# Model-Matching-Based Control  of  the Beverton-Holt  Equation in  Ecology

Authors: M. De La Sen; S. Alonso-Quesada
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

This paper discusses the generation of a carrying capacity of the environment so that the famous Beverton-Holt equation of Ecology has a prescribed solution. The way used to achieve the tracking objective is the design of a carrying capacity through a feedback law so that the prescribed reference sequence, which defines the suitable behavior, is achieved. The advantage that the inverse of the Beverton-Holt equation is a linear time-varying discrete dynamic system whose external input is the inverse of the environment carrying capacity is taken in mind. In the case when the intrinsic growth rate is not perfectly known, an adaptive law implying parametrical estimation is incorporated to the scheme so that the tracking property of the reference sequence becomes an asymptotic objective in the absence of additive disturbances. The main advantage of the proposal is that the population evolution might behave as a prescribed one either for all time or asymptotically, which defines the desired population evolution. The technique might be of interest in some industrial exploitation problems like, for instance, in aquaculture management.


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## Body

## 1. Introduction

The nonautonomous discrete so-called Beverton-Holt equation (BHE) is very common in Ecology and, in particular, in studying
the growth population dynamics (see, e.g., [1–9]). The equation is of great
importance in the fishery industry concerning the growth and exploitation of
species like, for instance, plaice, haddock and coho salmon, as well as other
bottom feeding fish populations [1]. It is proven that, for subcritical biomass,
according to a low value of the intrinsic growth rate of the studied
population, zero is a stable attractor if the carrying capacity is not
independent of the intrinsic growth rate in order that the model is well posed.
In particular, the carrying capacity should decrease as the spawning stock decreases
converging to zero. The BHE is a two-parameter widely used nonlinear equation
of the form [1–9] (1.1)xk+1=μkKkxkKk+(μk−1)xk,k∈N0:=N∪{0},with N being the set of natural numbers, where x0>0, μk∈R0+:=R+∪{0} (R+ being the set of positive real numbers) is the intrinsic growth rate of the
population, determined by life cycle and demographic properties (specie growth
rate, survivorship rate, etc.), and Kk∈R+,
so-called the environmentcarrying capacity, is a characteristic
of the habitat being dependent on resources availability, temperature, humidity,
and so on. Typically, μk>1 and Kk=Kk+p∈R0+,
for some p(≥2)∈N,
due to periodic fluctuations. It has been reported that a negative carrying
capacity is nonsense since a small carrying capacity is sufficient to interpret
a very unfavorable situation for the population [6], even to model the so-called
Allee effect, that is, the difficulty of finding mates due to the very scarce
number of population [7]. Note that Kk=0 is sufficiently unfavorable to extinguish the
population since, from (1.1), a recruitment xk+1=0 would occur for any spawning stock. The population extinction has been dealt with in
[10] for intrinsic growth rates less than unity by discussing the spawning
stock biomass conditions for the zero equilibrium to be a stable attractor. It
has been concluded that the environment carrying capacity is not independent of
the intrinsic growth rate for subcritical spawning stock biomass. Generalizations
of the BHE including bounded discontinuities at sampling points, which can be
interpreted as the consequence of impulsive effects [11], in the generating
continuous logistic equation, have been also studied.This paper is devoted to the model
matching of a prescribed reference model which is also defined in practice by a
BHE. The standard case of intrinsic growth-rate sequences being greater than
unity is considered. The environment carrying capacity is locally modified
around its reference values to achieve the prescribed behavior. Its inverse
plays the role of the control of the IBHE, namely, the inverse of the BHE which
is a linear dynamic system [9], and then much easier to deal with than the BHE.
The control law consists of calculating the appropriate carrying capacity
inverse so that the solution of the IBHE (namely, the inverse of the population
levels) coincides with the solution of the reference IBHE. In that way, the
population levels also coincide with the reference-suited ones. In the case
when the intrinsic growth rate is not exactly known, the scheme is extended by
incorporating parametrical estimation and a related adaptive control law. The
tracking objective becomes asymptotic as a result of the incorporation of the estimation
process. Since, in many practical situations, the reference model and the
current one are locally deviated from each other, the solution matches very
closely the reference solution for all time. The proposed technique is very feasible
in close or semi-open environments like, for instance, aquaculture industry,
for instance, by local modification of the temperature. In the case when the BHE
parameters are not exactly known, an adaptive extension of the method
incorporating parameter estimation is used instead. The paper is organized as
follows. Section 2 is devoted to the statement of a control law for the case of
known parameters. The BHE is generalized to include, if suited, additive
disturbances which may be useful to describe uncertainties in the parameters of
the BHE or phenomena which are not included in the nominal standard equation
like, for instance, local or global migrations of the population. The reference
model to be tracked (also often referred to as “matched” in the literature) is
also a BHE which potentially includes additive disturbances corresponding to
those present in the given BHE being defined as that possessing as its solution,
the IBHE of the given BHE. Such an inverse has a linear structure by nature so
that it is easy to deal with, and it describes the appropriate system behavior.
The control law is of a feedback type and consists of designing appropriately
the environment carrying capacity, within some prescribed margins, so that the
reference model is perfectly tracked (i.e., the solution of the controlled
equation and that of the reference model coincide for all samples). This is
very feasible in closed environments like, for instance, fisheries. It turns
out that the achievement of a perfect tracking of the IBHE corresponds to
achieving the objective of a perfect tracking of the corresponding reference
BHE. Section 3 extends the method to the adaptive case by incorporating an
adaptive scheme. It is assumed that certain a priori knowledge on the additive
disturbances is available so that a relative dead zone is used in the adaptation
algorithm to freeze the adaptation under small-adaptation errors to prevent
against stability caused by the disturbances when the parameters are not fully
known [12]. The adaptive control law is implemented by replacing the controller
parameters in the case of perfect knowledge by those computed based on the
estimates of the IBHE. The closed-loop system is proven to be stable if the
additive disturbances grow, at most linearly, with the maximum of the solution
of the IBHE. In the adaptive case, the tracking objective is only achievable asymptotically
provided that the contribution of the additive disturbances is zero or becomes
extinguished asymptotically. The method to discuss the stability and
convergence of estimates relies on the use of a Lyapunov sequence, [8, 13, 14]. Section 4 presents two numerical examples and,
finally, conclusions end the
paper.


## 2. Control Law for the Case of Known Parameters

Define an inverse system of (1.1) through
the change of variable sk=xk−1 resulting directly from (1.1) in the
time-varying discrete linear system (2.1)sk+1=aksk+bkuk,s0=x0−1>0,where ak=μk−1, bk=1−μk−1=1−ak, and uk=Kk−1,
that is, the inverse carrying capacity is the control action. If there exists an
additive disturbance sequence ηk in (2.1), one gets, instead of (2.1) [9], (2.2)sk+1=aksk+bkuk+ηk.The IBHE (2.2) corresponds, through the
change of variable xk=sk−1,
to an extended BHE which includes a sequence of additive disturbances: (2.3)xk+1=KkxkakKk+(bk+Kkηk)xk=μkKkxkKk+(μk−1+μkKkηk)xk=μkKkxkKk+(μk−1)xk+vk0=μkKkxkKk+(μk−1)xk+vk,where (2.4)vk0=μkKkηkxk,vk=−μkKkxkvk0(Kk+(μk−1)xk)(Kk+(μk−1+μkKkηk)xk)=−μk2Kk2xk2ηk(Kk+(μk−1)xk)(Kk+(μk−1+μkKkηk)xk).Note that the sequence {vk}0∞ is an additive
disturbance of the BHE which is
identically zero if the additive disturbance sequence {ηk}0∞ of its inverse is identically zero. The
disturbance may include the effects in the solution of parametrical
uncertainties, for instance, in the intrinsic growth rate, or effects like, for
instance, migrations or local migrations which are not taken into account in
the standard BHE. The following assumptions are concerned with the stability
and controllability of the IBHE and with the knowledge of an upper bound of the
disturbance term if such a disturbance is unknown.Assumption 2.1.
∞>μk≥1+εμ and ∞>Kk≥εK, for all  k∈N0, and some εμ∈R+,εK∈R+.Assumption 2.2.
ηk(∈R0+)≤η¯k,  for all k∈N0, and {η¯k}0∞ is known.Note that Assumption2.1 implies that
the BHE is stable since μk−1<1,
and also controllable since bk≥εμ′:=εμ/(1+εμ)>0 so that  (1−μk−1), for all k∈N0,
so that sk in (2.2), and then the BHE solution xk=sk−1 may be driven to sequences of prescribed
values. These two properties are crucial for the solution to track a prescribed
reference sequence. Note also that εμ may be chosen in the real interval (0,1) without any loss in generality since, if Assumption 2.1
holds for any given εμ>0,
then it always holds for some εμ∈(0,1).
This fact is used in the proof of the subsequent result, which is direct and
related to the positivity and boundedness of the solution.Assertion 2.3.
If Assumptions2.1 and 2.2 hold and {ηk}0∞ is uniformly bounded, then all solutions of
the BHE and the IBHE are uniformly bounded and positive, provided that ∞>x0>0.Proof.
μk≥1+εμ⇔bk∈[εμ′,1−εμ′].
Define ρ:=Max0≤k<∞(μk−1)∈(0,(1/(1+εμ′))]⊆(0,1).
Since s0=x0−1>0,
it follows from (2.2) that (2.5)0<xk−1=sk≤ρsk0+∑i=0k−1ρk−i−1(1−εμ′)(Ki−1+ηi)≤ρsk0+1−ρk1−ρ(1−εμ′)Max0≤i<∞(Ki−1+ηi)≤s0+(1−εμ′2)εμ′εKMax0≤i<∞(ηi)<∞⇔xk∈(0,∞),∀k∈N0,since11−ρ≤1−εμ′εμ′,Max0≤i<∞(Ki−1)≤1εK.The subsequent set of structures is
considered in order to be then able to formulate the control law.(1) A Reference
BHExk+1*=μk*Kk*xk*/(Kk*+(μk*−1)xk*), for all  k∈N0,
which defines the suitable solution through the appropriate reference values of
the intrinsic growth rate and the environment carrying capacity of its corresponding reference IBHE sk+1*=ak*sk*+bk*rk,
with reference input rk=(Kk*)−1>0,
and parameter sequences ak*=(μk*)−1<1 and bk*=(μk*)−1>0.(2) The Current
BHExk+1=μkKkXk/(Kk+(μk−1)xk)+vk and its corresponding IBHE (2.6)sk+1=aksk+bkuk+ηk, whose parameters are ak=μk−1,  bk=1−μk−1 are deviated from the suited ones ak*=(μk*)−1 and bk*=1−(μk*)−1,
which may include disturbance terms grouped in a sequence {vk}0∞ with control input uk=Kk−1 obtained according to a feedback law.(3) The Control Law:
(2.7)uk=tkrk+fksk−ωk, provided
that ηk is known, which is a relaxation of Assumption
2.2 which is parameterized by (2.8)fk=ak*−akbk,tk=bk*bk,ωk=ηkbk,with bk=1−ak=1−μk−1.
Note that the control law generates the
inverse carrying capacity from the reference inverse carrying capacity and the
solution of the IBHE with two parametrical sequences {fk}0∞ and {tk}0∞ and a correcting sequence {ωk}0∞ calculated from (2.8). The subsequent constraint (2.9)rk=(Kk*)−1>ωkxk−fktkxk=1tk(ηk1−ak+ak−ak*xk)must hold for all samples to
guarantee that the environment carrying capacity is positive. The following
result establishes that the tracking error in between the reference sequence
and the current solution for the given control law is zero. In that way, the
perfect tracking objective is achieved if the disturbance sequence is known.Proposition 2.4.
Ifs0=s0*,
then the control law (2.7)-(2.8) achieves an identically zero tracking error {εk:=sk−sk*}0∞.Proof.
It
follows, by complete induction, by assumingsk=sk*⇔εk=0, for all k∈N0,
that (2.10)εk+1=(ak+bkfk−ak*)sk*+(bktk−bk*)rk=0,∀k∈N0.If the disturbance is unknown, but
Assumption2.2 still holds, then the last identity in the parameterization of the
control law in (2.8) is replaced, with similar expressions, with the disturbances
absolute upper bounds, namely, ωk=η¯k/bk,
where |ηk|≤η¯k.
Thus the tracking error is not identically zero but it is uniformly bounded. It
is also very close to zero, provided that the deviation of the upper bound of
the disturbance with respect to such a disturbance is very close to zero. Then
one has the subsequent result.Proposition 2.5.
If Assumption2.1 holds and |ηk|≤η¯k with ηk being unknown, but η¯k being known, then the control law (2.7)-(2.8) of
Proposition 2.4, with the modification ωk=η¯k/bk,
achieves a uniformly bounded tracking error {εk:=sk−sk*}0∞,
provided that the disturbance sequence is upper bounded by a bounded sequence.Proof.
Direct
calculations yield(2.11)sk+1=aksk+bkuk+ηk,sk+1*=ak*sk*+bk*rkso that, from (2.7)-(2.8), (2.12)εk+1=ak*εk+(ηk−η¯k)≤(a*)k+1ε0+mη1−a*<∞,∀k∈N0,where  0<ak*≤a*<1, for all k∈N0, and (2.13)mη:=Max0≤i<∞(|η¯i−ηi|)≤2Max0≤i<∞(η¯i)<∞.Since
the inverse of the environment carrying capacity is the control action, a large
deviation from its nominal values for tracking purposes may be not admissible. Note
that Assumption2.1 also applies to the reference to be tracked by the control
action consisting of appropriate achievement of an
environment carrying capacity, that is, ∞>μk*≥1+εμ>1, and ∞>Kk*≥εk>0,  for all k∈N0. That
means that the maximum deviation of the suited reference sequence, related to
the current solution, depends on the maximum allowed variations of the
environment carrying capacity with respect to its nominal value. More precisely,
the subsequent result follows.Proposition 2.6.
Assume that the maximum allowed absolute variation of the carrying capacity
with respect to its nominal value isΔK>0.
Then the following properties hold.
(i)
The control
parameter sequence{tk}0∞ is subject to the constraint   tk∈[1−ΔK/Kk*,1+ΔK/Kk*], for all  k∈N0, which induces the parametrical constraints (2.14)ak∈[Kk*ak*−ΔKKk*−ΔK,Kk*ak*+ΔKKk*+ΔK],bk∈[Kk*bk*Kk*+ΔK,Kk*bk*Kk*−ΔK].(ii)
The tracking-error
sequence{εk:=sk−sk*}0∞ is subject to the constraint   (2.15)(εk+1−akεk+(ak*−ak)sk*)∈[ηk−bk*ΔKKk*2,ηk+bk*ΔKKk*2],  for all k∈N0, which is trivially bounded, provided that the
disturbance sequence is bounded.(iii)
A
positive carrying capacity is obtained from the controller synthesis if(Kk*)−1>(ωkxk−fk)/tkxk=(1/tk)(η¯k/(1−ak)+(ak−ak*)/xk),  when ηk is unknown but η¯k≥|ηk| is known. The control parameter tk∈(0,1] and the calculated Kk from the control law are positive if tk is calculated satisfying  ΔK∈[0,Kk*−η¯k/(1−ak)+(ak*−ak)/xk) with  η¯k∈[0,(1−ak)(Kk*+(ak−ak*)/xk)).Proof.
(i) Note that(2.16)tk∈[1−ΔKKk*,1+ΔKKk*]⇔(1−ak)∈[Kk*(1−ak*)Kk*+ΔK,Kk*(1−ak*)Kk*−ΔK]⇔1−Kk*(1−ak*)Kk*+ΔK≥ak≥1−Kk*(1−ak*)Kk*−ΔK,which
yields directly Property (i) since bk=1−ak.
(ii)
One gets
from (2.6)–(2.8) and Property (i) that (2.17)εk+1=sk+1−sk+1*=akεk+(ak−ak*)sk*+(tk−1)bk*rk+ηk∈[akεk+(ak−ak*)sk*−ΔKKk*bk*rk+ηk,akεk+(ak−ak*)sk*+ΔKKk*bk*rk+ηk],and the result follows.(iii)
The
first part follows with the replacement ofηk→η¯k≥|ηk| in (2.9). The second part follows directly by
replacing the admissibility domain of the controller parameter tk in the above constraint.


## 3. Adaptive Control

For
the case when the parameters of the BHE are unknown, an estimation scheme with
adaptation dead zone for robust closed-loop stabilization is incorporated (see,
e.g., [12, 13, 15, 16]). Such a mechanism governs the evolution of the IBHE and
it is now discussed from a theoretical point of view. It will then corroborate,
through numerical simulation, the potential usefulness of numerical and
adaptive techniques as previously tested in some ecological models (see, e.g., [17–19]). One takes the advantage that, since the parameters fulfill
the constraint ak+bk=1, for all k∈N0,
the same constraint might be introduced in the estimation scheme, what reduces
it to the estimation of one parameter only. A modification would consist of
estimating both sequences of parameters separately. In that case, the
estimation is performed without taking into account the parametrical constraint
at the expense of an increase in computation time and memory storage. However,
it may be proven that all the properties of boundedness and convergence of the
estimates still hold if existing parametrical constraints are not taken into
account, provided that each parameter is updated with its corresponding
component in the set of measured data (see, e.g., [12, 13, 15]). In a first
step, a simple procedure is developed for the nominal BHE through controlling
its inverse, and then an extension is made to prevent a good operation in the
presence of additive disturbances. A relative dead zone to the estimation
scheme is added to prevent potential instability caused by those disturbances.


### 3.1. Adaptive Control approach Based on the Intrinsic Growth Rate Estimate

Additive Disturbances
The deviations of the intrinsic growth rate with respect to a
certain unknown constant valueμ=a−1,
implying b=1−a, are incorporated within
a disturbance sequence {ηk}0∞ which can also include other unstructured
disturbance contributions in a sequence {ηk0}0∞: (3.1)ηk:=(ak−a)sk+(bk−b)uk+ηk0=(ak−a)(sk−uk)+ηk0,and uk=Kk−1.
In this way, the nominal parameter vector of the IBHE, namely, θ=(a,b)T,
is constant, which then facilitates an updating algorithm that has an identical
structure for the parameter estimates as for the parametrical errors. The IBHE,
its estimate, the identification error, the reference IBHE, the tracking error,
and the parameter-adaptive law are, respectively, as follows.IBHE
(3.2)sk+1=ask+buk+ηk=a(sk−uk)+uk+ηk.If ηk≡0,
the resulting particular case of (3.2) is called the nominal IBHE, and its
corresponding inverse equation, whose solution is the sequence {xk≡sk−1}0∞,
is called the nominal BHE.Estimate of the IBHE
(3.3)s^k+1=a^ksk+b^kuk=a^k(sk−uk)+uk,where a^k and b^k=1−a^k are the estimates of a and b at the kth sample.Identification Error
(3.4)ek+1:=sk+1−s^k+1=−a˜ksk−b˜kuk+ηk=−a˜k(sk−uk)+ηk,  where a˜k:=a^k−a, b^k:=b^k−b=a−a^k=−a˜k are the parametrical errors. Note that
although ηk is unknown, the identification error is
available for measurement through the first identity in (3.4).Reference IBHE
It is assumed to be time invariant as the
nominal IBHE, namely,(3.5)sk+1*=a*sk*+b*rk=a*(sk*−rk)+rkwith b*=1−a* and rk=(Kk*)−1, is the reference environment carrying capacity
inverse.Tracking Error
(3.6)εk+1:=sk+1*−sk+1=(sk+1*−s^k+1)+(s^k+1−sk+1)=sk+1*−s^k+1−ek+1.Adaptation Algorithm 1:
(3.7)a^k+1=a^k+σk(sk−uk)ek+11+γk(sk−uk)2,a^0∈(0,1),∀k∈N0,is a recursive estimation of the
intrinsic growth rate, where {σk}0∞ and {γk}0∞ are
real sequences satisfying σk≥0, γk>0 to be specified later for purposes of
convergence of the algorithm. The control law has the same structure as
(2.7)–(2.9) by replacing the true parameters by its estimates, and by deleting the
correcting sequence ωk.
Instability drawbacks due to disturbances will be dealt with by using a specific
dead-zone updating rule for the sequence σk≥0 in the parameter-adaptive algorithm.Adaptive Control Law:
(3.8)uk=tkrk+fksk=11−a^k[(1−ak*)rk+(ak*−a^k)sk]with rk=(Kk*)−1 with the
sample-dependent controller parameters (3.9)fk=ak*−a^k1−a^k,tk=1−ak*1−a^k.The following result concerning
useful technical relationships in between the IBHE solution, its estimated IBHE,
and the reference IBHE independent of the parameter adaptive algorithm is
immediate.Proposition 3.1.
The adaptive control law (3.8)-(3.9) leads to s^k=sk* (i.e., the estimated solution equalizes the
reference solution), εk=−ek (i.e., the tracking error equalizes the minus-identification
error), for all k∈N,
for any finite s0, s^0=s0*.
The above results are independent of the parameter-adaptive algorithm.Proof.
Proceed, by induction, by assumings^k=sk* for any given k∈N0.
Then one gets, from (3.8)-(3.9) into (3.3) by using b^k=1−a^k, (3.10)s^k+1=a^ksk+(1−a^k)(tkrk+fksk)=[a^k(1−fk)+fk]sk+(1−a^k)tkrk=ak*sk+bk*rk=sk+1*if sk=sk*, and then using (3.6), (3.11)εk+1=(sk+1*−s^k+1)+(s^k+1−sk+1)=(s^k+1−sk+1)=−ek+1,∀k∈N0,what
completes the proof.


### 3.2. Boundedness and Convergence Results of the Adaptation Algorithm

In
the following, the boundedness and convergence properties of the adaptation
algorithm are investigated in the subsequent result.Theorem 3.2.
Assume that a sequence{η¯k}0∞ satisfying η¯k≥|ηk|, for all k∈N0,
is known that the adaptation algorithm
(3.7) is updated by choosing γk≥ς0,
and (3.12)σk={0if|ek+1|≤ςη¯k,2γk(ς−1−ς1)ςif|ek+1|>ςη¯k,for
all k∈N0, for some prefixed real constants ς>1,  ς−1>ς0>0,  and ς1∈[ς0,ς−1).
Then the following properties hold irrespective of the control law.
(i)
The sequences{a^k}0∞, {(σk/(1+γk(sk−uk)2))1/2|ek+1|}0∞,  and {(σk/(1+γk(sk−uk)2))ek+12}0∞ are bounded and have limits a^∞<∞,
and zero,
respectively.(ii)
There exist
the subsequent limits(3.13)limk→∞(∑i=0kσiηi21+γi(si−ui)2)≤limk→∞(∑i=0kσiη¯i21+γi(si−ui)2)≤limk→∞(∑i=0kσiei+121+γi(si−ui)2)<∞,limk→∞(∑i=0k(σiηi21+γi(si−ui)2)1/2)≤limk→∞(∑i=0k(σiη¯i21+γi(si−ui)2)1/2)≤limk→∞((σiei+121+γi(si−ui)2)1/2)<∞.Proof.
Let{Vk}0∞ be defined by Vk:=a˜k2.
Note, by subtracting a in both sides of (3.7), that a˜k+1=a˜k+σkφkek+1/(1+γkφk2),  where φk:=sk−uk.
Since (3.14)a˜kφk=ηk−ek+1≤ςη¯k−ek+1,from
(3.4), for any absolute upper-bound η¯k≥|ηk|,
one
gets (3.15)ΔVk:=a˜k+12−a˜k2≤σk2φk2ek+12(1+γkφk2)2+2σk|ek+1|(η¯k−|ek+1|)1+γkφk2=σk|ek+1|1+γkφk2(σkφk2|ek+1|1+γkφk2−2(|ek+1|−η¯k)).From (3.15), σk=0⇒ΔVk=0. Assume |ek+1|>ςη¯k so that for any real ς>1, one gets, from (3.15), (3.16)ΔVk≤σkek+121+γkφk2(σkφk21+γkφk2−2(1−ς−1))≤−2σkς1ek+12ς(1+γkφk2)≤0for all k∈N0, provided
that γk>0 and σk satisfies 0≤σkφk2/(1+γkφk2)≤σk/γk≤2(ς−1−ς1)/ς0 for any real ς1∈[ς0,ς−1).
Then Vk≤V0<∞,
what implies that the sequences {|ΔVk|}0∞,  {a˜k}0∞ and {a^k}0∞,
and (3.17){(σk1+γk(sk−uk)2)1/2|ek+1|}0∞,{(σk1+γk(sk−uk)2)1/2|η¯k|}0∞are
uniformly bounded and there exist limits  limk→∞(|ΔVk|)=limk→∞(σkek+12/(1+γk(sk−uk)2))=0, limk→∞(Vk)=V∞≤V0<∞,   limk→∞(a^k)=a^∞<∞, and limk→∞(a˜k)=a˜∞<∞.  limk→∞(σkek+12/(1+γk(sk−uk)2))=0⇒limk→∞((σk/(1+γk(sk−uk)2))1/2|ek+1|)=0 so that limk→∞((σk/(1+γk(sk−uk)2))1/2|η¯k|)=0 since either σk≥2ς0(ς−1−ς1)/ς
or σk=0, for all k∈N0, and Property (i) is proven. Since Vk converges to a finite limit, Property (ii)
follows from (3.18)∑i=0kσkη¯i21+γk(sk−uk)2≤∑i=0kσkek+121+γk(sk−uk)2=∑i=0k|ΔVi|=V0−Vk+1≤V0−V∞<∞as well as the boundedness of the square of each element of
the sequence associate with the above series.The boundedness of the estimates and
estimation and tracking errors as well as the convergence of the estimates to
finite limits are crucial issues to formulate a well-posed problem. The relative dead zone of the algorithm (3.12)
is relevant for adaptive stabilization under external disturbances. The
interpretation is direct; if the identification error is small related to the
known upper bound of the disturbances, the estimation is frozen (i.e.,
stopped). The reason is that the contribution to the identification error of
the parameterized part, which is being updated, may be smaller than that of the
disturbance and then to maintain the estimation, updating may lead to
instability. The incorporation of relative adaptation dead zones into the
estimation scheme is a standard mechanism to cope against adaptive instability
since the sequence {Vk}0∞ is bounded, see, for instance, [12, 13, 15]. Otherwise, the estimates computations and then
the control law are unfeasible. If
Assumption 2.1 is satisfied, then a projection method may be used for the
estimates to guarantee that they lie inside its definition domain. Theorem 3.2 is
then extended as follows.Theorem 3.3.
If both the nominal and current intrinsic  growth rates satisfy Assumption2.1, then there exist real constants εμ∈R0+, ε1>1,
such that (3.19)a,ak∈[11+ε1εμ,11+εμ],∀k∈N0,and the above adaptation algorithm is
replaced with the set of modified recursive equations (3.20)–(3.22) below as to include projection of the estimates on an admissibility domain.
Adaptation Algorithm 2
It consists
of two steps, namely, a priori and a
posteriori estimations as follows.
A Priori Estimation:
(3.20)a^k+10=a^k+1+σk(sk−uk)ek+11+γk(sk−uk)2,a^0∈(0,1),∀k∈N0,being updated by
choosing γk≥ς0,
and (3.21)σk={0if|ek+1|≤ςη¯k0,2γk(ς−1−ς1)ςif|ek+1|>ςη¯k0:ς(η¯k+(1−a^ka^k0)a^k0(sk−uk)),for all k∈N0,
for some prefixed real constants ς>1,  ς0>0,
and ς1∈[ς0,ς−1).
A Posteriori Estimation Via Projection on the Interval(1/(1+ε1εμ),1/(1+εμ)):
(3.22)a^k+1={11+ε1εμifa^k+10≤11+ε1εμ,a^k+10ifa^k+10∈(11+ε1εμ,11+εμ),11+εμifa^k+10≥11+εμ.Then Theorem 3.2(i)-(ii) holds. In addition, similar
properties hold by replacing (3.23)ηk⟶ηk0=ηk+(1−a^ka^k0)a^k0(sk−uk),η¯k⟶η¯k0=η¯k+(1−a^ka^k0)a^k0(sk−uk),ek+1[A posteriori identification error]:=sk+1−s^k+1=sk+1−a^k(sk−uk)⟶ek+10[A priori identification error]:=sk+1−s^k+10=sk+1−a^k0(sk−uk).Proof.
Define the sequence(3.24)0≤ΔVk0:=(a˜k+10)2−a˜k2≤σk|ek+1|1+γkφk2(σkφk2|ek+1|1+γkφk2−2(|ek+1|−η¯k0))≤0from (3.20)-(3.21) using a close
development to that in Theorem 3.2 so that it is monotonically decreasing (3.25)ΔVk:=a˜k+12−a˜k2=a˜k+12−(a˜k+10)2−|ΔVk0|≤−(1−(a^k+1a^k+10)2)(a˜k+10)2−|ΔVk0|≤0for all  k∈N0 since 0<a^k+1/a^k+10≤1, for all k∈N0, from the projection part (3.22) of the
estimation algorithm. Then {Vk}0∞ is bounded and converges to a finite limit,
and {ΔVk}0∞ is also bounded and converges to zero. Also |ΔVk0|=(1-a^k+1/a^k+10)<1 for all k∈N0, and |ΔVk0|=(1-a^k+1/a^k+10)<1. Since ΔVk0:=(a˜k+10)2-a˜k2=0, for all k∈N0,
then there exists a finite limit limk→∞(∑i=0k|ΔVi0|)<∞,  then ((a˜k+10)2−a˜k2) converges to a finite limit and |a˜k0| converges since |a˜k| converges (since Vk converges) to finite limits as k→∞.
Thus Theorem 3.2(i)-(ii) follows. On the other hand, note that (3.26)ek+1[A posteriori identification error]:=sk+1−s^k+1=ηk−a˜k(sk−uk)=ek+10+(a^k0−a^k)(sk−uk)=ηk0−a˜k0(sk−uk),ek+10[A priori identification error]:=sk+1−s^k+10=ηk−a˜k0(sk−uk)=ek+1+(a^k−a^k0)(sk−uk)=ηk0+(1−a^ka^k0)(2a^k0−a)(sk−uk).Since the normalized sequences {(σk/(1+γkφk2))1/2|ek+1|}0∞ and {(σk/(1+γkφk2))1/2η¯k0}0∞  are bounded, it follows that the reported replacements may be
performed leading to Theorem 3.3(iii).


### 3.3. Some Practical Design Constraints

The tracking error of the solutions of the IBHE and the reference IBHE are given by the set of equivalent expressions:(3.27)εk+1=sk+1*−sk+1=sk+1*−s^k+1−ek+1=sk+1*−s^k+1+a˜k(sk−uk)−ηk.The solution of the IBHE is equivalently calculated
either as a function of the nominal parameters a and b=1−a or as a function
of their estimates as follows, by considering the adaptive control law (3.8): (3.28a)sk+1=a(sk−uk)+uk+ηk=s^k+1+ek+1=a^k(sk−uk)+uk+ηk−a˜k(sk−uk)(3.28b)=s^k+1+ek+1=a^k(sk−uk)+uk+ηk−a˜k(sk−uk)(3.28c)=a^ksk+[(1−ak*)rk+(ak*−a^k)sk]+ek+1(3.28d)=ak*sk+(1−ak*)rk+ek+1.It
is important to discuss when the adaptive control law has no division by zero with
the property being numerically robust. This translates, in practice, to a^k≠1 being sufficiently deviated from unity and {sk}0∞ with all its elements positive and
sufficiently deviated from zero. This guarantees that the BHE does not diverge
when driven by an appropriate nonzero finitely upper-bounded environment
carrying capacity. More formally, a
reasonable assumption for practical design purposes is to extend the assumption of Theorem 3.3 to also consider
the same admissibility range referred to for the reference intrinsic growth-rate
inverse and the estimated intrinsic growth rate.Assumption 3.4.
a,ak,a^k,ak*∈[1/(1+ε1εμ),1/(1+εμ)], for all k∈N0, and some prefixed real constants εμ>0 and ε1>1. A
direct calculation with Assumption 3.4 yields (3.29)(1−a^k)∈[εμ1+εμ,ε1εμ1+ε1εμ]⊂(0,1),(11−a^k)∈[1+1ε1εμ,1+1εμ],|ak*−a^k|≤εμ(ε1−1)(1+ε1εμ)(1+εμ),(ak*−a^k)≤[−(ε1−1)εμ(1+ε1εμ)(1+εμ),(ε1−1)εμ(1+ε1εμ)(1+εμ)],while
similar expressions follow for replacements involving any two elements of the quadruplet (a,ak,a^k,ak*) as follows: (3.30)(1+εμ1+ε1εμ)[ε1rk+ε1−11+εμsk]≥uk=11−a^k[(1−ak*)rk+(a^k−ak*)sk]≥(1+εμ1+ε1εμ)[ε1(rk−11+εμsk)+11+εμsk]≥(1+εμ1+ε1εμ)[ε1(rk−sk)+11+εμsk].The
above expressions (3.29)-(3.30) lead to the following result referring to explicit positive lower and
upper bounds of uk=Kk−1=(Kk*+K˜k)−1 so that uk−1=Kk*+K˜k=Kk*+rk−1 since rk=(Kk*)−1.Proposition 3.5.
Assumesk<ε1(1+εμ)/(ε1−1)Kk*, for all k∈N0.
Then the following constraints hold: (3.31)1+εμε1(1+εμ)(Kk*)−1+(ε1−1)sk≤1uk≤1+εμε1(1+εμ)((Kk*)−1−sk)+sk,(3.32)1+εμε1(1+εμ)(Kk*)−1+(ε1−1)sk−Kk*≤K˜k≤1+εμε1(1+εμ)(Kk*)−1−(ε1−1)sk−Kk*.Proof.
It
follows directly from (3.29)-(3.30). The necessary condition sk<ε1(1+εμ)/(ε1−1)Kk* is needed to keep (ε1(1+εμ)(Kk*)−1−(ε1−1)sk)>0 in the right-hand-side upper bound of K˜k, for all  k∈N0 in
(3.32).


### 3.4. Closed-Loop Stability

The
boundedness and convergence properties of the parameter estimates have been
proven. In the following, the closed-loop stability is proven under a condition
of slow growing of the disturbances with respect to the solution of the IBHE if such a disturbance is unknown. In
particular, it is assumed thatη¯k=O[Max0≤i≤k(si)] with known constants of the related linear
upper bound and with sufficiently slow growth of η¯k with respect to Max0≤i≤k(si).
This is a reasonable assumption used in the background literature since a
complete lack of knowledge of disturbances makes impossible the stabilization
in the general case (see, e.g., [12, 13, 15] and references therein).Assumption 3.6.
There exist known finite nonnegative real constantsλ1,2 such that  (3.33)η¯k≤λ1+λ2Max0≤i≤k(si).Then, from (3.28b), (3.31)-(3.32),
and Assumptions 3.6–3.4, one gets (3.34)sk+1=a^k(sk−uk)+uk+ek+1.The
following results are preparatory for the stability theorem to be stated later
on. They basically establish that the absolute value of the identification error
grows not faster than linearly with the
upper-bounding sequence {η¯k}0∞ of the additive disturbances.Assertion 3.7.
If the Adaptation Algorithm 1 is used, then|ek+1|≤ς1η¯k+ς2, for all k∈N0, for some real constants ς1(≥ς)∈R+ and ς2∈R0+.Proof.
If|ek+1|≤ςη¯k,
the assertion is true with ς1=ς∈R+ and ς2=0. If |ek+1|>ςη¯k, then, from Theorem 3.2, (i) 0<(|ek+1|−ςη¯k)/(1+γk(sk−uk)2)1/2≤M<∞, for all k∈N0.
Thus (3.35)|ek+1|≤ςη¯k+M(1+γk|sk−uk|)≤(ς+Mγ)η¯k+Mγ|uk|+Mand
the assertion follows with (3.36)ς1=ς+Mγ,ς2=M(1+γ)Max0≤k<∞(|uk|)<∞since Min0≤k<∞(Kk)>0.Assertion 3.8.
If the Adaptation Algorithm 2 is used and Assumption3.4 holds, then |ek+1|≤ς1η¯k+ς2, for all k∈N0, with (3.37)ς1=1+2(ε1−1)εμ(1+ε1εμ)(1+εμ),ς2=Max0≤i<∞(|a˜i|)Max0≤i<∞(|ui|).Proof.
It follows from Assumption3.4, (3.4) and (3.29), and the fact that {Kk}0∞ is positively lower bounded, which yield
directly (3.38)|ek+1|≤|a˜k(sk−uk)|+η¯k≤(1+Max0≤i<∞(|a˜i|))η¯k+Max0≤i<∞(|a˜i|)Max0≤i<∞(|ui|)≤ς1η¯k+ς2,∀k∈N0.From
Assertions2.3 and 3.7, for the Adaptation Algorithms 1 and 2, respectively, Assumption
3.6 and (3.28c), (3.39)sk+1≤|ak*sk|+|1-ak*|rk+ς1η¯k+ς2≤ρks0+1−ρk1−ρ(ς1λ1Max0≤i≤k+1(sk)+ρ1ρ2+ς1λ2+ς2)≤ς1λ11−ρMax0≤i≤k+1(sk)+11−ρ(ρ1ρ2+ς1λ2+ς2)+s0,∀k∈N0,for any s0>0 and some positive finite constants  ρ,ρ1, and ρ2 satisfying 1>ρ≥Max0≤k<∞(ak*), ρ1≤1−Min0≤k<∞(ak*)<1, and ρ2≥Max0≤k<∞(rj).
The last two inequalities of (3.39) follow directly by proceeding recursively
with the preceding one by using ∑i=0k−1ρi=(1−ρk)/(1−ρ)≤1/(1−ρ) since ρ∈(0,1).
It is obvious that, for
each k∈N0,
it exists a unique k′(k)∈N0,
dependent in general on k, such that k′(k)≤k such that sk′=Max0≤j≤k′(sj)=Max0≤j≤k(sj) and there is no other j∈N0 such that k≥j>k′(k),
where the same maximum is reached. A precise definition is (3.40)k′(k):=Arg(j∈N0:0≤j≤k⋀sk′=Max0≤j≤k′(k)(sj)=Max0≤j≤k(sj),¬∃j(k′(k)<j≤k)∈N0:sj=Max0≤j≤k(sj))with   “⋀” and “¬” being the conjunction and negation logic
symbols. The substitution of (3.40) into (3.39) yields (3.41)Max0≤j≤k(sj)≤(1−ϑ)−1C<∞,∀k∈N0,provided that  ς1λ1/(1−ρ)<1,
which holds if and only if 0≤λ1<(1−ρ)/ς1 (i.e., if
Assumption 3.6 holds with sufficiently small λ1), where (3.42)ϑ:=ς1λ11−ρC:=11−ρ(ρ1ρ2+ς1λ2+ς2)+s0with C being a finite
positive real, which
depends on the initial conditions of the IBHE (and thus on the initial conditions of the BHE) on the available upper bound
of the additive disturbances, on the reference, and on the free-design
parameters of the adaptation algorithm, γk,  ς>1,  ξ0, ξ1,
and so on. The above result also holds for the Adaptation Algorithm 2 with
specific constants redefined by (3.43)ρ:=11+εμ<1,ρ1′:=ε1εμ1+ε1εμ,ρ1:=ρ1′1+ε1εμ=ε1εμ(1+ε1εμ)2,ρ2:=ε1(1+εμ)Max0≤k<∞(Kk*)−1<∞,and ξ1,2 defined in Assertion 2.3 for the Adaptation
Algorithm 1 and in Assertion 3.7 for the Adaptation Algorithm 2. The boundedness of the solution of the IBHE for finite initial conditions implies
that of the additive disturbances via Assumption 3.6, that of the identification error via Assertions 2.3–3.7, and
that of the tracking error since the solutions of the current and the reference one are both bounded. The
estimated parameters and the estimated solution are bounded from Theorems 3.2 and
3.3. Those conclusions are now summarized in the result below.Theorem 3.9 (closed-loop stability).
If Assumption3.6 holds with sufficiently small coefficient λ1,
then all the signals in the closed-loop system remain bounded for all time so
that the IBHE (and then the associate BHE) is globally
stable if any of the Adaptation Algorithms 1 or 2 is used.Remark 3.10.
Theorem3.9 becomes stronger than formerly stated in the absence of additive
disturbances (i.e., ηk≡0) since asymptotic perfect tracking is
furthermore  achieved. Another result is that the identification error converges
asymptotically to zero if the relative dead zone of the adaptation algorithms
is removed, that is, the {σk}0∞ is not zeroed at any sample as a result of a
comparison procedure of the identification error and the additive disturbances. These results follow directly from Theorems 3.2
and 3.3 and (3.39). Note that the results may be directly generalized to the
extension of using relative dead zones for adaptive algorithms by incorporating
the estimation of the constants λ1,2 characterizing {η¯k}0∞,
provided they are unknown [16].Remark 3.11.
The closeness between the IBHE and that of its reference model has been considered  for feasibility
reasons since, in many ecological problems, where such models are commonly of
interest, the environment characteristics, which directly influence the value
of the environment carrying capacity,
cannot be abruptly modified even in closed environments. The employed philosophy about relative dead-zone adaptation
seems to be promising to be also applied to other ecological controlled
problems involving, for instance, Kolmogorov-type ecological models or models for
a biochemical aquariums [17, 18]. The obtained results
seem to be promising for mathematical modeling and the use of adaptive control
in other kinds of problems related to biology and ecology (see, e.g., [20–22])
and to extend the research to governing extinction conditions [10].


## 3.1. Adaptive Control approach Based on the Intrinsic Growth Rate Estimate

Additive Disturbances
The deviations of the intrinsic growth rate with respect to a
certain unknown constant valueμ=a−1,
implying b=1−a, are incorporated within
a disturbance sequence {ηk}0∞ which can also include other unstructured
disturbance contributions in a sequence {ηk0}0∞: (3.1)ηk:=(ak−a)sk+(bk−b)uk+ηk0=(ak−a)(sk−uk)+ηk0,and uk=Kk−1.
In this way, the nominal parameter vector of the IBHE, namely, θ=(a,b)T,
is constant, which then facilitates an updating algorithm that has an identical
structure for the parameter estimates as for the parametrical errors. The IBHE,
its estimate, the identification error, the reference IBHE, the tracking error,
and the parameter-adaptive law are, respectively, as follows.IBHE
(3.2)sk+1=ask+buk+ηk=a(sk−uk)+uk+ηk.If ηk≡0,
the resulting particular case of (3.2) is called the nominal IBHE, and its
corresponding inverse equation, whose solution is the sequence {xk≡sk−1}0∞,
is called the nominal BHE.Estimate of the IBHE
(3.3)s^k+1=a^ksk+b^kuk=a^k(sk−uk)+uk,where a^k and b^k=1−a^k are the estimates of a and b at the kth sample.Identification Error
(3.4)ek+1:=sk+1−s^k+1=−a˜ksk−b˜kuk+ηk=−a˜k(sk−uk)+ηk,  where a˜k:=a^k−a, b^k:=b^k−b=a−a^k=−a˜k are the parametrical errors. Note that
although ηk is unknown, the identification error is
available for measurement through the first identity in (3.4).Reference IBHE
It is assumed to be time invariant as the
nominal IBHE, namely,(3.5)sk+1*=a*sk*+b*rk=a*(sk*−rk)+rkwith b*=1−a* and rk=(Kk*)−1, is the reference environment carrying capacity
inverse.Tracking Error
(3.6)εk+1:=sk+1*−sk+1=(sk+1*−s^k+1)+(s^k+1−sk+1)=sk+1*−s^k+1−ek+1.Adaptation Algorithm 1:
(3.7)a^k+1=a^k+σk(sk−uk)ek+11+γk(sk−uk)2,a^0∈(0,1),∀k∈N0,is a recursive estimation of the
intrinsic growth rate, where {σk}0∞ and {γk}0∞ are
real sequences satisfying σk≥0, γk>0 to be specified later for purposes of
convergence of the algorithm. The control law has the same structure as
(2.7)–(2.9) by replacing the true parameters by its estimates, and by deleting the
correcting sequence ωk.
Instability drawbacks due to disturbances will be dealt with by using a specific
dead-zone updating rule for the sequence σk≥0 in the parameter-adaptive algorithm.Adaptive Control Law:
(3.8)uk=tkrk+fksk=11−a^k[(1−ak*)rk+(ak*−a^k)sk]with rk=(Kk*)−1 with the
sample-dependent controller parameters (3.9)fk=ak*−a^k1−a^k,tk=1−ak*1−a^k.The following result concerning
useful technical relationships in between the IBHE solution, its estimated IBHE,
and the reference IBHE independent of the parameter adaptive algorithm is
immediate.Proposition 3.1.
The adaptive control law (3.8)-(3.9) leads to s^k=sk* (i.e., the estimated solution equalizes the
reference solution), εk=−ek (i.e., the tracking error equalizes the minus-identification
error), for all k∈N,
for any finite s0, s^0=s0*.
The above results are independent of the parameter-adaptive algorithm.Proof.
Proceed, by induction, by assumings^k=sk* for any given k∈N0.
Then one gets, from (3.8)-(3.9) into (3.3) by using b^k=1−a^k, (3.10)s^k+1=a^ksk+(1−a^k)(tkrk+fksk)=[a^k(1−fk)+fk]sk+(1−a^k)tkrk=ak*sk+bk*rk=sk+1*if sk=sk*, and then using (3.6), (3.11)εk+1=(sk+1*−s^k+1)+(s^k+1−sk+1)=(s^k+1−sk+1)=−ek+1,∀k∈N0,what
completes the proof.


## 3.2. Boundedness and Convergence Results of the Adaptation Algorithm

In
the following, the boundedness and convergence properties of the adaptation
algorithm are investigated in the subsequent result.Theorem 3.2.
Assume that a sequence{η¯k}0∞ satisfying η¯k≥|ηk|, for all k∈N0,
is known that the adaptation algorithm
(3.7) is updated by choosing γk≥ς0,
and (3.12)σk={0if|ek+1|≤ςη¯k,2γk(ς−1−ς1)ςif|ek+1|>ςη¯k,for
all k∈N0, for some prefixed real constants ς>1,  ς−1>ς0>0,  and ς1∈[ς0,ς−1).
Then the following properties hold irrespective of the control law.
(i)
The sequences{a^k}0∞, {(σk/(1+γk(sk−uk)2))1/2|ek+1|}0∞,  and {(σk/(1+γk(sk−uk)2))ek+12}0∞ are bounded and have limits a^∞<∞,
and zero,
respectively.(ii)
There exist
the subsequent limits(3.13)limk→∞(∑i=0kσiηi21+γi(si−ui)2)≤limk→∞(∑i=0kσiη¯i21+γi(si−ui)2)≤limk→∞(∑i=0kσiei+121+γi(si−ui)2)<∞,limk→∞(∑i=0k(σiηi21+γi(si−ui)2)1/2)≤limk→∞(∑i=0k(σiη¯i21+γi(si−ui)2)1/2)≤limk→∞((σiei+121+γi(si−ui)2)1/2)<∞.Proof.
Let{Vk}0∞ be defined by Vk:=a˜k2.
Note, by subtracting a in both sides of (3.7), that a˜k+1=a˜k+σkφkek+1/(1+γkφk2),  where φk:=sk−uk.
Since (3.14)a˜kφk=ηk−ek+1≤ςη¯k−ek+1,from
(3.4), for any absolute upper-bound η¯k≥|ηk|,
one
gets (3.15)ΔVk:=a˜k+12−a˜k2≤σk2φk2ek+12(1+γkφk2)2+2σk|ek+1|(η¯k−|ek+1|)1+γkφk2=σk|ek+1|1+γkφk2(σkφk2|ek+1|1+γkφk2−2(|ek+1|−η¯k)).From (3.15), σk=0⇒ΔVk=0. Assume |ek+1|>ςη¯k so that for any real ς>1, one gets, from (3.15), (3.16)ΔVk≤σkek+121+γkφk2(σkφk21+γkφk2−2(1−ς−1))≤−2σkς1ek+12ς(1+γkφk2)≤0for all k∈N0, provided
that γk>0 and σk satisfies 0≤σkφk2/(1+γkφk2)≤σk/γk≤2(ς−1−ς1)/ς0 for any real ς1∈[ς0,ς−1).
Then Vk≤V0<∞,
what implies that the sequences {|ΔVk|}0∞,  {a˜k}0∞ and {a^k}0∞,
and (3.17){(σk1+γk(sk−uk)2)1/2|ek+1|}0∞,{(σk1+γk(sk−uk)2)1/2|η¯k|}0∞are
uniformly bounded and there exist limits  limk→∞(|ΔVk|)=limk→∞(σkek+12/(1+γk(sk−uk)2))=0, limk→∞(Vk)=V∞≤V0<∞,   limk→∞(a^k)=a^∞<∞, and limk→∞(a˜k)=a˜∞<∞.  limk→∞(σkek+12/(1+γk(sk−uk)2))=0⇒limk→∞((σk/(1+γk(sk−uk)2))1/2|ek+1|)=0 so that limk→∞((σk/(1+γk(sk−uk)2))1/2|η¯k|)=0 since either σk≥2ς0(ς−1−ς1)/ς
or σk=0, for all k∈N0, and Property (i) is proven. Since Vk converges to a finite limit, Property (ii)
follows from (3.18)∑i=0kσkη¯i21+γk(sk−uk)2≤∑i=0kσkek+121+γk(sk−uk)2=∑i=0k|ΔVi|=V0−Vk+1≤V0−V∞<∞as well as the boundedness of the square of each element of
the sequence associate with the above series.The boundedness of the estimates and
estimation and tracking errors as well as the convergence of the estimates to
finite limits are crucial issues to formulate a well-posed problem. The relative dead zone of the algorithm (3.12)
is relevant for adaptive stabilization under external disturbances. The
interpretation is direct; if the identification error is small related to the
known upper bound of the disturbances, the estimation is frozen (i.e.,
stopped). The reason is that the contribution to the identification error of
the parameterized part, which is being updated, may be smaller than that of the
disturbance and then to maintain the estimation, updating may lead to
instability. The incorporation of relative adaptation dead zones into the
estimation scheme is a standard mechanism to cope against adaptive instability
since the sequence {Vk}0∞ is bounded, see, for instance, [12, 13, 15]. Otherwise, the estimates computations and then
the control law are unfeasible. If
Assumption 2.1 is satisfied, then a projection method may be used for the
estimates to guarantee that they lie inside its definition domain. Theorem 3.2 is
then extended as follows.Theorem 3.3.
If both the nominal and current intrinsic  growth rates satisfy Assumption2.1, then there exist real constants εμ∈R0+, ε1>1,
such that (3.19)a,ak∈[11+ε1εμ,11+εμ],∀k∈N0,and the above adaptation algorithm is
replaced with the set of modified recursive equations (3.20)–(3.22) below as to include projection of the estimates on an admissibility domain.
Adaptation Algorithm 2
It consists
of two steps, namely, a priori and a
posteriori estimations as follows.
A Priori Estimation:
(3.20)a^k+10=a^k+1+σk(sk−uk)ek+11+γk(sk−uk)2,a^0∈(0,1),∀k∈N0,being updated by
choosing γk≥ς0,
and (3.21)σk={0if|ek+1|≤ςη¯k0,2γk(ς−1−ς1)ςif|ek+1|>ςη¯k0:ς(η¯k+(1−a^ka^k0)a^k0(sk−uk)),for all k∈N0,
for some prefixed real constants ς>1,  ς0>0,
and ς1∈[ς0,ς−1).
A Posteriori Estimation Via Projection on the Interval(1/(1+ε1εμ),1/(1+εμ)):
(3.22)a^k+1={11+ε1εμifa^k+10≤11+ε1εμ,a^k+10ifa^k+10∈(11+ε1εμ,11+εμ),11+εμifa^k+10≥11+εμ.Then Theorem 3.2(i)-(ii) holds. In addition, similar
properties hold by replacing (3.23)ηk⟶ηk0=ηk+(1−a^ka^k0)a^k0(sk−uk),η¯k⟶η¯k0=η¯k+(1−a^ka^k0)a^k0(sk−uk),ek+1[A posteriori identification error]:=sk+1−s^k+1=sk+1−a^k(sk−uk)⟶ek+10[A priori identification error]:=sk+1−s^k+10=sk+1−a^k0(sk−uk).Proof.
Define the sequence(3.24)0≤ΔVk0:=(a˜k+10)2−a˜k2≤σk|ek+1|1+γkφk2(σkφk2|ek+1|1+γkφk2−2(|ek+1|−η¯k0))≤0from (3.20)-(3.21) using a close
development to that in Theorem 3.2 so that it is monotonically decreasing (3.25)ΔVk:=a˜k+12−a˜k2=a˜k+12−(a˜k+10)2−|ΔVk0|≤−(1−(a^k+1a^k+10)2)(a˜k+10)2−|ΔVk0|≤0for all  k∈N0 since 0<a^k+1/a^k+10≤1, for all k∈N0, from the projection part (3.22) of the
estimation algorithm. Then {Vk}0∞ is bounded and converges to a finite limit,
and {ΔVk}0∞ is also bounded and converges to zero. Also |ΔVk0|=(1-a^k+1/a^k+10)<1 for all k∈N0, and |ΔVk0|=(1-a^k+1/a^k+10)<1. Since ΔVk0:=(a˜k+10)2-a˜k2=0, for all k∈N0,
then there exists a finite limit limk→∞(∑i=0k|ΔVi0|)<∞,  then ((a˜k+10)2−a˜k2) converges to a finite limit and |a˜k0| converges since |a˜k| converges (since Vk converges) to finite limits as k→∞.
Thus Theorem 3.2(i)-(ii) follows. On the other hand, note that (3.26)ek+1[A posteriori identification error]:=sk+1−s^k+1=ηk−a˜k(sk−uk)=ek+10+(a^k0−a^k)(sk−uk)=ηk0−a˜k0(sk−uk),ek+10[A priori identification error]:=sk+1−s^k+10=ηk−a˜k0(sk−uk)=ek+1+(a^k−a^k0)(sk−uk)=ηk0+(1−a^ka^k0)(2a^k0−a)(sk−uk).Since the normalized sequences {(σk/(1+γkφk2))1/2|ek+1|}0∞ and {(σk/(1+γkφk2))1/2η¯k0}0∞  are bounded, it follows that the reported replacements may be
performed leading to Theorem 3.3(iii).


## 3.3. Some Practical Design Constraints

The tracking error of the solutions of the IBHE and the reference IBHE are given by the set of equivalent expressions:(3.27)εk+1=sk+1*−sk+1=sk+1*−s^k+1−ek+1=sk+1*−s^k+1+a˜k(sk−uk)−ηk.The solution of the IBHE is equivalently calculated
either as a function of the nominal parameters a and b=1−a or as a function
of their estimates as follows, by considering the adaptive control law (3.8): (3.28a)sk+1=a(sk−uk)+uk+ηk=s^k+1+ek+1=a^k(sk−uk)+uk+ηk−a˜k(sk−uk)(3.28b)=s^k+1+ek+1=a^k(sk−uk)+uk+ηk−a˜k(sk−uk)(3.28c)=a^ksk+[(1−ak*)rk+(ak*−a^k)sk]+ek+1(3.28d)=ak*sk+(1−ak*)rk+ek+1.It
is important to discuss when the adaptive control law has no division by zero with
the property being numerically robust. This translates, in practice, to a^k≠1 being sufficiently deviated from unity and {sk}0∞ with all its elements positive and
sufficiently deviated from zero. This guarantees that the BHE does not diverge
when driven by an appropriate nonzero finitely upper-bounded environment
carrying capacity. More formally, a
reasonable assumption for practical design purposes is to extend the assumption of Theorem 3.3 to also consider
the same admissibility range referred to for the reference intrinsic growth-rate
inverse and the estimated intrinsic growth rate.Assumption 3.4.
a,ak,a^k,ak*∈[1/(1+ε1εμ),1/(1+εμ)], for all k∈N0, and some prefixed real constants εμ>0 and ε1>1. A
direct calculation with Assumption 3.4 yields (3.29)(1−a^k)∈[εμ1+εμ,ε1εμ1+ε1εμ]⊂(0,1),(11−a^k)∈[1+1ε1εμ,1+1εμ],|ak*−a^k|≤εμ(ε1−1)(1+ε1εμ)(1+εμ),(ak*−a^k)≤[−(ε1−1)εμ(1+ε1εμ)(1+εμ),(ε1−1)εμ(1+ε1εμ)(1+εμ)],while
similar expressions follow for replacements involving any two elements of the quadruplet (a,ak,a^k,ak*) as follows: (3.30)(1+εμ1+ε1εμ)[ε1rk+ε1−11+εμsk]≥uk=11−a^k[(1−ak*)rk+(a^k−ak*)sk]≥(1+εμ1+ε1εμ)[ε1(rk−11+εμsk)+11+εμsk]≥(1+εμ1+ε1εμ)[ε1(rk−sk)+11+εμsk].The
above expressions (3.29)-(3.30) lead to the following result referring to explicit positive lower and
upper bounds of uk=Kk−1=(Kk*+K˜k)−1 so that uk−1=Kk*+K˜k=Kk*+rk−1 since rk=(Kk*)−1.Proposition 3.5.
Assumesk<ε1(1+εμ)/(ε1−1)Kk*, for all k∈N0.
Then the following constraints hold: (3.31)1+εμε1(1+εμ)(Kk*)−1+(ε1−1)sk≤1uk≤1+εμε1(1+εμ)((Kk*)−1−sk)+sk,(3.32)1+εμε1(1+εμ)(Kk*)−1+(ε1−1)sk−Kk*≤K˜k≤1+εμε1(1+εμ)(Kk*)−1−(ε1−1)sk−Kk*.Proof.
It
follows directly from (3.29)-(3.30). The necessary condition sk<ε1(1+εμ)/(ε1−1)Kk* is needed to keep (ε1(1+εμ)(Kk*)−1−(ε1−1)sk)>0 in the right-hand-side upper bound of K˜k, for all  k∈N0 in
(3.32).


## 3.4. Closed-Loop Stability

The
boundedness and convergence properties of the parameter estimates have been
proven. In the following, the closed-loop stability is proven under a condition
of slow growing of the disturbances with respect to the solution of the IBHE if such a disturbance is unknown. In
particular, it is assumed thatη¯k=O[Max0≤i≤k(si)] with known constants of the related linear
upper bound and with sufficiently slow growth of η¯k with respect to Max0≤i≤k(si).
This is a reasonable assumption used in the background literature since a
complete lack of knowledge of disturbances makes impossible the stabilization
in the general case (see, e.g., [12, 13, 15] and references therein).Assumption 3.6.
There exist known finite nonnegative real constantsλ1,2 such that  (3.33)η¯k≤λ1+λ2Max0≤i≤k(si).Then, from (3.28b), (3.31)-(3.32),
and Assumptions 3.6–3.4, one gets (3.34)sk+1=a^k(sk−uk)+uk+ek+1.The
following results are preparatory for the stability theorem to be stated later
on. They basically establish that the absolute value of the identification error
grows not faster than linearly with the
upper-bounding sequence {η¯k}0∞ of the additive disturbances.Assertion 3.7.
If the Adaptation Algorithm 1 is used, then|ek+1|≤ς1η¯k+ς2, for all k∈N0, for some real constants ς1(≥ς)∈R+ and ς2∈R0+.Proof.
If|ek+1|≤ςη¯k,
the assertion is true with ς1=ς∈R+ and ς2=0. If |ek+1|>ςη¯k, then, from Theorem 3.2, (i) 0<(|ek+1|−ςη¯k)/(1+γk(sk−uk)2)1/2≤M<∞, for all k∈N0.
Thus (3.35)|ek+1|≤ςη¯k+M(1+γk|sk−uk|)≤(ς+Mγ)η¯k+Mγ|uk|+Mand
the assertion follows with (3.36)ς1=ς+Mγ,ς2=M(1+γ)Max0≤k<∞(|uk|)<∞since Min0≤k<∞(Kk)>0.Assertion 3.8.
If the Adaptation Algorithm 2 is used and Assumption3.4 holds, then |ek+1|≤ς1η¯k+ς2, for all k∈N0, with (3.37)ς1=1+2(ε1−1)εμ(1+ε1εμ)(1+εμ),ς2=Max0≤i<∞(|a˜i|)Max0≤i<∞(|ui|).Proof.
It follows from Assumption3.4, (3.4) and (3.29), and the fact that {Kk}0∞ is positively lower bounded, which yield
directly (3.38)|ek+1|≤|a˜k(sk−uk)|+η¯k≤(1+Max0≤i<∞(|a˜i|))η¯k+Max0≤i<∞(|a˜i|)Max0≤i<∞(|ui|)≤ς1η¯k+ς2,∀k∈N0.From
Assertions2.3 and 3.7, for the Adaptation Algorithms 1 and 2, respectively, Assumption
3.6 and (3.28c), (3.39)sk+1≤|ak*sk|+|1-ak*|rk+ς1η¯k+ς2≤ρks0+1−ρk1−ρ(ς1λ1Max0≤i≤k+1(sk)+ρ1ρ2+ς1λ2+ς2)≤ς1λ11−ρMax0≤i≤k+1(sk)+11−ρ(ρ1ρ2+ς1λ2+ς2)+s0,∀k∈N0,for any s0>0 and some positive finite constants  ρ,ρ1, and ρ2 satisfying 1>ρ≥Max0≤k<∞(ak*), ρ1≤1−Min0≤k<∞(ak*)<1, and ρ2≥Max0≤k<∞(rj).
The last two inequalities of (3.39) follow directly by proceeding recursively
with the preceding one by using ∑i=0k−1ρi=(1−ρk)/(1−ρ)≤1/(1−ρ) since ρ∈(0,1).
It is obvious that, for
each k∈N0,
it exists a unique k′(k)∈N0,
dependent in general on k, such that k′(k)≤k such that sk′=Max0≤j≤k′(sj)=Max0≤j≤k(sj) and there is no other j∈N0 such that k≥j>k′(k),
where the same maximum is reached. A precise definition is (3.40)k′(k):=Arg(j∈N0:0≤j≤k⋀sk′=Max0≤j≤k′(k)(sj)=Max0≤j≤k(sj),¬∃j(k′(k)<j≤k)∈N0:sj=Max0≤j≤k(sj))with   “⋀” and “¬” being the conjunction and negation logic
symbols. The substitution of (3.40) into (3.39) yields (3.41)Max0≤j≤k(sj)≤(1−ϑ)−1C<∞,∀k∈N0,provided that  ς1λ1/(1−ρ)<1,
which holds if and only if 0≤λ1<(1−ρ)/ς1 (i.e., if
Assumption 3.6 holds with sufficiently small λ1), where (3.42)ϑ:=ς1λ11−ρC:=11−ρ(ρ1ρ2+ς1λ2+ς2)+s0with C being a finite
positive real, which
depends on the initial conditions of the IBHE (and thus on the initial conditions of the BHE) on the available upper bound
of the additive disturbances, on the reference, and on the free-design
parameters of the adaptation algorithm, γk,  ς>1,  ξ0, ξ1,
and so on. The above result also holds for the Adaptation Algorithm 2 with
specific constants redefined by (3.43)ρ:=11+εμ<1,ρ1′:=ε1εμ1+ε1εμ,ρ1:=ρ1′1+ε1εμ=ε1εμ(1+ε1εμ)2,ρ2:=ε1(1+εμ)Max0≤k<∞(Kk*)−1<∞,and ξ1,2 defined in Assertion 2.3 for the Adaptation
Algorithm 1 and in Assertion 3.7 for the Adaptation Algorithm 2. The boundedness of the solution of the IBHE for finite initial conditions implies
that of the additive disturbances via Assumption 3.6, that of the identification error via Assertions 2.3–3.7, and
that of the tracking error since the solutions of the current and the reference one are both bounded. The
estimated parameters and the estimated solution are bounded from Theorems 3.2 and
3.3. Those conclusions are now summarized in the result below.Theorem 3.9 (closed-loop stability).
If Assumption3.6 holds with sufficiently small coefficient λ1,
then all the signals in the closed-loop system remain bounded for all time so
that the IBHE (and then the associate BHE) is globally
stable if any of the Adaptation Algorithms 1 or 2 is used.Remark 3.10.
Theorem3.9 becomes stronger than formerly stated in the absence of additive
disturbances (i.e., ηk≡0) since asymptotic perfect tracking is
furthermore  achieved. Another result is that the identification error converges
asymptotically to zero if the relative dead zone of the adaptation algorithms
is removed, that is, the {σk}0∞ is not zeroed at any sample as a result of a
comparison procedure of the identification error and the additive disturbances. These results follow directly from Theorems 3.2
and 3.3 and (3.39). Note that the results may be directly generalized to the
extension of using relative dead zones for adaptive algorithms by incorporating
the estimation of the constants λ1,2 characterizing {η¯k}0∞,
provided they are unknown [16].Remark 3.11.
The closeness between the IBHE and that of its reference model has been considered  for feasibility
reasons since, in many ecological problems, where such models are commonly of
interest, the environment characteristics, which directly influence the value
of the environment carrying capacity,
cannot be abruptly modified even in closed environments. The employed philosophy about relative dead-zone adaptation
seems to be promising to be also applied to other ecological controlled
problems involving, for instance, Kolmogorov-type ecological models or models for
a biochemical aquariums [17, 18]. The obtained results
seem to be promising for mathematical modeling and the use of adaptive control
in other kinds of problems related to biology and ecology (see, e.g., [20–22])
and to extend the research to governing extinction conditions [10].


## 4. Numerical Examples

Example 4.1.
The objective is that
the BHE solution{xk}0∞ tracks a reference sequence {xk*}0∞ by synthesizing a control system in order that
IBHE matches a reference model. Such a reference model is another IBHE defined
by the appropriated sequences {μk*}0∞ and {Kk*}0∞ to generate the sequence {sk*}0∞ associated to the desired {xk*}0∞ to be tracked by the BHE solution. The
reference values are μk*=1.55,
and Kk*=170, for all  k∈N0, are used for the reference model and BHE,
respectively, with the period T=1 (one year). It is assumed that small local
variations of the carrying capacity of the habitat are allowed. The evolution
data of the population of cod in the North Atlantic Sea in the period 1952–1992
are given in [4], and exhibit a quasiperiodic behavior of the intrinsic growth
rate. Based on these data, the intrinsic growth rate is modeled as a sinusoid
defined by μk=2.3+0.15sin(KT/100),
which adjusts closely to those experimental data. The average value of Kk is 200, what is used as a fixed uncontrolled
carrying capacity for the whole set of data. The initial population of the cod
is x0=500 being identical to the reference value. Figure 1 displays the population of cod if the BHE is controlled and if it is not
controlled.
The population of the
controlled BHE tracks perfectly the reference sequence as in the transient as
in the stationary regime for all time since the true parameterization of the
BHE is known, and then the control law is not adaptive. Note, however, the uncontrolled BHE does not track the reference. Figure2 displays the controlled
carrying-capacity sequence, which leads to a perfect tracking of the reference.Figure 1
Evolution of the population of cod fromx0=500.Figure 2
Evolution of the control sequence for the solution of Figure1.Example 4.2.
A numerical adaptive example is now discussed.
It is assumed that the environment carrying capacity of the reference model-environment
carrying capacity is a constant value of 200, and the reference intrinsic growth
rate is constant of value 2. The objective is to design the environment
carrying-capacity sequence via estimation and feedback so that asymptotic
tracking of the reference solution is achieved with positive solutions. It is
assumed that no unmodeled dynamics or parametrical uncertainties are present so
that a standard recursive least-squares algorithm is used for parametrical
estimation of the intrinsic growth rate whose initial value is chosen and the
initial value of the time-decreasing covariance gain is fixed to 1000. Figure3
displays the reference BHE, the uncontrolled BHE, and the controlled one. It is
seen that the reference BHE is asymptotically tracked, while the uncontrolled
one differs from the reference. The carrying capacity of the unknown
uncontrolled BHE varies from 180 to 220 according to the shape of Figure 4, and
its intrinsic growth rate is constant, equal to 1.6. The carrying capacity
exerting the control action is also displayed on Figure 4 with maximum and
minimum admissible values being constrained by the adaptive controller to 280
and 200, respectively.BHE Solutions: (a) reference, (b) uncontrolled, and (c) controlled for initial covariance gain 1000.
(a)(b)(c)Environment carrying capacities: (a) uncontrolled BHE and (b) controlled.
(a)(b)


## 5. Conclusions

This
paper has considered the well-known BHE used in ecology within a control
problem context where the control action on the linear IBHE is the inverse of
the carrying capacity. An extended version of the standard BHE has been
considered by incorporating additive disturbances. The overall control problem
is firstly stated on the IBHE by taking advantage of its linear nature. This
point of view is very feasible in close or semi-open environments where
humidity, temperature, and other factors of the environment may be selected
within certain margins. A reference is also defined, which describes the suited
behavior for the system, the control action having an objective that the
solution of the current BHE is able to perfectly track that of the reference (a).
For feasibility purposes, the overall problem is stated in terms of a local variation
of the inverse carrying capacity to perform the control action on the IBHE so
that the reference (a) is fully tracked. This implicitly means that, in practice,
the current parameters of the BHE and then those of the IBHE are locally
deviated from those of its reference model. Then the method has been extended by
incorporating an adaptive version for the case when the BHE parameterization is
partly or fully unknown. The use of a relative adaptation dead zone freezes the
adaptation when the identification error is sufficiently small according to an
available absolute upper bound of the disturbances sequence so that the
algorithm is proven to prevent potential instability caused by the presence of
such additive disturbances. The second one incorporates to the estimation dead
zone and estimates projection procedure by using a priori knowledge on the parameters of the BHE and the
second algorithm, which have been presented. It has been proven that both
adaptation algorithms and associate control law stabilize the current, provided that the absolute value of the
additive disturbances grows not
faster than linearly with the
maximum of the solution with a sufficiently small slope. Some related numerical examples have
been discussed to corroborate the theoretical results. The potential generalization
to the general case, where the parameterizations of the current BHE and its
reference one to be tracked are not close to each other, is direct although it may be not feasible in some
practical cases, since the adaptation algorithms and the main stability results
are formulated in a general way.



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