Beyond age‐structured single‐species management: Optimal harvest selectivity in the face of predator–prey interactions

In the single‐species literature, it is widely acknowledged that conserving young fish for future harvesting is beneficial. This finding holds great significance in fisheries economics and has garnered substantial attention over the years. In this study, a full‐blown age‐structured predator–prey model is developed and used to demonstrate that multispecies considerations may shift the optimal selection of predators towards smaller individuals, providing valuable counteractive insights. These new results offer a fresh perspective highly relevant to regulation and choice of selectivity patterns.

changes in the absolute and relative strengths of the interspecies predator-prey interactions, ranging from nonexistent to strong.By comparing the NPV of the entire multispecies fishery, as well as its predator and prey components, across the different predator-prey conditions, we can identify any discrepancies between the optimal predator selectivity for the overall fishery and that which is most beneficial to the predator fishery alone.This, in turn, allows us to validate or refute the hypothesis regarding potential shifts in optimal target age and size for a fish stock when transitioning from a single-species perspective to a multispecies perspective.
The remainder of the paper is organized as follows.First, we describe the materials and methods employed in this paper, detailing the model, the different scenarios we consider, and our approach to solving them.Next, we share our findings, followed by a comprehensive sensitivity analysis to ensure the robustness of these results.Lastly, we delve into a discussion of our model and results and draw conclusions.

| Bioeconomic model
The model presented here considers a sole owner who manages two fleets and two interacting commercial fish stocks-one fleet targets a predator stock while the other targets a prey stock.The biological submodel describes the processes of natural mortality (here to be understood as natural mortality excluding predation-induced natural mortality), predation mortality, growth, maturation, and recruitment, while the economic submodel describes fishing effort, costs, revenue, and profits associated with harvest.The harvest functions bridge the biological and economic dimensions.Parts of the model are inspired by cod and capelin in the North-East Arctic (NEA).However, the inspiration from cod and capelin is mainly motivated by the model complexity, the consequential reliance on numerical solution approaches, and the consequential need for sensible structural and numerical specifications-the model is not intended for a case study of said fisheries, but rather as a tool for conceptual study (Table 1).
The sole owner's objective is to maximize the NPV of harvest (Equation 1 in Table 2) from the two fish stocks with respect to the effort of the two fleets (E pred and E prey ) (Equation 2 in Table 2) subject to the system (Equations 3-15 in Table 2).
In relation to the NPV, we assume size-independent and constant prices (P pred and P prey ) and constant costs per unit effort (C pred and C prey ).Regarding harvest, we apply the Beverton-Holt age-structured framework, in which the Baranov catch equation represents an integral part.In line with the framework, the harvest by each fleet from each age group in each stock (y pred,a,t and y prey,i,t ) is modeled as a function of effort and the number of individuals in each age group (N pred,a,t and N prey,i,t ) (Equations 3 and 4 in Table 2) (Baranov, 1918).Harvest is density-dependent and increases linearly in the sizes of the stocks-that is, catch per unit effortwise, there are benefits to having access to and maintaining abundant fish stocks.
The control variables are restricted such that E t = E t+1 for both fleets in all periods t = 0, 1, …, T − 1.Although effort remains constant over time, it is important to note that this does not imply a constant harvest over time.Constant effort and linear density-dependent harvest imply harvest feedback policies that are linear in the number of individuals in each age group and fish stock, meaning that harvest will be lower when the stocks are small compared with when they are large.This combination of constraints and assumptions is useful as it prevents the possibility of pulse fishing schemes, which are rarely of practical interest.Moreover, by preventing the possibility of pulse fishing schemes, that is, enforcing steady-state fishing schemes, it becomes possible to present results without the time dimension, which may be convenient in many circumstances, including ours.That said, it is important to note that this combination of assumptions does not come without a cost.First, constraining solutions to adhere to linear harvest feedback policies may result in second-best solutions, as compared with solutions that could be obtained under circumstances with more freedom.Second, the resulting steady states may exhibit sensitivity to the initial state of the fish stocks.It is imperative to bear this in mind when interpreting the findings.It also underscores the necessity of conducting sensitivity analyses concerning the initial stock values.
Inspired by state-of-the-art modeling of NEA cod, the predator stock is split into 12 age groups, ranging from age group 3 to age group 14+, in which the first age group represents all individuals that are 3 years of age, while the last represents all individuals that are 14 years and older (Diekert et al., 2010;ICES, 2021;Kovalev & Bogstad, 2005).Similarly, inspired by state-of- the-art modeling of capelin, the prey stock is split into four age groups, ranging from age group 1 to age group 4+ (ICES, 2021).
In this model, both species are commercial (represented by positive prices), and thus potentially subject to fishing mortality (q pred,a E pred and q prey,i E prey ) (Equations 6, 7, 9, and 10 in Table 2).Of course, the predator and prey individuals are both subject to natural mortality (M pred,a and M prey,i ).In addition, the prey species is subject to predation mortality (λ prey,i,t ) (Equations 9 and 10 in Table 2).In the real world, young predator individuals could also be subject to predation mortality in terms of cannibalism mortality.However, the latter is left out of the scope of the model as the focus of our study rests on the effects of interspecies predation.
The natural mortality rates are considered exogenous and constant.Bang and Steinshamn (2022) show that assumptions of exogenous natural mortality rates can lead to significant overestimation of the biological and economic potential of fish stocks, which makes it clear why such factors should be considered endogenously in models that are intended for applied and practical analysis.However, the focus in this study is more conceptual and theoretical than practical, and we choose to treat these factors exogenously to allow clear focus on the objectives of this study, which encompass the effects of predation, weight-conversion, and relative prices on optimal harvesting schemes and preferred selectivity.
The number of prey individuals that die from predation is determined by scaled age-specific predation coefficients (SC p p pred,a,i ), the number of predator individuals at age, and the number of prey individuals at age (Equation 13 in Table 2).For intuition, the reader can think of the predator as a competing fishing fleet consisting of several vessel groups (age groups), each with its own efficiency and selection pattern in harvesting the prey (predation coefficients), and an employed effort (number of predator individuals at age).The predation functions have the same structure as the harvest functions, but as opposed to harvest, the predation functions generate no direct value for the fishing industry.The choice of density-dependent predation functions is motivated by the fact that cod has been shown to shift to alternative prey such as amphipods and krill when the capelin stock is low (Dalpadado, 2001;Holt et al., 2019)-that is, when less capelin is available, cod is likely to base more of its food consumption on alternative prey.
The weight at age for the predator in the current year (W pred,a,t ) is determined by the weight at age of that cohort in the previous year (W pred,a−1,t−1 ), scaled weight-conversion rates for that cohort in the previous year (SC w Υ a−1 ), the biomass consumption of prey per predator individual in the previous year (φ pred,a−1,t−1 /N pred,a−1,t−1 ), and an exogenous growth factor (w pred,a ) (Equation 11in Table 2).It is well known that predator species can have reduced feeding levels and smaller growth rates when the prey stock is at low levels, and vice versa (Gjøsaeter et al., 2009;Holt et al., 2019;Mehl & Sunnana, 1991).The weight functions capture this through the life history of the fish.The exogenous growth factor is meant to represent other factors that contribute to growth, such as predation on species and climatic factors that are not part of the model.
The weight at age for the prey (W prey,i ) is assumed exogenous and constant.Like assumptions of exogenous natural mortality, this assumption can lead to overestimation of biological and economic potential (Bang & Steinshamn, 2022).In the real world, weight at age for prey could depend on the density of the stock, climate, and evolutionary changes (Diekert, 2013;Pardoe et al., 2009;Zimmermann & Jørgensen, 2015).However, again, our intention here is not to provide accurate estimates on the biological and economic potential of either stock, but rather to provide a conceptual, theoretical, and broader contribution.As such, the simplification can be well defended.
The recruitment to the predator and prey stocks (R pred,t and R prey,t ) is determined by Beverton-Holt recruitment functions, that is, the recruitment to each stock is a concave function of their respective spawning stock biomass (SSB pred and SSB prey ), with positive horizontal asymptotes (Equations 5 and 8 in Table 2) (Beverton & Holt, 1957).The spawning stock biomass is calculated according to Equations ( 14) and ( 15) in Table 2.
Tables S5-S19 give insight into the numerical specification of the model.In relation to numerical specifications, it should be noted that these are only partly inspired by cod and capelin.Some of the numerical values, like, base weights and predator recruitment, are set based on Bang andSteinshamn (2022) andICES (2021).Other values, like those pertaining to predation and weight-conversion, are based on reason for conceptual and illustrative purposes.

| Scenarios
We will solve the optimization model for two fixed predator selectivity modes, namely, "Target big" and "Target small," across three different scenarios, namely, "The simple case," "Disproportional predation," and "Nonuniform weight-conversion," as outlined in Table 3, with interspecies interactions ranging from nonexistent to strong.The details concerning the selectivity, predation, and weight-conversion modes are specified in Figure 1 and outlined in the following.With our primary focus on predator selectivity and whether the preferred selectivity might change in response to multispecies considerations, we note that prey gear selectivity is kept constant across all scenarios, at levels defined in Table S16.
The "Target big" and "Target small" predator selectivity modes are designed such that "Target big" represents a more advantageous selectivity mode than "Target small" in the context of single-species management.More precisely, "Target big" is tailored to facilitate efficient utilization of predator growth potential at both individual and stock level, while "Target small" is intentionally designed to lead to predator growth overfishing.In the "Target big" mode, predator individuals aged 9 and above are targeted, while "Target small" mode targets those aged 7 and above.Preliminary analysis by the authors confirms this design can serve its intended purpose, which is to illustrate the classical single-species finding and establish a baseline to explore the impact of multispecies interactions and considerations.In the real world, selectivity is a variable

Scenario pair Selectivity mode Predation mode Conversion mode
The that fishery managers and fishermen can influence to a considerable extent, although not with perfect accuracy.It is determined by factors such as the choice of fishing gear and, depending on definition, fishing location (and the age-and size-composition of the fish located in that area).An effective approach to managing selectivity is by adjusting the mesh size in trawl nets, a practice that is often reflected in regulations specifying minimum mesh sizes for various fisheries.Another vital strategy, especially pertinent for NEA cod and likely applicable to many other fish stocks, involves increasing fishing pressure in spawning grounds, where mature fish gather for reproduction purposes, while decreasing it in feeding areas, which may contain a mixture of immature and mature fish, and vice versa.
The first scenario pair, labeled "The simple case," is designed to allow for a comparison of the performance of the two selectivity modes under assumptions of predation coefficients that increase proportionally with base predator weights at age and weight-conversion rates that are uniform for all predator age groups above a certain age.These assumptions are intuitive, simple, and reasonable, providing a solid foundation for exploring the effects we are interested in.The simplicity of this scenario pair also creates a baseline for comparison, enabling us to identify the effects of more nuanced assumptions regarding interspecies predator-prey interactions with greater accuracy.
The second scenario pair, labeled "Disproportional predation," is designed to allow for a comparison of the performance of the two selectivity modes under assumptions of predation that increases more than proportionally with base predator weights at age and weight-conversion rates that are uniform for all predator age groups above a certain age.Mehl (1986) states that "With increasing predator length fish prey become more and more important.For sizegroup 20-39 cm fish were the major prey in 2/3 of the investigated areas and periods.while for cod 60 cm fish always were the dominating prey category.And with increasing predator length the size and importance of larger fish prey increased gradually."On the basis of this, disproportional predation appears more realistic than proportional predation.By comparing the results from this scenario pair to the results from the first scenario pair, we can isolate the effect of disproportional predation on the performance of the two selectivity modes.
The third scenario pair, labeled "Nonuniform weight-conversion," is specifically designed to compare the performance of the two selectivity modes under assumptions of disproportional predation and nonuniform weight-conversion.It is well known that older fish allocate less energy intake towards growth, which makes decreasing weight-conversion rates with age, followed by stabilization at a certain level, more realistic than uniform weight-conversion rates.By comparing the results from this scenario pair to the second scenario pair, we can identify the impact of nonuniform weight-conversion on the performance of the two selectivity modes with greater accuracy.
To enable performance comparison across various levels of predation and weightconversion within and across all scenario pairs, we solve each scenario pair for 72 different combinations of the scaling factors SC P and SC W , which determine the strengths of interspecies interactions.Table 4 displays the employed combinations of scaling factors.

| Solution approach
When possible and tractable, it is preferable to solve optimization problems analytically without numerical specifications of parameters that may be subject to change in analyses, that is, in general notation.This is because analytical expressions for the solutions of generalized optimization problems can give explicit insight into how the solutions depend on different factors.However, as highlighted by Clark (2010), it is often impossible or intractable to solve age-structured optimization problems analytically.Even the simplest age-structured models with Beverton-Holt population dynamics are difficult to solve analytically.With simpler population dynamics on the other hand, like in Helgesen et al. (2018), it is possible to derive analytical expressions.
T A B L E 4 Overview of combinations of scaling values for use in solving each scenario.
The GRG Nonlinear solver may get stuck on locally optimal solutions.Thus, to ensure that we report results that are globally optimal, we have solved the problems several times with different initial search values for the control variables.Using this procedure, we observe that the solver converges towards the same solutions regardless of initial search values.The observation from the repetitive solving procedure goes a long way in validating the global optimality of the results.To further validate our findings, we employed the nonlinear interior point trust region optimization (KNITRO) solver, available through the Analytic Solver add-in for MS Excel, for selected model scenarios.The results obtained using KNITRO matched those produced by the GRG Nonlinear solver.This consistency further reaffirms the reliability of our results.

| RESULTS
In this section, we present the results from the main scenarios.To provide a clear presentation, we divide the section into three subsections, each detailing the outcomes of a scenario pair.

| The simple case
We begin by investigating the results for the first scenario pair.That is, the scenarios with predation coefficients that increase proportionally to base predator weights at age and uniform weight-conversion rates (ref.Figure 1).This scenario pair allows a concise analysis of the effects of predation and weight-conversion.It also establishes a baseline for investigating how changes in characteristics of interspecies predator-prey interactions affect the optimal age-specific selectivity for the predator.
Result I: Increasing levels of predation can shift the optimal predator harvest selectivity towards younger and smaller predator individuals First, consider the left plots in Figures 2 and 3.These parts of the figures give high level, but key novel insight.Starting with zero predation and zero weight-conversion, the plots show that it is most beneficial to apply the "Target big" selectivity mode.This result reflects the classical finding in the single-species literature; spare the young fish and target the older and bigger ones (Bang & Steinshamn, 2022;Diekert et al., 2010;Helgesen et al., 2018;Reed, 1980;Skonhoft et al., 2012).However, when predation increases (right-wards movement on the axis labeled "Level of predation," which is defined by the scaling factor SC P , which along with the baseline predation coefficients determine the scaled predation coefficients in Equation 13, as illustrated in Figure 1), the net benefit of applying the "Target big" selectivity mode shrinks relative to the alternative "Target small" mode, where younger and smaller predator individuals are also targeted (ref.Figure 1).And after a certain point, it becomes optimal to apply the "Target small" mode.
F I G U R E 2 Net present values (NPVs) under optimal harvesting with the "Target big" and "Target small" selectivity modes, proportional predation and uniform weight-conversion, and varying levels of predation and weight-conversion.A shift in optimal selection pattern occurs for higher levels of predation.Predation-weightconversion can to some extent counter the shift.
F I G U R E 3 Net present values (NPVs) under optimal harvesting with the "Target big" and "Target small" selectivity modes with proportional predation and uniform weight-conversion, varying levels of predation, and three different fixed values of the scaling factor for weight-conversion.A shift in optimal selection patterns occurs for higher levels of predation.Higher levels of predation-weight-conversion can to some extent counter the shift.Result II: The optimal predator harvest selectivity for the predator-prey fishery need not align with what is optimal for the predator fishery alone The shift in optimal selectivity for high levels of predation occurs despite the finding depicted in the top-right plots of Figures 2 and 3, which shows that the "Target big" approach consistently yields a higher NPV from the predator stock compared with the "Target small" approach.These findings offer robust evidence in support of the hypothesis that the optimal selection pattern for a particular species within a commercial multispecies fishery need not remain optimal when the impacts on other species and fisheries are considered.This, in turn, underscores the importance of taking a holistic approach to fisheries management, which considers the interdependent relationships between different species.
Result III: Weight-conversion can to some extent counter the effect of predation on optimal selectivity, but not enough to eliminate the shift towards younger and smaller predator individuals for all levels of predation As per expectation and reason, when the predation is zero, an increasing level of weight-conversion (right-wards movement on the axis labeled "Conversion rate," which is defined by the scaling factor SC W , which along with the baseline weight-conversion coefficients determine the weightconversion rate in Equation ( 11), as illustrated in Figure 1) has no effect upon the NPV of either selectivity modes.However, as the level of predation increases, the plots in Figures 2 and 3 show that increasing weight-conversion dampens the negative effect of predation upon the NPV and delays the shift in optimal selectivity.These results and insights are intuitive because any additional mortality has a negative impact on the potential of the prey stock, and with no weight-conversion, there will be no counteracting positive effect with the predator.However, with weight-conversion, an increasing level of predation means that the predator will consume more, and gain weight as a result, which means a higher potential harvest from the predator stock.
Result IV: When shifting from targeting old and big predators to also targeting younger and smaller predators, the sole owner sacrifices utilization of individual predator growth potential not just to limit the overall predation of the prey stock, but also to improve the utilization of individual prey growth potential Figure 4 shows the optimal age compositions and harvest profiles for the first scenario pair with the level of predation SC P set to 3 and the conversion rate SC W set to 0. The figure clearly shows how the optimal steady-state age composition and harvest profile respond to changes in the selection pattern for high levels of predation.It is shown that shifting the selection pattern towards smaller predator individuals leads to a reduction in the overall size of the predator stock and a reduction in the relative number and harvest of large to small predator individuals, which also implies reduced harvest efficiency in terms of catch per unit effort.This result may be obvious to the reader.However, the figure also shows something more, which is less obvious.
The changes in the overall size and age composition of the predator stock yield an increase in the overall size of the prey stock and the relative number and harvest of large to small prey individuals.As such, the changes in the selectivity and harvesting policy do not only increase the gross harvest and catch per unit effort for prey, but it also improves the utilization of individual prey growth potential.In other words, the sole owner sacrifices the utilization of individual predator growth potential not just to limit the overall predation of the prey stock, but also to improve the utilization of individual prey growth potential.This is an interesting detail and insight which cannot be gained from biomass predator-prey models.
Figures 5 and 6 give further and more general insight into how optimal harvesting for each of the selectivity modes changes with the levels of predation and weight-conversion.The figures show that increasing predation yields higher optimal fishing pressure on the predator stock-harvest relative to spawning stock biomass increases.This is done to limit predation mortality in the prey stock, and thereby limit the reduction in the NPV of the prey stock.Meanwhile, the harvest from the prey stock is reduced to compensate somewhat for the predation effect on the size of the stock.Overall, these changes in the optimal harvesting strategy correspond to findings in biomass predator-prey models.

| Disproportional predation
In this section, we will address the results from the second scenario pair.That is, the scenarios with predation coefficients that increase disproportionally to base predator weights at age, with uniform weight-conversion rates (ref.Table 3 and Figure 1).Result V: When compared with proportional predation, disproportional predation expedites the shift in optimal predator harvest selectivity towards younger and smaller predator individuals When compared with the results from the simple case in Figure 2, the results in Figure 7 show that disproportional predation leads to an earlier shift in preferred selectivity.Figure 8 displays this in a clearer manner through a direct comparison of the scenario pair with proportional predation and uniform weight-conversion and the scenario pair with disproportional predation and uniform weightconversion, for varying levels of predation and a fixed scaling factor for the weight-conversion.The key explanation for the earlier shift is that the indirect cost of having many large predators in terms of the reduced potential of the prey stock increases when large predator individuals eat relatively more prey than small predator individuals.This discovery is intriguing as it demonstrates that incorporating greater realism into the model can further increase the significance of harvest selectivity for a single fish stock within the context of a commercial multispecies fishery.

| Nonuniform weight-conversion
This section outlines the results from the third scenario pair.That is, the scenarios with predation coefficients that increase disproportionally to base predator weights at age, and with nonuniform weight-conversion rates (ref.Table 3 and Figure 1).
Result VI: When compared with uniform weight-conversion, nonuniform weight-conversion limits the countereffect of weightconversion on the shift in optimal predator harvest selectivity When compared with the case with disproportional predation and uniform weight-conversion, the results in Figure 9 show an earlier shift in preferred selectivity for higher levels of weightconversion.As such, when compared with uniform weight-conversion, nonuniform weightconversion can accelerate the shift to the "Target small" selectivity mode.This is perhaps best seen through Figure 10, which compares all three scenario pairs for different values of the scaling factor for the predation and a fixed value of the scaling factor for weight-conversion.
F I G U R E 7 Net present values (NPVs) under optimal harvesting with the "Target big" and "Target small" selectivity modes, disproportional predation and uniform weight-conversion, and varying levels of predation and weight-conversion.When compared with Figure 2, the shift in optimal selection pattern occurs at lower levels of predation.
F I G U R E 8 Net present values (NPVs) under optimal harvesting with the "Target big" and "Target small" selectivity modes with proportional predation and uniform weight-conversion versus disproportional predation and uniform weight-conversion, varying levels of predation and fixed values for the weight-conversion.It is shown that the shift in optimal selection pattern occurs at lower levels of predation when assuming disproportional predation instead of proportional predation.
The key explanation here is that the benefit of having access to large predator fish becomes relatively smaller when compared with having access to small predator fish because the relative differences in weights between the small and large predator fish decrease for high levels of predation and weight-conversion.Just as the discovery regarding disproportional predation, the findings pertaining to nonuniform weight-conversion are also significant, as they reinforce the idea that increasing realism in models can increase the significance of harvest selectivity for a specific fish population in a commercial multispecies fishery.

| SENSITIVITY ANALYSIS
Sensitivity analysis is important because it helps assess the robustness of the model and findings.There are many aspects that could be interesting to investigate in this model.Some of them, like changes in the strengths and characteristics of predation and weight-conversion, are already considered through the main analysis.Other interesting aspects include harvesting costs, prices, discounting, recruitment, and initial values.
We find it particularly interesting to investigate whether our findings remain robust to significant changes in the relative price of predator to prey.It is quite intuitive and demonstrated formally by Hannesson (1983) and others, that the relative price between predator and prey may have important implications for the optimal harvest pattern and stock levels.In the main analysis, we assume that predator and prey biomass are of equal value in terms of price.Therefore, as a sensitivity analysis, we also check how the results change when we move from a high-valued predator and a low-valued prey to the opposite.
In a single-species perspective, Zimmermann et al. (2011) demonstrate that positively sizedependent pricing clearly shifts optimal harvest strategies towards lower harvest rates and higher mean body size of caught fish.As such, it is reasonable to expect that positively sizedependent prices for the predator may significantly affect our results.Therefore, we will also investigate what happens when we introduce a price premium for large predator individuals, that is, increase the price only for the large predator individuals.
As the solutions to the optimization problems may depend significantly on the initial values of the fish stocks, especially considering the control variables are constrained, we will also check whether our findings remain robust to changes in the initial values of the fish stocks.In the main scenarios, the model is initialized with a medium-sized predator stock and a medium-sized prey stock.To test the robustness of our findings, we investigate what happens when the predator stock is initialized at a low level.Specifically, we investigate what happens when the stock is initialized at 10% of the size used in the main scenarios.To confirm the explanation of a perhaps counterintuitive finding, we also run this sensitivity scenario with a lower discount rate.

| Changes in the prices of fish
In the following, we present the results from our sensitivity analyses concerning price changes.
F I G U R E 9 Net present values (NPVs) under optimal harvesting with the "Target big" and "Target small" selectivity modes, disproportional predation and nonuniform weight-conversion, and varying levels of predation and weight-conversion.When compared with Figure 7, the shift in optimal selection pattern occurs at lower levels of predation for high levels of predation-weight-conversion.Result VII: An increase in the relative price of prey to predator expedites the shift in predator selectivity Figure 11 shows the results from increasing the price of the prey by 20% in a setting where the predation coefficients increase disproportionally to base predator weights at age and weight-conversion is nonuniform.When compared with the results from the baseline scenario pair with disproportional predation and nonuniform weight-conversion, the results in Figure 11 show an earlier shift to the "Target small" selectivity mode.This makes sense because high predator harvest and good utilization of the individual predator growth potential become relatively less valuable when the price of the predator becomes relatively lower.
F I G U R E 10 Net present values (NPVs) under optimal harvesting with the "Target big" and "Target small" selectivity modes with proportional predation and uniform weight-conversion versus disproportional predation and uniform weight-conversion versus disproportional predation and nonuniform weight-conversion, varying levels of predation and fixed values of the scaling factor for weight-conversion.It is shown that nonuniform weight-conversion can expedite the shift in selection pattern when compared with uniform weight-conversion.
F I G U R E 11 Net present values (NPVs) under optimal harvesting with the "Target big" and "Target small" selectivity modes, disproportional predation and nonuniform weight-conversion, and varying levels of predation and weight-conversion, with a 20% increase in prey price from the main scenarios.
Result VIII: An increase in the relative price of predator to prey delays the shift in predator selectivity due to the time preference Figure 12 shows the results from increasing the price of the predator by 20% in a setting where the predation coefficients increase disproportionally to base predator weights at age and weightconversion is nonuniform.As per expectation, when compared with the results from the baseline scenario pair with disproportional predation and nonuniform weight-conversion, the results show that an increase in the relative price of the predator counteracts the shift in optimal selection pattern observed in the main scenarios.Intuitively, this is reasonable because high predator harvest and good utilization of the predator growth potential become relatively more valuable when the price of the predator becomes relatively higher compared with the price of prey.
Result IX: Size-dependent pricing of the predator, with price premiums for larger individuals, can make the shift in selectivity disappear.In other words, price increases only applying to large predator individuals have a stronger effect upon the shift than price increases applying to all predator individuals Figure 13 shows the results of increasing the price of the predator of ages 10-14+ by 20% from the main scenarios.The results show that the shift to "Target small" now disappears.The price premium for large predator individuals makes it much more beneficial to spare young predator fish for future harvest, and it is worth taking the costs of doing so by lowering the economic potential of the prey stock.This is in line with expectations based on the findings of Zimmermann et al. (2011).However, even with the size-dependent price of the predator, a shift could still occur if the price of prey increases significantly.
In general, the sensitivity analyses exploring changes in price demonstrate that the shift in preferred selectivity remains robust even in the face of significant price fluctuations.At the same time, we show that positively size-dependent pricing of the predator can make the shift disappear.Further, it should also be noted that the shift will disappear also in extreme price settings, without size-dependent prices, in which one species is much more valuable than the other.Consequently, the findings from this study may not be as applicable to multispecies fisheries where the economic value of one species greatly exceeds that of the other(s).

| Changes in the initial stock sizes
Motivated by the fact that our results may be sensitive to initial stock values, this section outlines how our results would respond to changes in these.
Result X: Reducing the initial predator stock expedites the shift in predator harvest selectivity due to the time preference of the manager and the trade-off between early and late harvest Figure 14 shows the results from initializing the predator stock at 10% of the size used in the baseline scenarios, in a setting where the predation coefficients increase disproportionally to base predator weights at age and weight-conversion is nonuniform.Intuitively, one could expect a delayed shift in F I G U R E 12 Net present values (NPVs) under optimal harvesting with the "Target big" and "Target small" selectivity modes, disproportional predation and nonuniform weight-conversion, and varying levels of predation and weight-conversion, with a 20% increase in predator price from the main scenarios.
the optimal selectivity pattern, as this could contribute to rebuilding the stock quickly and robustly.Instead, we witness the opposite-the shift to the "Target small" selectivity mode occurs at much lower levels of predation and weight-conversion.The explanation for this is that it takes a too long time to build up the stock.Waiting comes at a higher cost, in terms of missing out on early, and therefore less discounted, net revenue, when compared with the benefits of enjoying a more substantial yield from a healthier predator population in the future, where net revenues are more heavily discounted.In other words, harvesting predators at a younger age, alongside a robust harvest from a healthy prey population exposed to low predation mortality, yields greater discounted benefits than waiting for the predator population to grow.Naturally, this trade-off depends on the chosen discount rate: a higher rate would make the manager even less inclined to wait for the predator stock's recovery, while a lower rate would increase their willingness to trade early revenue for the promise of greater future earnings.
Result XI: A reduction in the discount rate of the manager delays the shift in predator harvest selectivity as the trade-off between early and late harvest becomes less significant To confirm the explanation provided above, we solve the very same sensitivity scenario with a discount rate of 3% instead of 5%, which is applied in the baseline scenarios.Figure 15 shows the results.Because of the reduced cost of waiting for the stock to build up, the shift to the "Target small" occurs first at significantly higher levels of predation and weight-conversion.
F I G U R E 13 Net present values (NPVs) under optimal harvesting with the "Target big" and "Target small" selectivity modes, disproportional predation, and nonuniform weight-conversion, and varying levels of predation and weight-conversion, with a 20% increase in the price of predators of age 10-14+ from the main scenarios.
Overall, the sensitivity scenarios dealing with initial values show that the model and results are sensitive to changes in the initial conditions.However, the shift in optimal selection pattern remains robust to significant changes in the initial values.While we only tested this for a significant change in the predator stock, it seems plausible that this would also apply to significant changes in the initial value of the prey stock.

| LIMITATIONS
Through the modeling process, some limitations were imposed, and aspects excluded.To prevent the potential for pulse fishing schemes and avoid the necessity of representing our results over time, we implemented certain fishing effort constraints.Specifically, we maintained a constant level of effort throughout the entire study period.Given our other assumptions, most importantly, linear density-dependent harvest, this led to the adoption of harvest feedback policies that are linear in relation to the sizes of the fish stocks, ensuring steady-state fishing schemes.Despite the benefits of this approach, it is important to acknowledge that it comes with certain drawbacks.Notably, it makes the resulting steady states somewhat sensitive to the initial values of the stocks.Further sophistication could possibly be achieved, while still preventing pulse fishing, by dividing the planning horizon into multiple phases (allowing for varying effort in different time segments) or allowing for piecewise linear harvest feedback policies (permitting different effort levels for different stock states).Nonetheless, it is worth noting that such sophistication is unlikely to alter the fundamental qualitative insights provided by this study, a conclusion supported by our earlier sensitivity analyses.
F I G U R E 14 Net present values (NPVs) under optimal harvesting with the "Target big" and "Target small" selectivity modes, disproportional predation and nonuniform weight-conversion, and varying levels of predation and weight-conversion, with the predator stock initialized at 10% of the size employed in the main scenarios.
Another consideration was whether to use age-or size-based selectivity.Although sizebased selectivity would align better with the fishing gear's selection criteria, we chose age-based selectivity for practical reasons.When combining age-based selectivity with endogenous weight dynamics, like we do, our selectivity modes display small built-in shifts towards bigger predator fish as predation and weight-conversion increase (because age-based selectivity remains constant while the individuals at age weigh more when predation and weight-conversion are high than when they are low).However, these small built-in shifts in size-selectivity do not change the fact that the selectivity mode that maximizes NPV for the predator-prey fishery is not the same as the selectivity mode that maximizes NPV for the predator part of the fishery for all predator-prey conditions, and that the multispecies considerations shift the optimal predator harvest selectivity towards younger individuals, which are also smaller in size.Although the incorporation of size-based selectivity would result in a more sophisticated model, it appears safe to say that it would not significantly affect our results and findings.
To narrow the focus of this study, we assumed predation coefficients that are age-specific for predator, but age-unspecific for prey-that is, a predator of age a has the same selectivity on prey of age 1 as prey of age 2, and so forth.In the real world, a predator of age 3 may, for example, have a higher selectivity for prey of age 1 than prey of age 4, while a predator of age 10 may have a higher selectivity for prey of age 4 than of age 1. Predation coefficients that are agespecific for prey could produce more nuanced results.
Finally, another consideration was whether to incorporate cannibalism mortality.Although this is a relevant factor, we decided not to include it in our model, and to focus only on interspecies predator-prey interactions.We believe that cannibalism would have reinforced our hypothesis, as allowing more individuals to grow larger would further increase the indirect costs associated with maintaining a large stock of big predator individuals.
F I G U R E 15 Net present values (NPVs) under optimal harvesting with the "Target big" and "Target small" selectivity modes, disproportional predation and nonuniform weight-conversion, and varying levels of predation and weight-conversion, with the predator stock initialized at 10% of the size employed in the main scenarios, and a lower discount rate.
This study applies dynamic optimization in an age-structured, multifleet, and predator-prey model.While we reproduce insight from age-structured single-species bioeconomic models, we also show that preferred selectivity and optimal harvesting change with the level of predation and weight-conversion rates.
In single-species age-structured models, a classical and recurring finding is that it is optimal to spare young fish for future harvest.For zero predation, our results confirm this.However, for increasing levels of predation, the benefits of targeting only large predators are counteracted by disadvantages in terms of higher prey mortality, worsened utilization of individual growth potential for the prey, and lower catch per unit effort for the prey.At some point, the disadvantages can outweigh the benefits of targeting only large predators, thereby making it optimal to target smaller predator individuals and increase the overall fishing pressure for the predator.This occurs despite the NPV of the predator part of the fishery being consistently higher when large predator fish is targeted, which means that single-species management could lead to overall suboptimal management.
Assumptions of predation that increases more than proportionally with age and size can make the shift in optimal selectivity towards smaller predator individuals happen at lower levels of predation.It is also shown that increasing weight-conversion rates can counteract the effect of predation on the shift in the optimal selection pattern, more so when assuming uniform weight-conversion rates than when assuming weight-conversion rates that decrease with age, which is more realistic.
Further, it is shown that the shift in optimal selection pattern remains robust to significant changes in price, initial values, and discounting.However, if the predator becomes significantly more valuable than the prey, the shift will disappear-indicating that our results are of most relevance to commercial predator-prey systems where the prey has significant value relative to the predator.
The findings are interesting and important because they bring new and further awareness to why managers should think twice before changing gear restrictions in the direction of targeting bigger fish on the basis of single-species analyses.Moreover, they display the usefulness and value of age-structured multispecies modeling, which has not received much attention in the research literature.
For future research, we may suggest investigating the mechanisms put forward in this paper in different empirical contexts.In relation to that, we suggest focusing on commercial predator-prey fisheries where the prey is valuable compared with the predator.Further, we would like to suggest allowing more flexibility in harvest control rules and using a size-based approach to selectivity rather than an age-based approach since this would bring about further sophistication.Otherwise, we may suggest studying the effects of predation coefficients that are age-specific for both predator and prey.It could be interesting to study the implications of this for optimal selectivity and harvesting.

| Natural Resource Modeling
BANG and STEINSHAMN

| 5 of 28 T
A B L E 2 Model objective, control variables, and equations.

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I G U R E 1 Selectivity, predation, and conversion modes.The selectivity modes are defined by selectivity coefficient values.The predation modes are illustrated through scaled predation coefficients SC P = 1.The weight-conversion modes are illustrated through scaled weight-conversion rates with SC W = 0.5.

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I G U R E 4 Optimal age compositions and harvest profiles for "The simple case" with the level of predation SC P set to 3 and the conversion rate SC W set to 0. The relative abundance and harvest of large predator individuals decline, while the relative abundance and harvest of large prey individuals increase, when the predator harvest selectivity switches from targeting only big predator individuals to targeting big and smaller predator individuals.F I G U R E 5 Harvest and spawning stock biomass (SSB) results from the first scenario pair (proportional predation and uniform weight-conversion) with the "Target big" selectivity mode.F I G U R E 6 Harvest and spawning stock biomass (SSB) results from the first scenario pair (proportional predation and uniform weight-conversion) with the "Target small" selectivity mode.
Model sets, parameters, and variables.
T A B L E 1 , Predator maturity rate at age a Dimensionless υ prey i , Prey maturity rate at age i Dimensionless