On an Extended Time-Varying Beverton–Holt Equation Subject to Harvesting Monitoring and Population Excess Penalty

Research Article On an Extended Time-Varying Beverton–Holt Equation Subject to Harvesting Monitoring and Population Excess Penalty Manuel De la Sen , 1 Santiago Alonso-Quesada , 1 Asier Ibeas , 2 and Aitor J. Garrido 3 1 Institute of Research and Development of Processes, Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country (UPV/EHU), Leioa 48940, Bizkaia, Spain 2 Department of Telecommunications and Systems Engineering, Universitat Aut` onoma de Barcelona, UAB 08193, Barcelona, Spain 3 Department of Automatic Control and Systems, Institute of Research and Development of Processes, Faculty of Engineering of Bilbao, University of the Basque Country (UPV/EHU), Po. Rafael Moreno, 3, Bilbao 48013, Spain Correspondence should be addressed to Manuel De la Sen; [email protected] Received 22 December 2022; Revised 15 March 2023; Accepted 4 April 2023; Published 28 April 2023 Academic Editor: Ewa Pawluszewicz Copyright ©2023 Manuel De la Sen et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tis paper considers a more general eventually time-varying Beverton–Holt equation for species evolution which can include a harvesting action and a penalty for overpopulation numbers. Te harvesting action may be positive (typically consisting of hunting or fshing) or negative which refers to repopulation within the environment. One considers also a penalty of quadratic type on the overpopulation and the introduction of a term related to Allee efect to take account of small levels of population. Te intrinsic growth rate is assumed either to exceed unity or to be under unity. In the second case, the extinction point is a locally stable attractor while the other positive equilibrium point is unstable contrarily to the commonly studied case of intrinsic growth rate exceeding unity where the above roles are inverted. Tis consequence implies that the extinction point is also globally asymptotically stable for any given fnite initial condition. In the case when the eventual overpopulation is penalized with a sufciently large coefcient which exceeds a prescribed threshold, to quantify such an excess, only a globally asymptotically stable extinction attractor is present and no other positive equilibrium points exist. In the case of a positive moderate quadratic evaluation term for such an overpopulation, one or two positive equilibrium points coexist with the extinction one. Te smaller one is unstable contrarily to the extinction equilibrium which is locally asymptotically stable. If it exists a second largest positive equilibrium point, being distinct to the above-given one, then it can be unstable or locally stable depending on the parameterization. Also, some methods of monitoring the population evolution through control laws on the harvesting action are discussed. 1. Introduction Beverton–Holt equation is an useful discrete equation for modelling the evolution of species which reproduce by eggs such as birds, fshes and insects [1]. It can be considered the counterpart of the logistic equation in the Verhulst´s continuous-time model. Te basic Beverton–Holt model is parameterized by two positive sequences, namely, the carrying capacity of the environment which depends on resources availability, temperature, humidity, etc., and the intrinsic growth rate which is associated with the species reproduction capability, the survivorship chance etc. Te intrinsic growth rate has typically to exceed unity to avoid extinction. In the most general case, those parameters can be changed to sequences to describe potential diferent behaviors of the population evolution in diferent periods, for instance, seasonality. Tere are other two typical parameters to be eventually considered which generalize the model such as the independent consumption which describes recruitment variations due to unforeseen disturbances and eventual repopulation or interchange of population with neighbour environments and the harvesting process Hindawi Discrete Dynamics in Nature and Society Volume 2023, Article ID 5052799, 21 pages https://doi.org/10.1155/2023/5052799 associated with fshing or hunting and which depends on regulation based on the available spawning stock and foreseen recruitment. Te basic time-invariant Beverton–Holt equation has two equilibrium points which are the extinction point, which is locally unstable, and the carrying capacity level which is globally asymptotically stable. Te so-called Cushing–Henson conjecture established that, if the equation is modelled by periodic parameterizing sequences of carrying capacities and intrinsic growth rates, then the averaged periodic sequence of population lies below the average of the corresponding average of the carrying capacities. Te conjecture has been rigorously proved to be true by Stevic [2]. Some extensions of the basic model concerning the Cushing–Henson conjectures for the Beverton–Holt q-diference equation have been discussed in [3]. A control theory point of view on the Beverton–Holt equation has been adopted and discussed in [4–6] while giving a design procedure of the environment carrying capacity for monitoring the suitable sequence of values to follow of the population evolution. Te applicability of the proposed method is claimed for semiopen environments such like certain fsheries. It is discussed in [7] how, in practice, the intrinsic growth rate can be dependent on the environment carrying capacity. Also, it is discussed in [8] an impulsive extended competition Beverton–Holt model between species from the stability point of view. Te usefulness of the Beverton–Holt and other mathematical models in maritime biology is described in [9]. In [10–13], the harvesting action is investigated in an extended Beverton–Holt model. Normally, harvesting refers to fshing or hunting which is subjected to authorities regulation but it can also be total or partially illegal while associated to furtive uncontrolled actions. See also some references therein and [6, 8]. Other related studies consider alternative generalizations concerned with periodic behaviors, associated, for instance, to seasonality [14, 15], global dynamics analysis of some extended equation versions [16], presence of bifurcations [17, 18] or resonances [19], and perturbations of the basic model. See, for instance [18]. On the other hand, an extended Beverton model on isolated time scales is analyzed in detail in [20]. Also, an extension of the Beverton–Holt model including discrete delays in the evolution dynamics has been investigated in [21]. A Beverton–Holt model extension including discrete delays in the evolution dynamics has been investigated in [21]. On the other hand, it can be pointed out that Beverton–Holt-based models are used also by biologists when monitoring fshing stock availability and fshing migrations to evaluate the recommended maximum number of captures (or recommended harvesting action) to avoid the environment degradation and species extinction. See, for instance [22, 23], and some of the references therein. In this paper, we focus on a generalized Beverton–Holt equation which assumes a quadratic-type penalty for the population excess describing the potential internal competence between the individuals for food, refuge, etc. Te harvesting action is considered jointly with eventually present independent consumption if necessary. It is seen that the presence of such a term can translate into the presence of two other equilibrium points. Te paper also designs species evolution control laws by monitoring the harvesting action and the infuence in the results of considering a modelling function of Allee’s efect which makes difcult growing or even can cause extinction for small numbers of reproductive individuals. Te paper is organized as follows. Section 2 deals with the equilibrium points in the presence and absence of harvesting action, considered together with eventual independent consumption, in the case when the intrinsic growth rate elements exceed unity. Te harvesting sequence can have positive, negative or null elements. Te local asymptotic stability of each feasible (that is being real and non-negative) equilibrium points is characterized in the case when the parameterizing sequences converge to limits. Section 3 develops two methods to derive control laws for the harvesting actions again if the intrinsic growth rate exceeds unity for all time. Te frst proposed method is based on the convergence of the solution sequence of the population to a prescribed targeted equilibrium point of the population value by choosing the harvesting sequence. Classical criteria for convergence of sequences, such as D’Alembert, Cauchy, and Raabe criteria, are involved in the respective monitoring rules of the harvesting action. Te second method relies on a sample-to-sample monitoring of the solution sequence to target a prescribed evolution pattern by designing the harvesting sequence. Next, Section 4 relies on introducing Allee’s efect to modify the basic Beverton–Holt equation to describe the situation arising under small numbers of individuals which make difcult the reproductive action and can lead to extinction even the intrinsic growth rate exceeds unity for all time. Te equilibrium points, their stability conditions as well as extinction conditions are investigated if the intrinsic growth rate exceeds unity for all time. Te second part of this section proposes a penalty term in the Beverton–Holt equation for high levels of population in the absence of harvesting. Te resulting equilibrium points and their stability issues are also investigated if the intrinsic growth rate exceeds unity for all time. On the other hand, Section 5 relies on extinction conditions and the local asymptotic stability of the extinction equilibrium point if the intrinsic growth sequence has elements being less than unity in the absence of harvesting by considering the modifed model with quadratic penalty time for high levels of population. Te local asymptotic stability of the positive equilibrium points is also investigated. Section 6 is devoted to discuss some numerical examples and, fnally, some conclusions end the paper. In the following, the subsequent notation is used: Z0+and Z0−denote, respectively, the sets of nonnegative and nonpositive integer numbers Z+and Z−denote, respectively, the sets of positive and negative integer numbers R0+and R0−denote, respectively, the sets of nonnegative and nonpositive real numbers R+and R−denote, respectively, the sets of positive and negative real numbers 2Discrete Dynamics in Nature and Society 2. Equilibrium Points and Their Local Asymptotic Stability under Positive, Null, or Negative Harvesting “Harvesting” is referred to fshing and hunting subject to administrative regulation which depends on the species population. On the other hand, “independent consumption” refers to positive of negative supplies of extra populations due to migrations from outside of the environment under consideration [10, 13, 14]. In the sequel, we consider both efects together integrated in the same additive perturbation sequence to that of the standard population evolution sequence. Consider the following, in general, time-varying Beverton–Holt equation subject to harvesting action eventually combined with independent consumption: yn+1�anKnyn Kn+an−1  􏼁yn−hn,∀n∈Z0+,(1) with initial condition y0≥0, where an 􏼈 􏼉∞ n�0is the intrinsic growth rate of the species and Kn 􏼈 􏼉∞ 0is the environment carrying capacity sequence, both being positive real sequences. If a constraint yn+1≥zn+1≥0; ∀n∈Z0+is prefxed for some sequence zn 􏼈 􏼉∞ n�0⊂R0+so that the harvesting sequence hn 􏼈 􏼉∞ n�0has to fulfl: hn∈−∞,anKnyn Kn+an−1  􏼁yn−zn+1 􏼠 􏼣,∀n∈Z0+.(2) Te harvesting sequence defnes the population amount which is not related to the dynamics evolution within the environment because of natural reproduction and dead concerns. It is related to an increase or decrease of individuals due to population fux either from or to the habitat plus eventual decrease of population due to hunting or fshing. In this way, the sequence can take negative values at a particular sampling instant because of the sign in (1), this situation will correspond to an increase of the amounts of individuals), positive values (that is, decrease of population), or zero (that is, the population is just modifed by the natural reproduction and dead within the considered habitat). According to that philosophy, the harvesting is considered in this paper as the eventual combination of an eventual traditional harvesting (that is, hunting/fshing) and eventual migrations in both senses from or to the habitat under study. Also, the hunting or fshing includes, in general, legal or illegal actions (poaching). It turns out that any equilibrium point needs to be real non-negative in order to be feasible (that is, real and positive) as it is addressed in the subsequent result. Theorem 1. Assume that an 􏼈 􏼉∞ 0(⊂[1, a]) ⟶a,Kn 􏼈 􏼉∞ 0 (⊂[0, K]) ⟶K,zn 􏼈 􏼉∞ 0(⊂[0, z]) ⟶zand hn 􏼈 􏼉∞ 0⟶h with hn∈(− ∞,(anKnyn)/(Kn+(an−1)yn)− zn+1]; ∀n∈Z0+. Ten, the solution of (1) has: (i) A real positive equilibrium point y�K>0and a null equilibrium point (extinction) at y0�0if h�0. (ii) If h≥Kthen there is no nonextinction feasible equilibrium point and the nonextinction equilibrium points are feasible if h<K. (iii) If h≠0then the potential nonextinction equilibrium points y1>0and y2≥y1>0are given by y1,2�(a−1)(K−h)∓������������������������� (a−1)2(K−h)2−4(a−1)Kh 􏽱 2(a−1), (3) which are real if and only if h∈((− ∞,((a+1− 2�� a √)/(a−1))K)]∪[((a+1+2�� a √)/(a−1))K, + ∞). Te equilibrium point y2is feasible if and only if h∈[− ∞, K]which restricts the above-given inequality for realness. Also, the equilibrium point y1is feasible if and only if h∈[0, K]. If h�K(a+1∓2�� a √)/(a−1) then y1�y2>0. In terms of the intrinsic growth rate, the nonextinction equilibrium points y1and y2are both feasible if h∈[0, K]and (a−1)(K−h)2≥4Kh, equivalently, if a≥((K+h)/(K−h))2(a>1if h�0). Also, y1is not feasible for h<0and y2is feasible for h<0irrespective of a(>1)and K(>0). Proof. Note directly that the extinction level y0�0 is an equilibrium point. Also, by replacing in (1) the limits of the various sequences, one gets a single root y�Kif h�0 and, if h≠0, then (a−1)y2+(a−1)(h−K)y+Kh �0.(4) Tus, since a>1, if h≥Kthen there is no real nonnegative solution to (1) since (4) fails for y>0 and h≥K. If 0 <h<K, or if h<0, then the roots of (2) are y1�(a−1)(K−h)− ������������������������� (a−1)2(K−h)2−4(a−1)Kh 􏽱 2(a−1), y2�(a−1)(K−h)+ ������������������������� (a−1)2(K−h)2−4(a−1)Kh 􏽱 2(a−1). (5) Properties [(i)-(ii)] have been proved. On the other hand, the nonextinction equilibrium points y1and y2are both feasible for h∈[0, K)if (a−1)(K−h)2≥4Kh, equivalently, if a≥(K+h)/(K−h)2(a>1 if h�0). Also, y1 is not feasible for h<0 from (5) and y2is feasible, also from (5), for h<0 irrespective of a(>1)and K(>0). Discrete Dynamics in Nature and Society 3 Note that, in order for the roots of (4) to be real, since a>1, it is needed that θ(h)≥0, where θ(h) �(a−1)(K−h)2−4Kh �(a−1)h2+(a−1)K2−2(a−1)Kh −4Kh �(a−1)h2+(a−1)K2−2aKh +2Kh −4Kh �(a−1)h2+(a−1)K2−2Kh(a+1).(6) Note that, y1,2are real if and only if the zeros of θ(h), that is, h1,2�K(a+1∓2�� a √)/(a−1), are non-negative real since a>1. Since θ(h)is a convex parabola, θ(h)≥0, and both equilibrium points y1and y2are real (and they can be positive) if and only if h∈(− ∞, K(a+1−2�� a √/ (a−1)]∪[K(a+1+2�� a √)/(a−1),+∞). Contrarily, if h∈(K((a+1−2�� a √))/(a−1)), K(a+1+2�� a √)/(a−1)) then the zeros are not real and the nonextinction equilibrium points y1,2never exist. Since, their feasibility implies that h≤Kthen the equilibrium point y2is feasible if and only if h∈[− ∞, K]. Since (a+1−2�� a √)/(a−1)<1for a>1. Also, the equilibrium point y1is feasible if and only if h∈[0, K(a+1−2�� a √)/(a−1)]. It is obvious that h�h1,2� K(a+1∓2�� a √)/(a−1)then y1�y2. Property (iii) is proved. Te use of the inverse sequence of that of a the population evolution sequence is of interest to derive easily some interesting results concerning the stability and the asymptotic boundedness of the solution as it is addressed in the subsequent result: □ Theorem 2. Defne the inverse sequence of the solution of (1) as xn�y−1 n;∀n∈Z0+. Te following properties hold: (i) Te inverse sequence xn 􏼈 􏼉∞ n�0of the solution yn 􏼈 􏼉∞ n�0is given by the discrete equation: xn+1�μnxn+]n+hI n;∀n∈Z0+,(7) where μn�a−1 n;]n� (1−μn)K−1 n� (an−1)a−1 nK−1 n, subject to hn∈(− ∞,(anKnyn)/(Kn+(an−1)yn)]; ∀n∈Z0+, and hI nis zero if hn�0for any n∈Z0+, with hn�anKnhI n 2 1 −a−1 n 􏼐 􏼑+hI n+a−1 nxn 􏼐 􏼑Kn 􏼐 􏼑xn+K−1 nan+a−1 n−2 􏼐 􏼑+hI nan−1  􏼁;∀n∈Z0+,(8) for x0�y−1 0>0. Te solution is equivalently expressed from given initial conditions as follows: xn+1�􏽙 n i�0 μi 􏼂 􏼃⎞ ⎠x0+􏽘n i�0􏽙 n j�i+1 μj 􏽨 􏽩⎞ ⎠]i+hI i 􏼐 􏼑. ⎛ ⎝ ⎛ ⎝ (9) (ii) xn 􏼈 􏼉∞ n�0is bounded, equivalently, yn 􏼈 􏼉∞ n�0does not vanish neither at any sample nor asymptotically (and then the population does not extinguish either in fnite time or asymptotically) if lim supn⟶∞ 􏽐n i�0(􏽑n j�i+1[μj])(]i+hI i)<∞. In particular, if an 􏼈 􏼉∞ n�0⊂[a, a]and lim infn⟶∞an≥a>1then xn≤− a 􏼐􏼑−nx0+1−− a 􏼐􏼑−n 1−asup 0≤i≤n−1 ]i+hI i 􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌;∀n∈Z+, (10) is bounded for any given fnite x0≥0for all n∈Z0+ if sup0≤i≤n−1|]i+hI i|<+∞. For any fnite x0>0, so that y0�x−1 0, the sequence yn�x−1 n 􏼈 􏼉∞ n�0 satisfes: yn≥1−− a − a 􏼐􏼑−n1−− a 􏼐 􏼑x0+1−− a 􏼐􏼑−n 􏼐 􏼑 sup 0≤i≤n−1 ]i+hI i 􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌 >0; ∀n∈Z+. (11) (iii) Defne bn�1−μn� (an−1)/anby assuming that an 􏼈 􏼉∞ n�0⊂(1,∞]subject to 􏽐∞ n�0bn�􏽐∞ n�0(an−1)/ an�∞. Assume that |hI n+(an−1)/(Knan)| ≤ε(an−1)/anfor any integer n≥n0, some n0∈Z0+ and some ε∈R+. Ten, lim supn⟶∞xn ≤ε⇔lim infn⟶∞yn≥ε−1. Te constraint |hI n+(an−1)/Knan|≤ε(an−1)/anis satisfed under any of the subsequent stipulations for each n∈Z0+: (1) 0≤hI n≤(ε−K−1 n)(an−1)/anrequiring that the harvesting sequence hn≥0and the carrying capacity Kn≥1/ε (2) hI n<0and (an−1)/anK−1 n≥|hI n|≥(K−1 n−ε) (an−1)/an≤hI n<0requiring that hn<0and Kn<1/ε (3) hI n<0and (an−1)/Knan<|hI n|≤(an−1)/an (ε+K−1 n)requiring that hn<0 (iv) Te extinction equilibrium point y0�0is unstable. Te two positive equilibrium points y1,2in (5) arising 4Discrete Dynamics in Nature and Society when the parametrical sequences an 􏼈 􏼉∞ n�0⟶a, Kn 􏼈 􏼉∞ n�0⟶K,hn 􏼈 􏼉∞ n�0⟶hfulfl the following theories: (a) y2(>y1)is jointly feasible and locally asymptotically stable if −∞<h<(( �� a √−1)/(�� a √+ 1))K (b) If y1(<y2)is feasible then it is not locally asymptotically stable (c) If h� (( �� a √−1)/(�� a √+1))Kthen the unique nonextinction equilibrium point y1�y2� (K −h)/2�K/(�� a √+1)is jointly feasible and locally asymptotically stable Proof. One obtains (7) directly from the following equivalent expression to (1): xn+1�1 yn+1�1 anKnyn/Kn+an−1  􏼁yn  􏼁−hn�Kn+an−1  􏼁yn anKnyn−hnKn+an−1  􏼁yn  􏼁 �a−1 nxn−a−1 n−1 􏼐 􏼑K−1 n+hI n;∀n∈Z0+, (12) subject to hn∈(− ∞, anKnyn/(Kn+(an−1)yn)];∀n∈Z0+ for keeping the sample-to-sample non-negativity of yn 􏼈 􏼉∞ n�0, where xn�y−1 n;∀n∈Z0+, and hI n�xn+1−a−1 nxn+a−1 n−1 􏼐 􏼑K−1 n�Knxn+an−1 anKn−gn−a−1 nxn+a−1 n−1 􏼐 􏼑K−1 n;∀n∈Z0+,(13) with gn�hn(Knxn+an−1);∀n∈Z0+. One gets from (13) that K−1 n+hI n−a−1 nK−1 n−xn 􏼐 􏼑􏼐 􏼑gn�anKnhI n;∀n∈Z0+, (14) which leads to gn�anKnhI n K−1 n+hI n−a−1 nK−1 n−xn 􏼐 􏼑 �hnKnxn+an−1  􏼁;∀n∈Z0+, (15) so that hn�anKnhI n K−1 n+hI n−a−1 nK−1 n−xn 􏼐 􏼑􏼐 􏼑 Knxn+an−1  􏼁 �anKnhI n 2xn+K−1 nan+a−1 n−2 􏼐 􏼑+hI nKnxn+an−1  􏼁−2a−1 nxn+a−1 nKnx2 n �anKnhI n 2 1 −a−1 n 􏼐 􏼑+hI n+a−1 nxn 􏼐 􏼑Kn 􏼐 􏼑xn+K−1 nan+a−1 n−2 􏼐 􏼑+hI nan−1  􏼁;∀n∈Z0+, (16) which leads directly to (8). Equation (9) follows directly from recursive calculations with (7). Property (i) has been proved. Property (ii) is a direct consequence of Property (i) since 􏽑n+1 i�0[μi]<􏽑n i�0[μi]<1; ∀n∈Z0+and limn⟶ ∞􏽑n i�0 [μi] � 0 since lim infn⟶∞an≥a>1. To prove Property (iii), we rewrite an upper-bounding expression of (7) as xn+1�1−bn  􏼁xn+bn Kn+hI n≤1−bn  􏼁xn+bn Kn+hI n 􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌≤1−bn  􏼁xn+εnbn;∀n∈Z0+,(17) Discrete Dynamics in Nature and Society 5 since an 􏼈 􏼉∞ n�0⊂[1,∞]implies that bn�1−μn� (an−1)/ an∈[0,1];∀n∈Z0+subject to 􏽐∞ n�0bn�􏽐∞ n�0(an−1)/an� ∞. Since it is also assumed that |hI n+bn/Kn|≤εbn� ε(an−1)/an;∀n∈Z0+, this constraint is achieved if |hI n+bn/Kn| � |hI n+(an−1)/Knan|≤εnbn≤ε(an−1)/an;∀n (≥n0)∈Z0+, and some n0∈Z0+, where εn 􏼈 􏼉∞ n�0⊂[0,ε]for any integer n≥n0, some n0∈Z0+and some ε∈R+then it follows from [Lemma 1.2 (i), [24]] that 0 ≤lim sup n⟶∞ xn≤ε. Te stipulation 1 is got from the constraints hI n≥0 (implying that hn≥0) and then |hI n+(an−1)/Knan| � |hI n+(an−1)/ Knan|≤ε(an−1)/an,∀n∈Z0+. Te stipulation 2 follows for hI n<0 (implying that hn<0) and |hi n|≤bnK−1 nso that the subsequent constraint holds |hI n+(an−1)/Knan| � (an−1)/Knan− |hI n|;∀n∈Z0+. Te stipulation 3 follows for hI n<0 and |hi n|≤bnK−1 nso that the following constraint holds |hI n+(an−1)/Knan| � |hI n|− (an−1)/Knan;∀n∈Z0+. Property (iii) has been proved. Property (iv) follows from a local perturbation analysis. It is assumed that the invariant equation (1) perturbed from any equilibrium point yas yn�y+δyn;∀n∈Z0+. Te linearized perturbation transmitted to the next sample is y+δyn+1�aK/K/((y+ δyn)+a−1)− h;∀n∈Z0+which leads to |δyn+1|≤K2a/ (K+y(a−1))2|δyn|+o(|δyn|);∀n∈Z0+and, for sufciently small |δyn|,|δyn+1/δyn|<1, ∀n∈Z0+, so that yis locally asymptotically stable, if and only if K�� a √/(K+ y(a−1))<1. Equivalently, if and only if y>K(�� a √ −1)/(a−1) � K/(�� a √+1). Contrarily, the equilibrium point yis not locally asymptotically stable (that is, either critically stable or unstable) if and only if K�� a √/(K+y(a−1))≥1, that is, if and only if y≤K/(�� a √+1). In particular, it is unstable if K�� a √/(K+y(a−1))>1, that is, if y<K/(�� a √+1). Te local asymptotic stability constraint fails and the instability constraint holds if y�y0�0 (extinction equilibrium point) since a>1. Ten, the extinction equilibrium point is unstable. For addressing the local asymptotic stability of the other two equilibrium points y�y1and y�y2, provided they are feasible and distinct (that is, the radicand of (5) is real positive), note that equilibrium points are locally asymptotically stable if and only if y1,2�(a−1)(K−h)∓������������������������� (a−1)2(K−h)2−4(a−1)Kh 􏽱 2(a−1)>K �� a √+1.(18) Considering y2under the above constraint, that one becomes equivalent to ������������������������� (a−1)2(K−h)2−4(a−1)Kh 􏽱 >(�� a √−1)[2K− ( �� a √+1)(K−h)].(19) Te above-given constraint (19) holds if u>v, where u�(a−1) (a−1)(K−h)2−4Kh 􏽨 􏽩 �(a−1)a(K−h)2− (K+h)2 􏽨 􏽩 �α1h2+β1h+c1, (20) where α1�(a−1)2β1� − 2K a2−1 􏼐 􏼑;c1�(a−1)2K2,(21) and v�(( �� a √−1)[2K− ( �� a √+1)(K−h)])2 �(a+1−2�� a √)4K2+(a+1+2�� a √)K2+h2−2Kh 􏼐 􏼑−4K(�� a √+1)(K−h) 􏼐 􏼑 �α2h2+β2h+c2, (22) where α2�α1�(a−1)2;β2�2K(a−1)[2�� a √−a−1];c2�K2a2+6a−4a�� a √−4�� a √+1 􏼐 􏼑.(23) 6Discrete Dynamics in Nature and Society Tus, u>v, as necessary condition a≥((K+h)/(K− h))2, equivalently, h≤(( �� a √−1)/(�� a √+1))Kin order for the radicand in the defnition of y1,2to be non-negative (equilibrium feasibility condition), with a>((K+h)/ (K−h))2if h≠K(�� a √−1)/(�� a √+1)and h≠0. For h�0, y2�Kis locally asymptotically stable from u>vsince �� a √+ 1>�� a √−1 holds trivially. Now (20)–(23), one gets that y2is locally asymptotically stable, that is, u>v, if and only if: h<c1−c2 β2−β1�(a−1)2−a2+6a−4a�� a √−4�� a √+1 􏼐 􏼑 2(a−1)[2�� a √−a−1]+2a2−1 􏼐 􏼑 K�(a+1)�� a √−2a (a−1)�� a √K��� a √−1 �� a √+1K, (24) which coincides with the feasibility condition. Tus, y2is both feasible and locally asymptotically stable if and only if −∞<h<(( �� a √−1)/(�� a √+1))K. For the local asymptotic stability of y1, equation (19) becomes modifed as follows: −������������������������� (a−1)2(K−h)2−4(a−1)Kh 􏽱 >(�� a √−1)[2K− ( �� a √+1)(K−h)],(25) so that, one gets the two associated constraints: (1) (�� a √−1)[2K− ( �� a √+1)(K−h)]≤0 (2) �� u √�������������������������� (a−1)2(K−h)2−4(a−1)Kh 􏽱<� v √� − (�� a √−1)[2K− ( �� a √+1)(K−h)]⇔u<v Te above-given frst condition is equivalent to h≤(( �� a √−1)/(�� a √+1))Ksince a>1. Tis constraint is the feasibility constraint for harvesting also already needed for y →2. Te above-given second condition is equivalent to just to reverse the equality in the constraint (24), that is, h>(a+1)�� a √−2a (a−1)�� a √K��� a √−1 �� a √+1K, (26) so that y1≠y2is both feasible and locally asymptotically stable if and only if �� a √−1 �� a √+1K≥h>�� a √−1 �� a √+1K, (27) which is a contradiction. Tus, y1is unstable if feasible and distinct of y2. If y1�y2is feasible, that is, the radicand of (5) is null so that h� (( �� a √−1)/(�� a √+1))K, then the equilibrium point is given by y1�y2� (K−h)/2 �K/(�� a √+1)which satisfes trivially the above given local asymptotic stability condition y1�y2≥K/(�� a √+1)so that the confuent nonextinction equilibrium point y1�y2� (K−h)/2 �K/(�� a √+1)resulting with h� (( �� a √−1)/(�� a √+1))Kis locally asymptotically stable. Property (iv) has been proved. Concerning Teorem 2(i), note that the denominator in the right-hand-side of (8) cannot be zero at any sample since the the value of the sequence hn 􏼈 􏼉∞ n�0is bounded by hypothesis. □ Remark 1. Note that, the admissible harvesting sequence of Teorem 2(iii) can be generated from (8) by generating hI n 􏽮 􏽯∞ n�0as follows by fulflling one of the stipulations 1–3 for each n∈Z0+: (a) Trough the stipulation 1 in the proof of Teorem 2: hI n� (ε−K−1 n)(an−1)/an−σn≥0; ∀n∈Z0+, where ε∈R+is chosen such that Kn≥ε−1;∀n∈Z0+and the sequence σn 􏼈 􏼉∞ n�0is generated subject to 0 ≤σn ≤(ε−K−1 n)(an−1)/an;∀n∈Z0+. Note that, hI n≥0 and hn≥0; ∀n∈Z0+. (b) Trough the stipulation 2 in the proof of Teorem 2: hI n� (1−an)/anKn+σn, where 0 ≤σn≤((an−1) /an) (2K−1 n−ε);∀n∈Z0+and ε∈R+is chosen such that Kn<ε−1. Note that, hI n<0 and hn<0. (c) Trough the stipulation 3 in the proof of Teorem 2: hI n� ((1−an)/an)(1/Kn+ε)+σn, where 0≤σn≤((an−1)/an)ε;∀n∈Z0+and ε∈R+. Note that, hI n<0 and hn<0. Remark 2. Te local asymptotic stability of the equilibrium points addressed in Teorem 2 (iv) relies to the cases of absence of harvesting in the steady state dynamics (h�0) or in the cases of stationary fshing/hunting (h>0)or stationary repopulation actions (h<0). Tose cases correspond to constant values of the harvesting sequence in fnite time or asymptotically. In the paper, the dynamics is globally stable if the population solution sequence is bounded for any given fnite initial condition. Tis circumstance might be compatible with the event that some of the equilibrium points be locally unstable, stable, or critically stable if there are more than one equilibrium points. An equilibrium point is said to be globally asymptotically stable if it globally stable and all solution converges asymptotically to such a point for any given fnite initial conditions. In the third case, the largest positive equilibrium point is larger under negative stationary harvesting (having a meaning of stationary repopulation and/or immigration to the habitat from outside), than the equilibrium point K arising in the absence of harvesting. In the second case, the global stability condition leads to the conclusion that the larger equilibrium point y2is locally asymptotically stable and the smaller one y1is not locally asymptotically stable unless they are coincident for a stationary harvesting efort h� (( �� a √−1)/(�� a √+1))K. Discrete Dynamics in Nature and Society 7 Te following result proves that, if the harvesting action sequence has a limit h, then a limit point of the solution cannot exceed the amount K−h. Proposition 1. If an 􏼈 􏼉∞ 0⊂(1, a)⟶a,Kn 􏼈 􏼉∞ 0(⊂[0, K]) ⟶K,hn 􏼈 􏼉∞ n�0⟶hand yn 􏼈 􏼉∞ n�0⟶ythen h�0⟺(y�0)∨(y�K) h>0⟺y<K−h h<0⟺y>K+|h|.(28) Proof. On gets from (1) that hn�anKnyn Kn+an−1  􏼁yn−yn+1;∀n∈Z0+.(29) If yn 􏼈 􏼉∞ n�0⟶y, one gets by taking limits in (29) as n⟶ ∞ that h�aK K+(a−1)y−1 􏼠 􏼡y�(a−1)(K−y) K+(a−1)yy, (30) and, equivalently, hK �(a−1)(K−y−h)y, (31) |h|K� − hK �(a−1)(y− |h|− K)if h<0.(32) Ten, the given properties follow directly from (31) and . Te following result establishes the boundedness of the solution sequence under bounded non-negative harvesting. □ Proposition 2. Assume that an 􏼈 􏼉∞ 0⊂[1, a],Kn 􏼈 􏼉∞ 0⊂[0, K], hn 􏼈 􏼉∞ n�0(⊂R0+)If hn 􏼈 􏼉∞ n�0⊂([0, anKnyn/(Kn+(an− 1)yn)]);∀n∈Z0+then yn 􏼈 􏼉∞ 0(⊂R0+)is bounded. Proof. Assume on the contrary that yn 􏼈 􏼉∞ n�0⟶+∞.Ten, from L´Hopital rule for quotients with numerator and denominator tending to infnity, since the harvesting sequence is non-negative, lim yn⟶ ∞n⟶ ∞ sup yn+1−anKnyn Kn+an−1  􏼁yn 􏼠 􏼡 �lim n⟶ ∞ sup yn+1−anKn an−1 􏼠 􏼡≤0, (33) so that lim sup n⟶∞ yn+1�lim n⟶ ∞ yn+1≤lim inf n⟶∞ anKn an−1<+∞. (34) A contradiction for the sequence yn 􏼈 􏼉∞ n�0to diverge, which completes the proof. □ 3. Control Laws for Monitoring the Harvesting Action Te frst part of this section is addressed to derive harvesting control laws based on Teorem 2(iv) guaranteeing the convergence to a prescribed equilibrium point x∗under the assumption 􏽐∞ n�0(an−1)/an�∞on the intrinsic growth sequence an 􏼈 􏼉∞ n�0. Known criteria for absolute convergence of series or for convergence of series of non-negative elements to a prescribed limit can be used to calculate the harvesting control sequence hn 􏼈 􏼉∞ n�0based on the previous calculation of hI n 􏽮 􏽯∞ n�0, which refects the harvesting efect in the inverse of the solution sequence, so as to satisfy Teorem 2(iv). Te last part of the section proposes harvesting control laws which make the solution sequence of the population evolution to sample-to-sample, rather than asymptotically, behave according to a prescribed suitable pattern. Now, rewrite the population solution sequence xn�y−1 n 􏼈 􏼉∞ n�0as an equilibrium perturbation in the form 􏽥 xn+x∗ 􏼈 􏼉∞ n�0, where x∗is the suited equilibrium point and 􏽥 xn�xn−x∗;∀n∈Z0+. Tus, one can rewrite from (6) the one-step ahead evolution of the incremental sequence 􏽥 xnin the form of Lemma 1.2 (iii) of [24] as follows: 􏽥 xn+1�1−bn  􏼁􏽥 xn−bnx∗+]n+hI n �1−bn  􏼁􏽥 xn+ωn+cn;∀n∈Z0+,(35) where for each n∈Z0+, bn�1−μn�1−a−1 n�an−1 an ,(36) ωn�]n+βn�an−1 anKn+βn,(37) cn�hI n−bnx∗−βn,(38) for any βn 􏼈 􏼉∞ 0⊂Rchosen such that hI n−bnx∗≥ βn≥(1−an)/anKn;∀n∈Z0+which guarantees that ωn≥0 and cn≥0; ∀n∈Z0+. Note that, bn∈[0,1];∀n∈Z0+. Suffcient conditions for convergence 􏽥 xn 􏼈 􏼉∞ 0(⊂R0+)⟶0⟺ xn 􏼈 􏼉∞ 0⟶x∗are 􏽥x0≥0, and 􏽘∞ n�0bn�􏽘∞ n�0 an−1 an�∞,(39) ωn�o bn  􏼁⇔lim n⟶ ∞ βn an−1+1 anKn 􏼠 􏼡�0,(40) 􏽘∞ n�0cn<∞.(41) Te condition (40) is equivalent to limn⟶ ∞(βn− (1− an)/anKn) � 0 and the condition (41) is guaranteed if 8Discrete Dynamics in Nature and Society 􏽘∞ n�0hI n+1−an an x∗−K−1 n 􏼐 􏼑􏼠 􏼡<∞,(42) since, from (38) and the condition hI n−bnx∗≥ βn≥((1−an)/an)Kn;∀n∈Z0+, one gets: 􏽘 ∞ n�0 cn�􏽘 ∞ n�0 hI n−bnx∗−βn 􏼐 􏼑 ≤􏽘 ∞ n�0 hI n−an−1 an x∗+an−1 anKn 􏼠 􏼡 �􏽘 ∞ n�0 hI n+1−an an x∗−K−1 n 􏼐 􏼑􏼠 􏼡<∞, (43) which induces also the further necessary condition lim n⟶ ∞(hI n+((1−an)/an)(x∗−K−1 n)) � 0, since cn⟶0 as n⟶ ∞, guaranteed in turn if hI n 􏽮 􏽯∞ n�0⟶0 (equivalently, if hn 􏼈 􏼉∞ n�0⟶0) and K−1 n 􏼈 􏼉∞ n�0⟶x∗, equivalently, if Kn 􏼈 􏼉∞ n�0⟶y∗, or if hI n 􏽮 􏽯∞ n�0⟶0 and an 􏼈 􏼉∞ n�0⟶1 3.1. Harvesting Control Law Based on d´ Alembert Convergence Criterion. a) 􏽐∞ n�0cn<∞ with cn≥0; ∀n∈Z0+is guaranteed under d´ Alembert convergence criterion in order to xn 􏼈 􏼉∞ n�0⟶x∗�1/y∗>0, equivalently, yn 􏼈 􏼉∞ n�0⟶y∗>0, if hI n+1−bn+1x∗−βn+1 hI n−bnx∗−βn�cn≤c<1; ∀n∈Z0+,(44) if hI n≠bnx∗+βnand hI n≥bnx∗+βn;∀n∈Z0+. Ten, since bn� (an−1)/an;∀n∈Z0+, equation (44) leads to bn+1x∗+βn+1�an+1−1 an+1 x∗+βn+1 ≤hI n+1�bn+1x∗+βn+1+cnhI n−bnx∗−βn 􏼐 􏼑 �an+1−1 an+1−cn an−1 an 􏼠 􏼡x∗+βn+1+cnhI n−βn 􏼐 􏼑;∀n∈Z0+, (45) with limn⟶ ∞(βn− ((1−an)/anKn) � 0. One gets from (12), by using xn�y−1 n;∀n∈Z0+, that yn+1�x−1 n+1�anKnx−1 n−hnKn+an−1  􏼁x−1 n 􏽨 􏽩 Kn+an−1  􏼁x−1 n�anKn−hnKnxn+an−1 􏼂 􏼃 Knxn+an−1;∀n∈Z0+,(46) so that, equivalently, xn+1�Knxn+an−1 anKn−hnKnxn+an−1 􏼂 􏼃;∀n∈Z0+,(47) which equalized to (7) in Teorem 2 (i) leads to: hI n�hI nhn  􏼁�Knxn+an−1 anKn−hnKnxn+an−1  􏼁 +1 an 1−an Kn−xn 􏼠 􏼡;∀n∈Z0+. (48) Ten, combining (45) and (48) hI n+1�an+1−1 an+1−cn an−1 an 􏼠 􏼡x∗+cn Knxn+an−1 anKn−hnKnxn+an−1  􏼁+1 an 1−an Kn−xn 􏼠 􏼡􏼢 􏼣+βn+1−cnβn;∀n∈Z0+.(49) Equivalently, Discrete Dynamics in Nature and Society 9 In the same way, in order for y2to be locally asymptotically stable, |2−a− (1−a)/Ky2|<awhich is fulflled under two conditions, namely, (c) 0 ≤2−a− ((1−a)/K)y2<awhich holds if and only if y2∈(2K,(2−a)/(1−a)K]; (d) 0 ≤((1−a)/K)y2− (2−a)<awhich holds if and only if y2∈[(2−a)K/(1−a),2K/(1−a)) which are equivalently combined into the local stability condition y2∈(2K, 2K/1 −a). Taking into account that y2�1−a/c(1+����������� 1−4cK/1 −a √), the condition y2∈[(2− a)K/(1−a),2K/(1−a)) becomes equivalent to 2c(2−a) (1−a)2−1 􏼠 􏼡2≤1−4cK 1−a<4Kc (1−a)2−1 􏼠 􏼡2 ,(87) subject to 0 ≤c<(1−a)/4Kand defne parameters λc�c/a>0 (note that λc�0 does not need to be considered since then y2�y1�Kis stable) and λK�K/a>0. Ten, the constraint c<(1−a)/4Ktakes the form 4λcλKa2+a−1<0 and the above local stability constraint (87) of y2takes the form (76). □ Remark 4. Note from Teorem 4 (ii) that any nonzero equilibrium point is less than the carrying capacity if c>0 contrarily to the case when c�0 where y�Kis an equilibrium point. Remark 5. Note that, contrarily to Teorem 4(iv), if a≥1and c�0, then, in the absence of harvesting, yn 􏼈 􏼉∞ n�0cannot be strictly decreasing converging to zero since, for the limit solution sequence to be strictly decreasing, it is necessary that (yn+1/yn) � a/(1+((a−1)/K)yn)<1 what implies that yn>Kso that yn 􏼈 􏼉∞ n�0⟶0 is impossible for any given positive fnite initial condition if an≥1; ∀n∈Z0+. Tus, the extinction equilibrium point is unstable if an≥1; ∀n∈Z0+. 6. Numerical Simulations 6.1. Example 1. Tis example illustrates the results of Teorem 1. Te sequences an 􏼈 􏼉∞ 0,Kn 􏼈 􏼉∞ 0and zn 􏼈 􏼉∞ 0are respectively generated by means of the following diference equationsc: an+1�ε1an+ρ1, Kn+1�ε2Kn+ρ2, zn+1�ε3zn+ρ3,(88) with the following values for the parameters: ε1�0.9,ρ1�0.4,ε2�0.8,ρ2�200,ε3�0.75 and ρ3�100,(89) and the following initial conditions: a0�1.5, K0�500 and z0�10,(90) In this way the conditions of Teorem 1 about the sequences an 􏼈 􏼉∞ 0and Kn 􏼈 􏼉∞ 0are fulflled since a�limn⟶ ∞ an 􏼈 􏼉�ρ1/(1−ε1) � 4 and K�limn⟶ ∞ Kn 􏼈 􏼉�ρ2/(1−ε2) �1000. In a frst simulation, the harvesting sequence hn 􏼈 􏼉∞ 0with hn�5∗(0.7)nis considered so that h�limn⟶ ∞ hn 􏼈 􏼉�0 and then the conditions of Teorem 1(i) are satisfed. Figure 1 shows the evolution of the species population yn 􏼈 􏼉and that of the environment carrying capacity Kn 􏼈 􏼉 if the population is initially y0�200. One can see that y�limn⟶ ∞ yn 􏼈 􏼉�limn⟶ ∞ Kn 􏼈 􏼉�K�1000 as Teorem 1(i) points out. On the other hand, Figure 2 displays the evolution of the species population and that of the environment carrying capacity if the population is initially y0�10. Again, one can see that y�limn⟶ ∞ yn 􏼈 􏼉�limn⟶ ∞ Kn 􏼈 􏼉�K�1000 although the initial population is close to the equilibrium point y�0. Te results displayed in Figures 1 and 2 illustrate the fact that the equilibrium point y�0 is globally asymptotically unstable while the equilibrium point y�Kis globally asymptotically stable. In a second simulation the same values of (89) and (90) are maintained but the sequence hn 􏼈 􏼉∞ 0converges to a nonzero value h. Concretely, the time evolution of hn 􏼈 􏼉∞ 0is displayed in Figure 3 while that of yn 􏼈 􏼉and Kn 􏼈 􏼉if the population is initially y0�145 is shown in Figure 4. In this example the fact that 0≠h�limn⟶ ∞ hn 􏼈 􏼉<limn⟶ ∞ Kn 􏼈 􏼉�Kis observed so that the conditions of Teorem 1(ii) and Teorem 1(iii) are satisfed. In fact, the Beverton–Holt equation (1) possesses two equilibrium points given by (1). Concretely, such equilibrium points are y1�142.855 and y2�600. One can see that limn⟶ ∞ yn 􏼈 􏼉�y2�600 although the initial condition y0�145 is close to the equilibrium point y1. Such a fact illustrates that y1is unstable while y2is globally asymptotically stable in the sense that all solutions generated by fnite initial conditions converge to such an equilibrium. 6.2. Example 2. Tis example illustrates the results of Teorem 3 related with the Allee efect for small number of individuals in the species. Te intrinsic growth rate and carrying capacity sequences, an 􏼈 􏼉∞ 0and Kn 􏼈 􏼉∞ 0, are given by (88) with the same values for the parameters ε1,ε2,ρ1, and ρ2 than those pointed out in (89). Moreover, the harvesting sequence hn 􏼈 􏼉∞ 0is zero for all nεZ0+and the function f(yn) appearing in the modifed Beverton-Holt equation (58) is given by the following equation: f yn  􏼁�yn+α1 anyn+α2 ,(91) with α1�0.1 and α2�1. In this way, f(yn)<a−1 nfor all nεZ0+and the conditions of Teorem 3(i) are fulflled. Figure 5 shows the evolution of the species population yn 􏼈 􏼉if the population is initially y0�15. Figure 6 displays the evolution of the function f(yn)and the inverse of the sequence an 􏼈 􏼉∞ 0. In Figure 5, one can see that the species population converges to the extinction as Teorem 3(i) establishes since f(yn)<a−1 nfor all nεZ0+as it is shown in Figure 6. 16 Discrete Dynamics in Nature and Society Now, the function f(yn)appearing in the modifed Beverton–Holt equation (58) is given by the following equation: f yn  􏼁�αnyp n+βn,(92)with αn� (0.7(an−1)/anKn)y1−p nand βn�0.8a−1 nfor all nεZ0+. In this way, the conditions of Teorem 3(ii) are fulflled. Figure 7 shows the evolution of the species 5 1015202530354045500 n (days) 0 200 400 600 800 1000 yn Kn Figure 1: Time evolution of the species population and that of the environment carrying capacity if y0�200. 5 1015202530354045500 n (days) yn Kn 0 200 400 600 800 1000 Figure 2: Time evolution of the species population and that of the environment carrying capacity if y0�10. 5 1015202530354045500 n (days) -200 -100 0 100 200 300 hn Figure 3: Time evolution of the harvesting sequence. 10 20 30 40 500 n (days) 0 200 400 600 800 1000 yn Kn Figure 4: Evolution of the species population and that of the environment carrying capacity if y0�145. yn 0 2 4 6 8 10 12 14 16 10 20 30 40 50 60 700 n (days) Figure 5: Time evolution of the species population if y0�15 and f(yn)<a−1 n. 10 20 30 40 50 60 700 n (days) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1/an f (yn) Figure 6: Time evolution of the inverse of an 􏼈 􏼉∞ 0and f(yn)if y0�15. Discrete Dynamics in Nature and Society 17 population yn 􏼈 􏼉if the population is initially y0�15. One can see that the species population converges to the extinction as Teorem 3(ii) establishes. 6.3. Example 3. Te following two examples illustrate the results of Proposition 5 about the modifed Beverton–Holt equation (63). (i) Te sequences an 􏼈 􏼉∞ 0,Kn 􏼈 􏼉∞ 0and cn 􏼈 􏼉∞ 0are respectively defned as an�a01+0.5 sin 2πfan  􏼁􏼂 􏼃,(93) Kn�K01+0.02 sin 2πfKn  􏼁􏼂 􏼃,(94) cn�c01+0.3 sin 2πfcn  􏼁􏼂 􏼃,(95) to illustrated the result (i) of such a proposition with a0�1.8, fa�0.01, K0�500, fK�0.02, c0�0.03 and fc�0.03. Figure 8 shows the evolution of the species population yn 􏼈 􏼉if the population is initially y0�100. One can see that the species population neither extinguishes nor increases in an unboundedness way as Proposition 5 (i) establishes. (ii) Te sequences an 􏼈 􏼉∞ 0,Kn 􏼈 􏼉∞ 0, and cn 􏼈 􏼉∞ 0are, respectively, generated by means of the following diference equations: an+1�ε1an+ρ1, Kn+1�ε2Kn+ρ2, cn+1�ε3cn+ρ3,(96) with the following values for the parameters: ε1�09,ρ1�0.4,ε2�0.8,ρ2�200, ε3�0.75 and ρ3�0.005,(97) and the following initial conditions: a0�1.5, K0�500 and c0�0.01.(98) In this way the conditions of Proposition 5(ii) about the sequences an 􏼈 􏼉∞ 0,Kn 􏼈 􏼉∞ 0and cn 􏼈 􏼉∞ 0are fulflled since a�limn⟶ ∞ an 􏼈 􏼉�ρ1/(1−ε1) � 4, K�limn⟶ ∞ Kn 􏼈 􏼉� ρ2/(1−ε2) � 1000 and c�limn⟶ ∞ cn 􏼈 􏼉� ρ3/(1−ε3) � 0.02. Figure 9 shows the evolution of the species population yn 􏼈 􏼉and that of the environment carrying capacity Kn 􏼈 􏼉if the population is initially y0�200. One can see that y�limn⟶ ∞ yn 􏼈 􏼉� ((a−1)/2c)( ������������� 1+4Kc/(a−1) 􏽰−1)≈319 < min(K, ��������� K(a−1)/c 􏽰) � min(1000,��������� K(a−1)/c 􏽰)≈387 as Proposition 1(ii) points out. On the other hand, Figure 10 displays the evolution of the species population and that of the environment carrying capacity if the population is initially y0�10. Again, one can see that y�limn⟶ ∞ yn 􏼈 􏼉≈319 although the initial population is close to the equilibrium point y�0. Te results displayed in Figures 9 and 10 illustrate the fact that the equilibrium point y�0is globally asymptotically unstable while the equilibrium point y�Kis globally asymptotically stable for c>0. 6.4. Example 4. Te following examples illustrate the results of Section 5 about the modifed Beverton–Holt equation (63). Te sequences an 􏼈 􏼉∞ 0and Kn 􏼈 􏼉∞ 0are, respectively, generated by means of the following diference equations: an+1�ε1an+ρ1, Kn+1�ε2Kn+ρ2,(99) with the following values for the parameters: ε1�0.8,ρ1�0.1,ε2�0.9 and ρ2�50,(100) and the following initial conditions: a0�1.5 and K0�300.(101) In this way the conditions of Section 5 about the sequences an 􏼈 􏼉∞ 0and Kn 􏼈 􏼉∞ 0are fulflled since a�limn⟶ ∞ an 􏼈 􏼉�ρ1/(1−ε1) � 0.5 and K�limn⟶ ∞ Kn 􏼈 􏼉�ρ2/(1− ε2) � 500. Several choices for the sequence cn 􏼈 􏼉∞ 0are considered to illustrate the results of Section 5: (i) In the frst case cn 􏼈 􏼉∞ 0is given by the diference equation: cn+1�ε3cn+ρ3,(102) with the values for the parameters ε3�0.75 and ρ3�0 and the initial condition c0�0.01. In this way, c�limn⟶ ∞ cn 􏼈 􏼉�ρ3/(1−ε3) � 0 and then, the conditions that a<1, K>0 and c�0 are fulflled so that the modifed Beverton–Holt equation has two equilibrium points, namely, the stable point y1�0 (extinction) and the unstable one y2�K. Figure 11 shows the evolution of the species population yn 􏼈 􏼉and that of the environment carrying capacity Kn 􏼈 􏼉if the population is initially y0�250. 5 101520253035400 n (days) 0 5 10 15 y n Figure 7: Time evolution of the species population if y0�15 and f(yn) � αnyp n+βn. 18 Discrete Dynamics in Nature and Society Te results displayed in Figure 11 illustrates the fact that the equilibrium point y1�0 is globally asymptotically stable while the equilibrium point y2� 500 is unstable for c�0 as Section 5 points out. (ii) In the second case cn 􏼈 􏼉∞ 0is given by (102) with the values for the parameters ε3�0.75 and ρ3� (1− ε3)(1−a)/4K�6.25x10−5and the initial condition c0�0.01. In this way, c�limn⟶ ∞ cn 􏼈 􏼉�ρ3/(1− ε3) � 2.5x10−4and then, the conditions that a<1, K>0 and c� (1−a)/4Kare fulflled so that the modifed Beverton–Holt equation has two equilibrium points, namely, the stable point y1�0 (extinction) and the unstable one y2� (1−a)/2c�1000. Figure 12 shows the evolution of the species population yn 􏼈 􏼉and that of the environment carrying capacity Kn 􏼈 􏼉 if the population is initially y0�1050. Te results displayed in Figure 12 illustrate the fact that the equilibrium point y1�0 is globally asymptotically stable while the equilibrium point y2�1000 is unstable as Teorem 4 (i) establishes. (iii) In the third case cn 􏼈 􏼉∞ 0is given by (102) with the values for the parameters ε3�0.75 and ρ3�0.001 >(1−ε3)(1−a)/4K�6.25x10−5and the initial condition c0�0.01. In this way, c�limn⟶ ∞ cn 􏼈 􏼉�ρ3/(1−ε3) � 0.004 and then, the conditions that a<1, K>0 and c>(1−a)/4K� 2.5x10−4are fulflled so that the modifed Beverton–Holt equation has only one equilibrium point, namely, the stable point y1�0 (extinction). Figure 13 shows the evolution of the species population yn 􏼈 􏼉and that of the environment carrying capacity Kn 􏼈 􏼉if the population is initially y0�250. Te results displayed in Figure 13 illustrate the fact that the unique equilibrium point y1�0 is globally y n 0 50 100 150 200 50 100 150 200 250 3000 n (days) Figure 8: Time evolution of the species population if y0�100 and the species evolution is given by (64). 0 200 400 600 800 1000 10 20 30 40 50 60 700 n (days) yn Kn Figure 9: Time evolution of the species population and that of the environment carrying capacity if y0�200 and the species evolution is given by (63). 0 200 400 600 800 1000 10 20 30 40 50 60 700 n (days) yn Kn Figure 10: Time evolution of the species population and that of the environment carrying capacity if y0�10 and the species evolution is given by (63). 0 100 200 300 400 500 10 20 30 40 50 60 700 n (days) yn Kn Figure 11: Time evolution of the species population and that of the environment carrying capacity if y0�250 and the species evolution is given by (64) with a<1, K>0, and c�0. Discrete Dynamics in Nature and Society 19 asymptotically stable, that is asymptotically stable for any fnite initial condition, as Teorem 4 (ii) establishes. (iv) In the fourth case cn 􏼈 􏼉∞ 0is given by (102) with the values for the parameters ε3�0.75 and ρ3�3x10−5<(1−ε3)(1−a)/4K�6.25x10−5and the initial condition c0�0.01. In this way, c�limn⟶ ∞ cn 􏼈 􏼉�ρ3/(1−ε3) � 1.2x10−4and then, the conditions that a<1, K>0 and 0<c<(1−a)/4K�2.5x10−4are fulflled so that the modifed Beverton–Holt equation has three equilibrium points, namely, the stable point y1�0 (extinction) and the unstable ones y2� (1−a− ������������������ (1−a)2−4ck(1−a) 􏽱)/2c≈581 >K�500 and y3 � (1−a−������������������ (1−a)2−4ck(1−a) 􏽱)/2c≈3586 < (1−a)/c≈4167. Figure 14 shows the evolution of the species population yn 􏼈 􏼉and that of the environment carrying capacity Kn 􏼈 􏼉if the population is initially y0�700. Te results displayed in Figure 14 illustrate the fact that the equilibrium point y1�0 is globally asymptotically stable while the other ones are unstable as Teorem 4 (ii) establishes. 7. Conclusions Tis paper has discussed a generalized time-varying Beverton–Holt equation which considers the presence of positive or negative harvesting and, eventually, a quadratic-type penalty for the population excess. Such a term takes account for the potential internal competence between the cohort individuals for food, refuge, etc. Te harvesting action (describing hunting/fshing actions) is considered jointly with eventually present independent consumption (describing migrations from outside of the habitat to inside or vice-versa). It is seen that the presence of the penalty term can translate into the presence of two other positive equilibrium points. Some particular stability results have been also derived for the stationary equation, which arises when its parameterizing sequences converge, for the case of small levels of population by introducing a term taking account for the Allee efect. Te paper has also designed some species evolution control laws by monitoring the harvesting action and has discussed the infuence in the stability results of considering a modelling function of Allee efect which makes difcult growing or even can cause extinction for small numbers of reproductive individuals. Te equilibrium points of the stationary solution in the presence and absence of harvesting action have been characterized and their local asymptotic stability properties have been investigated in the case of intrinsic growth rate exceeding unity and eventual execution of harvesting actions 0 100 200 300 400 500 600 700 10 20 30 40 50 60 700 n (days) yn Kn Figure 14: Time evolution of the species population and that of the environment carrying capacity if y0�700 and the species evolution is given by (63) with a<1, K>0, and 0 <c<(1−a)/4K. 10 20 30 40 50 60 700 n (days) 0 200 400 600 800 1000 1200 yn Kn Figure 12: Time evolution of the species population and that of the environment carrying capacity if y0�1050 and the species evolution is given by (63) with a<1, K>0, and c� (1−a)/4K. 10 20 30 40 50 60 700 n (days) yn Kn 0 100 200 300 400 500 Figure 13: Time evolution of the species population and that of the environment carrying capacity if y0�250 and the species evolution is given by (63) with a<1, K>0, and c>(1−a)/4K. 20 Discrete Dynamics in Nature and Society and in the case of the intrinsic growth rate being less than unity. 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