Hybrid dynamics of multi-species resource exploitation

Decisions in Economics and Finance https://doi.org/10.1007/s10203-021-00336-9 Hybrid dynamics of multi-species resource exploitation Davide Radi1,2 ·Fabio Lamantia1,3 ·Tomáš Tichý1 Received: 1 February 2021 / Accepted: 25 May 2021 © The Author(s) 2021 Abstract In this paper, we analyze a bio-economic model of exploitation of renewable commercial resources. To take into account the typically continuous-time modeling of biological species and, instead, of the specialized harvesting activities, which by its nature cannot change continuously, the resulting dynamic system is of the hybrid type, i.e. continuous for biological variables and discrete for the economic ones. Through a discretization of the continuous variables, the problem is then reformulated by means of a three-dimensional map. Of this map, we study the dynamic properties to understand how economic parameters influence the long-run availability of resources. Keywords Fisheries management ·Mathematical bioeconomics ·Heterogeneous agents ·Evolutionary game theory ·Hybrid dynamical systems JEL Classifications C62 ·C73 ·Q57 1 Introduction The balance between the need to conserve resources and their use for economic purposes represents a challenging problem. To avoid over-exploitation that leads to the “tragedy of the commons” (see Hardin (1968)), forms of harvesting regulation are BFabio Lamantia [email protected] 1Department of Finance, V˘ SB–Technical University of Ostrava, Ostrava, Czech Republic 2Department of Economics and Management, University of Pisa, Pisa (PI), Italy 3Department of Economics, Statistics and Finance, University of Calabria, Rende (CS), Italy 123 D. Radi et al. usually employed, mainly based on optimal control techniques.1In spite of the various forms of regulation employed, most natural resources are over-exploited, with many species being extinct or on the verge of extinction, see, for instance, McWhinnie (2009). One of the main reasons for this failure relates to the difficulty in monitoring and enforcing optimal rules. For this reason, one could replace those forms of rigid control, difficult to impose, with milder forms of regulation. An example relating to the possible dynamics of these rules is provided in Bischi et al. (2013a) and concerns the harvesting of two species of shellfish from the Adriatic Sea (Venerupis aurea and Callista chione). These species do not interact biologically with each other and, on the basis of a proposal to regulate their exploitation, fishers must only commit themselves to harvest only one of the two resources for a given period of time. This can occur because an authority imposes that fishers have to stick to the decided strategy for a given period of time after which they can reconsider their decisions on the basis of observed profits. The dynamics of the resources, modeled by means of a continuoustime system, are therefore coupled with a discrete-time dynamics on the species to be taken, based on the profits from fishing. The model that follows is then a hybrid system that combines continuous and discrete time dynamics, see Aubin et al. (2002), Goebel et al. (2009) and Haddad et al. (2006). From a mathematical point of view, even if hybrid models are more complicated to study, they are closer to a more correct description of the reality being analyzed (see also Bischi et al. (2013b), Lamantia and Radi (2015) and Bischi et al. (2015) for related models). In this paper, we provide a dynamic analysis of the hybrid dynamical system proposed in Bischi et al. (2013a). In addition to the reasons indicated in Bischi et al. (2013a), this type of modeling is coherent to all those situations of specialized fisheries, in which a particular species is targeted and for a certain period of time, due to technical requirements of the harvesting (types of gears) or rules, it is not possible to change the species taken, see French et al. (2019) for an analogous example. The purpose of this paper is twofold. On the one hand, we extend the model to include financial evaluation of the interests on profits in the various harvesting periods. In addition, we provide an equivalent reformulation of the model in Bischi et al. (2013a) by means of a discretization of the continuous-time dynamics of the resources in the periods in which there may be a change in the choices of the species to be taken.2These choices are then modeled by means of an evolutionary dynamic (discrete-time replicator type), which is based on the profits obtained over the previous periods (see Weibull (1995) and Hofbauer and Sigmund (1998)). The discrete-time representation of the hybrid dynamical system proposed in Bischi et al. (2013a) allows us to underline interesting aspectsoftheglobaldynamicsofthemodelthatwereneglectedinBischietal.(2013a). The first relevant aspect that we are able to underline is the type of bifurcation, that is period-doubling or flip, through which the inner equilibrium of the system loses stability when a different time scale between the decision process of quantity to harvest and the one of the species to harvest is introduced. A second relevant aspect that we 1We refer to Clark (1990) and Conrad and Smith (2012) for comprehensive reviews on the subject. 2A similar approach for studying hybrid dynamical systems related to oligopoly models is employed in Cavalli and Naimzada (2016), Cavalli et al. (2018) and Tichý et al. (2020). 123 Hybrid dynamics of multi-species resource... underline by means of numerically generated bifurcation diagrams is the existence of chaotic dynamics. Specifically, we show a sequence of period-doubling bifurcations leading to chaotic dynamics. A third relevant aspect that we underline is the presence of coexisting attractors, that introduces path dependence and the related economic issues. Therefore, the current work adds to the existing literature by explaining for the first time the mechanisms behind the global dynamics of the hybrid bio-economic model proposed in Bischi et al. (2013a). The structure of the paper is as follows. In Sect. 2, we summarize the essential elements of the bio-economic model considered in Bischi et al. (2013a). In Sect. 3, we reformulate that hybrid model through a three-dimensional map and propose a possible extension to take into account the compounding of risk-free interests in harvesting profits. In Sect. 4, we present the main results on the global dynamics of the model, in particular, regarding the influence of economic parameters on the dynamics of the species subject to exploitation. Section 5concludes. All the proofs are in Appendix A. 2 The bio-economic setup Here we briefly recap the model proposed in Bischi et al. (2013a) for the particular case of interest for this paper and present a possible extension of the model. Consider two non-interacting species subject to commercial harvesting and denote their biomass measuresby X1and X2.Biomassifollowslogisticnaturalgrowth(seeMeliàandGatto (2005)) and its time evolution can be described by an ordinary differential equation (ODE) of the type ˙ Xi(t)=Xi(t)ρi1−Xi(t) ki−Hi(Xi(t),ri(t)) ,i=1,2,(1) where riis the share of fishers that harvest species iwhile ρi>0 and ki>0are, respectively, the intrinsic rate of growth and the carrying capacity of species i.In(1), the term Hidenotes the instantaneous harvesting of species i=1,2. ThemodelinBischietal.(2013a)describesthe exploitationofspecialized fisheries, in the sense that N∈N\{0,1}fishers are present but, given the specialized nature of the fishery, for a certain period of time each fisher can only harvest one species at once. This can be due to technical constraints linked to the adopted fishing technique (e.g. types of gears) or to the legislation (i.e. authority restricting each agent to catch only one species at a time). As a consequence, it is not possible to carry out a continuous change of the species caught but the targeted species can change only at the beginning of a fishing period. For the purpose of this paper, let us consider here the particular case of the model in Bischi et al. (2013a) with constant selling price ai>0 for species i. Thus, if species iis targeted, the choice on quantity hito harvest is carried out by maximizing profits of the form 123 D. Radi et al. πi(Xi(t),hi)=aihi−γi h2 i Xi(t),i=1,2,(2) where the term Ci(Xi(t),hi)=γi h2 i Xi(t),(3) denotes harvesting costs and γi>0 represents an inefficiency parameter for catching species i, see Clark (1990), Conrad and Smith (2012) and the Appendix in Bischi et al. (2013a) for details. Profit maximization leads to a myopic Markovian harvesting rule of the type h∗ i(Xi(t)) =aiXi(t) 2γi ,i=1,2,(4) By inserting (4)into(2), individual instantaneous profits assume the following form π∗ i(Xi(t)) =γi Xi(t)h∗ i(Xi(t))2=a2 iXi(t) 4γi ,i=1,2,(5) As mentioned before, fishers target only one species during a certain period of time. Therefore, considering the set of Nagents, in a given time interval, a fraction ri specializes in the harvesting of species i, so that total instantaneous harvesting in (1) can be written as follows Hi(Xi(t),ri(t)) =Nri(t)h∗ i(Xi(t)) =Nri(t)aiXi(t) 2γi .(6) In the next section, we describe how fractions riare updated over time according to evolutionary pressure. 3 Switching mechanism and bio-economic model In this section, we describe in detail the evolutionary mechanism that determines the temporal evolution of the fraction of agents that target each species by following and presenting a possible extension of the bio-economic model in Bischi et al. (2013a). The main reason for this choice is to model a specialized fishery, characterized by harvesting that concentrates on a particular species for a time interval. As mentioned above, a share riof fishers catches only speciesiduring an interval of time. Harvesting takes place continuously and agents select the harvesting according to (4). Thus, during a period of time, a fixed share riof fishers harvest only species i characterized by logistic growth, so that the time evolution of species iin (1) can be rewritten as 123 Hybrid dynamics of multi-species resource... ˙ Xi(t)=Xi(t)ρi1−θi(ri(t)) −Xi(t) ki,i=1,2,(7) where the term ρiθi(ri), with θi(ri)=Nri ρi ai 2γi, is the instantaneous harvesting for unit of biomass of species i. In Bischi et al. (2013a), the continuous-time dynamics of resources in (7) is coupled withadiscrete-timedynamicsfortheevolutionoftheshareoffishersdevotedtoharvest only species i,i=1,2, that is ri. This choice is carried out by considering the average profit π∗ iover a period of length s>0 for a representative agent targeting species i. Specifically, the value of π∗ iat time t+sis given by π∗ i(Xi(t),ri(t)) = s  0 π∗ i(Xi(t+τ)) dτ s,i=1,2,(8) By assuming that the magnitude of π∗ ican be assessed by agents of either group, Bischi et al. (2013a) proposes a hybrid dynamical system obtained by coupling continuous-time growth and harvesting of the fish species in (7) with a discrete (or pulse)fishingstrategyswitching,modeledbydiscretereplicatordynamics(seeWeibull (1995), Hofbauer and Sigmund (1998) and Cressman (2003)) of the form r(t+s)=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r(t)π∗ 1(X1(t),r(t)) r(t)π∗ 1(X1(t),r(t)) +(1−r(t)) π∗ 2(X2(t),r(t)) if t+s s=t+s s rt+s ssotherwise , (9) where xis the largest integer not greater than x(i.e. the floor of x), π∗ i,i=1,2are given, respectively, in (8). Differently from Bischi et al. (2013a), here we limit the attention to describe the state of the system at the switching times S={0,s,2s,...,ns,... }. Specifically, consider a time tat which fishers decide the biomass to target, that is t∈S. Then, at time tthe shares ri,i=1,2, are determined and remain fixed during the next interval of time [t,t+s). Moreover, if at time tthe biomass of species imeasures Xi(t),at time Xi(t+s)itsvaluecan be obtained by solvingthe correspondent Cauchy problem with dynamics (7) and initial condition Xi(t). Thus, assuming that specialized fishers do not change harvested species in the interval of time [t,t+s), the total biomass of species iat time t+sreads Xi(t+s)=(1−θi(r(t))) kiXi(t) Xi(t)−[Xi(t)−(1−θi(r(t))) ki]e−(1−θi(r(t)))ρis.(10) The switching mechanism that describes how the fraction of specialized fishers changes targeted species is based on profit comparison on the relevant interval 123 D. Radi et al. [t,t+s), with t∈S. Specifically, extending the main framework in Bischi et al. (2013a), we assume that agents make an assessment of the total profits obtained by either strategy during the time interval [t,t+s), introducing a parameter δ∈(−1,1) that weighs the profits on the basis of the moment of realization. The particular case of δ=0 is the setup employed in Bischi et al. (2013a). Therefore, considering instantaneous profits in (5), at the end of each non-switching period [t,t+s), with t∈S,a representative agent who targeted species iassesses the evaluation of average profits π∗ i(compounded to time t+swhen δ<0 or with more weight on recent profits when δ>0) over the last period, by calculating π∗ i(Xi(t),ri(t)) =1 s s  0 eδ(τ−s)π∗ i(Xi(t+τ)) dτ =a2 i 4γis s  0 eδ(τ−s)Xi(t+τ)dτ;i=1,2 (11) where the integrand function is given by the product between the biomass of species iat time t+τ∈[t,t+s),givenin(10), times the weighting term eδ(τ−s). Clearly when δ<0, eδ(τ−s)can be interpreted as a standard compounding factor, which takes into account the interests on the profits obtained by harvesting species iin the time interval, where the evaluation time is the end of each period. The case δ> 0 corresponds to negative interests: we consider this possibility because it has two interesting interpretations. Firstly, the phenomenon of having negative interests has been recently observed in the financial world. Secondly, it can be interpreted as a term of fading memory, which places more weight on more recent profits than on older ones, as described in Bischi et al. (2020a) (see also Bischi and Merlone (2017) and Bischi et al. (2020b) for evolutionary games with fading memory). In the particular case of δ=0 we have that π∗ iin (11) is equal to π∗ iin (8), which is the original setup in Bischi et al. (2013a). Note also that as the integrand in (11) is non-negative so are average profits π∗ i. Expressing Xi(t+τ), with τ∈[0,s], as a function of Xi(t)using the formula in (10) and computing the integral in (11), the (discounted/capitalized) average profits in the period [t,t+s)can be assessed as π∗ i(Xi(t),ri(t)) =e−sδa2 i(1−θi(ri(t))) ki 4γiδseδs2F1(1, i(ri(t)) ;1+i(ri(t)) ;i(Xi(t),ri(t))) −2F11, i(ri(t)) ;1+i(ri(t)) ;e−s(1−θi(ri(t)))ρii(Xi(t),ri(t)),(12) where i(ri(t)) =δ ρi(1−θi(ri(t))) i(Xi(t),ri(t)) =Xi(t)−ki(1−θi(ri(t))) Xi(t) ,(13) 123 Hybrid dynamics of multi-species resource... and 2F1(·,·; ·; ·)is the Gaussian, or ordinary, hypergeometric function. In the particular case δ=0 (setup in Bischi et al. (2013a)), average profits over the non-switching period [t,t+s)are given by π∗ i(Xi(t),ri(t)) =a2 iki 4γiρis{(1−θi(ri(t))) ρis +log (Xi(t)−(1−θi(ri(t))) ki)e−(1−θi(ri(t)))ρis−Xi(t) −log ((θi(ri(t)) −1)ki)}.(14) Notice that average profits over the period [t,t+s)depend on Xi(t), the biomass of species iat time t,onθi(ri(t)) =Nri(t) ρi ai 2γi, where ri(t)is the share of fishers harvesting species iheld constant in [t,t+s), and on the length sof the period in which the two shares of fishers are fixed. Consider the average profits over the period [t,t+s)defined as in (11), where r1=rand r2=1−r. At each time t+s, with t∈S, the fraction ris updated accordingly. In fact, average profits are taken as a measure of fitness for the performance of each strategy, so that shares are updated according to replicator dynamics in discrete time, see Weibull (1995) and Hofbauer and Sigmund (1998), that is at each time t∈{s,2s,3s,...,ns,... },wehave: r(t+s)=r(t)π∗ 1(X1(t),r(t)) r(t)π∗ 1(X1(t),r(t)) +(1−r(t)) π∗ 2(X1(t),r(t)).(15) All in all, with a slight abuse of notation,3the original bio-economic hybrid model is given by ˙ X1(t),˙ X2(t),r(t)=L(X1(t),X2(t),r(t),X1(t−s),X2(t−s),r(t−s)) , (16) where L(X1(t),X2(t),r(t),X1(t−s),X2(t−s),r(t−s)) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ X1(t)ρ11−θ1(r(t)) −X1(t) k1 X2(t)ρ21−θ2(1−r(t)) −X2(t) k2 R(X1(t−s),X2(t−s),r(t−s)) (17) and R(X1(t),X2(t),r(t)) 3As the model is not differentiable at the switching times. 123 D. Radi et al. := ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ r(t)π∗ 1(X1(t),r(t)) r(t)π∗ 1(X1(t),r(t)) +(1−r(t)) π∗ 2(X2(t),r(t)) if t+s s=t+s s rt+s ssotherwise (18) to which the following parameter restrictions apply. It is possible to represent this hybrid model at times {0,s,2s,3s,...,ns,... } through a three-dimensional map by coupling the dynamics of the resources in (10) sampled at the discrete times of switching with the discrete replicator dynamics in (15). Then, the hybrid dynamical system can be reformulated recursively as a threedimensional map of the form: (X1(t+s),X2(t+s),r(t+s)) =G(X1(t),X2(t),r(t)) ,(19) where the function Gis defined as follows G(X1(t),X2(t),r(t)) := ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1−Nr(t) ρ1 a1 2γ1k1X1(t) X1(t)+1−Nr(t) ρ1 a1 2γ1k1−X1(t)e −1−Nr(t) ρ1 a1 2γ1ρ1s 1−N(1−r(t)) ρ2 a2 2γ2k2X2(t) X2(t)+1−N(1−r(t)) ρ2 a2 2γ2k2−X2(t)e −1−N(1−r(t)) ρ2 a2 2γ2ρ2s r(t)π∗ 1(X1(t),r(t)) r(t)π∗ 1(X1(t),r(t))+(1−r(t))π∗ 2(X2(t),r(t)) , (20) where t∈{0,s,2s,3s,...,ns,... }and with π∗ 2,i=1,2, defined as in (12)ifδ= 0 and as in (14)ifδ=0. Notice that the length sof the harvesting period in which shares of fishers are fixed (non-switching period) is a parameter of the model. By taking sas the unit of time or by operating the time rescaling z=t s, we can write the dynamical system (19)asa discrete dynamical system in standard form. By construction of the map G, the related discrete-time dynamical system (19) represents the hybrid dynamical system Lat any time t∈S. This proves the following proposition: Proposition 1 At any time t ∈S, the hybrid dynamical system (16)-(17) admits the discrete time representation given by map G (X1(t),X2(t),r(t)) in (19)-(20). Our conjecture is that any hybrid dynamical system admit a discrete time representation given a suitable construction of S. At the same time, it is easy to conclude that a hybrid dynamical system cannot have a representation in terms of a continuous-time dynamical system. Indeed, the function Lis not differentiable in r. Therefore, the 123 Hybrid dynamics of multi-species resource... continuous-time version of the discrete-time system (19), given by ˙ X1(t),˙ X2(t),˙r(t)=F(X1(t),X2(t),r(t)) ,(21) where Fis the vector of partial derivatives of G, cannot replicate the value of the hybrid model at any time t∈R+. The advantage of using a discrete-time replicator dynamics to represent the value of our hybrid dynamical system at time t∈Sis given by the possibility to explain the local and global dynamics of the hybrid model by means of the bifurcation theory for discrete-time dynamical systems. Therefore, in the next section we can provide a global analysis of the dynamics of the hybrid model and explain the mechanism through which equilibria lose stability. This analysis extends the results in Bischi et al. (2013a), where to provide some analytical results the dynamics of the hybrid system was approximated by means of the continuous-time dynamical system (21) to which we refer in the following as the benchmark model with continuous-time replicator dynamics. 4 Dynamic analysis The dynamical system (19) is analyzed in the set A=([0,+∞)×[0,+∞)×[0,1])/({(0,X2,0)|X2∈[0,+∞)} ∪{(X1,0,1)|X1∈[0,+∞)}∪{(0,0,r)|r∈[0,1]}).(22) The region of interest Ais an invariant set for the dynamical system (19), which admits some boundary equilibria under specific conditions on the value of the parameters. A boundary equilibrium is an equilibrium of the system that lies in one of the borders of A. Possible boundary equilibria of the dynamical system (19)are E1 1=0,k21−Na2 2γ2ρ2,0and E1 2=k11−Na1 2γ1ρ1,0,1, (23) characterizedbytheextinctionofthenon-harvestedspecies.Theseboundaryequilibria E1 i, with i=1,2, exist only under the specific condition Nai<2γiρi, with i=1,2, so that E1 ihas positive biomass in the harvested species i. The same condition Nai<2γiρi, with i=1,2, ensures the existence of another boundary equilibrium, which is characterized by harvesting of species ionly, and with the other species being unexploited and at carrying capacity. Consider i=1,2, these further boundary equilibria are E2 1=k1,k21−Na2 2γ2ρ2,0and E2 2=k11−Na1 2γ1ρ1,k2,1. (24) 123 D. Radi et al. temporal instants. The model takes its cue from Bischi et al. (2013a) by introducing a weighting function (that can be interpreted as a risk-free interest rate or a rate of fading memory) in the profit accumulation and discretizing the continuous dynamics of logistic growth by reformulating the model through a three-dimensional map of which the main dynamic properties are presented here. In particular, it is possible to determine the bifurcation values of the inner equilibrium and the paths that lead to greater dynamic complexity. As is natural, greater pressure in terms of a greater commercial value of a resource leads to greater exploitation which, therefore, leads to chaotic fluctuations of all resources involved in the specialized harvesting. From this point of view, even indirectly, the increase in the price of a resource can lead to fluctuations in another resource whose price remains constant if the latter is included withinthesamespecializedfishery. Alongside the phenomena of coexistenceand pathdependency, we find in this reformulation that the increase in the width of the interval in which harvesting remains focused on a species does not have a unique impact on the conservation of the resource, but its increase could be beneficial. Furthermore, we find thatinvestingprofitsat a risk-freeratecanhavea less conservativeandmorefluctuating effect on resources, whereas the opposite occurs when fading memory is assumed. The model in question is a possible starting point for further investigations on hybrid modeling in the exploitation of resources and provides a possible methodological contribution for the reformulation of some existing models in a more coherent way with the underlying bio-economic reality. Acknowledgements The Authors would like to thank Gian Italo Bischi for having introduced them to the study of hybrid dynamical systems and their applications and for having inspired the model that has been studied in this work. All remaining mistakes are our own. Funding Open access funding provided by Universitá della Calabria within the CRUI-CARE Agreement. The research was supported by the Czech Science Foundation (GACR) under Project 20-16701S and by VŠB-TU Ostrava under the SGS Project SP2021/15. Declarations Conflict of interest The authors declare that they have no conflict of interest. Open Access ThisarticleislicensedunderaCreativeCommonsAttribution4.0InternationalLicense,which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Appendix A Proof of Proposition 2Let us prove the stability conditions for equilibria E1 1and E2 1, as for equilibria E1 2and E2 2it is only necessary to swap the indices. Equilibrium 123 Hybrid dynamics of multi-species resource... E1 1=0,k21−Na2 2γ2ρ2,0is feasible when biomass of species 2 is nonnegative, that is when Na2 2γ2ρ2<1. In E1 1the Jacobian matrix of the system (19)is JE1 1= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ esρ100 0e−ρ2s+Na2s 2γ2 Na2k21−e Na2s 2γ2−ρ2s 2γ2ρ2 00 π∗ 1(0,0) π∗ 2k21−Na2 2γ2ρ2,0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,(30) where π∗ 1and π∗ 2are defined as in (11). Being the Jacobian matrix upper triangular, the eigenvalues are λE1 1 1=esρ1, which is always greater than one since sand ρ1 are positive parameters, λE1 1 2=e−ρ2s+Na2s 2γ2, which is always in (0,1)as E1 1∈A implies Na2<2γ2ρ2, and λE1 1 3=0, since π∗ 1=0 any time X1=0. This proves that equilibrium E1 1whenever exists is always a saddle point. Concerning equilibrium E2 1=k1,k21−Na2 2γ2ρ2,0is meaningful when E1 1exists, that is when Na2 2γ2ρ2< 1. At this equilibrium the Jacobian matrix of system (19)is JE2 1= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ e−sρ10a1k1N(e−sρ1−1) 2γ1ρ1 0e−ρ2s+Na2s 2γ2 Na2k21−e Na2s 2γ2−ρ2s 2γ2ρ2 00 π∗ 1(k1,0) π∗ 2k21−Na2 2γ2ρ2,0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,(31) where π∗ 1and π∗ 2are defined as in (11). Being the Jacobian upper triangular, the eigenvalues are λE2 1 1=esρ1, which belongs to (0,1)since sand ρ1are positive parameters, λE2 1 2=e−ρ2s+Na2s 2γ2, which is always in (0,1)as E2 1∈Aimplies Na2< 2γ2ρ2, and λE2 1 3=π∗ 1(k1,0) π∗ 2k21−Na2 2γ2ρ2,0, where π∗ 1(k1,0)=a2 1k1esδ−1 4γ1δ,(32) and π∗ 2k21−Na2 2γ2ρ2,0=a2 2k2(2γ2ρ2−a2N)esδ−1 8γ2 2ρ2δ>0,(33) 123 D. Radi et al. which is positive any time E2 1∈A. Therefore, λE2 1 3is in modulus less than one and E2 1 is a stable node provided that condition (25) holds. Clearly, when in (25) the reverse inequality holds, equilibrium E2 1is a saddle. This completes the proof.  Proof of Proposition 3From the equilibrium condition of the biomass, it is immediate to observe from (10) that the species imust be at the modified carrying capacity X∗ i=ki1−θir∗ i=ki1−Nr∗ i ρi ai 2γi,i=1,2,(34) where r∗ iis the equilibrium share of harvesters targeting species i. Employing the equilibrium biomass in the calculation of average profits in (11), it is then immediate to obtain the corresponding equilibrium profits for species i, that are π∗ iX∗ i,r∗= a2 11−Nr∗ i ρi ai 2γiki 4γi ,(35) when δ=0, and π∗ iX∗ i,r∗= a2 i1−Nr∗ i ρi ai 2γikieδs−1 4γiδs=π∗ iX∗ i,r∗eδs−1 δs,(36) when δ>0. By (15), an equilibrium share r∗ iof harvesters for species iis obtained by imposing the iso-profit condition π∗ 1X∗ 1,r∗=π∗ 2X∗ 2,r∗which shows the uniqueness of the equilibrium (27) for the dynamical system (19). This completes the proof.  References Aubin, J.P., Lygeros, J., Quincampoix, M., Sastry, S.S., Seube, N.: Impulse differential inclusions: a viability approach to hybrid systems. IEEE Trans. Autom. Control 47(1), 2–20 (2002) Bischi, G.I., Merlone, U.: Evolutionary minority games with memory. J. Evol. Econ. 27(5), 859–875 (2017) Bischi, G.I., Lamantia, F., Radi, D.: Multi-species exploitation with evolutionary switching of harvesting strategies. 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