK/(μ−1)+d<∞, for all k1∈θ¯.
Now, build the real vector xk(0):=(xk,xk−1,…,xk−θ+1)T→x¯k(0) as k→∞.
The linearization about the limit oscillation (limit cycle) is defined by the
dynamics zk+1(0)=Azk(0) where A∈Rθ×θ is block-partitioned as A:=[ζ(x¯)ψ′(x¯)ψ(x¯)ζ′(x¯)0⋯00Iθ−1]. 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)
Ifd=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,xj>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+ζ(0)ψ′(0)=1>0, |ψ(0)ζ′(0)|+ζ(0)ψ′(0)=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).


## 4. 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 (4.1)0<sk=∏i=0k[ρi]s0+∑i=0k−1∏j=i+1k−1[ρj]ηi=∏i=0k[ρi]s0+∑i=0k−1∏j=i+1k−1[ρj](vi−ωi)≤∞,∀k∈N, where ρi:=1/γiμi−1ζi,ωi:=Ki−1si−1δidi/ρi−1(ρi−1+δidi) and δi:=Ki−1si−1+μi−1−1,
which implies and it is implied by xk∈[0,∞); for all k∈N. Furthermore,
if dk>0; for all k∈N then the above upper-bound is finite which implies and it is implied
by xk∈[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 χx:N02→R and χx¯:N0×N2→R as follows: (4.2)χx(j,k):=xk−xj,χ¯x(k,k+j,k+j+ℓ):=χx(k,k+j)χx(k,k+j+ℓ)=(xk+j−xk)(xk+j+ℓ−xk), 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∈N0, there exist two finite natural numbers Ni(k);i=1,2, depending on k, such that (4.3)χ¯x(k,k+N1(k),k+N1(k)+N2(k))<0.Definition 4.2.
A particular solution of (1.2) is weakly oscillatory if for any k∈[k¯1,k¯2]=k¯2∖k1−1¯⊂N0 and some k¯1∈N0 and k2(≥k1)∈N0,
there exist two finite natural numbers Ni(k);i=1,2, depending on k, such that (4.4)χ¯x(k,k+N1(k),k+N1(k)+N2(k))<0. 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+N1(k),k+N1(k)+N2(k)], only one oscillation test is needed. If xk=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.Assertion 4.3.
χ¯x(k,k+N1(k),k+N1(k)+N2(k))<0 for k∈N0,Ni(k)∈N(i=1,2)⇔χ¯s(k,k+N1(k),k+N1(k)+N2(k))<0 for k∈N0,Ni(k)∈N(i=1,2).Proof.
It is
direct fromsk=xk−1, for all k∈N0 if xk>0.
If xk=0,
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.Assertion 4.4.
The subsequent technical result holds:(4.5)χ¯x(k,k+N1(k),k1)<0∧χ¯x(k1(k),k1(k)+N1(k1(k)),k1(k)+N1(k1(k))+N2(k1(k)))<0 with k1(k):=k+N1(k)+N2(k) for
some k∈N0,Ni(k)∈N,Ni(k1)∈N(i=1,2)(4.6)∃Ni(k+j)∈N(i=1,2):χ¯x(k+j,k+N1(k+j)+N2(k+j))<0, for all j(≤N1(k)+N2(k))∈N such that xk+j>0,Ni(k)∈N.Proof.
Note that(4.7)sign(xk+N1(k)−xk)≠sign(xk+k1(k)−xk),sign(xk+N1(k)−xk)≠sign(xk+k1(k)−xk+N1(k)),sign(xk1(k)+N1(k1(k))−xk1(k))≠sign(xk1(k)+N1(k1(k))+N2(k1(k))−xk1(k)+N1(k1(k))),sign(xk1(k)+N1(k1(k))−xk1(k))≠sign(xk1(k)+N1(k1(k))+N2(k1(k))−xk1(k)+N1(k1(k))). Thus,
it is obvious that there exist Ni(k+j)∈N(i=1,2):χ¯x(k+j,k+N1(k+j)+N2(k+j))<0, for all j∈N such that xk+j>0, for all j(≤N1(k)+N2(k))∈N,
since it always exist by construction at least one N1(k+j)≤N1(k1(k)) and at least one N2(k+j)≤N2(k1(k)), for all j∈N such that xk+j>0.
If xk+j=0, for all j≠k,k+N1(k),k+N1(k)+N2(k),
then there are always at least three sign changes in the interval [k,k+N1(k)+N2(k)] and the proof also follows.Assertions4.3-4.4 imply directly the following result which is concerned with the
simplification of the test for strong oscillatory solutions.Assertion 4.5.
Assume that there exists a subset
of infinite cardinal of the set of natural numbersSj:={kj∈N0:1≤kj+1−kj<∞}⊂N0 such that (4.8)χ¯x(k,k+N1(k),k+N1(k)+N2(k))<0∀k∈SjforsomeNi(k)∈N(i=1,2)χ¯s(k,k+N1(k),k+N1(k)+N2(k))<0fork∈N0,Ni(k)∈N(i=1,2) and
some solution {xk}1∞ generated by
initial conditions x−1≥0,x0>0. Then, such a solution is strongly
oscillatory. The converse is also
true.The
main result of this section follows.Theorem 4.6.
A solution of the modified
Beverton-Holt equation (1.2), subject to initial conditions x−1≥0,x0>0, is strongly oscillatory if and only if there exists an infinite sequence of triples {ki,N1(ki),N2(ki)}i∈N0∈N3, where ki+1=ki+N1(ki)+N2(ki)∈S⊂N for some finite k0∈N0, for all i∈N0 with N∋Nj(ki)<∞(j=1,2) such that (4.9)skiski+N1(ki)ski+N1(ki)+N2(ki)>0 and, furthermore, (4.10)ski+N1(ki)+N2(ki)<ski(ski+N1(ki)−ski)ski+N1(ki)−ski=ski+N1(ki)−skiκi−1xki+N1(ki)+N2(ki)>xki−xki+N1(ki)xki(xki−xki+N1(ki))(ski+N1(ki)+N2(ki)<ski<ski+N1(ki))∨(ski+N1(ki)+N2(ki)>ski>ski+N1(ki))(xki+N1(ki)+N2(ki)>xki>xki+N1(ki))∨(xki+N1(ki)+N2(ki)<xki<xki+N1(ki)), where κi:=ski+N1(ki)/ski.Proof.
Expandχ¯s(ki,ki+N1(ki),ki+N1(ki)+N2(ki)) using Definition 4.1 to obtain (4.11)χ¯s(ki,ki+N1(ki),ki+N1(ki)+N2(ki))=ski(ski−ski+N1(ki)−ski+N1(ki)+N2(ki))+ski+N1(ki)ski+N1(ki)+N2(ki)<0.
Then,
the proof follows directly from Assertions4.4-(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 Theorem4.6 for existence of strong oscillation may be
expanded as follows: (4.12)1γki+N1(ki)−1μki+N1(ki)−2ζki+N1(ki)−2(∏k=kiki+N1(ki)−2[ρj])+∑j=0ki+N1(ki)−1∏ℓ=j+1ki+N1(ki)−1[ρℓ](ηjski)>1>ski+N1(ki)+N2(ki)=1γki+N1(ki)+N2(ki)−1μki+N1(ki)+N2(ki)−2ζki+N1(ki)+N2(ki)−2=(∏k=kiki+N1(ki)+N2(ki)−2[ρj])+∑j=0ki+N1(ki)+N2(ki)−1∏ℓ=j+1ki+N1(ki)+N2(ki)−1[ρℓ](ηjski) 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.


## 5. 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 bacteriaBuchnera 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: (5.1)ak+1=fr(1−m)ak=[fr(1−m)]k+1a0, where ak is the number of adult female aphids
in the kth generation, pk=fak 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.Figure1 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.Figure 1
μ=2,K=200,d=0,p=0.01,α=0.002, and γ=0.8.Figure 2
μ=2,K=200,d=0,p=0.01,α=0.001, and γ=0.8.Figure 3
μ=2,K=200,d=0,p=0.01,α=0.001, and γ=0.8.Figure 4
Relative comparison with the
standard Beverton-Holt equation.On
the other hand, Figures5 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.Figure 5
μ=2,K=200,γ=0.8,d=0.1×(0),α=0.1,p=10.Figure 6
μ=2,K=200,γ=0.4,d=0.1×(0),α=0.1,p=10.


## 6. 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.



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*Source: 592950-2008-07-15.xml*