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.

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., [

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 [

Define an inverse system of (

Note that Assumption

If Assumptions

The subsequent set of structures is considered in order to be then able to formulate the control law.

BHE

BHE

If

It
follows, by complete induction, by assuming

If the disturbance is unknown, but
Assumption

If Assumption

Direct
calculations yield

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 Assumption

Assume that the maximum allowed absolute variation of the carrying capacity
with respect to its nominal value is

The control
parameter sequence

The tracking-error
sequence

A
positive carrying capacity is obtained from the controller synthesis if

(i) Note that

One gets
from (

The
first part follows with the replacement of

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., [

The deviations of the intrinsic growth rate with respect to a
certain unknown constant value

It is assumed to be time invariant as the
nominal IBHE, namely,

The adaptive control law (

Proceed, by induction, by assuming

In the following, the boundedness and convergence properties of the adaptation algorithm are investigated in the subsequent result.

Assume that a sequence

The sequences

There exist
the subsequent limits

Let

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 (

If both the nominal and current intrinsic growth rates satisfy Assumption

It consists of two steps, namely, a priori and a posteriori estimations as follows.

Define the sequence

The tracking error of the solutions of the IBHE and the reference IBHE are given by the set of equivalent expressions:

Assume

It
follows directly from (

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 that

There exist known finite nonnegative real constants

Then, from (

If the Adaptation Algorithm 1 is used, then

If

If the Adaptation Algorithm 2 is used and Assumption

It follows from Assumption

From
Assertions

If Assumption

Theorem

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 [

The objective is that
the BHE solution

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. Figure

Evolution of the population of cod from

Evolution of the control sequence for the solution of Figure

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. Figure

BHE Solutions: (a) reference, (b) uncontrolled, and (c) controlled for initial covariance gain 1000.

Environment carrying capacities: (a) uncontrolled BHE and (b) controlled.

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.

The authors are very grateful to the Spanish Ministry of Education for its partial support of this work through Project DPI 2006-00174. They are also grateful to the referees for their useful comments.