Issue 
Aquat. Living Resour.
Volume 36, 2023



Article Number  22  
Number of page(s)  10  
DOI  https://doi.org/10.1051/alr/2023016  
Published online  25 July 2023 
Research Article
Marine protected areas for resilience and economic development
^{1}
Arba Minch University, Department of Mathematics, Arba Minch, Ethiopia
^{2}
State University of New York, Department of Mathematics, Farmingdale, NY, USA
^{3}
Addis Ababa University, Department of Mathematics, Addis Ababa, Ethiopia
^{*} Corresponding author: biteww@farmingdale.edu; fitih.fikru@amu.edu.et
Handling Editor: Dr. Olivier Thebaud
Received:
10
January
2023
Accepted:
9
June
2023
In this research, we attempt to give a comparative analysis of the space allocation of multipleuse marine protected areas (MPAs) including but not limited to the introduction of aquaculture in the area. Specifically, we consider the case where there is a need to develop MPAs for the conservation of the environment and ecological diversity. There is also a prevailing call for the establishment of aquaculture activities within the area to meet societal demands. Although aquaculture has negative externalities on MPAs, it helps to reduce the pressure on the capture fishery and increases the supply of fish. We develop a deterministic bioeconomic model that describes the transition dynamics and interrelationships of the systems. We find an optimal aquaculture size relative to the optimal size of MPAs that maximizes the overall economic and ecological benefits. Using numerical methods we determine the trajectory of optimal solutions, the recovery rate of the stocks in and outside the MPAs, and the expansion rate of the aquaculture. Sensitivity analysis was also performed to see the effect of a change in the parameters on the optimal solutions. The numerical results show that MPAs are resilient after the implementation of aquaculture. Moreover, the effectiveness of the optimized management system mainly depends on the cooperative planning between the capture fishery and aquaculture managers.
Key words: Marine protected areas / aquaculture production / space allocation / growth rate
© F.F Hailu et al., Published by EDP Sciences 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Marine protected areas (MPAs) are areas placed in the ocean or in a water body where human activities are restricted to varying degrees and are often established with multiple objectives in mind such as ecosystem protection, sustainable use of the natural resource, food security, economic development, etc (O'Leary et al., 2018). There is a continuing dialogue between conservationists and fishers on creating a common understanding of the conservation objectives of MPAs and their business contributions (Westlund et al., 2017). For many fishers and aquaculture producers, MPAs are viewed as places where no farming is allowed, notake and nouse zones, which is a misconception. The International Union for the Conservation of Nature (IUCN) has defined six categories of MPAs where the two most commonly applied types allow some aquaculture activities (Le Gouvello et al., 2017). For this study, we consider the notake multipurpose MPAs that allow aquaculture. This type of MPAs aims to preserve biodiversity and enhance a sustainable economy by managing related impacts and synergies. During the recent decades, around 85% of the world's fisheries are either being fished at full capacity or already more exploited (Le Gouvello et al., 2017; Sampantamit et al., 2020). Global fishery stocks are declining rapidly and are no longer capable of producing a sustainable amount due to overfishing and habitat degradation. Thus, aquaculture gradually becomes an option to meet the shortfall and the increasing demand for fish (Sampantamit et al., 2020; Tewabe, 2015; Armstrong et al., 2016; Le Gouvello et al., 2017).
Aquaculture continues to grow at a faster rate than other agricultural sectors, but growth would not be sustainable if planning and management are not improved significantly (Hai et al., 2018). The performance of the fisheries mainly relies on strategic management and regulatory mechanisms that integrate the biological and ecological behavior of the resource in space and time with economic factors (Anderson and Seijo, 2010). In this research, we will investigate the advantage of developing aquaculture farms in multipleuse MPAs. We consider the case where there is an existing multipleuse MPAs and a call to develop aquaculture within the area or a creation of new multipleuse MPAs with aquaculture operations. Fish cultivation in MPAs must take into account the demand of the community and compatibility with the management objectives of the MPAs and free fishing area activities (Le Gouvello et al., 2017). MPAs under certain conditions and circumstances may be preferable for fish cultivation compared to openaccess areas (Le Gouvello et al., 2017). In this situation, the type of aquaculture systems, rate of expansion, and the maximum optimal size of the area in the MPAs dedicated to fish farming would need careful selection. Management guidelines would have to account for the changes in biological and economic factors. For the rest of the paper, MPAs refer to multiuse MPAs.
One of the main factors in evaluating the effectiveness of MPAs on capture fisheries and conservation includes activities outside MPAs (Westlund et al., 2017). Most fishery researchers consider one of the following assumptions. First, some ecologybased papers focus on the assumption that all fish outside the MPAs, up to the maximum sustainable yield, would be harvested by using the available maximum effort (Cabral et al., 2019). The second common approach assumes MPAs allow fixed fishing harvest rates based on quota or effortlimiting policy (Akpalu and Bitew, 2014). The third approach, the one in which we use for this paper assumes the formation of MPAs with aquaculture activities as part of an optimized fishery management strategy (Le Gouvello et al., 2017). This plan still would be overly optimistic in the presence of poor fishery management. Considering the optimal size of MPAs, Cabral et al. (2019) developed a bioeconomic model of a fishery under open access, which is meant to represent a broad class of fisheries in developing countries. In that setting, they derived the MPAs size that maximizes food security (catch), recognizing that individual fishers respond to realtime economic conditions. On the other hand, the research conducted by Akpalu and Bitew (2014) studies the use of marine reserves as a management tool for minimizing the negative effects of fishing in areas where the species are biologically diverse. On resource allocation, Pichika and Zawka (2019) studied a harvesting effort for renewable resources in the presence of environmental pollution. The research showed the stock of pollution was assumed to affect both the saturation level and intrinsic per capita growth rate of the resource. In our paper, like Akpalu and Bitew (2018) and Pichika and Zawka (2019), we consider the negative externalities to the environment due to fishing and aquaculture activities. Moreover, we assume aquaculture activities are solely implemented in MPAs. Then we determine the optimal aquaculture size that can be implemented in an MPAs relative to its optimal size and optimal harvest effort outside the reserve. This dictates the coexistence of the MPAs, aquaculture production, and openaccess fishery that preserves the environment and species diversity as well as supports economic development.
The remainder of the paper is organized as follows. In the next section, we set up deterministic dynamic equations of the stock of fish in and outside the MPAs without aquaculture and determine the Maximum Sustainable Yield (MSY) and optimal harvest with corresponding effort levels. In Section 3, we extend the model further by including aquaculture in the MPAs. In both cases, we analyze the control version of the models and find steadystate optimal solutions numerically for both the control and state variables. Section 4 presents the discussion and Section 5 concludes the paper.
2 Dynamic models and optimal management with MPAs
In this section, we describe the dynamics of capture fishery where fishing activities affect the habitat. Then we designate multipleuse MPAs for the purpose of protecting the environment and restocking the fish population in the fishing area. We also consider the control version of the model to determine the optimal size of the MPAs.
2.1 Capture fishery problem without MPAs
For this study, we start with the conventional ecological transition model given by
where X is the size of the fish stock, r is the natural growth rate, K is the carrying capacity of the fishing environment, and σEX is the harvest, where σ is the catchability coefficient and E is the effort.
Recent studies show habitat degradation is more responsible for the sharp decline of the stock than overfishing (Tewabe, 2015; Kahui et al., 2016). Foley et al. (2012) studied the influence of habitat on intrinsic growth and carrying capacity. Armstrong et al. (2016) analyzed the consequences of exogenous habitat loss and the disproportionate impact of habitat degradation upon profits when the habitat is at a low level. Pichika and Zawaka (2019) formulated a dynamic equation of the stock where the growth rate of the population is affected by pollution. Following their models, we extend the above dynamic equation, equation (1), further by assuming that harvesting activity or effort negatively affects the fishing ground or the habitat, and hence the growth rate of the stock. Let its impact increases linearly with E. Therefore, we set up the dynamic equation for the fish stock as
where 0 ≤ r_{0} − ϵ_{1}E ≤ r, r_{0} is the measure of the current growth rate, and ϵ_{1} ≥ 0 is the conversion factor of fishing effort level, E, to its impact on the growth rate.
To align the transition equation with our densitybased analysis in the next section, we divide both sides of equation (2) by K and rewrite the equation as
where is the density of the stock.
2.1.1 Steady state analysis
The trivial equilibrium (x = 0) is unstable, and the nontrivial equilibrium is stable whenever r_{0} − ϵ_{1}E > 0. The corresponding sustainable harvest (yield) function for each level of effort, h_{s} (E), is given by:
The maximum sustainable yield (MSY) is attained at the effort level E where That is at
Substituting E_{MSY} into equation (4), we get
The sustainable profit is given by
and the maximum is attained at an effort level where which implies
provided pσ (_ 1 + σ) ≥ _ 1c, where p is the unit price and c is the cost of fishing per unit effort per unit carrying capacity. While the maximum sustainable yield reflects the potential of the fishing area, the maximum profit is the gap between the revenue and cost.
It may be noted that it is difficult to have the values of the parameters including the conversion factors involved in the model based on realworld observations. Some of the values are taken from Akpalu and Bitew (2018) and we choose reasonable values for other parameters based on the conditions we impose on the model for the purpose of numerical illustrations.
Using the values from Table 1 the maximum sustainable yield is MSY = 0.342945 attained at effort level E_{MSY} = 40.5726. The maximum sustainable profit is attained at a lower optimal effort, E* = 39.9488. Moreover, E* declines with ε_{1}, permitting less fishing when the fishing impact is higher (see Fig. 1).
Parameters and their values used for numerical simulations.
Fig. 1 The optimal effort as a function of the level of fishing activities impact on the growth rate. 
2.2 Capture fishery with MPAs problem
Yamazaki et al. (2015) and Akpalu and Bitew (2014) studied harvest and/or effort control rules and notake marine reserves as effective management tools for rebuilding depleted fish stocks and averting the collapse of fisheries. Following their formulation let M be part of the carrying capacity set aside for marine protected areas (MPAs) from the total capacity, K. Then the carrying capacity of the remaining area, the fishing area, becomes 1–m, where . The reserved environment gradually improves and starts positively impacting the growth of the stock of fish in the MPAs. We assume that the current growth rate increases with the size of m. Moreover, we consider that the density of fish in the MPAs over time becomes higher than the density of the stock outside the MPAs because of the notake policy. Therefore, it initiates a densitybased net influx of fish into the fishing ground, , where y is the density of fish in the marine protected area, , and d_{1} is the dispersion rate (Akpalu and Bitew, 2014). Under these assumptions, we describe the dynamic equation of the stock in the MPAs as:
where 0 ≤ r_{0} + ϵ_{2}m ≤ r and ϵ_{2} is the measure of the positive impact of the MPAs on the growth rate of the fish in the area, and the stock dynamics outside the MPAs becomes^{1}
2.2.1 Steady state analysis
The equilibrium or steady state solution for the above system, equations (6) and (7), is
and
The positive steady statestate stock outside the reserve then is
See eq. (10a) Below
provided r_{0} − (ϵ_{1} + σ) E > 0
Substituting the steadystate solution x_{s} (E, m) in the harvest function, the sustainable yield function is
See eq. (10b) Below
Then we solve the equation to determine the maximum sustainable effort, E_{MSY}, that corresponds to the MSY in terms of the parameters. Since the expression for the solution is very long, for numerical exposition, we substitute the values included in Table 1 and a reserve size, m = 0.06, to find maximum sustainable effort, E_{MSY} = 42.6252 with maximum sustainable yield MSY = 0.337241 . Compared to this effort level, the maximum profit is attained at an effort level E such that
where Π (E, m) = ph_{s} (E, m) − cE is the net profit at time t, for unit price, p, and the cost per unit effort per unit carrying capacity, c. The analytic solution for equation (11) is very long and complicated. For numerical illustration, we use values given in Table 1 and m = 0.06 to find the optimal effort level, E* = 41.897 with corresponding optimal harvest h* = 0.337119 . Observe that the optimal effort and harvest are lower than the E_{MSY} and MSY. We can verify that the optimal solutions are stable for appropriate values of the parameters. For example, using the values given in Table 1 and assigning m = 0.06 and E = 41.897, and linearizing equations (6) and (7) about the optimal solution x* = 0.5364 and y* = 0.0062, we find the two roots corresponding to the characteristic polynomial, r_{1} = − 0.7746 and r_{2} = − 0.1933, which are real and negative. Therefore, (x*, y*) is a stable equilibrium.
2.2.2 Optimal capture fish management with MPAs
Suppose the sole manager of the fishery decides on harvesting effort, E, and the size of the MPAs, m. For simplicity, we assume that the unit price, p, and cost per unit effort per unit carrying capacity, c, are constants. If all future costs and benefits are discounted at a positive social discount rate of δ, the objective of the manager is to maximize the overall present value of the discounted stream of surpluses (which we denote by MP (E, m), i.e.,
subject to the dynamic equations (6) and (7).
The current value Hamiltonian corresponding to this problem is given by
where λ_{1} and λ_{2} are the shadow value of the stock in the fishing area and in the MPAs, respectively.
Using equations (6) and (7), the first conditions for optimality with respect to the flow variables, E and m, and the costate equations to each state variable, at the steady state we get the following system of nonlinear equations
By assigning the values for the parameters from Table 1 and the combination of the conversion factors given in Table 2, we solve the above system of equations using MATHEMATICA 13.0™ for the steadystate optimal solutions, the optimal effort and optimal stock including the optimal harvest summarized in Table 2. We also perform a sensitivity analysis of the model with respect to the conversion factors (see Tab. 2).
From Table 2 we observe that when the impact of harvesting effort on the environment becomes more severe, the optimal steadystate effort decreases and the size of the MPAs slightly increases. Moreover, if the benefit of the MPAs to the reserved environment (to the growth rate) is better than expected (i.e., when ε_{2} increases), the optimal steadystate size of the MPAs decreases, and the effort level increases. And if the dispersion rate is lower than predicted, we must reduce the size of the MPAs and increase effort.
Capture fish and MPAs numerical results and sensitivity analysis.
3 Capture fishery and MPAs with aquaculture problem
In this section, we consider the case where we introduce aquaculture within the MPAs. Although aquaculture activity negatively impacts the MPAs, compared to the fishing ground, we assume that over time the stock in the MPAs will still have a better density. The size of the aquaculture must be small compared to the size of the MPAs so that the MPAs serve their intended purposes (i.e., environmental and ecological protections). We will find an optimal aquaculture size relative to the optimal size of the MPAs that maximizes the overall economic and ecological benefits.
As mentioned earlier, like the fishing activities outside the MPAs, farming activities inside the MPAs affects the reserved environment and consequently lower the per capita growth rate of the stock in the area. We assume the extent of the impact is directly proportional to the aquaculture size, A, hence the carrying capacity of the MPAs becomes m − γa, where and γ is a conversion factor. In addition, the spillover factor depends on the relative size of a, that is on . Therefore, we set up the biomass dynamics in and outside the MPAs as
and
The main purpose of aquaculture production in MPAs is to meet societal demands while balancing economic benefits and environmental protection. Moreover, aquaculture is more productive in MPAs than in openaccess areas because of the clean environment or water quality (Le Gouvello et al., 2017). Aquaculture helps to redirect and ease the pressure on the fishing environment caused partly by the allocation of some areas for MPAs and by intense fishing activity due to the decline of the stock. We assume that aquaculture is developed solely in MPAs and the expansion rate of aquaculture depends on the wildcatch harvest level and the proportion of the area in the MPAs dedicated to aquaculture. Hence, we set up the aquaculture dynamics as
where v is a control variable related to the expansion rate and .
3.1 Steady state analysis
The nonnegative equilibrium solution/point for equations (19) through (21), (a_{s}, y_{s}, x_{s}), is
provided m(r_{0} + ϵ_{2}m)) > d_{2} (1 − v), and
See eq. Above
Substituting x_{s} (E,m,v) into the harvest function, we get the equilibrium harvest (yield) function:
By substituting the values for σ, r_{0}, ϵ_{1}, ϵ_{2}, γ, and d_{2} from Table 1 in (22) and assigning a reserve size, m = 0.06, and v = 0.47, and solving the equation , the maximum sustainable effort, E_{MSY} = 42.7769 with MSY = 0.338603 . And the maximum profit is attained at effort level E such that , where p is the per unit price and c cost per unit harvest/carrying capacity. For p = $15 per kilogram, c=$0.005, and the values from Table 1 the maximum profit attained at optimal effort level E* = 41.6757 with optimal harvest, h* = 0.338327. Moreover, the Jacobian matrix that corresponds to the dynamical system, at the optimal solution, (a* = 0.0329, x* = 0.5412, y* = 0.0134),
has all negative eigenvalues
Therefore, the optimal solution is stable.
3.1.1 Comparison of the three sustainable yield trajectories
In Figure 2, we compare the trajectories of the steadystate yield functions with respect to the level of fishing effort in the three scenarios.
From Figure 2, even though the maximum sustainable yield is lower after the implementation of the MPAs, the fish species are protected from a possible collapse caused by overfishing and other factors. Moreover, we observe a further reduction of the maximum sustainable yield after the development of aquaculture in the MPAs due to an adjustment of the effort level (less effort) outside the MPAs.
Fig. 2 The yield versus effort curve for the basic model, with MPAs, and with MPAs and aquaculture in MPAs. 
3.2 Optimal capture fish effort and optimal aquaculture size within the MPAs
Suppose the production function for aquaculture is Z (A) = P_{a}A and the cost of production is quadratic Φ (A) = c_{a}(P_{a})^{2}A, where P_{a} is per unit area production of farmed fish and c_{a} is a cost parameter (Mykoniatis and Ready, 2016). We assume that buyers cannot distinguish between a species that is farmed or caught in the wild. Let the unit price of both fish be constant, p. Hence the net profit from aquaculture at time t is pP_{a}a − c_{a}(P_{a})^{2}a. In our analysis, we include a separate onetime cost of acquiring extra units of the MPAs. Let the cost per unit area/carrying capacity be c_{3}.
We can rewrite the production and cost functions in terms of a as z (a) = P_{a}a and φ (a) = c_{a}(P_{a})^{2}a, where and , K is the carrying capacity. Then the optimization problem is
subject to the dynamic equations (19), (20), and (21).
The current value Hamiltonian for this problem is
where . where c_{1} = P_{a} production per unit aquaculture area and c_{2} = c_{a}(P_{a})^{2} is the cost per unit area, and λ_{1}, λ_{2}, and λ_{3} are the shadow values of stock outside and inside the MPAs and the aquaculture area, respectively.
Using the first conditions for optimality with respect to the flow variables, E, m, and v:
and the costate equations to each state variable:
At the steady state, the state and costate variables including the control variables satisfy the following equations: , and equations (23) to (25). This is a system of nine nonlinear equations and the formulas for the analytic solutions in terms of the parameters are beyond comprehension.
3.1.2 Numerical illustration
For the purpose of numerical solutions, we assign the values given in Table 3 and the combination of the conversion factors given in Table 4. Then using MATHEMATICA 13.0^{TM} , we find the corresponding optimal solutions presented in Table 4. We also perform a sensitivity analysis to see the effects of the change in the conversion factors and dispersion rate on the steadystate solutions.
From the steadystate optimal numerical solutions summarized in Table 4
i) As the impact of effort on the fishing environment worsens, the size of the MPAs, as well as the effort, must be lowered to ease the pressure on the wild capture fish. This creates a scarcity of wild fish in the market. However, the optimal management strategy implies that the decline in the supply of wild fish can be matched with an expansion of aquaculture production.
ii) If the environmental recovery rate of the reserved area is better than expected or the dispersion rate is lower, the MPAs could be set to a smaller size, and the effort level in the fishing area can be relaxed.
iii) If the impact of fishing farming on the growth rate of the fish stock in MPAs is more severe, the reserve area should be expanded and we reduce the aquaculture activity and lower the effort level outside MPAs.
Parameters and their values used for numerical solutions.
Capture fishery, MPAs, and aquaculture numerical results and sensitivity analysis.
3.3 Transition dynamics and stability
3.3.1 Transition dynamics
Because of the nonlinear nature of the functional forms of the equations used in the dynamic analysis and the number of equations, it is not easy to find analytic solutions. Therefore, we use the fourthorder RungeKutta forwardbackward sweep method to solve the system of equations numerically. First, we approximate the state equations, (19) through (21), by firstorder forward difference and the corresponding costate equations, (28) through (30), by firstorder backward difference equations. Then by substituting the values of the parameters given in Table 3, using initial values, x (0) = 0.27, y (0) = 0.006, and a (0) = 0, and using a guess for optimal control say the steadystate values, we solve the state equations forward for the discretetime interval of [0, t_{f}] partitioned into n parts using a time step h such that t_{f} = hn. Then using the state values and the transversality condition at t_{f}, we solve the costate equations backward. After each forwardbackward computation, we update the control values using the state and costate values and repeat the process until the control values become sufficiently close. The accuracy or convergence of the iterative method is based on Hackbusch (1978). Figure 3 displays the trajectories of the numerical solutions generated using Hertmite interpolation of order 3. Note that for all combinations of the conversion factors and the dispersion rate in Table 3, the trend is the same except they converge to different state and control values.
Fig. 3 The recovery rates of the open access and MPAs stocks and the expansion rate of the aquaculture within the reserve for the optimal reserve size, m = 0.04789. 
3.3.2 Stability
Following the computation of the optimal trajectories of the system, it is necessary to analyze the reaction of the system when an unexpected change or shock occurs due to exogenous factors. In particular, to investigate the stability of the equilibrium, we linearized the equations (19) through (21) and (28) through (30) around the equilibrium points/values. Then using the parameter values and the corresponding steadystate values of the variables x, y, a, λ_{1}, λ_{2}, λ_{3}, E, m, and v, the Jacobian matrix of the model is
The corresponding eigenvalues are
The real parts of the eigenvalues have positive and negative values. The signs of the eigenvalues are consistent across all the other combinations of the values of the parameters used for the empirical analysis. As a result, the control version of the system is unstable. It implies that any deviation from the steady state equilibrium due to say changes in the price of the fish, the cost per unit effort, or the cost of aquaculture production will lead to further departure from the equilibrium. Since all the eigenvalues are nonzero real, the equilibrium is hyperbolic (i.e. it is robust, a small perturbation displaces the equilibrium by a small amount). This shows that economic factors play a significant role in the profitability and stability of the sector. Especially, in developing countries, the relationship between production cost and revenue determines the feasibility of aquaculture.
4 Discussion
The effectiveness of the optimized management plan depends on the cooperation and informationsharing practices between the capture fishery and aquaculture managers. Moreover, in marine natural resource management, it is vital to understand the current status of the ecological environment before introducing a new management strategy. Then we plan accordingly toward achieving a sustainable system over time following the predefined fastapproaching path. For this reason, we solve the system of nonlinear differential equations derived from the necessary conditions for optimality. We find the expansion rate of the aquaculture and hence the recovery rate of MPAs and the wild fishery stocks. We also perform sensitivity analysis to show the necessary adjustments we must make to the paths when the performance of the growth rate parameters and the impact of aquaculture is different from the predicted values. It may be noted that the values of the parameters and other values are not based on real observations. Therefore, the numerical results and the corresponding recovery and expansion rate curves should be considered from a qualitative point of view.
The optimal population growth curve in the reserved area, Figure 3, shows that the stock recovers rapidly while gradually restocking the fishing ground. We also observed that the implementation of aquaculture doesn't change the trend of the trajectories of the stocks and the MPAs still shield the stock (see Fig. 2). However, integrating aquaculture in marine reserves slows down the fishery rebuilding process inside and outside the reserved area. When the impact of aquaculture is higher than expected (i.e., when ϵ_{2} increases), the recovery rate of the reserved area (hence the open access) converges to a lower optimal equilibrium (see Fig. 4).
If an unexpected shock, such as an oil spill and chemical contamination due to human activity, affects the open access (which can be interpreted as a higherlevel impact of effort on the habitat, i.e., larger ϵ_{1}), the optimal effort will be reduced significantly. This helps to minimize the impact of the effort and facilitates the remediation process. The effortlimiting policy creates a scarcity of wild fish in the market. In this case, we can increase the size of the aquaculture to satisfy the demand for fish. We assumed earlier that consumers can't distinguish between wild catch and farmed fish (this is a common practice in most developing countries) and both are sold at the same price. Therefore, the decline in the wild catch can be matched with its substitute or alternative, farmed fish, by increasing aquaculture production.
As the dynamic optimization model implies, one of the management strategies is to lower the wild catch effort level if the intensity of its impacts on the habitat is higher than predicted. This restriction has implications for the fisheries community. In this situation, to divert the effort from open access and to ease the tension between the two sectors, the concerned group (the government) must encourage some of the fishers to get involved in the aquaculture industry. This can be done by prioritizing aquaculture permits for capture fishers and providing incentives, like monetary and technical support.
In addition to the optimized space allocation practices, the profitability and sustainability of the aquaculture sector depend on price to perunit production and cost of production. By assumption, the net profit from aquaculture at time t is given by Π (P_{a}) = pP_{a}a − c_{a}(P_{a})^{2}a, where P_{a} is per unit production and c_{a} is a cost parameter (Mykoniatis and Ready, 2016). Therefore, the feasibility of the business is determined by the relation Π (P_{a}) ≥ 0 ⇒ pP_{a}a ≥ c_{a}(P_{a})^{2}a. Most of the time this relationship is not easy to achieve, especially at the beginning of the investment. Therefore, we suggest that investors must get incentives such as subsidized loans and tax relief until their revenue becomes larger than the cost of production. If investors are encouraged and take part in the aquaculture industry, in the long run, we can maintain a sustainable supply of fish at the same time support the economy by creating jobs.
Fig. 4 The MPAs stock for γ = 1 and γ = 1.2 and m = 0.04789. 
5 Conclusion
MPAs have an advantage in the conservation of the natural environment and help to protect and sustain the fish population. In addition, they have economic benefits if aquaculture production is operating in the area even though it has negative externalities. In this paper, we attempt to determine the optimal size of fish farming in MPAs based on the level of food security and the ecological status of the fishing environment by allocating the right amount of area from the open access areas toward MPAs. This helps in restocking the fishing area at the same time to protect the species of fish from possible collapse.
Le Gouvello et al. (2017) in their study investigated how can MPAs support aquaculture development, how should aquaculture activities support MPAs and how can negative interactions be minimized and greater common trust between fisheries can be achieved. They recommend that a set of principles must accompany the process of setting up aquaculture for reconciling aquaculture and MPAs. The establishment of an aquaculture farm in the MPAs requires a longterm growth and monitoring plan. Therefore, it is necessary to decide the appropriate size of the aquaculture in MPAs for poverty alleviation and food security of MPA's local communities while supporting the recovery process and advancing the resilience of the environment. The results obtained from our study will fill this gap and provide insight into the optimal sizes of multipleuse MPAs and aquaculture size relative to the size of MPAs. It also determines the optimal harvesting effort outside MPAs for sustainable fisheries management. To the best of our knowledge, this is the first dynamic bioeconomic model that considers aquaculture in MPAs and attempts to determine the optimal size of aquaculture relative to reserved areas and fishing efforts.
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^{1}If m→1 the equilibrium stock size x^{x}(E,m)→0 and the corresponding yield h_{x}(E,m)→0 implying there will be no human harvest.
Cite this article as: Hailu FF, Bitew WT, Ayele TG, Zawka SD. 2023. Marine protected areas for resilience and economic development. Aquat. Living Resour. 36: 22
All Tables
Capture fishery, MPAs, and aquaculture numerical results and sensitivity analysis.
All Figures
Fig. 1 The optimal effort as a function of the level of fishing activities impact on the growth rate. 

In the text 
Fig. 2 The yield versus effort curve for the basic model, with MPAs, and with MPAs and aquaculture in MPAs. 

In the text 
Fig. 3 The recovery rates of the open access and MPAs stocks and the expansion rate of the aquaculture within the reserve for the optimal reserve size, m = 0.04789. 

In the text 
Fig. 4 The MPAs stock for γ = 1 and γ = 1.2 and m = 0.04789. 

In the text 
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