Meeting yield and conservation objectives by balancing harvesting of juveniles and adults
Niklas L.P. Lundström,
Department of Mathematics and Mathematical Statistics,
Umeå University, 90187 Umeå, Sweden,
Sorbonne Universités, UPMC Univ Paris 6, UPEC, Univ Paris Diderot, Univ Paris-Est Créteil, CNRS, INRA, IRD, Institute of Ecology and Environmental Sciences-Paris (IEES Paris),
place jussieu, 75005 Paris, France,
College of Information Science and Engineering,
Shandong University of Science and Technology, Qingdao 266590, China,
Department of Ecology and Environmental Science,
Umeå University, 90187 Umeå, Sweden,
Division of Applied Mathematics, School of Education, Culture and Communication,
Mälardalen University, Box 883, SE-72123 Västerås, Sweden,
Evolution and Ecology Program,
International Institute for Applied Systems Analysis, 2361 Laxenburg, Austria.
Sustainable yields that are at least of the maximum sustainable yield are sometimes referred to as pretty good yield (PGY). The range of PGY harvesting strategies is generally broad and thus leaves room to account for additional objectives besides high yield. Here, we analyze stage-dependent harvesting strategies that realize PGY with conservation as a second objective. We show that (1) PGY harvesting strategies can give large conservation benefits and (2) equal harvesting rates of juveniles and adults is often a good strategy. These conclusions are based on trade-off curves between yield and four measures of conservation that form in two established population models, one age-structured and one stage-structured model, when considering different harvesting rates of juveniles and adults. These conclusions hold for a broad range of parameter settings, though our investigation of robustness also reveals that (3) predictions of the age-structured model are more sensitive to variations in parameter values than those of the stage-structured model. Finally, we find that (4) measures of stability that are often quite difficult to assess in the field (e.g. basic reproduction ratio and resilience) are systematically negatively correlated with impacts on biomass and impact on size structure, so that these later quantities can provide integrative signals to detect possible collapses.
Keywords fisheries management; size-structure; maximum sustainable yield; Pareto frontier;
pretty good yield; risk measure; selective harvest; stage-structured; age-structured.
Almost one third of the world’s fished marine stocks are currently overexploited (FAO 2016). Some fish stocks have even collapsed, with examples including the Californian sardine (Sardinops sagax, Jenyns) fishery in the 1950s (Radovich 1982, ), the Atlanto-Scandian herring (Clupea harengus, Linnaeus) fishery in the late 1960s (Krovnin and Rodionov 1992, ), the Peruvian anchovy (Engraulis ringens, Engraulidae) fishery in the 1970s (Clark 1977, ), and the Northern cod (Gadus morhua, Gadidae) fishery of the east coast of Canada in the 1990s (Hannesson 1996, ; Olsen et al. 2004, ). The large proportion of overexploited marine fish stocks underscore the importance of implementing sustainable harvesting practices and for further improving modern fisheries-management methods.
Maximum sustainable yield (MSY) has for long been a central concept in population ecology (Smith and Punt 2001, ; Hilborn 2007, ; Mesnil 2012, ). While maximization of yield from harvested populations is economically desirable, there is a rich scientific literature that criticizes the MSY concept and highlights its shortcomings, including the difficulty of correctly estimating MSY, the inappropriateness of long-term yield maximization as the single management objective, and the practical difficulty of accurately implementing the required level of harvesting effort (Smith and Punt 2001, ). MSY has further been criticized for its inability to prevent the collapse of important fisheries (Beverton and Holt 1957, ; Larkin 1997, ; Mangel and Levin 2005, ; Hilborn 2010, ). As an example, Alaska’s Bering Sea Pollock fishery declined in 2009, and despite being known as a sustainable fishery which implements scientific recommendations, the management has been criticized for considering mainly MSY (Morell 2009, ).
MacCall and Hilborn have introduced the concept pretty good yield (Hilborn 2010, ) for sustainable yields that are at least of the MSY. In contrast to MSY harvesting-management objectives, PGY can be realized by a range of harvesting strategies and therefore leaves room to account for other desirable objectives in addition to the maximization of yield. The added value that PGY offers will depend on the extent to which the implemented harvesting strategies can successfully account for other desirable objectives beyond yield.
The aim of this paper is to investigate to which extent PGY harvesting strategies can simultaneously account for high yield and large conservation benefits. To increase the chances that our conclusions are valid over a broad range of circumstances, we base our study on two established population models. The first is a version of the model used by Hilborn 2010 to introduce the PGY concept. It is an age-structured model (henceforth age model) that is commonly used for modeling fish populations and evaluating fishing strategies (Punt 1994, ; Punt et al. 1995, ; Punt and Hilborn 1997, ; Francis 1992, ; Hilborn 2010, ). The second is a stage-structured consumer-resource population model (henceforth stage model) that has been introduced by de Roos et al. (2008). Both models are capable of describing a range of aquatic and terrestrial animal populations. The stage model is derived from a fully size-structured counterpart with food-dependent growth, fecundity and maturation, and accounts for feedbacks from resource depletion. In particular, it accounts for ontogenetic asymmetry, i.e., differential abilities of juveniles and adults to utilize available resources (de Roos and Persson 2013, ).
We extend both models by introducing selective harvesting of juveniles and adults, giving wide ranges of possible harvesting strategies with different consequences for yield and conservation. While it is straightforward to quantify the yield of a harvesting strategy, it is less obvious how the conservations benefits should be measured. Here, we consider four different measures of conservation benefits: two measures that capture the direct impacts on the harvested population (the impact on population biomass and the impact on the populations size structure) and two measures that capture the indirect risks of collapse due to changes in population dynamics (resilience and the basic reproduction ratio). We determine trade-offs between yield and conservation benefits by finding the so-called Pareto-efficient front; the set of strategies that cannot simultaneously be improved upon in both yield and conservation benefit. These trade-off curves allow us to assess how large conservation benefits can be gained while preserving pretty good yield. Finally, we determine the relationship between the direct impact measures and the indirect risk measures, with the idea that the former are likely to be more easily observable in the field while the latter better reflect the risks of collapse. Taken together, our results show that there are large potential gains of using specific PGY harvesting strategies over traditional MSY strategies. Moreover, among PGY strategies, the ones that balance harvesting of adults and juveniles often allow the best compromises between conservation and yield.
In this section we first present the two population models, one age-structured and one stage-structured model. We extend both models by introducing selective harvesting of juveniles and adults, giving wide ranges of possible harvesting strategies with different consequences for the realized yield and for conservation. We next present our methods of stability analysis involving the impact measures and risk measures that we will use to evaluate different harvesting strategies. Finally, we recall the concept of maximum sustainable yield (MSY) and the economic concept of Pareto-efficiency which we will use to determine trade off curves between yield and conservation.
The age model
We adopt a simple age-structured population model,
which is similar to what has been widely used when modeling fish populations and evaluating fishing strategies
(Punt 1994, ; Punt et al. 1995, ; Punt and Hilborn 1997, ; Francis 1992, ; Hilborn 2010, ). The model relates recruitment to stock size by means of a Beverton-Holt stock-recruit relationship and accounts for recruitment variability explicitly. Natural mortality is assumed to be independent of age and time, and age-specific harvesting is assumed constant over time. The central elements of the model, which are mainly derived from Hilborn (2010), are described below.
The total egg production in year , , is
where is the number of individuals of age in year , is the fraction of the population of age which are mature females and is the number of eggs per mature female of age . The number of individuals of each age is updated according to
where is the survival from natural mortality, is the vulnerability of individuals to harvesting and is the recruitment in year , as described further below. We adopt a von Bertalanffy (1957) growth curve to describe individual length as a function of age, and assume that individual mass is proportional to the cube of individual length, i.e.
where is the mass of an individual at age , is the asymptotic maximum body mass, is a growth rate parameter and is the hypothetical negative age at which the individual has zero length.
We assume that individuals mature at age after which they reproduce at an age-dependent rate proportional to their body size. The age-dependent fraction of mature females and their corresponding fecundity then becomes
where is a positive constant and is the age where individuals mature and become adults. Individuals younger than are considered juveniles, while individuals older than or with age equal to are considered adults.
We incorporate stage-selective harvesting by allowing separate constant fractions harvested of juveniles () and adults () by setting the vulnerability of individuals to
The yield (catch expressed in biomass) in year is defined as
We assume Beverton-Holt recruitment (Beverton and Holt 1957, ),
where are independent and normally distributed random variables with mean 0 and standard deviation . The parameters and are given by
where is the steepness and sets the sensitivity of recruitment with respect to the egg production. The parameters and are, respectively, the average egg production and recruitment at equilibrium in the absence of harvesting mortality. The relation between and is given by
The parameter scales the number of individuals in the population as well as the size of the environment in which they live.
Concerning the parametrization of the model, we first note that without loss of generality we can choose , as well as the maximum size of an individual, , to 1. The results for arbitrary and can then be obtained by multiplying the yield and biomass, obtained for , by these quantities. Moreover, from (5) it follows that the proportionality constant in (The age model) can also be set to without loss of generality. In the results and discussion sections, we will consider variations of the values of the remaining parameters (age at which an individual has zero length), (growth rate parameter), (age at maturation), (sensitivity of recruitment with respect to egg production), (survival from natural mortality) and (randomness in recruitment).
The stage model
We adopt an archetypal consumer-resource model that has recently been introduced by de Roos et al. (2008) as a reliable approximation of a fully size-structured population model. The model is stage-structured and incorporates key aspects of individual life history such as food-dependent growth, maturation and fecundity. The central elements of this model are described below, with the detailed model formulation given in de Roos et al. (2008) and de Roos and Persson (2013). Individuals are composed into two stages; juveniles and adults, depending only on their size. Both juveniles and adults foraging on a shared resource . The juvenile biomass is denoted by while adult biomass is denoted by . Juveniles are born with size and grow until they reach the size at which they cease to grow, mature, and become adults. Juveniles use all available energy for growth and maturation, while adults do not grow and instead invest all their energy in reproduction. The juveniles growth rate, and adults reproduction rate, depend on resource abundance. In accordance with metabolic theory of ecology, foraging ability and metabolic requirements increase with individual body size (Brown et al. 2004, ). Juveniles and adults do not produce biomass when the energy intake is insufficient to cover maintenance requirements. The net biomass production per unit biomass for juveniles and adults equals the balance between ingestion and mass-specific metabolic rate according to
Here, represents the efficiency of resource ingestion, and the maximum juvenile and adult ingestion rates per unit biomass equal and , respectively. is half-saturation constant of the consumers. The factor describes the difference in ingestion rates between juveniles and adults.
We incorporate stage-selective harvesting on the stage model by allowing separate constant harvesting rates on juveniles and on adults , in addition to the natural mortality rates and . Finally, let be resource turn-over rate and the maximum resource density. The stage-structured population model consists of the following three ordinary differential equations.
for and . Moreover, the term represents biomass production through the growth of juveniles, is maturation of juveniles to adults (The function describes the maturation rate by determining how fast juvenile biomass is transferred to adult biomass) and represents reproduction from adults.
We have nondimensionalized and parameterized the stage model as in de Roos et al. (2008), to which we refer the reader for details and motivations. Here, we briefly note that without loss of generality we can choose , which fixes the environmental volume in which the population is assumed to live and thus scales the biomass densities , and . The time variable we scale with by introducing the new time variable . The resulting set of equations for the stage model is largely identical to (The stage model) except for that and the rate parameters , , , , and are expressed as multiples of .
The size at birth and the size at maturation appears only as the fraction in the model, and, therefore, we let . By following the proof of Meng et al. (2013, Theorem 3.3) we see that all results determined completely from the relative changes of the equilibrium due to harvesting do not depend on and , when keeping the product constant. Thus, in such case, only the resilience (which we measure from transient dynamics) may be affected by and , and, in particular, the equation for the resource reveals that resource dynamics will be slow for small and , and fast for large and . Roughly speaking, this means that resilience will be determined by the resource itself as , . This will only marginally affect our conclusions and hence we do not consider variations and , only in the product .
In the results and discussion sections, we will consider variations of the values of the remaining parameters , (natural mortality), (maximum resource density), (efficiency of resource ingestion times maximum ingestion rate), (ratio of mass of newborn and maximum sized individual) and (difference in ingestion rate between juveniles and adults).
Both the age model and the stage model are nonlinear dynamical systems and therefore complicated dynamics can not be ruled out simply. However, extensive numerical simulations indicate that solution trajectories end up, after sufficient time, at a globally stable equilibrium. This equilibrium is therefore the only attractor which is either an interior (positive) equilibrium or an extinction equilibrium depending on the harvesting rates. We refer the reader to Meng et al. (2013) and Roos et al. (2008) for more on dynamic properties and mathematical analysis of the stage model.
Stability analysis: measures of conservation
Stability of ecological systems is important for both conservation and harvesting purposes. In unstable systems, population dynamics may transiently go to low biomass values where the populations become vulnerable to demographic stochasticity or other factors. Hence, lack of stability promotes extinction. Stability is desirable also for harvest managers as it ensures stable yield. There are many definitions of stability, see e.g. McCann (2000), and we propose here to study the consequences of harvesting on stability through four different measures of conservation. The first two are impact measures; impact of harvesting on the population biomass and impact of harvesting on the population size structure. The second two measures have a natural link to the risk of extinction and will be referred to as risk measures. These are the resilience and the basic reproduction ratio (the recovery potential in case of the stage model).
We define the resilience as the reciprocal of the time needed for the population to recover from a perturbation, and we consider here both small and large perturbations.
The basic reproduction ratio/recovery potential describes the populations’s rate of increase from very low abundances, and thus could be construed as the likelihood of population rebound,
following a crash (e.g. due to a large disturbance).
Measures of impact on biomass and size structure
Let and denote the juvenile and adult biomass at equilibrium, respectively, of the harvested population. Let also and denote the juvenile and adult biomass at equilibrium in case of no harvesting pressure . Moreover, let and . We measure impact on biomass of harvesting through the expression
Similarly, we consider impact on size-structure through the expression
which considers the fraction of juvenile biomass in the population.
Resilience and basic reproduction ratio as risk measures
Resilience as a risk measure is increasingly used in ecology
(Pimm and Lawton 1977, ; Loreau and Behera 1999, ; Petchey et al. 2002, ; Montoya et al. 2006, ; Loeuille 2010, ; Valdovinos et al. 2010, ). Resilience is now also increasingly discussed in a fishery management context (Hsieh et al. 2006, ; Law et al. 2012, ; Fung et al. 2013, ). The higher the resilience, the smaller the risk of extinction due to random drift.
We consider resilience of the population as a measure of the risk of collapse by measuring the reciprocal of the time needed for the population to recover the positive equilibrium given a random perturbation. To describe the procedure, let denote the equilibrium of biomass in the stage model, and let , denote the equilibrium of number of individuals in the age model. For a given constant that scales the maximum displacement of the population from the equilibrium, we start a trajectory from a random point uniformly distributed in the cube
We then find the return time as the time needed for this trajectory to be close enough to the equilibrium in the sense that
for some small . After repeating this procedure times we find the resilience as a function of the harvest rates, , by taking the reciprocal of the mean value of the corresponding return times. It turns out that our results are not sensitive to the choices of and . In our simulations, we have used , and . Contrary to many works on resilience (that assess resilience based on eigenvalues of the Jacobian matrix) our approach is not limited to the immediate neighborhood of the equilibrium, but can also tackle large distributions.
We also consider the basic reproduction ratio as a risk measure, which represents the average number of offspring produced over the lifetime of an individual in the absence of density-dependent competition, i.e., when the population abundance is very low. For the age model, we derive the following expression for the basic reproduction ratio:
The derivation of expression (9) can be found in the appendix.
In case of the stage model we use the recovery potential introduced in Meng et al. 2013 (), which is the following analogue of the basic reproduction ratio for this continuous model:
The recovery potential is the generational net biomass production (per unit body mass) in a pristine environment (free from density-dependent mortality) and is therefore closely related to the basic reproductive ratio.
A basic reproduction ratio (recovery potential) larger than one ensures that the biomass of an initially small population increases on average, while a basic reproductive ratio (recovery potential) less than one implies that the population will eventually become extinct.
The basic reproduction ratio (recovery potential) is directly linked to the probability of surviving a period of low population abundance during which random drift caused by demographic stochasticity can lead to extinction.
We further motivate and discuss our choices of conservation measures in the discussion section.
Maximum sustainable yield and trade-off through Pareto efficiency
Recalling that and denote the juvenile and adult biomass at equilibrium, for any given harvest rates , , the yield objective function is given by
Moreover, the maximum sustainable yield (MSY) is obtained by taking the maximum of across all harvesting strategies . In addition to the yield function we are, in case of both the age model and the stage model, armed with four measures of conservation as functions of the harvest rates . Using these objective functions we can calculate both the yield and the conservation for given harvesting strategies, see Fig. 1 in the Results section.
To determine the trade-off between the two objectives yield and conservation, we plot the yield as a function of each conservation measure in the results section and apply the economic concept of Pareto efficiency to evaluate different harvesting strategies. A harvesting strategy is Pareto efficient if it cannot be improved upon without trading off one of the considered objectives against the other, see e.g. Karpagam (1999, page 11). The Pareto front is the set of all Pareto efficient harvesting strategies. Hence, managers can restrict the choice of harvesting strategy to this set, rather than considering the full range of possible harvesting strategies. The closer a strategy is to the Pareto front, the more efficient it is.
Figure 1 shows the four measures of conservation and the yield as functions of the harvest rates considering equal harvesting rates on juveniles and adults (henceforth equal harvesting), i.e. . As harvesting pressure increases, the yield first increases to reach MSY, after which it decreases as the population becomes “overexploited”. The impact on biomass and the impact on size-structure increase with harvesting pressure, while the basic reproduction ratio and the recovery potential decrease. The resilience decreases with harvest pressure in case of the age model, but first increases to a maximum and then decreases in case of the stage model.
We show trade-offs between yield and the four measures of conservation in Figs. 2 and 3. Figure 2 represents results from the age model, while Fig. 3 gives the corresponding results for the stage model. The dashed-dotted red curves correspond to harvesting adults only and the dashed yellow curves juveniles only. The solid black curves result from equal harvesting on both juveniles and adults. The solid green curve represents the Pareto front, while the dotted blue lines give the border for PGY, i.e. of MSY.
Pretty good yield allows large conservation benefits
Focusing on the stage model, Figs. 3 (c) and (d) show that harvesting for MSY (which is obtained by harvesting only adults) gives a resilience and a recovery potential that is only a tiny fraction of the unexploited state and is close to the boundary of extinction. Fig. 3 (b) shows also that the impact on size structure is at a maximum at MSY. Hence, harvesting for MSY may substantially increase the risk of stock collapse. Focusing on the age model, Figs. 2 (b)-(d) show slightly different results; the basic reproduction ratio is relatively low and the impact on size structure is also relatively large at MSY, while the resilience is relatively high at MSY. Common for both models and all four measures is, see Figs. 3 (a)-(d) and Figs. 2 (a)-(d), that by stepping back in yield by into the range of PGY, we can find harvesting strategies with nearly half the impacts on biomass and half the impact on size structure, and also with nearly twice the basic reproductive ratio (age model) and a much higher recovery potential (stage model). Resilience can also be improved in case of both models, though the difference in resilience is most impressive for the stage model, see Figs. 3 (c). Hence, both the age model and stage model give the result that PGY allows for large conservation benefits. Varying parameter values show that this conclusion is robust in both models.
However, stepping back in yield into the range of PGY does not automatically ensure conservation in terms of any of the measures we consider. To exclude the non-optimal harvesting strategies and to find the best ones within the range of PGY, we apply the economic concept of Pareto efficiency, as introduced in the previous section. Following the Pareto front (the set of all Pareto efficient harvesting strategies shown as the green curves in Figs. 2 and 3) reveals these preferable harvesting strategies. In the following, we will discuss simple harvesting strategies which are relatively close to the Pareto fronts in all cases.
Equal harvesting rates on juveniles and adults is often a good strategy
Figs. 2 and 3 show that equal harvesting , performs well with respect to both models and all four measures of conservation. In particular, in the range of PGY, the black curves come rather close to the Pareto front in all subfigures. Therefore, we can harvest juveniles and adults at equal rates, which should be strategies that are rather easy to implement, without losing too much yield or conservation. The black dots in Figs. 2 and 3 show one such strategy. Indeed, harvesting only adults is costly on some aspects, particulary in terms of resilience (stage model) and the impact on size structure as well as basic reproduction/recovery potential (both models). Harvesting only juveniles is costly in terms of resilience (both models) and in terms of impact on biomass (age model).
Varying parameter values show that this conclusion is very robust in the stage model, where it seems to remain in the wide ranges However, equal harvesting is often but not always suggested by the age model. Here, the efficient strategies seem to depend on the fraction of juveniles at the unharvested equilibrium, , as well as on the survival from natural mortality, . We proceed by investigatng this dependence by comparing pure adult harvesting, equal harvesting and pure juvenile harvesting for a wide range of parameter values in the age model. Figure 4 gives an approximation of regions in which the age model suggest pure adult harvesting, equal harvesting and pure juvenile harvesting. Pure juvenile or adult harvesting is suggested only if such strategies are the most Pareto-efficient once, within the range of PGY, with respect to all four conservation measures. The borders in Figure 4 are approximations which are produced by examining a large number of variants of Figure 2 for parameter values in the intervals The parameter intervals include several relevant parametrizations, e.g., , and which is used by Punt et al. (1995, page 290) studying the albacore (Thunnus alalunga, Scombridae), giving results in the pure adult harvesting range for most values of .
The age model is more sensitive to variations in parameter values than the stage model
Focusing on the age model we first note that for the parameter values used in Figs. 1 and 2 we have a survival from natural mortality of (Mills et al. 2002, ) and the fraction of juveniles in an unharvested population, . We conclude that in this case equal harvesting is a good strategy. Varying the parameter values, it turns out that an increase in the fraction of juveniles implies an increase in the yield obtained when harvesting only juveniles, i.e. the yellow curves will be lifted in Fig. 2. Similarly, a decrease in the fraction of juveniles implies an increase in the yield obtained when harvesting only adults, i.e. the red curves will be lifted in Fig. 2. This dependence, which is expected and natural, can be observed in both models, but it is much stronger in the age model. While Fig. 4 gives an approximation of the borders between pure adult harvesting, equal harvesting and pure juvenile harvesting, a similar investigation on the stage model gives a much larger region suggesting equal harvesting. In particular, in the stage model, the most Pareto-efficient strategies, within the range of PGY, seems to be dominated equal harvesting as long as . (For the parameter values used in Figs. 1 and 3, we have .)
In conclusion, for populations in the region where the age model suggests equal harvesting, the age model and the stage model agree on similar results. For populations outside of this region the age model suggests pure adult harvesting, or, for some rare parameter settings, pure juvenile harvesting.
Impact on size structure and impact on biomass serve as warning signals
As neither the resilience nor the basic reproduction ratio (recovery potential) can be directly measured in the field, it is important to identify reliable proxies for conservation management that can be measured in field surveys. Figs 5 and 6 show that a harvesting strategy with a high impact on population size structure, or a high impact on biomass, implies a low basic reproductive ratio (recovery potential) and a low resilience and hence a high risk of collapse. Indeed, we find that resilience and basic reproduction ratio (recovery potential) are systematically negatively correlated with impacts on biomass and size structure, so that these later quantities, which should be relatively easy to measure in field surveys, can provide integrative signals to detect possible collapses.
We have investigated how well stage-dependent harvesting strategies that qualify for pretty good yield (PGY) can account for conservation as a second objective. To increase the chances that our results apply to a broad range of populations, we have studied two established population models and reported conclusions that are common to both. We have also investigated a wide range of parameter values for both models. To incorporate conservation as a second objective for our optimization procedure, we have used four different measures of conservation applied to both the age model and the stage model. First, this extended analysis allows us to conclude strong robustness of the results where all measures agree for both models; e.g., that there are large potential gains of using specific PGY harvesting strategies that often, but not always, correspond to equal harvesting rates of juveniles and adults. Second, we are able to discuss and compare both the two models and the four measures of conservation with each other.
Implications for management of harvested populations
We advocate for management policies that explicitly accounts for conservation as a secondary objective besides high yield. Our study supports the implementation of PGY, which can readily be achieved through equal harvesting of juveniles and adults, in conjunction with regular surveys that aim to detect changes in population biomass and size structure. Managers aiming to implement optimal regulations rather than rely on such rules of thumb may want to parameterize the age and stage model (or other suitable population models) for the specific species in question. A similar analysis as the one presented here can then be carried out and will give the specific harvesting strategy that maximizes conservation benefits, e.g. as described by the four conservation measures considered here, for a given target-value of sustainable yield.
Managers relying on other management approaches may still be interested in assessing changes in the size structure of a population, as well as changes in biomass, as these are strongly linked to our risk measures and may thus serve as warning signals for an impending collapse. In fisheries management, changes in size structure and biomass can be measured through trial fishing, reinforcing our conclusion that size-structure and biomass are appropriate proxies for the risk of collapse and possible extinction.
Why harvest juveniles? Differences and similarities between the age model and the stage model
While harvesting individuals before they mature is a debated topic, we have seen that both the age model and the stage model give arguments for equal harvesting rates of juveniles and adults. Indeed, relying on the stage model this argument is robust with respect to variations in parameters values. The age model is more sensitive and the suggested harvesting strategy varies between mainly equal harvesting and pure adult harvesting as a function of parameter values. To understand these results we first recall that if it is possible to obtain pretty good yield for a wide range of harvesting strategies, then our conservation measures are in favor of equal harvesting. Therefore, we can focus the following discussion on when and why the models allow for such wide range of harvesting strategies.
By extensive numerical experiments we illustrate this dependence for the age model in Figure 4. Varying the parameters values in the age model, it turns out that an increase in the fraction of juveniles implies an increase in the yield obtained when harvesting only juveniles, and that a decrease in the fraction of juveniles implies an increase in the yield obtained when harvesting only adults. This is natural; when consider harvesting of a population consisting of mainly juveniles it is not possible to obtain a good yield by harvesting only adults. From Fig. 4 we also see that as the survival from natural mortality increases, the recommendation goes towards including more juveniles in the harvesting strategy. A reason for this is that for small additional mortality through harvesting on young individuals implies that too few individuals survive and become adults and the population declines.
A corresponding parameter dependence, as described in Fig. 4, is much weaker in case of the stage model. Indeed, a version of Fig. 4 for the stage model would almost always suggest equal harvesting. One reason for this difference in sensitivity of the recommended harvesting strategies between the two models, with respect to parameter values, is as follows. The stage model explicitly models the resource through the third equation in (The stage model), and reproduction, growth and maturity are assumed to be increasing functions of the resource. Therefore, removing adult or juvenile biomass through harvesting results in more resource available for the remaining population, which in turn increases biomass production trough all three mechanisms reproduction, growth and maturity of juveniles. This feedback implies that the dynamics of the stage model allows for wide ranges of efficient harvesting strategies.
On the other hand, the age model incorporates the Beverton-Holt spawner recruit curve in (5) for reproduction, and, independent of the recruitment, individuals are assumed to grow following the Bertalanffy growth curve in (2). Growth and recruitment are thus assumed to be independent in the age model, while they are dependent through the resource in the stage model. This means that if some juveniles are removed by harvesting it will not be in favor of the recruitment of newborns in case of the age model, as it would be in case of the stage model. Thus, it is more costly to harvest juveniles in the age model than in the stage model, and therefore the age model more often suggests to leave small individuals, let them grow, and catch them as adults.
In conclusion, the more extensive population-level feedbacks in the stage model makes the population productive for a wider range of harvesting strategies than the age model does, and the age model is more restrictive to juvenile harvesting than the stage model. This explains why equal harvesting performs well through wider ranges of parameter values in the stage model, than in the age model.
Importance of preserving population size structure
Our advice is based on our finding that large impacts on size-structure generally implies a high risk of collapse as captured by our risk measures, see Fig. 5 and 6. To reduce the impact of harvesting on population size structure, it seems advisable to harvest juveniles as well as adults, see Fig. 2 and 3. Thus, equal harvesting is more likely to preserve the size structure than single-stage harvesting. (A similar conclusion was reached by Jacobsen et al. 2014.) We have shown that large impacts on size structure typically indicate unfavorable readings of our risk measures. Our work thus reinforces the conclusions from a large and growing number of studies (considering both ecological and evolutionary aspects) that argue for the importance of preserving the size structure of harvested populations. These studies, which we discuss below, reinforce the importance of including impact on size structure explicitly as an important conservation measure when discussing harvesting strategies. In fact, not accounting for the impact on size structure explicitly in our analysis means that we should find the recommended harvesting strategies from Figs. 2 and 3 (a), (c) and (d) only, not including the Pareto efficiency in Figs. 2 (b) and 3 (b). This would result in a shift towards recommending adult harvesting, especially in case of the age model.
From an ecological point of view, Anderson et al. (2008) show that populations with a larger fraction of juveniles have less stable population dynamics because of changes in demographic parameters, and, therefore, suffers a larger risk of extinction. (In this context, see also Wikström et al. 2012.) Moreover, changes in the size-structure of the population affect the balance of intra- and interspecific competition (Loreau and Mazancourt 2013, ).
From an evolutionary point of view, affecting the size-structure of a population can potentially induce changes in biological traits such as size-at-age and age-at-maturation. One reason is that harvesting only large individuals creates a large mortality selective pressure so that only adults that reproduce early, and at small size, pass their genes (Dunlop et al. 2015, ). A consequence may be evolution toward small individuals reproducing early, which is generally not desirable from an ecological (low reproduction) nor from an economical (too small to be valuable) point of view (Grift et al 2003, ; Olsen et al. 2004, ). Evolutionary changes may have large impact on economic profit and future management (Conover and Munch 2002, ; Jorgensen et al. 2007, ; Belgrano and Fowler 2013, ), and may be difficult to reverse. Studying the collapse of the northern cod (Gadus morhua, Gadidae), it has been shown that, before government imposed a moratorium, the life history shifted towards maturation at earlier ages and at smaller sizes (Olsen et al. 2004, ), suggesting fisheries-induced evolution of maturation patterns. Moreover, a recent study provides experimental evidence for rapid evolution induced by changes in the population size-structure of a fished population (van Wijk et al. 2013, ). Significant genetic variation for production-related traits is also present in fished populations (Law 2000, ), and Cameron et al. (2013) experimentally demonstrate evolutionary changes, in response to harvesting juveniles or adults. In Kuparinen and Merilä (2007) the authors argue that we should stop targeting only large individuals to avoid evolutionary impact on fisheries. See also Garcia et al. (2012) and Law et al. (2012) for more arguments for balanced harvesting preserving populations size structure. In conclusion, we recommend that managers consider the impact on size structure and that they avoid large deviations from the size structure of a pristine, unharvested population.
Relations between the four measures of conservation
We have considered conservation as a second objective, beyond yield, in our optimization procedure. To quantify conservation we have chosen two impact measures; impact on biomass and impact on size structure, and two “risk” measures; resilience and basic reproduction ratio (recovery potential). It is not obviously true, even though it is expected, that the impact measures relate simply to the risk measures. Therefore, we present Figs 5 and 6 which show that a large impact on biomass, or size structure, implies a low basic reproductive ratio (recovery potential) and also a low resilience. From this fact we concluded that it is important to preserve size structure and biomass in order to preserve stability of the population, and that impact on biomass and impact on size structure work as warning signals for a collapse.
From Figs. 5 and 6 it is clear that the relations between the four measures of conservation are not simple. This is clearest from Fig. 6 (a) and (c) representing the stage model; resilience can vastly differ from the other measures by increasing with harvest pressure for some harvesting strategies. This phenomena, which bears resemblances to the paradox of enrichment (Rosenzweig 1971, ; Rip and McCann 2011, ), deserves attention since it is very strong in the stage model under equal harvesting, but not present at all under pure adult harvesting. This thus provides a substantial argument in favour of equal harvesting. Fig. 7 shows some trajectories for the stage model, as functions of resource and juvenile biomass , similar to those used for estimating resilience. In Fig. 7 (a) there is no harvest pressure, i.e. , giving a resilience of approximately 0.03. In Fig. 7 (b) giving a much higher resilience of . The trajectories in Fig. 7 (b) give nearly optimal resilience, see Fig. 3 (c). The dots on the black trajectories in Fig. 7 are located at equidistant timesteps. Therefore, the time needed for the trajectories, in both subfigures, to enter the circle surrounding the equilibrium is proportional to the number of dots, which can easily be compared. It is clear that the resource converges fast in both cases, but the juveniles biomass converges much slower when no harvest is present. In particular, it seems that without harvesting, the fraction of juveniles in the population, , converges rather slowly to its equilibrium, causing the increase in resilience with harvesting.
Comparing resilience simulations in Figs. 2 (c) and 3 (c) we conclude that harvesting of only adults is among the best strategies in the age model, while such strategy is among the worst using the stage model. The stage model instead suggest equal harvest rates. A figure similar to Fig. 7 but for the age model will show that trajectories will return faster to the equilibrium when harvesting only adults. Hence, transient behavior, and therefore the resilience, behaves different in the models.
Understanding relations between different conservation measures are interesting in general, and would be an interesting topic for future research. Indeed, measures of stability in ecological systems is today an active research area, see e.g. Neubert and Caswell (1997) for alternatives to resilience, Nimmo et al. (2015) for discussions of resistance and resilience, Isbell et al. (2015); Dunne et al. (2002) for stability and its relations to biodiversity, and Donohue et al. (2013) for discussions about how different measures are related or not.
Motivations of our choice of conservation measures
As our result may depend on our chosen conservation measures, we consider here additional motivations and discussions concerning this topic. First, the measures impact on biomass and impact on size structure are important to consider simply since they can be measured in reality. Second, these measures are natural, simple and easy to interpret and a large impact on biomass would certainly imply impact on the surrounding ecosystem. Moreover, in the subsection Importance of preserving population size structure, we further motivate impact on size structure as a central measure, based on the fact that populations size structure is important to preserve from both an ecological and evolutionary point of view.
To motivate the basic reproductive ratio (recovery potential) as a risk measure, we note that, as already mentioned in the methods section, these measures are directly linked to the probability of surviving a period of low population abundance during which random drift caused by demographic stochasticity can lead to extinction. To see this, consider a small population in a pristine environment in which all individuals are, for simplicity, assumed to be identical. In this case, the basic reproductive ratio (recovery potential) is simply the ratio between birth-rate and death-rate , that is . As proved by Grimmett and Stirzaker (1992, page 272), the probability of avoiding extinction through random drift is given by if , and zero if . Hence, there is a direct link between the basic reproduction ratio (recovery potential) and the probability of surviving a period of low population abundance; a high basic reproduction ratio (recovery potential) ensures a high probability of surviving. To justify the investigation of the effects of large disturbances that bring the population to small numbers, so that density dependence can be ignored, placing individuals in a pristine environment, we mention mass mortality events (Fey et al. 2015, ), drastic climate variability such as heat waves, storms, and floods (Reusch et al. 2005, ), and heavily exploited ecosystems (Jones and Schmitz 2009, ).
To motivate our choice of using resilience as a risk measure, we first note that when dealing with stability in nonlinear dynamical systems one usually first considers local stability. This can be done by locally linearizing the equations near the equilibrium, telling immediately if the equilibrium is stable or not. However, such approach gives only information arbitrary close to the equilibrium, saying little about the basin of attraction (the set of initial conditions attracted by the equilibrium) to the equilibrium. If the equilibrium is locally stable but the basin of attraction is small, then only a tiny perturbation can force the system to jump to another attractor, having a completely different behavior. Therefore, it is preferable if the basin of attraction is large and convex and it is natural to consider the size and the shape of the basin of attraction as stability/risk measures for nonlinear systems, see e.g., Lundström and Aidanpää (2007), Menck et al. (2013) and Lundström (2017). However, extensive numerical investigations indicate that, in both the age model and the stage model, the positive equilibrium is the unique globally stable attractor, which means that it attracts all positive initial conditions. Therefore, the basin of attraction is the whole positive space and a stability/risk measure based solely only on the size and shape of the basin of attraction does not give us any information here. We therefore proceed to the next natural candidate; the returntime to the equilibrium given a perturbation, and define (nonlocal) resilience as the reciprocal of the expected time needed for a trajectory to retain the equilibrium, given a perturbation. We consider both small and (relatively) large perturbations in terms of initial conditions (see Methods). In contrary to our approach, resilience is often defined through the largest eigenvalue of the locally linearized system. Such definition gives, however, as discussed above, only information arbitrary close to the equilibrium and can not invoke effects from large perturbations. We refer the reader to Lundström (2017) for further discussions, definitions and applications of similar nonlocal resilience measures. For further discussion on the use of local resilience in ecology and the fact that it can be difficult to assess from an empirical point of view, see Haegeman et al. (2016).
Topics for future research
In both the age model and the stage model we considered individuals in only two stages, as juveniles or as adults. We then considered harvesting strategies that allow for different mortality rates in these two stages. Using slightly generalized versions of the age model and the stage model, many more possible harvesting strategies can be explored. A natural first step is to consider harvesting on a size interval, and this can later be extended to include several size intervals as well as more realistic descriptions of harvesting mortality as a function of size. Classical works of Beverton and Holt (1957) and Holt (1958) consider separate harvest rates on each year/size class and show that given a fixed harvesting effort, the yield is maximized if fish are caught at the size or age where cohort biomass is maximum. Extending our modeling to allow for different harvesting rates on each year/size class would allow for evaluating such result with respect to our suggested measures of conservation.
Another promising extension of the work presented here is to move beyond single-species management towards ecosystem-based management. We believe in a trend from single-objective towards multi-objective approaches using single-species population models, strengthened by the present paper. This trend may evolve towards multi-objective approaches using multi-species population models, that is, towards ecosystem management. Studies in this direction already exists, see e.g. White et al. (2012). The present method using Pareto frontiers to find sustainable harvesting strategies can be applied also in general multi-species settings. For example, harvesting on a set of species in a food web with respective harvesting rates (), we can for any desired yield determine the harvesting strategy that offers the highest conservation benefits. Hence, the methods presented here open a door for reconciling economic and conservation issues in ecosystem management and can be extended to more complex scenarios including for example management of multiple fisheries and maintaining species diversity.
ÅB and XM gratefully acknowledge financial support from the Wenner-Gren Foundations, and XM acknowledge financial support from the National Natural Science Foundation of China (No. 11371230). The authors declare no conflict of interest.
Appendix: Derivation of the basic reproduction ration for the age model
To calculate the basic reproduction ratio for the age model, we consider first the expected egg production of one single individual during the individual’s life span. To remove the density-dependent competition for this single individual, which is induced by the Beverton-Holt recruitment assumption (5), we blow up the size of the environment by taking the limit . (The Beverton-Holt recruitment is then completely unsaturated which means that no density-dependent competition is present.)
From (1) the egg production of one individual over its lifetime can be expressed as
and since according to (The age model) we conclude that
Now, since is normally distributed with mean and standard deviation , we obtain, where denotes expectation,
This expression is identical to (9) and gives the basic reproduction ratio for the age model.
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