A Control Theory point of view on Beverton–Holt equation in population dynamics and some of its generalizations

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Abstract

This paper is devoted to develop some “ad hoc” Control Theory formalism useful for the famous Beverton–Holt equation arising in population dynamics. In particular, the inverse equation is redefined for a finite set of consecutive samples under the equivalent form of a discrete linear dynamic system whose input sequence is defined by the sequence of carrying capacity gains and the unforced dynamics is directly related to the intrinsic growth rate. For that purpose, the environment carrying capacity gains are allowed to be time-varying and designed for control purposes. The controllability property is also investigated on this dynamic extended system as well as the stability, equilibrium points and attractor oscillating trajectories. The properties of the dynamic system associated with the Beverton–Holt inverse equation allow extrapolate in a simple dual way the above properties to the standard Beverton–Holt equation. Some generalizations are given for the case when there are extra parameters in the equation or when the system is subject to the presence of additive disturbances. In all cases, a reference model being also of Beverton–Holt type is proposed to be followed by the control system.

Introduction

The non-autonomous discrete Beverton–Holt equation (BHE) is very useful in Ecology and, in particular, in studying the growth population dynamics, (see, for instance [1], [2], [3], [4], [5]). 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 has also been reported to be useful to describe the behavior in the population evolution of the acorn wood pecker. In the last years, an important effort has been devoted to investigate the mathematical properties of such an equation concerning the properties of the equilibrium points, attractors and some conjectures concerning positive solutions, stable oscillations of the solution and joint properties of the equilibrium points and some of the constants, or sequences of parameters, defining the dynamic evolution problem, (see, for instance [6], [7], [8], [9], [10], [11], [12], [13]). In particular, an important effort has been addressed to solve the so-called Cushing–Henson conjectures [7]. The standard non-autonomous BHE is:BHE:xk+1=μkKkxkKk+(μk-1)xk,kN0N{0},where x0 > 0, μk  R+ 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 carrying capacity, is a characteristic of the habitat being dependent on resources availability, temperature, humidity, etc. Typically, μk > 1 and Kk = Kk+p  R+ for some p(⩾2)  N, the period p being a consequence of the fact that periodic fluctuations are common in biological problems where parameters fluctuate and usually are periodic functions of time. In the BHE, biologists call xk the spawning stock at continuous time kTs, Ts being the sampling period in-between consecutive samples of (1), providing a recruitment xk+1, and x¯limxk{xk+1}=αK=αx=αlimk{xk}, with αμμ-1=limkμkμk-1 being the limit recruitment provided that such a limit exists and that K=limk{Kk}>0 exists and μk > 1 [4]. In this context, it has been reported that the BHE, often referred to also under the alternative name of Pielou logistic difference equation is equivalent to the discrete Verhulst logistic equation xk+1 = rkxk/(xk + νk) under a given initial condition x0. The particular time-invariant Verhulst equation is the discretization of the logistic continuous-time time-invariant equation x˙(t)=rx(t)(1-x(t)/K);x(0)=x0,x(t) being the size of the resource population [12]. In a recent work, Stevic has proved analytically the conjectures of Cushing–Henson [11], for periodically varying carrying capacities of the BHE. In this way, the conjectures have been rigorously proved and then confirmed. The first conjecture establishes that there is a periodic steady-state regime if the carrying capacity varies in a periodic fashion which is a global attractor of all the positive solutions. The second one establishes that the average value of the periodically varying values of the solution is not larger than the average value of the carrying capacity over one such period. Another attractive point of Stevic’s analysis in [11] is the use of the inverse of Beverton–Holt equation (IBHE) for the evolution of the sequence sk=xk-1. This inverse equation is linear and then much easier to analyze than (1). Furthermore, this inverse equation is driven by the inverse of the carrying capacity. In other words, the carrying capacity inverse is the control sequence for the non-autonomous IBHE. It is well-known that Control Theory is an important mathematical theoretical tool, useful in a lot of applications in Milling Industry, Economic Models, Robotics, Electrical Machine Regulation, etc., which allows monitoring the solution of differential, difference and hybrid systems in a prescribed way. In this paper, the BHE is discussed through the evolution of its inverse from a Control Theory point of view. In particular, the properties of controllability and stability are discussed when the equation is driven by the carrying capacity. Another issue is the study of the equation when subject to additive disturbances and its control in order to match a prescribed reference model describing the suitable prescribed behavior for the case of known parameters. This implies that the equation solution is identical to a prescribed one established by a reference model. The paper is organized as follows. Section 2 is concerned with the stability of the BHE, and a discrete dynamic systems associate with its inverse which is appropriate for the study of the equilibrium points, stability and a modeling tool for focusing on a Control Theory approach with the inverse of the carrying capacity being the control action. Some extensions are given to the use of multirate sampling with the input being sampled at fast sampling rate than the solution. This is a useful technique to stabilize the zeros of discrete time-invariant plants, to improve the controllability performance and to accommodate the various sampling rates to the various physical signals presented in a particular problem, [14]. Section 3 is devoted to the property of controllability. It is seen that controllability holds with the intrinsic growth of the population exceeding unity with a prescribed margin for all samples, but the property can be lost if the intrinsic growth tends to unity, which describes the case of zero growth of the specie. It is, however, seen that the multirate approach with fast control sampling can keep controllability even if the controllability threshold of the intrinsic growth rate is violated at some samples. Section 4 presents some generalizations of the BHE and discusses the existence of periodic solutions being asymptotic attractors in the case when the intrinsic growth, carrying capacities and extra parameters, if any, converge asymptotically to a finite set of constant positive values. The design of “ad hoc” values of the carrying capacity so that the overall solution behaves as a prescribed suited solution is also focused on. The problem is stated as a local modification of the carrying capacity inverse, so that the solution of the IBHE is corrected so as to behave as a prescribed suitable solution. Section 5 discusses some illustrative examples and, finally, conclusions end the paper.

Section snippets

Basics BHE, IBHE, associate dynamic systems, stability and multirate sampling

The change of variable sk=xk-1 in the BHE (1) [11], yields the IBHE provided that μk  0, Kk  0, ∀k  N0:IBHE:sk+1=qksk+(1-qk)uk,where qk=μk-1 and uk=Kk-1 under initial condition s0=x0-1>0. Also, for any k  N0, a, in general, time-varying linear dynamic discrete system IBHS of any dimension n may be associated to the IBHE as follows:IBHS:zk+1=Akzk+bkuk,yk=Czk,where zk  z(kTs) = [sk, sk−1, …, skn+1]T  Rn is the n-dimensioned state vector sequence subject to initial conditions z0  [s0, s−1, …, sn]T with Ts

Controllability results

This section discusses controllability (i.e. the solution sequences or the states of the associated dynamic systems are carried to prefixed values in a finite number of samples under the appropriate control action; i.e. the appropriate values of the sequence of carrying capacity). In particular, it is seen that controllability of the IBHS stands if the intrinsic growth rate sequence does not take values in the neighborhoods of values zero and unity. However, it is proved that MIBHS (FIS) may

Generalizations of the BHE and IBHE

The IBHE is now generalized to possess two independent parametrical sequences {αk}0 and {βk}0 related to the population growth and another one uk=Kk-10 related to the carrying capacity which exerts as its control input. Thus, the generalized IBHE is:sk+1=αksk+βkuk,s0=s>0,If {sk}0 has no zero value, through the change of variable xk=sk-1, one obtains the following generalization of the BHE:xk+1=xkαk+βkxkuk,x0=s-1>0,which yields the following particular cases:

  • 1.

    The standard BHE (1) with

Numerical examples

Some simulation results which illustrate the effectiveness of the proposed method to control the BHE solution are shown in the current section. The control objective is that the BHE solution {xk}0 tracks a reference sequence xk0. Such an objective is obtaining by synthesizing a control system in order to the IBHE matches a reference model. Such a reference model can be another IBHE parameterized by sequences μk0 and Kk0 being considered the most appropriated ones to obtain a desired

Conclusions

This paper has discussed the well-known classical Beverton–Holt equation of Ecology from a Control Theory point of view. The main tools for appropriate modeling have been to build an inverse Beverton–Holt equation and an associate discrete dynamic system whose control action is the inverse of the carrying capacity. The controllability property has been discussed. Extensions to the use of different sampling periods for the solution and the input have been discussed in a multirate sampling

Acknowledgements

The authors are very grateful to the Spanish Ministry of Education by its partial support of this work through Project DPI2006-00174.

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