Altenate stable states in coupled fishey-aquacultue systems by Melissa Oobko A thesis submitted in confomity with the equiements fo the degee of Maste of Science Depatment of Ecology & Evolutionay Biology Univesity of Toonto c Copyight 2016 by Melissa Oobko
Abstact Altenate stable states in coupled fishey-aquacultue systems Melissa Oobko Maste of Science Depatment of Ecology & Evolutionay Biology Univesity of Toonto 2016 Stagnating captue fisheies poduction has led to apid aquacultue development due to an inceasing shotfall of wild fish in meeting demand. Famed fish can negatively ecologically affect neaby wild fish and they can inteact economically by supplying simila poducts to the maket, but geneal theoy on how these inteactions affect fishey-aquacultue population dynamics and system esilience is lacking. I develop simple theoetical models integating the ecology, economics, and management of captue fisheies and aquacultue and demonstate that altenate stable states and hysteesis of equilibium population abundances can exist, indicating deceased system esilience. Abupt tansitions fom a captue fishey-dominated state to an aquacultue-dominated state with few o no wild fish can occu that ae difficult to evese. Empiical pattens of salmonid captue fisheies and aquacultue poduction data ae examined consideing the developed theoy, which may apply to othe systems with wild and domesticated species that ae coupled ecologically and economically. ii
Acknowledgements Fist and foemost, I would vey much like to thank my supeviso, Maty košek, and my co-supeviso, Péte Molná, fo thei invaluable contibutions to this eseach and fo thei vey helpful and positive guidance and suppot thoughout my gaduate pogam. Thei supevision has substantially contibuted to my academic and pofessional development and to a vey fulfilling gaduate education. I would like to thank the othe membes of my supevisoy committee, Maie-Josée Fotin and Nicole Mideo, fo thei insightful comments and questions that helped me appoach the eseach fom a boade pespective. I thank my examining committee, Maty, Pete, Nicole, Bian Shute, and Nick Mandak, fo thei thoughtful questions and fo the inteesting discussion about fisheies and aquacultue. I thank my košek and Molná lab mates fo thei feedback on my eseach, witing, and pesenting and fo inteesting convesations about science and consevation: Luke Roges, Lindsey Ogston, Stephanie Peacock, Andew Bateman, Sean Godwin, Pepijn Luijckx, Devin ik, Abby Daigle, Jessica Phillips, Dylan Shea, Juan Sebastián Vagas Soto, Alex Nascou, and Zach Mooe. I am also vey gateful to have had the oppotunity to spend two months at the wondeful Salmon Coast Field Station in BC whee I was exposed to eal fishey-aquacultue systems and the people who study them. The ideas that aose duing duing this time substantially impoved this eseach. I thank all of the people thee with whom I discussed my eseach; Pete Haington, in paticula, helped impove the obustness of the theoy developed in this wok. I would like to thank the Mideo lab fo thei vey helpful feedback on my eseach and fo eminding me to think of the biology of the models and not just the math: Nicole, Dave Smith, Cylita Guy, Megan Geischa, Philip Geenspoon, and Alison Wadlaw honouay membe). Finally, I thank my paents, the est of my family, and my good fiends fo thei suppot and enthusiastic encouagement of my academic studies thoughout the yeas. iii
Contents 1 Intoduction 1 2 Supply and Demand Model 8 2.1 Model Development.................................. 8 2.2 Model Analysis..................................... 14 3 Economic Rent Models 25 3.1 Model Development.................................. 25 3.2 Open Access Model................................... 28 3.2.1 Model Development.............................. 28 3.2.2 Model Analysis................................. 29 3.3 Rent-Optimization Model............................... 33 3.3.1 Model Development.............................. 33 3.3.2 Model Analysis................................. 35 4 Empiical Pattens 40 5 Discussion 46 A Appendix 55 A.1 Poof that the gowth function of the wild fish population is less than o equal to zeo when famed fish abundance is geate than o equal to /α........ 57 A.2 Deivation of MSYF)................................. 57 A.3 Deivation of fishing effot ate, EF)......................... 59 A.4 Deivation of df /dt in the supply and demand model............... 61 A.5 Deivation of wild and famed fish equilibium abundances, W and F, in the supply and demand model............................... 62 A.6 Deivation of the total poduction of fish, b 1 c d), in the ent models...... 71 A.7 Deivation of ent, RF,W), in the economic ent models.............. 71 A.8 Deivation of maginal pofitability, R F, in the ent-optimization model..... 73 A.9 Deivation of wild and famed fish equilibium abundances, W and F, in the ent-optimization model................................ 74 Bibliogaphy 83 iv
List of Tables 2.1 Vaiables, paametes, and aconyms in all models.................. 9 2.2 Wild and famed fish equilibium abundances, W and F, espectively, in the supply and demand model see A.5 fo deivations). In each ow, if the stength of the negative ecological effect of famed fish on wild fish, α, meets the given condition, then W and F occu ove the given egion of demand, D. These equilibia ae pesented gaphically in figues 2.2 and 2.4, whee equilibium banches match the numbeed expessions hee.................... 15 2.3 Citical values fo the diffeent models of coupled wild and famed fish dynamics when thee is a negative ecological effect of famed fish on wild fish, α > 0. SD = supply and demand, OA = open access, and RO = ent-optimization. Fo deivations of citical points, see sections A.5.2.2.2 and A.5.2.2.5 fo the supply and demand and open access models, and sections A.9.2.2.2 and A.9.2.2.5 fo the ent-optimization model.............................. 16 3.1 Wild and famed fish equilibium abundances, W and F, espectively, in the open access model. Equilibia ae the same as in the supply and demand model table 2.2), with the addition of equilibium banch 6. In each ow, if the stength of the negative ecological effect of famed fish on wild fish, α, meets the given condition, then W and F occu ove the given egion of demand, D. These equilibia ae pesented gaphically in figues 3.1 and 3.2, whee equilibium banches match the numbeed expessions hee.................... 30 3.2 Wild and famed fish equilibium abundances, W and F, espectively, in the ent-optimization model see A.9 fo deivations). In each ow, if the stength of the negative ecological effect of famed fish on wild fish, α, meets the given condition, then W and F occu ove the given egion of demand, D. These equilibia ae pesented gaphically in figues 3.3 and 3.4, whee equilibium banches match the numbeed expessions hee.................... 35 v
List of Figues 2.1 When famed fish have a negative ecological effect on the gowth ate of wild fish i.e., α > 0), the maximum sustainable yield of the wild fish population, M SY F ) equation 2.3), deceases quadatically to zeo as famed fish abundance, F, inceases to α...................................... 11 2.2 Altenate stable states of equilibium abundances of wild fish a) and famed fish b) can exist in the supply and demand model in esponse to demand fo fish, D. Numbeed equilibium banches coespond to the numbeed equilibium expessions in table 2.2. Thee is a captue fishey-dominated stable state of a lage wild fish population and no famed fish banch 1), an aquacultuedominated stable state of a lage famed fish population and eithe no o few wild fish banch 3), and an unstable state of modeate abundances of wild and famed fish banch 2). The equilibium abundances of wild fish and famed fish at the lowe bifucation point, demand = D bif, ae W bif and F bif, espectively see table 2.3). Wild fish ae extinct in the aquacultue-dominated stable state when demand exceeds D ext, which occus when F > α. Paamete values used wee = 0.01, = 10,000, h = 0.07, and α = /200 = 5x10 5.......... 17 2.3 Phase planes show the tansient dynamics of the system and the bounday dak gay line) between the altenate stable states solid cicles) at diffeent levels of demand within the egion that altenate stable states and hysteesis occu, D = 13.55 a), D = 17.1 b), and D = 20.1 c). The unstable states open cicles) ae depicted as dashed lines and stable states as solid lines in figues 2.2 and 2.4. The size of the basin of attaction and the esilience of the captue fisheydominated state shaded gay) deceases as demand inceases. Othe paamete values used wee = 0.01, = 10,000, h = 0.07, and α = /200 = 5.0x10 5.. 19 vi
2.4 Equilibium abundances of famed fish a) and wild fish b) in esponse to demand fo fish, D, ove a ange of fam effect stengths, α = 0, α = 2h = 1.43x10 5, α = 400 = 2.5x10 5, α = 200 = 5.0x10 5, and α = 50 = 2.0x10 4 in the supply and demand model. Othe paamete values that wee used wee = 0.01, = 10,000, and h = 0.07. Numbeed equilibium banches coespond to the numbeed equilibium expessions in table 2.2. When α > 2h, altenate stable states and hysteesis occus with thee possible solutions to meeting demand between the two bifucation points: a captue fishey-dominated stable state banch 1), an aquacultue-dominated stable state banch 3), and an intemediate unstable state banch 2). All aquacultue-dominated stable equilibium banches convege to F = D h and W = 0. When α 2h, changes in demand always esults in continuous changes in equilibium abundance banches 4 and 5). When α not shown), thee is a stable equilibium of no wild fish and a famed fish abundance of D h at all values of demand................. 21 2.5 Phase planes show the tansient dynamics of the system and the bounday dak gay line) between the altenate stable states solid cicles) at diffeent stengths of the ecological inteaction, α = 400 = 2.5x10 5 a), α = 200 = 5.0x10 5 b), and α = 50 = 2.0x10 4 c)fo a level of demand at which altenate stable states and hysteesis occu D = 21.9). The unstable states open cicles) ae depicted as dashed lines and stable states as solid lines in figue 2.4. The size of the basin of attaction and the esilience of the captue fishey-dominated state shaded gay) deceases as α inceases. Othe paamete values used wee = 0.01, = 10,000, and h = 0.07.................................. 22 3.1 Altenate stable states and hysteesis of equilibium abundances of wild fish a) and famed fish b) can exist in the open access model in esponse to demand fo fish, D. Numbeed equilibium banches coespond to the numbeed equilibium expessions in table 3.1. Thee is a captue fishey-dominated stable state of a lage wild fish population and no famed fish banch 1), an aquacultuedominated stable state of a lage famed fish population and eithe no o few wild fish banch 3), and an unstable state of modeate abundances of wild and famed fish banch 2). The equilibium abundances of wild fish and famed fish at the lowe bifucation point, demand = D bif, ae W bif and F bif, espectively see table 2.3). Wild fish ae extinct in the aquacultue-dominated stable state when demand exceeds D ext, which occus when F > α. An unstable equilibium state of no famed fish exists when D > 4, which esults in a conditionally stable wild fish equilibium state of 2 if and only if F = 0 banch 6). Paamete values used wee = 0.01, = 10,000, h = 0.07, and α = /200 = 5x10 5... 31 vii
3.2 Equilibium abundances of famed fish a) and wild fish b) in esponse to demand fo fish, D, ove a ange of fam effect stengths, α = 2000 = 5.0x10 6, α = 2h = 1.43x10 5, α = 400 = 2.5x10 5, α = 200 = 5.0x10 5, and α = 50 = 2.0x10 4 in the open access model. Othe paamete values used wee = 0.01, = 10,000, and h = 0.07. Numbeed equilibium banches coespond to the numbeed equilibium expessions in table 3.1. When α > 2h, altenate stable states and hysteesis occu with thee possible solutions to meeting demand between the two bifucation points: a captue fishey-dominated stable state banch 1), an aquacultue-dominated stable state banch 3), and an intemediate unstable state banch 2). All aquacultue-dominated stable equilibium banches convege to F = D h and W = 0. When α 2h, changes in demand always esult in continuous changes in equilibium abundance banches 4 and 5). When α not shown), thee is a stable equilibium of no wild fish and a famed fish abundance of D h at all values of demand. An unstable equilibium state of no famed fish exists when D > 4, which esults in a conditionally stable wild fish equilibium state of 2 if and only if F = 0 banch 6). This state exists fo all values of α............................. 32 3.3 Altenate stable states and hysteesis of equilibium abundances of wild fish a) and famed fish b) can exist in the ent-optimization model in esponse to demand fo fish, D. Numbeed equilibium banches coespond to the numbeed equilibium expessions in table 3.2. Thee is a captue fishey-dominated stable state of a lage wild fish population and no famed fish banch 1), an aquacultue-dominated stable state of a lage famed fish population and eithe no o few wild fish banch 3), and an unstable state of modeate abundances of wild and famed fish banch 2). Rent of the aquacultue industy is negative in the egion shaded gay. The equilibium abundances of wild fish and famed fish at the lowe bifucation point, demand = D bif, ae W bif and F bif, espectively see table 2.3). Wild fish ae extinct in the aquacultue-dominated stable state when demand exceeds D ext, which occus when F > α. Paamete values used wee = 0.01, = 10,000, h = 0.07, and α = /200 = 5x10 5.......... 36 viii
3.4 Equilibium abundances of wild fish fist column) and famed fish second column) in esponse to demand fo fish, D, ove a ange of fam effect stengths, α = 0 a - b), α = 2h = 1.43x10 5 c - d), α = 300 = 3.33x10 5 e - f), and α = 100 = 1.0x10 4 g - h) in the ent-optimization model. Othe paamete values used wee = 0.01, = 10,000, and h = 0.07. Rent of the aquacultue industy is negative in the egions shaded gay. Numbeed equilibium banches coespond to the numbeed equilibium expessions in table 3.1. When α > 2h e - h), altenate stable states and hysteesis occu with thee possible solutions to meeting demand between the two bifucation points: a captue fishey-dominated stable state banch 1), an aquacultue-dominated stable state banch 3), and an intemediate unstable state banch 2). All aquacultue-dominated stable equilibium banches convege to F = D 2h and W = 0. When α 2h a - d), changes in demand esult in continuous changes in equilibium abundance banches 4 and 5). When α not shown), thee is a stable equilibium of no wild fish and a famed fish abundance of D 2h at all values of demand........ 38 4.1 Poduction of captue fisheies poduction dak gay) and aquacultue poduction light gay) in the fist six counties of inteest fom 1950 to 2013. Aquacultue poduction is stacked on top of captue fisheies poduction in the thid column. The fouth column displays captue fisheies and aquacultue poduction as popotions of total poduction........................ 41 4.2 Poduction of captue fisheies poduction dak gay) and aquacultue poduction light gay) in the last six counties of inteest fom 1950 to 2013. Aquacultue poduction is stacked on top of captue fisheies poduction in the thid column. The fouth column displays captue fisheies and aquacultue poduction as popotions of total poduction........................ 42 ix
1 Intoduction Global fish poduction is cuently in the midst of a damatic tansition fom captue fisheiesto aquacultue-based poduction Natale et al., 2013; Bostock et al., 2010). Most wild fish stocks ae cuently fully exploited o oveexploited and substantial inceases in captue fisheies poduction ae unlikely Wom et al., 2009; Godfay et al., 2010; FAO, 2014). Demand fo fish, howeve, has been inceasing and will continue to do so with inceasing human population gowth and affluence FAO, 2014). Stagnating captue fisheies poduction in combination with inceasing demand fo fish has stimulated massive gowth of aquacultue ove the past seveal decades Duate et al., 2009; Naylo et al., 2000; FAO, 2014); it is the fastest gowing food industy and apid gowth is pojected to continue Subasinghe et al., 2009; Bostock et al., 2010; Béné et al., 2015). Coastal ecosystems, in paticula, habou lage aquacultue activity due to favouable envionmental conditions fo fish gowth Pimavea, 2006). Aquacultue has the potential to elieve havesting pessue on wild fish populations and incease nouishment and food secuity woldwide Naylo et al., 2000; Subasinghe et al., 2009; Duate et al., 2009). Howeve, lage populations of famed fish in maine ecosystems can inteact with wild fish populations ecologically and economically Natale et al., 2013). Captue fishey-aquacultue systems ae, theefoe, coupled human and natual systems, which ae known to have complex system dynamics and ae expeiencing inceasingly stong levels of human influence Liu et al., 2007). 1
1 Intoduction 2 Wild and famed fish can inteact ecologically in a vaiety of ways Pimavea, 2006). Lage stocks of famed fish can have negative ecological effects on neaby wild fish Natale et al., 2013), especially when famed fish ae in open-net sea cages though which wate can feely flow between the nets and the envionment. Disease can spead between wild and famed fish, which can lead to outbeaks on fish fams due to high host densities, and spill-back of these diseases to wild fish populations can occu košek et al., 2007). Famed fish can escape and intebeed with wild fish, which can esult in genetic intogession and educed adaptiveness of wild fish to thei envionment Naylo et al., 2005). Lage famed fish populations can deposit lage amounts of waste into the envionment, which has been linked to toxic algal blooms Pimavea, 2006). Thee is evidence that these types of ecological effects can tanslate to population-level negative impacts of fish faming on neaby wild fish populations, such as in wild salmon populations in many egions aound the wold that ae exposed to salmon fams Fod & Myes, 2008). Faming canivoous fish species can also negatively affect wild fish populations though a eliance on wild fish fo feed, with wild fish inputs often exceeding aquacultue outputs Naylo et al., 2000; Pimavea, 2006; Natale et al., 2013). Aquacultue has been a contibuting facto to lage human impacts on coastal ecosystems Halpen et al., 2008). Many stategies have been implemented to educe the ecological and envionmental impacts of aquacultue, impove sustainability of the industy, and educe effects on wild fish populations Bostock et al., 2010). Technology continues to develop that enhances spatial sepaation of wild and famed fish though closed containment systems and moving opeations offshoe Bostock et al., 2010), and that impoves feed-to-food efficiencies Tacon & Metian, 2008). Some aquacultue can potentially impove wild fish stocks though techniques such as fish anching and habitat modification, but it is not clea that these can have substantial net positive impacts on wild fish populations Duate et al., 2009). When consideing othe types
1 Intoduction 3 of food, maine aquacultue impacts ae modest in compaison to the impacts of teestial agicultue in tems of nutient unoff to the suounding envionment, habitat modification, nitogen-use efficiency, and antibiotic use Duate et al., 2009). Nonetheless, lage stocks of famed fish in maine ecosystems can potentially have negative ecological impacts on wild fish populations. Aquacultue and captue fishey industies ae also known to inteact economically and the stength of these inteactions ae inceasing ove time as aquacultue continues to apidly incease Natale et al., 2013). Wild and famed fish can inteact economically though supplying a simila poduct to the maket, whee poduction of each type of fish can affect the pice and stock size of the othe Andeson, 1985; Asche et al., 2001). Theoy has shown that aquacultue development can educe pessue on wild fish populations and potentially allow fisheies to ebuild Andeson, 1985; Ye & Beddington, 1996). Howeve, competition of wild and famed fish in the maket can decease the economic pofitability of captue fishey industies due to inceased total supply fom aquacultue gowth, which has occued in an Alaskan captue fishey industy Valdeama & Andeson, 2010). The gowth of the aquacultue industy is stongly affected by total demand fo fish and total supply of fish Bostock et al., 2010), which compises supply fom famed fish and fom wild fish populations. Fishey-aquacultue systems that ae coupled ecologically and economically ae coupled human and natual systems, which tend to be dynamically complex with nonlinea inteactions between the system components Liu et al., 2007). Thesholds, feedback loops, and time lags can exist that can lead to altenate stable states, hysteesis, and deceased system esilience Liu et al., 2007). A system exhibits altenate stable states and hysteesis when it can exist in fundamentally diffeent states fo a given set of envionmental conditions that ae stabilized by einfocing feedbacks Holling, 1973; Scheffe et al., 2001). These systems do not smoothly
1 Intoduction 4 change in esponse to envionmental change; instead, the system abuptly tansitions fom one stable state to anothe once theshold bifucation points of envionmental conditions ae supassed. These tansitions tend to be unexpected due to the discontinuous natue of the system esponse and ae difficult to evese. One of the chaacteistic featues of hysteesis is that evesing a tansition to a stable state equies a change in envionmental conditions that is lage than the initial change that caused the tansition Scheffe et al., 2001). Altenate stable states have been pedicted and obseved in many diffeent types of ecosystems including coal eef communities, feshwate lakes, and desets Sutheland, 1990; Capente et al., 1999; Scheffe et al., 2001; Beisne et al., 2003; Folke et al., 2004), and many studies have focused on thei existence in maine ecosystems nowlton, 2004; Petaitis & Dudgeon, 2004; DeYoung et al., 2008; ing et al., 2014; Gådmak et al., 2014; Convesi et al., 2015). The existence of altenate stable states and hysteesis in a system means that it has low esilience, which can be defined as the capacity of the system to maintain its fundamental stuctue, function, and feedbacks afte a distubance Walke & Meyes, 2004). Deceased esilience in ecosystems means the system is moe vulneable to expeiencing damatic changes in esponse to even small petubations Folke et al., 2004). Reseach on coupled human and natual systems demonstates the necessity of integating diffeent types of inteactions between system components to fully undestand the dynamics of the system Liu et al., 2007). I hypothesize that ecological and economic inteactions between wild and famed fish can lead to the existence of altenate stable states and hysteesis in coupled captue fisheyaquacultue systems. Aquacultue gowth is at least patially diven by the diffeence between demand fo fish and total supply of fish Bostock et al., 2010), which can be met by both wild and famed fish. Stagnating captue fisheies poduction thus stimulates the gowth of aquacultue. Howeve, if thee is a net negative ecological effect of famed fish on wild fish,
1 Intoduction 5 then aquacultue development will decease the gowth of wild fish populations, deceasing the supply of wild fish, and would esult in futhe incentive fo aquacultue gowth to meet demand. Eventually, an aquacultue-dominated stable state may be eached whee famed fish supply most demand and suppess the population size of wild fish. Howeve, if the supply fom wild fish is sufficient to meet demand, then thee is no economic incentive fo aquacultue development. A small o nonexistent famed fish population would not substantially decease the supply fom wild fish, and incentive fo aquacultue development would emain low, esulting in a captue fishey-dominated stable state whee wild fish supply most o all demand and famed fish ae not needed. If these altenate stable states occu in coupled fishey-aquacultue systems, then abupt tansitions between the states can occu that ae difficult to pedict and evese and that may diffe in thei ecological, economical, and social desiability e.g., Scheffe et al., 2001). We cuently lack foundational theoy to undestand how inteactions in coupled fisheyaquacultue systems affect wild and famed fish population dynamics and system esilience. Pevious studies have investigated the effects of ecological inteactions between wild and famed fish in specific systems e.g., košek et al., 2007) and economic inteactions between wild and famed fish Andeson, 1985; Ye & Beddington, 1996). Seveal bioeconomic models have been developed that incopoate both ecological and economic inteactions along with the management contexts of wild and famed fish Hannesson, 2003; Hoagland et al., 2003; Mikkelsen, 2007; Pomeoy et al., 2008; Jiang, 2010; Liu et al., 2013). These models conside ecological effects of famed fish on wild fish though a eduction in the caying capacity Hoagland et al., 2003; Mikkelsen, 2007), a eduction in the intinsic ate of incease o catchability of wild fish Mikkelsen, 2007), genetic intogession of escaped famed fish Liu et al., 2013), o consumption of wild fish by canivoous famed fish Hannesson, 2003). They also conside local o maket
1 Intoduction 6 economic inteactions between the populations, and seveal investigate the effects of industy management on system dynamics. Howeve, none of these studies have included ecological and economic inteactions and sustainable havesting of wild fish in geneal and analytically tactable models of coupled fishey-aquacultue systems, and none have identified altenate stable states and hysteesis in these systems. Hee, I develop theoetical models that integate the ecology, economics, and management of wild and famed fish to exploe the dynamics and esilience of coupled fishey-aquacultue systems. I fist develop a model with a simple negative ecological effect of famed fish on wild fish, sustainable havesting of wild fish, and a simple economic inteaction between the populations whee the diffeence between demand and total supply of fish dives aquacultue gowth. I then use the famewok of economic ent to include explicit economic factos that dive aquacultue gowth and incopoate two diffeent stategies of aquacultue gowth, an open access scenaio whee pofits dissipate due to ational self-inteest of fish fames and a ent-optimization scenaio whee aquacultue pofits ae maximized. The models ae solved analytically and numeically fo equilibium abundances of wild and famed fish in esponse to demand, and to detemine how the stength of the ecological effect of famed fish on wild fish and the stategy of aquacultue gowh affect system dynamics and esilience. Unde cetain conditions, the coupled captue fishey-aquacultue system exhibits altenate stable states and hysteesis with high wild fish abundance and low famed fish abundance in one stable state, high famed fish abundance and a small o extinct wild fish population in the othe stable state, and a modeate fishey and small aquacultue industy in an unstable state. These altenate stable states indicate deceased esilience of these coupled fishey-aquacultue systems, whee abupt, unexpected, and difficult-to-evese tansitions between the states can occu. The existence of altenate stable states and system esilience is highly affected by the stength of
1 Intoduction 7 the ecological inteaction and only modeately affected by the stategy of aquacultue gowth. Using salmonid captue fisheies and aquacultue as a case study, global pattens of captue fisheies and aquacultue poduction ae obseved that ae consistent with the theoy developed hee.
2 Supply and Demand Model 2.1 Model Development I fist develop a simple deteministic model of a coupled fishey-aquacultue system with a population of wild fish and a population of famed fish that inteact ecologically and economically. Thee is a simple negative ecological effect of fish faming on wild fish and gowth of the fish faming industy is diven by the gap between demand and total supply of fish. Wild fish ae havested sustainably to eflect cuent fisheies management Wom et al., 2009). Rates of change of wild and famed fish ae descibed using continuous diffeential equations. Wild fish population dynamics ae epesented by a modified poduction model fom Clak 2010), whee time notation is suppessed fo bevity. dw dt = GF, W ) HF, W ) 2.1) Wild fish abundance, W, inceases accoding to a gowth function, GF,W), and deceases accoding to a havest function, HF,W), which ae both functions of the abundance of wild fish and famed fish, F. See table 2.1 fo all model vaiables, paametes, and abbeviations. The gowth function, GF, W ), is a modified vesion of the Vehulst logistic gowth model Vehulst, 1938), GF, W ) = W 1 W ) αf W 2.2) 8
2 Supply and Demand Model 9 Table 2.1: Vaiables, paametes, and aconyms in all models. Symbol Definition Units α Rate of incease of pe capita motality of W fo each F. tonnes F 1 time 1 b The maximum amount of fish demanded in the ent models. tonnes time 1 b 1 c ) d Total poduction of fish at maket equilibium in the ent models. tonnes time 1 c Cost of poducing one unit of F in the ent models. $ tonnes F 1 d The maximum pice that would be paid fo a unit of fish in the ent models. $ tonnes 1 D Rate of demand fo fish. tonnes time 1 E Rate of fishing effot; the popotion of W that is havested. time 1 F Abundance of famed fish. tonnes F G Gowth ate of W. tonnes W time 1 h Rate of havest of F. time 1 H Rate of havest of W. tonnes W time 1 Caying capacity of W. tonnes W MSY Maximum sustainable yield of W with no ecological effect of F, 4. tonnes W time 1 MSY F ) Maximum sustainable yield of W with an ecological effect of F. tonnes W time 1 p Pice of one unit of fish in the ent models. $ tonnes 1 Intinsic ate of incease of W. time 1 R Rent, o pofitability, of the F industy. $ S F, S T Supply of famed fish, total fish. tonnes time 1 W Abundance of wild fish. tonnes W whee the wild fish population undegoes density-dependent logistic gowth with an intinsic ate of incease,, and envionmental caying capacity,. Thee is a simple negative ecological effect of famed fish on wild fish whee the pe capita motality ate of wild fish inceases in popotion to famed fish abundance at ate α, which can be consideed the stength of the fam effect. This eflects a density-dependent effect of famed fish on wild fish whee gowth of the wild fish population is educed as the abundance of famed fish inceases. The mechanism though which this effect occus is not specified fo the puposes of simplicity and geneality but it could be due to disease tansmission fom famed to wild fish. I assume that thee is no spatial stuctue, wild and famed fish do not diectly compete fo esouces, pedato-pey inteactions do not occu, paametes ae constant ove time, and stochasticity does not occu. The gowth function is zeo o negative fo any population size of wild fish when famed fish abundance equals o exceeds α, which is the atio of the intinsic ate of incease of the wild fish population to the stength of the ecological effect see Appendix A.1 fo poof). When
2 Supply and Demand Model 10 F α, the maximum possible gowth ate of the wild fish population,, is not lage enough to compensate fo the loss of wild fish fom the fam effect. As the stength of the ecological effect inceases o the intinsic ate of incease of wild fish deceases, α deceases and the gowth ate of the wild fish population becomes negative at a smalle famed fish abundance. If the stength of the fam effect is geate than the intinsic ate of incease α > ), the abundance of famed fish at which wild fish expeience negative gowth becomes less than one, meaning that even one unit of famed fish added to the system esults in a negative gowth ate of wild fish fo any population size of wild fish. The wild fishey is havested to achieve its maximum sustainable yield MSY). Despite the limitations of havesting at MSY Lakin, 1977), MSY-based havest egulation povides a simple intuitive logical famewok fo managing a natual esouce and eflects effots aound the wold to sustainably manage captue fisheies Wom et al., 2009), including egulation such as the Magnuson-Stevens Fishey Consevation and Management Act in the United States NOAA, 2007) and the Convention on the Law of the Sea by the United Nations United Nations Teaty Seies, 1982). The MSY is the havest ate that exactly balances the maximum gowth ate of the wild fish population. If famed fish have a negative ecological effect on wild fish, the abundance of famed fish educes the gowth ate of the wild fish population as specified in equation 2.2. This educes the MSY of wild fish, and the MSY as a function of famed fish abundance is called MSY F ) equation 2.3; see A.2 fo deivation). MSY F ) deceases quadatically fom 4 when thee ae no famed fish to zeo when famed fish abundance eaches α figue 2.1). When F > α, the gowth function of the wild fish population is less than zeo fo any wild fish abundance and MSY F ) is zeo.
2 Supply and Demand Model 11 0 if F α MSY F ) = α 2 4 F 2 α 2 F + 4 if F < α 2.3) Figue 2.1: When famed fish have a negative ecological effect on the gowth ate of wild fish i.e., α > 0), the maximum sustainable yield of the wild fish population, MSY F ) equation 2.3), deceases quadatically to zeo as famed fish abundance, F, inceases to α. Captue fishey-aquacultue systems ae also coupled economically since the two industies supply simila poducts to the maket. The model fomalizes simple intuition that the fish faming industy gows if thee is a gap between total supply of fish and demand. The fish faming industy gows in diect popotion to the diffeence between total supply of fish, which compises both wild and famed fish, and demand fo fish. df dt = D HF, W ) }{{} Wild fish supply }{{} hf Famed fish supply 2.4) Famed fish ae havested at ate h, which is the popotion of the famed fish population that can be havested at any time. This eflects that thee ae biological constaints on famed fish havest ates, such as that a fish stocked into a fish fam equies a peiod of time to gow to
2 Supply and Demand Model 12 a havestable size Aye & Tyedmes, 2009). Demand, D, is an economic contol vaiable that epesents the ate of demand fo fish. It is met by the total supply of fish, which compises the supply fom both wild and famed fish, HF, W ) and hf, espectively. Famed fish poduction inceases when demand is geate than total supply, and vice vesa. The mechanism by which this occus is not specified fo geneality, but may occu by fish fames enteing o leaving the industy, adding o emoving fams, o inceasing o deceasing the size o density of existing fams. The tems in equation 2.4 do not sepaately epesent change in famed fish abundance as they do in the equation fo wild fish gowth equation 2.1). Rathe, the tems epesent incentives fo gowth o eduction of the size of the fish faming industy due to ational self inteest of fish fames. The model assumes that gowth of the aquacultue industy esponds instantaneously to the gap between supply and demand and that aquacultue manages have pefect knowledge on all model paametes and vaiables. The havest ate of wild fish is equal to the ate of fishing effot, EF), multiplied by the abundance of wild fish equation 2.5), which is substituted into equation 2.1 to yield equation 2.6. HF, W ) = EF ) W 2.5) dw dt = W 1 W ) αf W } {{ } Gowth EF ) W }{{} Havest 2.6) Havest is popotional to wild fish abundance to avoid ovefishing. The fishing effot ate is the popotion of the wild fish population that is havested and is dependent on famed fish abundance due to the ecological inteaction between the populations equation 2.2). Fishing effot is
2 Supply and Demand Model 13 adjusted in esponse to demand to pevent excess unnecessay havesting and oveexploitation. If demand is less than the wild fish population s MSY F ), fishing effot is deceased to a level that would balance demand at equilibium so excess wild fish ae not havested unnecessaily equation 2.7; see A.3 fo deivation). If demand is geate than the wild fish population s M SY F ), then to pevent oveexploitation, the havest of wild fish is constained to thei MSY F ) and the supply of wild fish is insufficient to fully meet demand. At any level of demand, when famed fish abundance exceeds α then no havesting of the wild fish population is sustainable since the gowth function is negative and fishing effot is zeo in this case. If thee is no negative ecological effect of famed fish on wild fish i.e. α = 0), α does not exist so only the fist two cases in equation 2.7 apply. EF ) = 1 2 ) αf αf ) 2 4D if D < MSY F ) & F < α αf 2 if D MSY F ) & F < α 2.7) 0 if F α The havest ate of wild fish, EF ) W, is substituted into the equation fo famed fish population ate of change equation 2.4) to detemine famed fish dynamics moe explicitly equation 2.8; see A.4 fo deivation). If demand is geate than o equal to MSY F ) and famed fish abundance is less than α, wild fish ae havested at MSY F ) to pevent oveexploitation. Since this supply of wild fish is not sufficient to meet demand, thee is incentive fo famed fish to gow. If famed fish abundance is equal to o exceeds α, then the gowth ate of the wild fish population is equal to o less than zeo and thus the MSY F ) of the wild fish population is zeo. Since thee is no supply fom wild fish, famed fish simply gow accoding to the diffeence between demand and the supply of famed fish. The model assumes that fishey
2 Supply and Demand Model 14 manages have pefect knowledge of all model paametes and vaiables and that havesting is adjusted instantaneously in esponse to changes in these factos. df dt = D W 2 α 2 4 D hf ) αf αf ) 2 4D hf ) F 2 + α 2 h) F + D 4 ) if D < MSY F ) & F < α if D MSY F ) & F < α if F α 2.8) 2.2 Model Analysis To exploe the dynamics of the coupled captue fishey-aquacultue system, the equilibium abundances of wild and famed fish ae detemined in esponse to the economic contol vaiable, demand fo fish, D. The simplicity of the model enhances tactability and enables detemination of explicit solutions fo equilibium population abundances. The population ates of change equations 2.6 and 2.8) ae set to zeo and ae solved fo the equilibium population abundances of wild fish, W, and famed fish, F table 2.2; see A.5 fo deivation). The equilibium states of the system change depending on the level of demand and the stength of the negative ecological effect of famed fish on wild fish. Altenate stable states and hysteesis In the coupled captue fishey-aquacultue system, altenate stable states of wild and famed fish abundances and hysteesis can exist in esponse to demand fo fish figue 2.2). These altenate stable states consist of a captue fishey-dominated state with no famed fish and all demand met by wild fish, and an aquacultue-dominated state with a small o extinct wild fish population and most o all demand met by famed fish. These two states ae stabilized by ecological and economic feedbacks between the wild and famed fish populations. The only
2 Supply and Demand Model 15 Table 2.2: Wild and famed fish equilibium abundances, W and F, espectively, in the supply and demand model see A.5 fo deivations). In each ow, if the stength of the negative ecological effect of famed fish on wild fish, α, meets the given condition, then W and F occu ove the given egion of demand, D. These equilibia ae pesented gaphically in figues 2.2 and 2.4, whee equilibium banches match the numbeed expessions hee. Banch α Demand W F ) 1 any α D < 4 2 1 + 1 4D 0 2 α > 2h D bif D < 4 h α + 1 α 3 α > 2h D bif D < D ext h α 1 α h 2 αh + α2 D α 2h α 2 2 α 2 h 2 αh + α2 D α 2h α 2 + 2 α 2 3 α > 0 D D ext 0 D 4 α = 0 D 4 5 0 < α 2h 4 < D < D ext 2 h α 1 α h 1 h ) D 4 h 2 αh + α2 D α 2h α 2 + 2 α 2 h 2 αh + α2 h 2 αh + α2 h 2 αh + α2 D D D solution to meeting a low level of demand, D < D bif see table 2.3), is the captue fisheydominated state. If the system stats in a state of low demand, then the wild fish population can fully meet demand and thee is no need fo famed fish until demand exceeds the MSY of the wild fish population, 4. The only solution to meeting demand that exceeds 4 is the aquacultue-dominated state. When D D ext, wild fish go extinct in this state and all demand is met fom famed fish. Bifucation of the system occus at the two theshold demand points, demand = D bif and D = 4, esulting in hysteesis whee the equilibium cuves fold between these points and abupt discontinuous tansitions between the altenate stable states occu with small changes in demand nea the bifucation points. These tansitions ae difficult to evese due to einfocing ecological and economic feedbacks between wild and famed fish in these stable states. Between the theshold levels of demand, thee ae thee possible solutions fo meeting demand fo fish. Two solutions ae the stable captue fisheyand aquacultue-dominated stable states. The thid solution is an intemediate equilibium whee both wild and famed fish coexist at modeate population sizes, but this state is unstable
2 Supply and Demand Model 16 and difficult to maintain. Stability of the states is not poven fomally, but phase planes of wild fish abundance vesus famed fish abundance confim the intuition that the uppe and lowe equilibium banches ae stable and the banch connecting them is unstable see figues 2.2, 2.3, and 2.5). Table 2.3: Citical values fo the diffeent models of coupled wild and famed fish dynamics when thee is a negative ecological effect of famed fish on wild fish, α > 0. SD = supply and demand, OA = open access, and RO = ent-optimization. Fo deivations, see sections A.5.2.2.2 and A.5.2.2.5 fo the supply and demand and open access models, and sections A.9.2.2.2 and A.9.2.2.5 fo the ent-optimization model. D bif D ext F bif W bif SD & OA RO model Desciption Units models ) ) h α 1 h 4h α 3α 1 h α 12 Demand at which the lowe bifucation occus. tonnes t 1 h α ) α 1 2h α h α 2h α ) 2 3α 1 2h α Demand at and above which wild fish ae extinct in the aquacultue-dominated stable state. Abundance of famed fish at D bif. tonnes t 1 tonnes F 2h 3α + 6 Abundance of wild fish at D bif. tonnes W Feedbacks The altenate stable states ae stabilized by einfocing feedbacks between the wild and famed fish populations. If the system is in the captue fishey-dominated state and wild fish abundance inceases efe to figue 2.3), inceased fishing effot and deceased gowth equation 2.6) will cause the population to decease back to the stable state. If wild fish abundance deceases, deceased fishing effot and inceased gowth will cause the population to incease back to the stable state. If famed fish abundance inceases, total supply of wild and famed fish will incease above demand. Thee is then no incentive to maintain a famed fish population and the population will etun to the stable state of zeo famed fish. If demand is geate than the bifucation level of demand, D bif, then a cetain level of famed fish can be added to the system that disupts the feedbacks stabilizing the captue fishey-dominated state. When this
2 Supply and Demand Model 17 Figue 2.2: Altenate stable states of equilibium abundances of wild fish a) and famed fish b) can exist in the supply and demand model in esponse to demand fo fish, D. Numbeed equilibium banches coespond to the numbeed equilibium expessions in table 2.2. Thee is a captue fishey-dominated stable state of a lage wild fish population and no famed fish banch 1), an aquacultue-dominated stable state of a lage famed fish population and eithe no o few wild fish banch 3), and an unstable state of modeate abundances of wild and famed fish banch 2). The equilibium abundances of wild fish and famed fish at the lowe bifucation point, demand = D bif, ae W bif and F bif, espectively see table 2.3). Wild fish ae extinct in the aquacultue-dominated stable state when demand exceeds D ext, which occus when F > α. Paamete values used wee = 0.01, = 10,000, h = 0.07, and α = /200 = 5x10 5
2 Supply and Demand Model 18 happens, the MSY F ) of the wild fish population is educed by the famed fish to such a degee that the additional supply fom famed fish is less than the loss of wild fish supply. This means that moe famed fish must be added to compensate fo the loss of wild fish supply, which continues until the aquacultue-dominated stable state is eached. Once the system is in the aquacultue-dominated state, the high abundance of famed fish maintains the small o extinct wild fish population though the negative ecological effect of famed on wild fish. If famed fish abundance inceases, supply becomes highe than demand and thee is incentive fo the population to decease to the stable state. If famed fish abundance deceases, supply becomes smalle than demand and thee is incentive fo the population to incease to the stable state. The bounday between the altenate stable states The system will tansition between the altenate stable states if it cosses the bounday between them. If demand is within the bifucation points whee thee ae thee possible equilibium states fo any given level of demand see figue 2.2), the system will tansition between the altenate stable states when the bounday between them is cossed. The bounday changes with wild fish abundance, famed fish abundance, and demand fo fish figues 2.3 and 2.5). When the bounday is cossed, feedbacks that einfoce the cuent stable state abuptly switch to einfoce the altenate stable state. The egions on eithe side of the bounday ae the basins of attaction of the altenate stable states since the system will move towad a stable state if it is in that state s basin of attaction. Hee, esilience of a stable state is appoximated by the size of its basin of attaction; a lage basin of attaction means that the system is less likely to tansition to the othe stable state in esponse to petubations o stochasticity. If demand is just above the lowe bifucation demand level, D bif figue 2.3a), then changing the abundance of famed fish does not have a lage effect on the location of the bounday between
2 Supply and Demand Model 19 Figue 2.3: Phase planes show the tansient dynamics of the system and the bounday dak gay line) between the altenate stable states solid cicles) at diffeent levels of demand within the egion that altenate stable states and hysteesis occu, D = 13.55 a), D = 17.1 b), and D = 20.1 c). The unstable states open cicles) ae depicted as dashed lines and stable states as solid lines in figues 2.2 and 2.4. The size of the basin of attaction and the esilience of the captue fishey-dominated state shaded gay) deceases as demand inceases. Othe paamete values used wee = 0.01, = 10,000, h = 0.07, and α = /200 = 5.0x10 5. the stable states. This means that the stable state that the system moves towad is mainly detemined by the abundance of wild fish, and the esilience of the captue fishey-dominated state is high. As demand inceases figue 2.3b and 2.3c), the bounday between the altenate stable states becomes moe stongly influenced by the abundance of famed fish. Fo any given wild fish abundance, fewe famed fish ae equied to tip the system into the aquacultuedominated state. The basin of attaction and the esilience of the captue fishey-dominated state theefoe decease as demand inceases. Sensitivity to the negative ecological effect of famed fish on wild fish A stonge negative ecological effect of famed fish on wild fish educes the esilience of the system and of the captue fishey-dominated stable state. Hee, esilience of the system can be measued by the ange of demand ove which altenate stable states exist, which is D bif to
2 Supply and Demand Model 20 4. The system is moe esilient if this ange of demand is small o non-existent because it is less likely to fundamentally change states in esponse to petubations. Altenate stable states and hysteesis occu in the system once the stength of the ecological effect of famed fish on wild fish exceeds a theshold value, α > 2h figue 2.4). This indicates that the theshold α value is educed if the abundance of wild fish at the MSY ) 2, is lage. The theshold is also lowe if the havest ate of famed fish, h, is smalle since this means thee needs to be moe famed fish in the fams to supply the same amount of famed fish, leading to highe motality of wild fish. As the ecological effect inceases beyond this value, altenate stable states exist ove a lage ange of demand since the lowe bifucation point occus at a lowe demand level. This indicates deceased esilience of the system. The basin of attaction of the captue fisheydominated state is smalle figue 2.5) with stonge ecological coupling of the populations, indicating educed state esilience since the system can moe easily coss the bounday into the aquacultue-dominated state. If the system is in the aquacultue-dominated state, a stonge ecological effect esults in a smalle wild fish abundance fo any demand level and extinction of wild fish occus at lowe levels of demand. This means that coexistence of wild and famed fish in the aquacultue-dominated state occus ove a much smalle ange of demand. If the stength of the ecological effect is geate than o equal to the intinsic ate of incease of the wild fish population i.e. α ), then wild fish gowth can neve compensate fo the motality fom the pesence of famed fish, and extinction occus with any famed fish pesent in the system. In this state, famed fish supply all of the demand fo fish, and famed fish abundance is equal to D h since only a popotion of the famed fish population can be havested at any time. If the stength of the negative ecological effect of famed fish on wild fish is less than o equal to the theshold value of 2h, altenate stable states and hysteesis do not occu figue 2.4).
2 Supply and Demand Model 21 Figue 2.4: Equilibium abundances of famed fish a) and wild fish b) in esponse to demand fo fish, D, ove a ange of fam effect stengths, α = 0, α = 2h = 1.43x10 5, α = 400 = 2.5x10 5, α = 200 = 5.0x10 5, and α = 50 = 2.0x10 4 in the supply and demand model. Othe paamete values that wee used wee = 0.01, = 10,000, and h = 0.07. Numbeed equilibium banches coespond to the numbeed equilibium expessions in table 2.2. When α > 2h, altenate stable states and hysteesis occus with thee possible solutions to meeting demand between the two bifucation points: a captue fishey-dominated stable state banch 1), an aquacultue-dominated stable state banch 3), and an intemediate unstable state banch 2). All aquacultue-dominated stable equilibium banches convege to F = D h and W = 0. When α 2h, changes in demand always esults in continuous changes in equilibium abundance banches 4 and 5). When α not shown), thee is a stable equilibium of no wild fish and a famed fish abundance of D h at all values of demand.
2 Supply and Demand Model 22 Figue 2.5: Phase planes show the tansient dynamics of the system and the bounday dak gay line) between the altenate stable states solid cicles) at diffeent stengths of the ecological inteaction, α = 400 = 2.5x10 5 a), α = 200 = 5.0x10 5 b), and α = 50 = 2.0x10 4 c)fo a level of demand at which altenate stable states and hysteesis occu D = 21.9). The unstable states open cicles) ae depicted as dashed lines and stable states as solid lines in figue 2.4. The size of the basin of attaction and the esilience of the captue fishey-dominated state shaded gay) deceases as α inceases. Othe paamete values used wee = 0.01, = 10,000, and h = 0.07. Thee is a continuous change in the equilibium abundances of wild and famed fish ove all values of demand. If thee is no ecological inteaction between the populations, then the wild fish population is maintained at half of its caying capacity and famed fish abundance inceases popotionally to demand at a ate of 1 h. Sensitivity to othe paametes Othe model paametes also affect the qualitative dynamics of the system. The intinsic ate of incease of the wild fish population,, affects the demand level at which wild fish go extinct in the aquacultue-dominated stable state, D ext, but does not affect the theshold value of the ecological effect above which altenate stable states occu, α > 2h. This indicates that a fastegowing wild fish population cannot eadicate the existence of altenate stable states, but it can incease the level of demand needed to cause extinction of wild fish in the aquacultue-dominated
2 Supply and Demand Model 23 state. Caying capacity,, has the opposite effect; it affects the theshold fam effect stength above which altenate stable states occu but does not affect D ext. The havest ate of famed fish, h, highly influences system dynamics since it affects both D ext and the theshold value of the ecological effect above which altenate stable states occu. This is biologically intuitive since a highe fam havest ate means that a smalle abundance of fish in the fams ae needed to supply the same amount of famed fish, theeby educing the negative ecological effect on wild fish gowth. Summay Oveall, the esults fom the supply and demand model indicate that incopoating a negative ecological effect of famed fish on wild fish in addition to a simple economic inteaction between wild and famed fish populations leads to altenate stable states of wild and famed fish abundances if the ecological effect is stong enough. These altenate stable states epesent a loss of system esilience since the system can abuptly tansition between two fundamentally diffeent stable states, which has implications fo management of coupled captue fishey-aquacultue systems. If altenate stable states exist and demand is between the bifucation points, the thee possible system states can epesent thee diffeent management stategies fo meeting demand. The two stable states epesent solutions that, once eached, ae elatively easy to maintain due to einfocing feedbacks in the system. Howeve, coexistence of the populations is not possible in the captue fishey-dominated state and esults in a small o extinct wild fish population in the aquacultue-dominated state. Coexistence of both the captue fishey and aquacultue industies at modeate population sizes is only possible in the unstable state, but this state is difficult to maintain due to its unstable natue. Thus, management must balance a tade-off between coexistence of the populations and the ease of maintaining the system state.
2 Supply and Demand Model 24 This model consides demand fo fish as a constant economic paamete that is exogenous to the system. This assumption is intuitively appealing but may be unealistic because demand is often elated to othe factos such as total supply and pice. Demand as a constant contol vaiable ignoes the economic feedbacks between demand, poduction, and pice. Theefoe, the model is elaboated in the next section to include explicit economic paametes that incopoate moe ealistic economic factos that dive the gowth of the aquacultue industy.
3 Economic Rent Models 3.1 Model Development I elaboate the pevious model to include explicit economic factos that dive the gowth of the fish faming industy in ode to test how diffeent stategies of aquacultue gowth may affect system dynamics and esilience. These new models include the elationships among demand, poduction, pice, and consume pefeences fo fish by utilizing the famewok of economic ent that is commonly used in bioeconomic models of natual esouces Clak, 2010). Rent is the net pofitability of the famed fish industy and is equal to the diffeence between total evenues and costs. These costs include the total money spent to poduce the havest of famed fish and the oppotunity costs to fames, which is the income that could be eaned in othe compaable employment Godon, 1954). The advantages of using the ent famewok ae that the economic factos diving the industy s gowth ae explicitly specified and diffeent aquacultue gowth stategies based on the economics of the industy can be compaed. The intepetation of this moe complex model is guided by the esults fom the simple supply and demand model. Rent, RF, W ), is equal to the diffeence between total evenue and total cost of the industy. RF, W ) = ps T ) S }{{ F } c S }{{} F Total Revenue Total Cost 3.1) 25
3 Economic Rent Models 26 Total evenue is the supply of famed fish, S F, multiplied by the pice that each unit of fish can be sold fo, p. Pice and total supply of fish, S T, ae linealy elated, whee ps T ) = ) d 1 S T b Clak, 2010). This functional fom assumes that d is the maximum pice that consumes would be willing to pay fo one unit of fish if supply is vey low, b is the maximum amount of fish that is demanded if fish ae fee, and these two values ae linealy connected. The elationship between pice and total supply is the demand schedule fo fish that specifies the total amount of fish that consumes will puchase, S T, at each pice level, ps T ) see Clak, 2010). Total cost linealy inceases with the amount of famed fish poduced, S F, at ate c. Costs include any oppotunity costs of leaving the industy and pusuing othe employment. The poduction of famed fish, S F, is equal to the havest ate multiplied by the abundance of fish in the fish faming industy, o hf. Maginal cost as a function of the poduction of famed fish epesents the supply cuve fo the fish faming industy, which specifies the total amount of famed fish that the industy will poduce at each pice level see Clak, 2010, pp. 106-115). A poduction equilibium will occu when the supply pice equals the demand pice and the total amount of fish demanded equals the total amount of fish supplied. The total poduction of fish at which this equilibium occus is detemined by setting the demand schedule equal to the supply schedule and solving fo the total amount of fish poduced. The solution is S T = b 1 d) c see A.6 fo deivation). This epesents the ealized demand fo fish by consumes, which is validated by a late equilibium analysis, and it changes depending on the explicit economic paametes b, c, and d. Total ealized demand fo fish inceases when the maximum amount of fish that consumes would buy at any pice b) inceases, the cost of poducing famed fish c) deceases, o the maximum pice that consumes would be willing to pay fo one unit of fish d) inceases. In othe wods,
3 Economic Rent Models 27 ealized demand inceases when the maximum possible demand and the maximum possible pice incease, and the cost of famed fish poduction deceases. Wild fish dynamics ae the same as in the supply and demand model see section 2.1) except that demand is eplaced by ealized demand, b 1 d) c. dw dt = W 1 W ) αf W }{{} Gowth EF )W = 0 3.2) }{{} Havest EF ) = 1 2 αf αf ) 2 4 b 1 d) ) c if b 1 c ) d < MSY F ) & F < α αf 2 if b 1 c ) d MSY F ) & F < α 3.3) 0 if F α As in the supply and demand model, the supply of wild fish is the havest ate of the population, EF ) W, and the supply of famed fish is hf. The fishing effot ate is substituted into equation 3.4, which is then simplified to yield expessions fo ent as functions of ealized demand and famed fish abundance see A.7 fo deivation). RF, W ) = d 1 1b ) EF )W + hf ) hf } {{ } Revenue c hf }{{} Costs 3.4)
3 Economic Rent Models 28 RF, W ) = dhf b dhf b [ ) b 1 c d W 2 αf [ αf ) 2 4 b 1 d) ) ] c hf α 2 4 ) F 2 + α 2 h) F + b 1 c d if b 1 c d) < MSY F ) & F < α ) ) ] 4 if b 1 c ) d MSY F ) & F < α dhf b [ ) ] b 1 c d hf if F α 3.5) I compae how two diffeent stategies of how an aquacultue industy may gow ove time, open access and ent-optimization, affect the dynamics and esilience of the coupled fisheyaquacultue system. The gowth of famed fish has a diffeent elationship with industy ent fo each scenaio. 3.2 Open Access Model 3.2.1 Model Development The fist stategy of how an aquacultue industy may gow ove time is an open access scenaio, which assumes that thee ae no egulatoy constaints on wokes enteing o leaving the industy. In this model, poduction inceases as long as the industy is pofitable, and vice vesa. This occus because oppotunity costs ae included in the total costs of the industy, so wokes will ente the industy and the abundance of famed fish will incease if the industy is pofitable since it povides a geate economic etun than othe industies Clak, 2010). The gowth ate of famed fish, df dt, is diectly popotional to the ent of the fish faming industy as defined in equation 3.5.
3 Economic Rent Models 29 df dt = RF, W ) = dhf b dhf b [ b 1 d) c W 2 αf [ α 2 4 αf ) 2 4 b 1 d) ) ] c hf ) F 2 + α 2 h) F + b 1 c d if b 1 c d) < MSY F ) & F < α ) ) ] 4 if b 1 c ) d MSY F ) & F < α dhf b [ ) ] b 1 c d hf if F α 3.6) This equation fo df dt is the same as the one in the supply and demand model equation 2.8) with two diffeences. The fist is that demand has been eplaced with the tem b 1 c d), which as descibed above is the ealized demand that esults fom a poduction equilibium between supply and demand. The second diffeence is that the ent model equation has a facto of dhf b that is not pesent in the supply and demand model equation. The fact that including explicit economic paametes esults in simila dynamics as the supply and demand model validates the simple intuition of that model and indicates that the supply and demand model esults can guide intepetations of the moe economically ealistic open access model. 3.2.2 Model Analysis I analytically solve the open access model by setting equation 3.6 to zeo and simultaneously solving it and equation 3.2 fo equilibium population abundances of wild fish, W, and famed fish, F. The esulting equilibium abundances table 3.1) and system dynamics ae the same as in the supply and demand model see section 2.2) with two diffeences. The fist is that
3 Economic Rent Models 30 demand, D, in the supply and demand model is analogous to ealized demand, b 1 c d), in this open access model. Fo simplicity, ealized demand is simply efeed to as demand, D, in the est of this section. Table 3.1: Wild and famed fish equilibium abundances, W and F, espectively, in the open access model. Equilibia ae the same as in the supply and demand model table 2.2), with the addition of equilibium banch 6. In each ow, if the stength of the negative ecological effect of famed fish on wild fish, α, meets the given condition, then W and F occu ove the given egion of demand, D. These equilibia ae pesented gaphically in figues 3.1 and 3.2, whee equilibium banches match the numbeed expessions hee. Banch α Demand W F ) 1 any α D < 4 2 1 + 1 4D 0 2 α > 2h D bif D < 4 h α + 1 α 3 α > 2h D bif D < D ext h α 1 α h 2 αh + α2 D α 2h α 2 2 α 2 h 2 αh + α2 D α 2h α 2 + 2 α 2 3 α > 0 D D ext 0 D 4 α = 0 D 4 5 0 < α 2h 6 any α D > 4 4 < D < D ext 2 h α 1 α h 1 h ) D 4 h 2 αh + α2 D α 2h α 2 + 2 α 2 2 0 h 2 αh + α2 h 2 αh + α2 h 2 αh + α2 D D D The second diffeence is that thee is now an unstable equilibium of no famed fish and a conditional stable equilibium of wild fish when demand exceeds 4 figue 3.1), which esults fom the facto of dhf b that is pesent in df dt equation 3.6. This unstable equilibium exists because thee must be at least one unit of famed fish in ode fo the fish faming industy to be pofitable. If no famed fish ae added to the system, then the system will emain at this equilibium because an industy cannot be pofitable if thee ae no poducts to sell, and gowth is detemined by pofitability. The wild fish population will emain at a stable equilibium of 2 if and only if famed fish abundance emains at zeo. If even one unit of famed fish is added, then fish faming becomes pofitable and the system moves towad the aquacultue-dominated stable state.
3 Economic Rent Models 31 Figue 3.1: Altenate stable states and hysteesis of equilibium abundances of wild fish a) and famed fish b) can exist in the open access model in esponse to demand fo fish, D. Numbeed equilibium banches coespond to the numbeed equilibium expessions in table 3.1. Thee is a captue fishey-dominated stable state of a lage wild fish population and no famed fish banch 1), an aquacultue-dominated stable state of a lage famed fish population and eithe no o few wild fish banch 3), and an unstable state of modeate abundances of wild and famed fish banch 2). The equilibium abundances of wild fish and famed fish at the lowe bifucation point, demand = D bif, ae W bif and F bif, espectively see table 2.3). Wild fish ae extinct in the aquacultue-dominated stable state when demand exceeds D ext, which occus when F > α. An unstable equilibium state of no famed fish exists when D > 4, which esults in a conditionally stable wild fish equilibium state of 2 if and only if F = 0 banch 6). Paamete values used wee = 0.01, = 10,000, h = 0.07, and α = /200 = 5x10 5.
3 Economic Rent Models 32 Figue 3.2: Equilibium abundances of famed fish a) and wild fish b) in esponse to demand fo fish, D, ove a ange of fam effect stengths, α = 2000 = 5.0x10 6, α = 2h = 1.43x10 5, α = 400 = 2.5x10 5, α = 200 = 5.0x10 5, and α = 50 = 2.0x10 4 in the open access model. Othe paamete values used wee = 0.01, = 10,000, and h = 0.07. Numbeed equilibium banches coespond to the numbeed equilibium expessions in table 3.1. When α > 2h, altenate stable states and hysteesis occu with thee possible solutions to meeting demand between the two bifucation points: a captue fishey-dominated stable state banch 1), an aquacultue-dominated stable state banch 3), and an intemediate unstable state banch 2). All aquacultue-dominated stable equilibium banches convege to F = D h and W = 0. When α 2h, changes in demand always esult in continuous changes in equilibium abundance banches 4 and 5). When α not shown), thee is a stable equilibium of no wild fish and a famed fish abundance of D h at all values of demand. An unstable equilibium state of no famed fish exists when D > 4, which esults in a conditionally stable wild fish equilibium state of 2 if and only if F = 0 banch 6). This state exists fo all values of α.
3 Economic Rent Models 33 Vaying the stength of the negative ecological effect, α, of famed fish on wild fish has the same effect as in the supply and demand model see section 2.2), except that the unstable equilibium of zeo famed fish and the conditionally stable equilibium of wild fish at 2 exists fo all α figue 3.2). Feedbacks, tansient dynamics, the bounday between the altenate stable states, esilience, and sensitivity of the esults to model paametes ae assumed to be the same as in the supply and demand model. 3.3 Rent-Optimization Model 3.3.1 Model Development The second stategy of aquacultue industy gowth is one whee ent is optimized, o the industy attempts to poduce the maximum possible economic yield. This can only occu by limiting enty to the industy to pevent ent dissipation Schaefe, 1957). The simplest possible example of this is an industy with a sole owne who has full contol ove the abundance of famed fish and changes it until ent is maximized. The industy can also attempt to manage itself to optimize ent by contolling the abundance of famed fish among a goup of fish fames. Regadless of the mechanism by which net pofitability is maximized, the undelying elationship between ent and famed fish abundance may not be known. Theefoe, to maximize pofitability, famed fish abundance may be adjusted accoding to net maginal pofitability. If adding a unit of famed fish to the cuent famed fish population size inceases pofitability, then it is logical to continue adding famed fish until pofitability eithe does not incease anymoe o becomes zeo. This holds fo emoving a unit of famed fish, as well. If adding a unit of famed fish to the cuent famed fish population size deceases pofitability, then it is logical to ty emoving a unit of famed fish to see if that inceases pofitability, and the
3 Economic Rent Models 34 opposite applies fo emoving a unit of famed fish. The abundance of famed fish should, theefoe, incease if maginal pofitability is positive and decease if maginal poductivity is negative until an equilibium is eached at maximal poductivity, which can be a local o global maximum. The ate of change of famed fish abundance in the ent-optimization model is diectly popotional to maginal pofitability of the fish faming industy, which is found by diffeentiating ent equation 3.5) with espect to famed fish abundance equation 3.7; see A.8 fo deivation). df dt = R F = dh b dh b [ b 1 c d) 2hF [ W F F 2 αf αf ) 2 4 b 1 d) ])] c if b 1 c ) d < MSY F ) & F < α [ 3α 2 4 ) F 2 + α 2h) F + b ] 1 d) c 4 if b 1 c d) MSY F ) & F < α dh b [ ) ] b 1 c d 2hF if F α 3.7) The equilibium poduction amount of fish, b 1 d) c, is the amount of demand that would be ealized in an open access scenaio whee total supply exactly meets demand see section 3.1). In a ent-optimization scenaio, howeve, supply is less than demand at equilibium so that a high pice of fish is maintained and the industy expeiences maximum pofitability see Clak, 2010). In the emaining text, b 1 d) c is efeed to as demand, D, since it epesents the demand that would esult fom a maket equilibium whee total supply meets demand. It also means that esults can be diectly compaed with those fom the pevious models.
3 Economic Rent Models 35 3.3.2 Model Analysis To exploe the dynamics of the coupled wild and famed fish populations in a scenaio of entoptimization of the fish faming industy, I detemine the equilibium abundances of wild and famed fish in esponse to demand. Equations 3.2 and 3.7 ae set to zeo and the esulting equilibium abundances of wild and famed fish in tems of the model paametes and demand ae detemined table 3.2; see A.9 fo deivation). These equilibium population abundances occu when maginal pofitability is zeo, which can epesent a local minimum o a local o global maximum. Table 3.2: Wild and famed fish equilibium abundances, W and F, espectively, in the ent-optimization model see A.9 fo deivations). In each ow, if the stength of the negative ecological effect of famed fish on wild fish, α, meets the given condition, then W and F occu ove the given egion of demand, D. These equilibia ae pesented gaphically in figues 3.3 and 3.4, whee equilibium banches match the numbeed expessions hee. Banch α Demand W F ) 1 any α D < 4 2 1 + 1 4D 0 2 α > 2h D bif D < 4 6 + 2h 3α + 1 3α [ 2 3α 1 2h 4h 2 4αh + α2 2 4 + 3α2 D 1 α α 4h 2 4αh α2 2 4 + 3α2 ] D 3 α > 2h D bif D < D ext 6 + 2h 3α 1 3α [ 2 3α 1 2h 4h 2 4αh + α2 2 4 + 3α2 D 1 α 3 α > 0 D D ext 0 D 4 α = 0 D 4 2 2h 1 2h α + 4h 2 4αh α2 2 ) D 4 4 + 3α2 ] D 5 0 < α 2h 4 < D < D ext 6 + 2h 3α 1 3α 2 3α 4h 2 4αh + α2 2 4 + 3α2 D 1 α [ 1 2h α + 4h 2 4αh α2 2 4 + 3α2 ] D The qualitative behaviou of the system is simila to the esults fom the pevious two models in that altenate stable states exist between theshold levels of demand figue 3.3). Since the fish faming industy gows accoding to maginal poductivity, a ent minimum acts as an unstable equilibium. If famed fish abundance is above the ent minimum, then adding
3 Economic Rent Models 36 Figue 3.3: Altenate stable states and hysteesis of equilibium abundances of wild fish a) and famed fish b) can exist in the ent-optimization model in esponse to demand fo fish, D. Numbeed equilibium banches coespond to the numbeed equilibium expessions in table 3.2. Thee is a captue fishey-dominated stable state of a lage wild fish population and no famed fish banch 1), an aquacultue-dominated stable state of a lage famed fish population and eithe no o few wild fish banch 3), and an unstable state of modeate abundances of wild and famed fish banch 2). Rent of the aquacultue industy is negative in the egion shaded gay. The equilibium abundances of wild fish and famed fish at the lowe bifucation point, demand = D bif, ae W bif and F bif, espectively see table 2.3). Wild fish ae extinct in the aquacultue-dominated stable state when demand exceeds D ext, which occus when F > α. Paamete values used wee = 0.01, = 10,000, h = 0.07, and α = /200 = 5x10 5.
3 Economic Rent Models 37 a unit of famed fish inceases pofitability and dives the system to the aquacultue-dominated stable state, while the opposite is tue below the ent minimum and the system is diven to the captue fishey-dominated stable state. Thee ae fou key diffeences between the ent-optimization model and the open access model. 1) Altenate stable states exist ove a lage ange of demand in the ent-optimization model, which means that system esilience is deceased since tansitions between the altenate stable states can occu at lowe levels of demand. 2) The unstable equilibium abundance of famed fish is less than in the open access model, meaning that fewe famed fish ae equied to coss the bounday dividing the altenate stable states, and system esilience is also deceased. 3) Thee ae fewe famed fish and moe wild fish in the aquacultue-dominated state in the ent-optimization model, and extinction of wild fish occus at a highe level of demand than in the pevious models. This is a mutually beneficial esult fo both industies. Diffeences 1) and 2) indicate that an open access aquacultue industy is moe beneficial fo wild fish if demand is less than the MSY of the wild fish population since esilience of the captue fishey-dominated state is geate. Howeve, diffeence 3) indicates that ent-optimization of the aquacultue industy is moe beneficial fo wild fish if demand is geate than the MSY of the wild fish population since it esults in moe wild fish in the only possible stable state. 4) In the aquacultue-dominated state in the ent-optimization model, demand is not completely met in ode to incease economic pofitability of the aquacultue industy by maintaining a highe pice of fish. This epesents a social loss since less fish potein is supplied to society, which is detimental fo global food secuity. The stength of the ecological coupling affects dynamics in simila ways as in the pevious two models. As the stength inceases, altenate stable states exist ove a geate ange of demand, the equilibium abundance of famed fish in the aquacultue-dominated stable state
3 Economic Rent Models 38 Figue 3.4: Equilibium abundances of wild fish fist column) and famed fish second column) in esponse to demand fo fish, D, ove a ange of fam effect stengths, α = 0 a - b), α = 2h = 1.43x10 5 c - d), α = 300 = 3.33x10 5 e - f), and α = 100 = 1.0x10 4 g - h) in the entoptimization model. Othe paamete values used wee = 0.01, = 10,000, and h = 0.07. Rent of the aquacultue industy is negative in the egions shaded gay. Numbeed equilibium banches coespond to the numbeed equilibium expessions in table 3.1. When α > 2h e - h), altenate stable states and hysteesis occu with thee possible solutions to meeting demand between the two bifucation points: a captue fishey-dominated stable state banch 1), an aquacultue-dominated stable state banch 3), and an intemediate unstable state banch 2). All aquacultue-dominated stable equilibium banches convege to F = D 2h and W = 0. When α 2h a - d), changes in demand esult in continuous changes in equilibium abundance banches 4 and 5). When α not shown), thee is a stable equilibium of no wild fish and a famed fish abundance of D 2h at all values of demand.
3 Economic Rent Models 39 deceases and the amount of wild fish in that stable state inceases, and fewe famed fish ae needed to tansition to the aquacultue-dominated stable state figue 3.4). Feedbacks, tansient dynamics, the bounday between the altenate stable states, esilience, and sensitivity of the esults to model paametes ae assumed to be the same as in the supply and demand model.
4 Empiical Pattens The theoy developed in this wok has showed that altenate stable states of wild and famed fish abundances can occu in captue fishey-aquacultue systems that ae coupled ecologically and economically. The existence of altenate stable states and the esilience of these systems wee stongly affected by the stength of the negative ecological effect of famed fish on wild fish and somewhat affected by the stategy of aquacultue gowth. I now exploe captue fisheies and aquacultue poduction data aound the wold to investigate how the developed theoy may be linked to empiical pattens. Salmonids ae used as a case study because thee ae known ecological impacts of salmonid aquacultue on wild salmonid populations aound the wold Taange et al., 2015), including inceased motality of wild fish due to disease tansmission between famed and wild salmonids košek et al., 2007; Naylo & Buke, 2005). Wild and famed salmonids ae also known to inteact economically since they ae at least patially substitutable poducts, and gowth of the salmonid aquacultue industy is affected by the supply fom wild salmonid populations Andeson, 1985; Ye & Beddington, 1996; Valdeama & Andeson, 2010). To obseve tends in wild and famed salmonid populations whee native wild salmonid populations can potentially ecologically inteact with famed salmonids, I obtained data fom the Food and Agicultue Oganization of the United Nations FAO, 2015) on captue fisheies and aquacultue poduction of salmonids in seveal counties of inteest fom 1950 to 2013 whee ecological inteactions ae possible figues 4.1 and 4.2). These counties wee Canada, Denmak, the Faoe Islands, Finland, Iceland, 40
4 Empiical Pattens 41 Figue 4.1: Poduction of captue fisheies poduction dak gay) and aquacultue poduction light gay) in the fist six counties of inteest fom 1950 to 2013. Aquacultue poduction is stacked on top of captue fisheies poduction in the thid column. The fouth column displays captue fisheies and aquacultue poduction as popotions of total poduction.
4 Empiical Pattens 42 Figue 4.2: Poduction of captue fisheies poduction dak gay) and aquacultue poduction light gay) in the last six counties of inteest fom 1950 to 2013. Aquacultue poduction is stacked on top of captue fisheies poduction in the thid column. The fouth column displays captue fisheies and aquacultue poduction as popotions of total poduction.