Dynamics and bifurcations of a family of piecewise smooth maps arising in population models with threshold harvesting

 View Online  Export Citation CrossMark RESEARCH ARTICLE | JULY 06 2020 Dynamics and bifurcations of a family of piecewise smooth maps arising in population models with threshold harvesting  Eduardo Liz  ; Cristina Lois-Prados Chaos 30, 073108 (2020) https://doi.org/10.1063/5.0010144 09 January 2024 11:29:54 Chaos ARTICLE scitation.org/journal/cha Dynamics and bifurcations of a family of piecewise smooth maps arising in population models with threshold harvesting Cite as: Chaos 30, 073108 (2020); doi: 10.1063/5.0010144 Submitted: 8 April 2020 ·Accepted: 17 June 2020 · Published Online: 6 July 2020 View Online Export Citation CrossMark Eduardo Liz1,a)and Cristina Lois-Prados2,b) AFFILIATIONS 1Departamento de Matemática Aplicada II, Universidade de Vigo, 36310 Vigo, Spain 2Instituto de Matemáticas, Universidade de Santiago de Compostela, Campus Vida, 15782 Santiago de Compostela, Spain a)Author to whom correspondence should be addressed: [email protected].es b)Electronic mail: [email protected] ABSTRACT We study a discrete-time model for a population subject to harvesting. A maximum annual catch His fixed, but a minimum biomass level T must remain after harvesting. This leads to a mathematical model governed by a continuous piecewise smooth map, whose dynamics depend on two relevant parameters Hand T. We combine analytical and numerical results to provide a comprehensive overview of the dynamics with special attention to discontinuity-induced (border-collision) bifurcations. We also discuss our findings in the context of harvest control rules. Published under license by AIP Publishing. https://doi.org/10.1063/5.0010144 In the context of sustainable exploitation of natural resources (fishing, forestry, hunting), it is crucial to reduce the exploitation rate at low stock size. Harvest policies aiming at protecting endangered species are based on reference points such as a maximum allowable catch or a minimum biomass level that must remain after harvesting. Discrete-time mathematical models for these harvesting strategies lead in a natural way to piecewise smooth maps, whose dynamics are challenging because multiple discontinuity-induced bifurcations may appear. In this paper, we provide a thorough analysis of a threshold harvesting model depending on two parameters: a maximum harvesting quota and a minimum population threshold with a twofold aim of understanding the interplay between both parameters and uncovering interesting features of the underlying piecewise smooth map. I. INTRODUCTION In the context of chaos control, some strategies based on thresholding or limiter controls have been proposed to stabilize a fixed point or a desired periodic orbit.8,15,32 The same mechanism determines effective harvest control rules to prevent population collapses in fisheries and forestry.10,26 A simple threshold mechanism to control a variable xis the following: choose a critical threshold Tin such a way that whenever the value of xexceeds T, it is reset to T.33 In the framework of sustainable harvesting, the same mechanism is known as threshold harvesting (TH); this harvest control rule removes the surplus of a population above a given threshold T(minimum biomass level) and takes no harvest below T. The recent paper (Ref. 18) includes a table collecting the many names of TH in different contexts. When TH is applied to a one-dimensional map f, the controlled system is governed by a flat-topped map F. The dynamics of Fis usually much simpler than the one of f, although it still has their own and interesting features (see, e.g., Refs. 32,36, and 39). In particular, Fis a non-smooth map, so some bifurcations different from the usual ones for smooth maps arise: the so-called border-collision bifurcations.2 Perhaps, the more relevant property for usual flat-topped maps is that almost all solutions converge to a periodic orbit containing T. Though this behavior can be seen as a nice property from a chaos control point of view, if TH is used as a harvesting rule, it opens the possibility of harvesting moratoria, with serious socioeconomic consequences. In the opposite side, a strategy of constant catch harvesting keeps the yield constant, rather than the escapement (biomass level after harvesting). It is well known that a serious drawback of Chaos 30, 073108 (2020); doi: 10.1063/5.0010144 30, 073108-1 Published under license by AIP Publishing. 09 January 2024 11:29:54 Chaos ARTICLE scitation.org/journal/cha FIG. 1. Different harvesting strategies illustrated by sketching catch (red solid lines) as a function of population density or biomass. (a) Threshold harvesting, (b) constant quota harvesting, and (c) precautionary threshold constant-catch harvesting. The blue dotted lines indicate the identity diagonal line, Tis the minimum biomass level, and His the maximum allowed catch. this method is that it highly elevates the risk of extinction, even leading to sudden collapses.26,30 Several combinations of the traditional harvesting strategies with protection rules based on thresholds have been proposed in order to minimize its drawbacks.10,12,13,17,26 In particular, a combination of threshold management and constant-catch harvesting was proposed by several authors.6,26,35 In the latter reference, this strategy was named “conditional fixed harvest.” The idea is simple: if population size is below a given threshold T, then no harvest is allowed; otherwise, we allow to catch a constant quota H but always ensuring that at least a minimum population stock Twill remain after harvesting. Thus, there are two reference breakpoints: the maximum allowed catch Hand the minimum biomass level T. In the spirit of Ref. 12, this harvesting rule can be seen as a precautionary modification of threshold constant-catch harvesting, and then we will refer to it as precautionary threshold constant-catch harvesting (PTCH for short). Figure 1 shows an illustration of the three mentioned harvesting policies: threshold harvesting, constant quota, and precautionary threshold constant-catch harvesting. Our aim in this paper is twofold: first, we combine analytical and numerical results to provide a rigorous mathematical analysis of PTCH using a discrete-time single-species model; second, we show that our model is a simple but insightful example of piecewise smooth dynamical system, complementing in this way the growing set of applications of this class of dynamical systems.2As far as we know, we formulate and study this harvesting rule in the framework of dynamical systems for the first time. Our study is focused on stability and bifurcations; in this regard, we consider the two relevant harvest parameters Tand Hand obtain 1-parameter and 2-parameter bifurcation diagrams that help one to understand how a continuous variation of any of them influence the dynamics, and the interplay between both parameters. We identify regions where global attraction, periodic attractors, bistability, or complex behavior are likely to occur, paying special attention at border-collision bifurcations. We organize the paper as follows: in Sec. II, we introduce the piecewise smooth one-dimensional map governing PTCH and the main assumptions on the involved functions. Section III is devoted to the study of fixed points: in Subsection III A, we give the location of positive equilibria depending on the relevant parameters; Subsection III B is devoted to provide stability results for the positive fixed points. In simple cases, all solutions converge to an equilibrium and in the general case, there are stability switches and the global picture is more complicated; we give some general results for global stability and study in more detail in the Ricker map, which is a prototype for discrete population models, especially in the context of fisheries;27,29,38 and finally, in Subsection III C, we describe all possible bifurcations of fixed points (smooth and border-collision bifurcations). In Sec. IV, we focus on a particular region of the parameter plane for which chaos and essential attraction can occur. The latter means that an equilibrium is not globally attracting, but solutions converge to it with probability one. In Sec. V, we address two case studies: a simple compensatory model, where only bifurcations of fixed points appear and an overcompensatory model that exhibits richer dynamics; in both cases, numerical bifurcation diagrams help one to understand the influence of the parameters. Finally, in Sec. VI, we discuss the obtained results in the framework of piecewise smooth dynamical systems (with special attention to those with maps containing flat branches) and in the framework of harvesting strategies. II. THE MODEL AND BASIC ASSUMPTIONS To carry out a theoretical study of the dynamics induced by PTCH, we assume that population growth in the absence of harvesting is governed by a one-dimensional difference equation xn+1=f(xn), (1) where xnis the population density at time step n,n=0, 1, 2 ..., starting at an initial value x0>0. The map fis the production (or stock–recruitment curve), for which we assume typical conditions for single-species population models, (A1) f: [0, ∞)→[0, ∞)is continuous, has a unique positive fixed point K,f(x) > xfor x∈(0, K), and 0 <f(x) < xfor x>K. Moreover, f(0)=0 and there exists limx→∞ f(x) < ∞. Most of the results will require additional conditions, which are summarized in the next hypothesis, (A2) fis a C2map having at most one critical point xc∈(0, ∞)such that f0(x) > 0 for all x∈(0, xc)and f0(x) < 0 for all x>xc. Moreover, f00(x) < 0on(0, xc). If fhas no critical points, then we consider xc= ∞, and hence, fis increasing. If fhas one critical point, we assume that limx→∞ f(x)=0. Sometimes, we will also assume that fis a C3map with negative Schwarzian derivative. We recall that the Schwarzian derivative is defined by the expression (Sf)(x)=f000(x) f0(x)−3 2f00(x) f0(x)2 , Chaos 30, 073108 (2020); doi: 10.1063/5.0010144 30, 073108-2 Published under license by AIP Publishing. 09 January 2024 11:29:54 Chaos ARTICLE scitation.org/journal/cha FIG. 2. Illustration of the graph of the piecewise smooth map F(blue solid line). We also plot the line y=x(red, dashed) and the graph of f(black, dashed). (a) Unimodal Ricker map f(x)=x e2.6(1−x), with H=0.5 and T=0.7. (b) Monotone Beverton–Holt map f(x)=2x/(1+x), with H=0.2 and T=0.7. whenever f0(x)6= 0. When (A2) holds with finite xc, the map fis unimodal; if, in addition, (Sf)(x) < 0 for all x6= xc, then fis an Sunimodal map.38 The concavity condition on (0, xc)is not usually assumed for this type of maps, but it is not very restrictive: on the one hand, it is satisfied by the models we consider; on the other hand, for S-unimodal maps, f00(0+) < 0 implies f00(x) < 0 for all x∈(0, xc).21 Both (A1) and (A2) hold for many compensatory and overcompensatory models usually employed in population dynamics. Throughout this paper, we understand that the model defined by (1) is compensatory if fsatisfies (A1) and (A2) with xc≥K(that is, f is nondecreasing at the positive equilibrium) and overcompensatory otherwise. For example, the compensatory Beverton–Holt model is defined by the map f(x)=r x/(1+x),r>1, which fulfills condition (A1) with K=r−1, and condition (A2) with xc= ∞. The Ricker model defined by f(x)=x er(1−x),r>0, satisfies (A1) with K=1 and (A2) with xc=1/r. It is compensatory if r≤1 and overcompensatory if r>1. Applying the precautionary threshold constant-catch harvesting rule to (1) and assuming that populations are measured after reproduction and before harvesting, we obtain the difference equation xn+1=F(xn):=     f(xn)if f(xn)≤T, Tif T<f(xn)≤T+H, f(xn)−Hif f(xn) > T+H. (2) For convenience, we define the map g(x)=f(x)−H, which allows us to write Fin the form F(x)=     f(x)if f(x)≤T, Tif g(x)≤T<f(x), g(x)if T<g(x). The piecewise smooth continuous map Fcan also be written in a line as F(x)=min f(x),f(x)−min{H,f(x)−T} =min f(x), max{g(x),T} and depends on the two harvesting parameters Hand T. The typical shape of Fcan be seen in Fig. 2. Roughly speaking, the graphs of fand gare joined by flat segments defined by T, which typically results in five intervals of smoothness for Fif fis unimodal and three if fis nondecreasing. Throughout the paper, we will assume that H>0 and T>0. It is clear that F≡fif H=0 or T≥sup{f(x),x≥0}. In the limit case T=0, (2) becomes the usual constant catch policy, defined by the map F(x)=max{f(x)−H, 0}. It is also worth noticing that if T<sup{f(x),x≥0} ≤ H+T, then the precautionary threshold constant-catch harvesting strategy (PTCH) becomes the pure threshold harvesting rule (TH), defined by the map F(x)=min{f(x),T}(see Ref. 18). For convenience of the readers, we include some important notations in Table I. Some of them have been already introduced, and others will appear in the following. TABLE I. Main notations. Symbol/concept Meaning H(Maximum) harvesting quota TThreshold harvesting parameter PTCH Precautionary threshold constant-catch harvesting TH (Pure) threshold harvesting fProduction map governing (1) g g(x)=f(x)−H FMap defining the PTCH rule (2) Ricker map f(x)=x er(1−x),r>0 xcCritical point of f(f0(xc)=0) ˜ xPoint such that f0(˜ x)=1 x(Smallest) point such that f0(x)= −1 KPositive fixed point of f p,qPositive fixed points of g(0 <p≤q<K) break point Point at which Fis not differentiable boundary fixed point Break point which is a fixed point of F admissible fixed point Fixed point of F virtual fixed point Fixed point of one map defining F but not of F BCB Border-collision bifurcation SB Smooth bifurcation Chaos 30, 073108 (2020); doi: 10.1063/5.0010144 30, 073108-3 Published under license by AIP Publishing. 09 January 2024 11:29:54 Chaos ARTICLE scitation.org/journal/cha III. FIXED POINTS: LOCATION, STABILITY, AND BIFURCATIONS In this section, we study the fixed points of Fdepending on the parameter values Tand H. In the first subsection, we focus on the number of fixed points and their location; in the second one, we study their stability properties; and in the third subsection, we describe the local bifurcations of fixed points, that is, we determine the critical values of the parameters for which fixed points are created or destroyed, or stability switches occur. One important consequence of conditions (A1) and (A2) is that there is a unique ˜ x>0 such that f0(˜ x)=1. Moreover, ˜ x∈(0, min{K,xc}). This property is a direct consequence of the mean value theorem, and the concavity of fon (0, xc). The point ˜ x plays an important role in the study of fixed points. A. Positive fixed points The following result provides the number of positive fixed points of the map Fdefined in (2). Notice that 0 is a fixed point of F, and it is always unstable. We assume that T>0 and H>0. Proposition 1: Assume that (A1) and (A2) hold and denote by ˜ x the unique solution of f 0(x)=1. Then, (i) If T ≥K,then K is the unique positive equilibrium of (2). (ii) If ˜ x≤T<K, then F has a unique positive fixed point,which is T if H ≥f(T)−T,and a fixed point q ∈(T,K)of g if H <f(T) −T. (iii) If 0<T<˜ x, then (a) F has a unique positive fixed point q ∈(˜ x,K)if H <f(T) −T. (b) F has three positive fixed points (T and two fixed points of g) if f(T)−T<H<f(˜ x)−˜ x. (c) T is the unique positive fixed point of F if H >f(˜ x)−˜ x. (d) F has two positive fixed points if either H =f(T)−T or H=f(˜ x)−˜ x. Proof. In view of (2),Kis a fixed point of Fif and only if K=f(K)≤T. Moreover, if K≤T, then Kis the only positive fixed point of Fbecause F(x)≥min{f(x),T}>xif x<K, and F(x) ≤f(x) < xif x>K. If T<K, then there are two possibilities for the positive fixed points of F: the threshold Tand the positive equilibria of g. It is obvious from the definition of Fthat Tis a fixed point if and only if f(T)≤T+H, that is, H≥f(T)−T. Since g0(x)=f0(x)and g00(x)=f00(x),gcan have at most two positive fixed points. Assume that p<qare two fixed points of g. Then, by the mean value theorem, p<˜ x<q, where ˜ xis the only point for which f0(˜ x)=1. It also follows that T<p<q <Kbecause p=F(p)=g(p)=f(p)−H>Tand g(x)=f(x) −H≤x−H<xfor all x≥K. Now, statements (ii)and (iii)follow easily. We include the proof of (ii)and leave the other to the reader. Assume ˜ x≤T<K. If H≥f(T)−T, then Tis a fixed point of F, and it is the only one because g(T)≤Tand g0(x) < 1 for all x>T. If H<f(T)−T, then Fhas a fixed point q∈(T,K)because g(T)=f(T)−H>Tand g(K)=f(K)−H=K−H<K. This FIG. 3. Number of positive fixed points of Fwith the Ricker map f(x)=x e2.6(1−x). There are two positive fixed points for the parameter values in the boundaries colored in red: H=f(T)−T, 0 <T<˜ xand H=f(˜ x)−˜ x, 0 <T<˜ x. There is only one positive fixed point for parameters at the boundaries colored in blue: H=f(T)−T,˜ x<T<1 and T=1. fixed point is unique because if there were two fixed points p<q, then T<pand g(x) < xfor all x<pwould imply that g(T) < T, a contradiction.  Figure 3 illustrates the number of fixed points in the parameter plane (H,T). B. Stability of fixed points This section is devoted to study the stability properties of the positive fixed points of F. We divide it into three parts: first, we deal with compensatory models; then, we give some global stability results; finally, we focus on the Ricker model. The points K,˜ x, and xchave the same meaning as in Sec. III A (see Table I). As usual, we say that a fixed point x∗of a map h:I→Iis stable if for each neighborhood Vof x∗in Ithere exists a neighborhood Uof x∗in Isuch that hn(x)∈Vfor all x∈Uand all n≥1. x∗is (locally) asymptotically stable (LAS for short) if it is stable and is a local attractor, that is, there exists an interval Jcontaining x∗ such that limn→∞ hn(x)=x∗for all x∈J. If J=I, we say that x∗ is a global attractor (or globally attracting). Since globally attracting fixed points of continuous interval maps must be stable, a global attractor is sometimes called globally asymptotically stable (GAS for short). Finally, we say that x∗is semistable if it is not stable but attracts either [x∗,x∗+ε) or (x∗−ε,x∗], for some ε > 0. We recall that, by Singer’s results,34 if fis an S-unimodal map with a unique fixed point x∗, then x∗is a global attractor if f0(x∗) ≥ −1, and unstable if f0(x∗) < −1. We will use the following more general result (Ref. 11, Corollary 2.9): Lemma 1: Let g :I→Cl(I)be a continuous function with a unique fixed point x∗such that g(x) > x if x <x∗, and g(x) < x if x>x∗. [Here, I is a real interval and Cl(I)denotes its closure.] Assume that there are points c,d∈I such that c <x∗<d and the restriction of g to (c,d)has at most one turning point and (whenever it makes sense) g(x)≤g(c)for every x ≤c and g(x)≥g(d) Chaos 30, 073108 (2020); doi: 10.1063/5.0010144 30, 073108-4 Published under license by AIP Publishing. 09 January 2024 11:29:54 Chaos ARTICLE scitation.org/journal/cha for every x ≥d. If g is decreasing at x∗, assume additionally that (Sg)(x) < 0for all x ∈(c,d)except at most one critical point of g and −1≤g0(x∗) < 0. Then, x∗is a global attractor of g. The following technical result will be also useful. It is a simple generalization of Ref. 4, Lemma 1. Lemma 2: Let f :(a,b)→(a,b)be a continuous function defined on a real interval (a,b)(−∞ ≤ a<b≤ ∞) such that f has a unique fixed point x∗with x <f(x) < x∗for all x ∈(a,x∗)and a<f(x) < x for all x ∈(x∗,b). Then, x∗is a global attractor of f. 1. Compensatory models Compensatory models exhibit simple dynamics. Recall that the model defined by (1) is compensatory if fsatisfies (A1) and (A2) with xc≥K. This framework includes monotone maps like in the Beverton–Holt model, and unimodal maps as the Ricker function f(x)=xer(1−x)with 0 <r≤1. Lemma 2 ensures that for compensatory models, Kis a global attractor of the model without harvesting (1). For the model with harvesting, we get the following result: Theorem 1: Assume that f satisfies (A1) and (A2) with T<K≤xc. Then, every solution of (2) converges to an equilibrium. More specifically, (a) If H <f(T)−T, then there is a unique positive equilibrium q∈(T,K)of F and it is globally attracting. (b) If H ≥f(T)−T and T ≥˜ x, or T <˜ x and H >f(˜ x)−˜ x, then T is the unique positive equilibrium of F and it is globally attracting. (c) If f(T)−T≤H≤f(˜ x)−˜ x, then there is bistability between two positive equilibria: T and q ∈(T,K). If H <f(˜ x)−˜ x, denote by p the largest fixed point of F smaller than q, and p∗=min {x∈F−1({p}):x>p}[p∗= ∞ if there are no x >p such that F(x)=p]. If f(T)−T<H<f(˜ x)−˜ x, then the basin of attraction of T is (0, p)∪(p∗,∞)and the basin of attraction of q is (p,p∗). If H =f(˜ x)−˜ x, then q =p is semistable and its basin of attraction is [p,p∗]. If H =f(T)−T, then T =p and the basin of attraction of T is (0, p]∪[p∗,∞). Proof. The proofs of (a)and (b)follow from Lemma 2. Next, we prove (c)when f(T)−T<H<f(˜ x)−˜ x. On the one hand, F maps the interval I=(p,p∗)into itself, and Lemma 2 ensures that qattracts I. On the other hand, Fmaps the interval J=(0, p)into itself, and Lemma 2 also applies to prove that Tattracts J. Finally, it is clear that Fmaps (p∗,∞)on J. The limit cases H=f(˜ x)−˜ xand H=f(T)−Tare left to the reader.  If T≥K, then the equilibrium Kis a global attractor (see Theorem 2 below). 2. Global stability results In this subsection, we provide some sufficient conditions under which the positive equilibrium is unique and globally attracting. The first case occurs when T≥K, which is considered in the next result. The proof is shifted to Appendix A 1. Theorem 2: Assume that f satisfies (A1) and K is a global attractor of f. If T ≥K, then K is a global attractor of F. In the following, we assume that (A1) and (A2) hold. Our next result deals with sufficient conditions for the global stability of Twhen T<K. Since the compensatory case was studied in Subsection III B 1, we deal with the overcompensatory case. Theorem 3: Assume that T <K and H ≥f(T)−T. If any of the following conditions holds, then T is the unique positive fixed point of F, and it is globally asymptotically stable. (a) T<˜ x and H >f(˜ x)−˜ x; (b) ˜ x≤T≤xc; (c) f(xc)−H≤T<K; and (d) f(g(xc)) ≥T and T >xc. Proof. In all cases, it follows from Proposition 1 that Tis the unique positive fixed point of F. In cases (a)and (b), it is easy to check that x<F(x)≤Tfor x<Tand 0 <F(x) < xfor x>T. Hence, the result follows from Lemma 2. Since scheme (2) becomes threshold harvesting when H+T ≥f(xc), in case (c), the result follows from Ref. 18, Proposition 2.1. Finally, conditions in (d)imply that the long-term behavior of the solutions of (2) and equation xn+1=G(xn),n≥0, is the same, where G(x)=(Tif g(x)≤T, g(x)if g(x) > T.(3) Since Tis stable for G, to prove the global stability it is enough to exclude periodic orbits of Gwith prime period 2 (see, e.g., Ref. 14, Proposition 1). But Gcannot have 2-periodic orbits because G(x)=Tfor all x≥T. In line with case (d)in the previous theorem, we can prove a global stability result for the fixed point qin case fis S-unimodal. Theorem 4: Assume that T <K, H <f(T)−T, and f is Sunimodal. If g0(q)≥ −1and f(g(xc)) ≥T, then q is globally asymptotically stable. Proof. Condition H<f(T)−Timplies that qis the unique positive fixed point of F. Since f(g(xc)) ≥T,Fcan be replaced by map Gdefined in (3) for the analysis of convergence of solutions, as in the proof of Theorem 3. Now, the proof easily follows from Lemma 1, with c=min g−1(T)and d=max g−1(T). We illustrate the results of Theorems 3 and 4 in Fig. 4. 3. Specific results for the Ricker map In this subsection, we focus our attention on the PTCH scheme (2) with the Ricker map f(x)=xer(1−x),r>1, in order to give a more comprehensive description of the stability properties of the positive equilibria. Recall that the case r≤1 has been addressed in Subsection III B 1. We recall some basic properties of the Ricker map. First, f is an S-unimodal map. In particular, ffulfills (A1) and (A2) with K=1 and xc=1/r, and there exists a unique ˜ x∈(0, 1/r)⊂(0, 1) such that f0(˜ x)=1. Since f0(x)=(1−rx)er(1−x)and f00(x)= −r(2−rx)er(1−x), it follows that f00(x) < 0 for all x∈(0, 2/r),f00 (2/r)=0 and f00(x) >0 for all x>2/r. Hence, f0attains its global minimum at 2/rand f0(2/r)= −er−2<−1 if and only if r>2. Thus, since fis an Sunimodal map, we have the following known result, which we state for later reference. Chaos 30, 073108 (2020); doi: 10.1063/5.0010144 30, 073108-5 Published under license by AIP Publishing. 09 January 2024 11:29:54 Chaos ARTICLE scitation.org/journal/cha FIG. 4. Regions of global stability for PTCH with the Ricker map f(x)=xe2.6(1−x) based on Theorems 3 and 4. The red solid curve is the graph of H=f(T)−T.T is GAS in regions defined by H>f(T)−Tand (a) T<˜ xand H>f(˜ x)−˜ x; (b) ˜ x≤T≤xc, (c) f(xc)−H≤T<K; and (d) f(g(xc)) ≥Tand T>xc. Notice that when H<f(T)−T, condition f(g(xc)) ≥Tholds to the right of the dashed line, so qis globally stable in region (e), determined by condition g0(q)≥ −1 (see Theorem 7). Proposition 2: If f(x)=xer(1−x)and 0<r≤2, then f 0(x) ≥ −1for all x >0. Moreover, f 0(x)= −1if and only if r =2and x=1. In particular, if r ≤2, then the positive equilibrium K =1is a global attractor of f in (0, ∞). If r >2, then f 0(K)=1−r<−1, and, therefore, K is unstable. If r>2, then there exists a unique x∈(xc, 2/r)such that f0(x)∈(−1, 0)for all x∈(xc,x),f0(x)= −1, and f0(x) < −1 for all x∈(x, 1). The value of xcan be numerically computed as the smallest root of equation f0(x)= −1, that is, the unique solution x∈(0, 1)of (rx −1)er(1−x)=1. We begin considering the case when the Ricker map has a stable equilibrium. The proofs of Theorems 5 and 6 are provided in Appendix A 2. They consider the cases when Fhas at least one positive fixed point different from T, or Tis the unique positive fixed point of F, respectively. Theorem 5: Assume 1<r≤2, T <1, and F has at least one positive fixed point different from T. Denote by p, q (p ≤q) the positive fixed points of g. Then, there are two possibilities: (i) If T is a fixed point of F (that is, H ≥f(T)−T), then there are three cases. If there are three positive fixed points of F (T <p <q), then q attracts (p,p∗)and T attracts (0, p)∪(p∗,∞), where p∗=max g−1(p). If there are two positive fixed points of F (T =p<q), then q attracts (T,T∗)and T attracts (0, T] ∪[T∗,∞), where T∗=min{x>T:g(x)=T}. If there are two positive fixed points of F (T <p=q), then q attracts [p,p∗]and T attracts (0, p)∪(p∗,∞). (ii) If H <f(T)−T, then q is the unique positive fixed point of F, and it is a global attractor. Theorem 6: If 1<r≤2, xc<T<1and H ≥f(T)−T, then T is the unique positive fixed point of F, and it is globally asymptotically stable. Now, we are in a position to state and prove the main result in this subsection. Theorem 7: Consider the map F defined by (2) with f(x) =xer(1−x), r >1. Denote by p,q, the fixed points of g(x)=f(x)−H, when they exist, with T <p≤q. (I) If T ≥1, then the positive fixed point K =1of f is the unique positive equilibrium of F. It is a global attractor if 1<r≤2and unstable if r >2. (II) If T <1and 1<r≤2, then the conclusions (a), (b), and (c) of Theorem 1 hold. In particular, every solution of (2) with initial condition x0>0converges to a positive equilibrium. (III) If T <1and r >2, then, • T is a fixed point of F if and only if H ≥f(T)−T and T≤1. Moreover, T is semi-stable if 0<T<˜ x and H=f(T)−T, and it is asymptotically stable otherwise. T is a global attractor in the cases stated in Theorem 3 (see Fig. 4). • The fixed point p is unstable if p <q, and semi-stable if p =q. • The fixed point q ∈(˜ x, 1)is unstable if H <f(x)−x, and asymptotically stable if H ≥f(x)−x. In the latter case, q is a global attractor if f(g(xc)) ≥T. Proof. The existence of the different fixed points of Fis given in Proposition 1. The proof of (I)follows from Theorem 2 and Proposition 2. The proof of (II)follows from Theorems 3, 5, and 6. Next, we consider the three cases of (III): • First, we deal with the stability of the threshold value T. If H>f(T)−T, then Tis asymptotically stable because Fis differentiable in an interval (T−ε,T+ε), and f0(T)=0. When H=f(T)−T, we distinguish several cases: First, if T>xc, then there exists m1>0 such that F(x)=Tfor all x∈[T,T+m1), which, by continuity, implies that there exists m2>0 such that F2(x)=Tfor all x∈(T−m2,T); therefore, Tis asymptotically stable. Next, if ˜ x≤T≤xc, then Tis the unique positive fixed point of F, and it is globally stable (see Theorem 3). When T∈(0, ˜ x),Tis semi-stable because there is η > 0 such that F(x)=Tfor x∈(T−η,T], and F0(T+) > 1. • As for the smallest fixed point of g,pis unstable if p<qbecause F(x)=g(x)in a neighborhood of pand p<˜ x. Hence, F0(p) =f0(p) > 1. When pand qcollide, there is a smooth tangent bifurcation and pis semistable. • Finally, we address the stability of q∈(˜ x, 1). Since F(x)=g(x) in a neighborhood of q, the asymptotic stability of qdepends on g0(q)=f0(q). Since q∈(˜ x,K), it is clear that qis asymptotically stable if q≤x(that is, H≥f(x)−x), and unstable if q>x (H<f(x)−x). The global stability result follows from Theorem 4.  C. Bifurcations of fixed points The boundaries found in Subsection III A help us to analyze the possible bifurcations of fixed points, using the bifurcation parameters Hand T. Since Fis a piecewise smooth continuous map, some bifurcations are non-smooth. We adopt some notations in Ref. 2; see Table I for quick reference in the future. The points where Fis not differentiable are called break points. When a break point x∗is a fixed point of the map F, then we call Chaos 30, 073108 (2020); doi: 10.1063/5.0010144 30, 073108-6 Published under license by AIP Publishing. 09 January 2024 11:29:54 Chaos ARTICLE scitation.org/journal/cha x∗aboundary fixed point. Boundary fixed points determine a kind of bifurcations that are called border-collision bifurcations since the work of Nusse and Yorke.24 Border-collision bifurcations (BCB for short) occur when, under variation of some parameters, a fixed point or a cycle collides with a break point, and this collision leads to a qualitative change in the dynamics. Sometimes, we use the term discontinuity-induced bifurcations for other type of bifurcations in which a break point is involved (for example, when the metric attractor is an interval, and one of its endpoints collides with a break point). For usual bifurcations where break points are not involved, we use the term smooth bifurcations (SB for short). We emphasize that Fis continuous, so discontinuity refers to its first derivative. According to Ref. 2, Sec. 3.1.2, there are four basic possible dynamical scenarios at a BCB defined by a boundary fixed point. We need the notions of admissible and virtual fixed points. For the PTCH rule (2), we recall that Fis a piecewise smooth map defined by three differentiable maps: f,g, and the constant function T. A fixed point of Fis called an admissible fixed point, whereas a virtual fixed point is a fixed point of one of the smooth maps defining Fbut not a fixed point of F. For example, Tand Kare either virtual or admissible fixed points of F; according to Proposition 1, Kis admissible if and only if T≥K, and Tis admissible if and only if H≥f(T)−Tand T≤K. • A fold BCB occurs when two coexisting admissible fixed points collide at a break point and become two virtual fixed points. • A flip BCB or period-doubling BCB occurs when an admissible fixed point collides with a break point and a 2-periodic orbit {p1,p2}appears, where p1and p2are at different sides of the break point. • A persistence BCB occurs when an admissible and a virtual fixed point collide at a break point and interchange their roles. No other periodic points are created or destroyed at the bifurcation point. • A period-multiplying BCB occurs when an admissible fixed point collides with a break point and an m-periodic orbit appears, with m>2. In our framework, these bifurcations can only occur when T becomes a boundary fixed point, that is, when g(T)=T[equivalently, H=f(T)−T] or f(T)=T(equivalently, T=K). When FIG. 5. Illustrations of BCBs arising when H=f(T)−Tin (2). The Ricker map f(x)=xer(1−x)is the black dashed curve, while the blue solid curve is the corresponding map F. The red dashed line is y=x. Top panel: fold BCB for r=2.1 and T=0.1 <˜ x≈0.353. (A1) H=0.3 <f(T)−T≈0.562; (B1) H=f(T)−T; (C1) H=0.9 >f(T) −T. Center panel: flip BCB for r=2.6 and T=0.7 >x≈0.485. The attracting 2-cycle is represented by a magenta box in (C2). (A2) H=1>f(T) −T≈0.827; (B2) H=f(T)−T; (C2) H=0.6 <f(T)−T. Bottom panel: persistence BCB for r=2.1 and T=0.7 (˜ x<T<x≈0.77). (A3) H=0.5 <f (T)−T≈0.614; (B3) H=f(T)−T; (C3) H=0.65 >f(T)−T. Chaos 30, 073108 (2020); doi: 10.1063/5.0010144 30, 073108-7 Published under license by AIP Publishing. 09 January 2024 11:29:54 Chaos ARTICLE scitation.org/journal/cha H=f(T)−T, the first three BCBs can occur as Tor Hare continuously changed. See Fig. 5 for illustrations. • A fold BCB occurs if H=f(T)−Tand T<˜ x:Tbecomes a boundary fixed point as Tis decreased or His increased; after this critical value, two admissible fixed points arise: an attractor T, and a repeller p. • A flip BCB occurs if H=f(T)−T,T>˜ x, and qis unstable for g: the attracting fixed point Tbecomes virtual after becoming a boundary fixed point as Tor Hdecrease, while an unstable fixed point qand an attracting 2-cycle {T,g(T)}appear. Notice that the flat shape of Fto the right of the break point prevents the possibility of a period-multiplying BCB in our framework. • A persistence BCB occurs if H=f(T)−T,T≥˜ x, and qis asymptotically stable for g: as Tor Hincrease, the virtual fixed point T and the admissible attractor qinterchange their roles. In view of Theorem 7, for a Ricker map f(x)=x er(1−x), with r>2, a flip BCB occurs when H=f(T)−Tand T>x, and a persistence BCB occurs when H=f(T)−Tand ˜ x≤T≤x. When Tbecomes a boundary fixed point at the critical value T=K, it only makes sense to choose Tas the bifurcation parameter; in this case, persistence or flip BCBs can occur as Tis continuously changed. These bifurcations are typical of threshold harvesting.18,32 • A persistence BCB occurs if T=Kand Kis asymptotically stable for f: as Tis increased, the fixed point Tbecomes virtual and the attracting fixed point Kbecomes admissible. • A flip BCB occurs if T=Kand Kis unstable for f: as Tis increased, the fixed point Tbecomes virtual, Kbecomes admissible, and an attracting 2-cycle {T,f(T)}emerges (see Fig. 6). For the Ricker map, a persistence BCB occurs if T=1 and 0<r≤2, and a flip BCB occurs if T=1 and r>2. Finally, smooth local bifurcations of fixed points may also occur for the bifurcation parameter H: a fold SB at H=f(˜ x)−˜ xif T<˜ x, and a flip SB when H=f(x)−x. In Figs. 7 and 9, we use solid lines for BCBs of fixed points and dashed lines for SBs of fixed points. Specifically, we use red color for fold bifurcations, orange for persistence BCBs, and brown for flip bifurcations. BCBs of periodic points and global bifurcations are shown in the case study considered in Subsection V B. FIG. 7. Stability diagram of PTCH with the Ricker map f(x)=xe0.3(1−x). There is a region of bistability, while in the others there is a unique equilibrium, which is globally attracting. We use orange color for persistence BCBs and red color for fold bifurcations (the solid curve corresponds to a fold BCB, and the dashed line to a fold SB). IV. ESSENTIAL ATTRACTION AND CHAOTIC BEHAVIOR In this section, we consider the regions of the parameter plane (H,T)in which complex dynamics is more likely to occur. First, we consider the case T<˜ xand H>f(T)−T. This region of the parameter plane (H,T)is represented in Fig. 9(b) for the Ricker map f(x)=xe2.6(1−x). Shifting the origin of coordinates from (0, 0)to (T,T), we show that the dynamics of (2) in this case can be derived from the results obtained by Schreiber30 for constant quota harvesting models. We introduce two definitions for later reference: an equilibrium x∗of a real map h:I→Idefined on an interval Iis essentially asymptotically stable if it is stable and limn→∞ hn(x)=x∗for Lebesgue almost all xin a neighborhood of x∗. If limn→∞ h(x)=x∗ for Lebesgue almost all x∈I, we say that x∗is an essential global attractor. A discussion of essential asymptotic stability22 and the relation to Milnor attractors23 can be found in Ref. 25, where the authors introduce the concept of the stability index. We state the main result of this subsection: Theorem 8: Let f : [0, ∞)→[0, ∞)be a C3unimodal map with negative Schwarzian derivative satisfying (A1) and (A2). If F FIG. 6. llustration of a flip BCB in (2) when T=K. The Ricker map f(x) =xe2.6(1−x)is the black dashed curve, while the blue solid curve is the corresponding map Fwith H=1.6. The red dashed line is y=x. The attracting 2-cycle {T,f(T)}is represented by a magenta box in panel (c). (a) T=0.95 <K=1; (b) T=K; (c) T=1.1 >K. Chaos 30, 073108 (2020); doi: 10.1063/5.0010144 30, 073108-8 Published under license by AIP Publishing. 09 January 2024 11:29:54 Chaos ARTICLE scitation.org/journal/cha play an essential role. Piecewise smooth maps with flat branches appear in different applications, ranging from chaos control8,15,32,36 to economic models,1,5,37 and sustainable harvest rules.18 PTCH can be seen as a combination of constant catch harvesting (CH for short) and threshold harvesting (TH) rules. The maps governing these two strategies are piecewise smooth: the graph of the map for constant harvesting has a flat part at the bottom,30 and the corresponding map for TH is flat-topped.18 The dynamics of PTCH is strongly influenced by both maps: results valid for CH help understanding the dynamics of PTCH for small values of T, and typical features of the bifurcation diagrams of TH are present in those of PTCH. As a result, the dynamics of PTCH is richer than and qualitatively different from the dynamics of TH and other threshold rules as proportional threshold harvesting (PTH, for short) (see, e.g., Ref. 17). We combine analytical and numerical results to provide a comprehensive overview of the dynamics of PTCH. For simple cases (compensatory models), the dynamics and bifurcations are thoroughly described analytically. As expected, overcompensatory models exhibit more complicated dynamics; in this case, some regions of the 2-parameter plane (H,T)can be completely understood; for example, some global stability results (Subsection III B 2), and regions where bistability or essential attraction occur (Sec. IV). We have chosen as case study a Ricker model which in the absence of harvesting is oscillatory but not chaotic. This choice allows us to provide good 2-parameter bifurcation diagrams (Fig. 9) and also to compare our results with those obtained for CH by Schreiber30 and for TH by Hilker and Liz.18 Selected numerical bifurcation diagrams illustrate the main dynamical features of PTCH, which include bistability regions created and destroyed by fold bifurcations or boundary collisions, sudden transitions between chaos and periodic attractors, and bubbles, among others. From a biological point of view, PTCH is a harvesting rule that helps preventing some undesirable consequences of traditional harvesting strategies, and, in particular, of constant quota harvesting, which can lead to sudden collapses due to overexploitation. The main drawback of a threshold approach is the fisheries moratoria associated with low stock size. A combined strategy like PTCH also helps to overcome this weakness of threshold harvesting rules. In this direction, a manager would be interested in situations where the long-term dynamics is governed either by g(usual constant harvesting) or by gand the threshold T. The reason is that periods without harvesting only appear when fplays a role in the definition of F. As it is expected, this undesirable situation corresponds to a combination of high values of Tand small values of H(more conservative strategies). In contrast with other threshold harvesting rules (TH, PTH), the fixed point of Fdoes not need to be unique for the usual unimodal maps. When two stable equilibria coexist, we observe a phenomenon similar to the so-called Allee effect: too low or too high population densities asymptotically approach the threshold value T, while intermediate population densities tend to a higher equilibrium. There are two different forms of increasing harvesting effort in PTCH: increasing the harvesting quota Hor decreasing the threshold value T. For large H, PTCH becomes TH, and, therefore, a stable equilibrium cannot be destabilized increasing harvesting by decreasing the threshold.18 However, fixing a lower value of H, decreasing Tcan produce several stability switches, some of them destabilizing. For the Ricker map, increasing Hcannot destabilize a stable equilibrium. However, it might occur for other population models governed by unimodal maps with negative Schwarzian derivative. Examples of such situations for a strategy of constant quota harvesting are provided by Jiménez López and Liz.19 ACKNOWLEDGMENTS The authors thank Professor F. M. Hilker and two anonymous reviewers for their constructive and useful comments and for pointing out some references. Eduardo Liz acknowledges the support of the Research Grant No. MTM2017–85054–C2–1–P (AEI/FEDER, UE). The research of Cristina Lois-Prados has been partially supported by the Ph.D. Scholarship No. FPU18/00719 (Ministerio de Ciencia, Innovación y Universidades, Spain) and Research Grant Nos. MTM2016-75140-P (AEI/FEDER, UE) and ED431C2019/02 (Xunta de Galicia). APPENDIX A: PROOFS 1. Proof of Theorem 2 The proof of Theorem 2 in the main text follows the proof of the analogous result for proportional threshold harvesting (Ref. 17, Proposition A.3). It is an easy application of the following lemma: Lemma A.3: (Ref. 11, Theorem B) Assume that f :(0, ∞) →(0, ∞)has a globally attracting equilibrium K, and let h : (0, ∞)→(0, ∞)be a continuous map satisfying that x <h(x) ≤max{f(x),K}for all x <K and x >h(x)≥min{f(x),K}for all x>K. Then, K is a globally attracting equilibrium of h. Proof. To prove the theorem, let us show that the conditions of Lemma 3 are fulfilled with h=F. First, by Proposition 1, Kis the unique positive equilibrium of (2). In particular, x<F(x)for all x<K. Hence, the result is trivial for x<Kbecause F(x)≤f(x)≤max{f(x),K}for all x≥0. Now, it is clear by (2) that F(x)≥min{f(x),T}, and therefore, since we are assuming that T≥K, it follows that x>F(x)≥min{f(x),T} ≥ min{f(x),K}for all x>K. 2. Proof of Theorems 5 and 6 Proof of Theorem 5. To prove (i) when f(T)−T≤H<f(˜ x)− ˜ x, notice that ghas negative Schwarzian derivative and by Proposition 2, g0(q)≥ −1. Hence, qis a local attractor of gand we can define Jas the connected component of the set S= {x∈(p,p∗): limn→∞ gn(x)=q}containing q. Clearly, J=(a,b)is an open interval. If J6= (p,p∗), then, since ghas no other fixed point than qin J, the only possibility is that g(a)=band g(b)=a. But this case is ruled out by Singer’s results (see, e.g., Ref. 34, Lemma 2.6). Finally, it is obvious that Tattracts [0, ∞)\[p,p∗] if T<pand attracts [0, ∞)\(p,p∗)if T=p. The case H=f(˜ x)−˜ xis left to the reader. To prove (ii), we use a different approach, following the ideas used in Ref. 17, Proposition A.4. Let us consider T∈(0, 1), H∈(0, f(T)−T)arbitrarily fixed; we prove that qis a global attractor using the enveloping method (Ref. 9, Theorem 3). For that purpose, we consider the linear map φ(x)=2q−xand show that F Chaos 30, 073108 (2020); doi: 10.1063/5.0010144 30, 073108-15 Published under license by AIP Publishing. 09 January 2024 11:29:54 Chaos ARTICLE scitation.org/journal/cha FIG. 15. Diagrams showing the map F (blue solid curve) enveloped by φ(red dot-dashed line). The original Richer map f(x)=xe1.8(1−x)is represented by the solid black curve, which is enveloped by the black dot-dashed line y=2 −x. The black dashed line is y=x. (a) φ(x)=2q−x. The equilibrium q is GAS for T∈(0, 1)and H∈(0, f(T) −T). (b) φ(x)=2T−x. The equilibrium Tis GAS for T∈(xc, 1)and H≥f(T)−T. fulfills the following inequalities: x<F(x) < φ(x)for all x∈(0, q), and x>F(x) > φ(x)for all x>q[see Fig. 15(a)]. We know by Proposition 1 that qis the unique fixed point of Fin (0, ∞); moreover, x<F(x)for all x∈(0, q)and x>F(x) for all x>q. Hence, it remains to prove that F(x) < 2q−xfor all x∈(0, q)and F(x) > 2q−xfor all x>q. Let us consider p1and p2the points such that 0<p1<T<q<p2,g(p1)=T=g(p2). Note that, for all x∈(p1,p2), we have F(x)=g(x), so Fis differentiable in (p1,p2). We distinguish three cases: • If x∈(0, p1], then F(x)=min{f(x),T} ≤ T<2T−x<2q−x. • If x∈(p1,p2), then F(x)=g(x). By Proposition 2, g0(x)≥ −1 for all x>0; moreover, g0(x) > −1 for all x6= 1. Hence, the mean value theorem guarantees that g(x)6= 2q−xfor all x6= q. Therefore, g(x) < 2q−xfor all x∈(p1,q)and g(x) > 2q−xfor all x∈(q,∞). • If x>p2, then we have F(x) > g(x) > 2q−x. An application of Cull’s theorem proves that qis globally asymptotically stable.  The proof of Theorem 6 follows from analogous arguments to those used in the proof of Theorem 5, using the line φ(x)=2T−x for enveloping, instead of 2q−x[see Fig. 15(b)]. 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