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Maintaining Ecosystem and Economic Structure in a Three-Species Dynamical System in Chesapeake Bay

We consider a three-species fish dynamical system in Chesapeake Bay consisting of the Atlantic menhaden as the prey and its two competing predators, the striped bass and the catfish. Building on our previous work in this system, we consider the issue of balancing economic harvesting goals (financial gain for fishermen) with ecological harvesting goals (non-extinction of species). In particular, we investigate the bionomic equilibria, maximum sustainable yield, and the maximum economic yield. Analytical computations and numerical simulations are employed to provide some mathematical guidance on fisheries management policies.

Recently, a fishing management strategy has been developed by NOAA and its local partner agencies to examine the potential impacts of commercial and recreational harvesting of one of Chesapeake Bay's species, the blue catfish, as "a means to reduce their abundance and mitigate their range expansion and ecological impacts" [7,1]. NOAA has reported that this particular fish species is an invasive species in the Chesapeake Bay. As a result, this agency has recommended various strategies to control the spread of the blue catfish, to include consumer-level management programs that promote harvesting and eating the invasive species.
Harvesting any fish species, however, must consider its social, ecological, and economic effects. It is important to balance and manage the population size of fish species (ecological), the revenues/costs/profits of fishing (economics), the amount of fishing effort and jobs (social), and the provision of food for people (social). In this paper, we will only consider the problem of managing the ecological and economic harvesting goals in a specific threespecies system in Chesapeake Bay using ordinary differential equations. Mathematically investigating the ecological, economic, and social impacts of over-harvesting may require a combination of dynamical system techniques, network analysis methods, and statistical perspectives.
In 1954, Schaefer [13] suggested and justified that a single species fishery will achieve maximum sustainable yield (MSY) and maximum economic yield (MEY), assuming logistic growth on the fish species. Maximum sustainable yield is the highest possible harvesting level at which a species may be harvested without the risk of extinction over time. Maximum economic yield is the highest possible harvesting level that provides maximum economic benefits over time. Balancing ecological and economic goals by computing and analyzing the MSY and MEY in fishery systems provides important theoretical insights that guide public policies on managing fisheries. Intuitively, it is relatively easy to understand that over-fishing at extreme levels in a short period of time may quickly result to depletion and extinction of fish species. However, it may not be apparent that harvesting beyond a certain level may not necessarily result to more economic gains, for example, higher profits for the fishermen.
Later, Legovic [5] showed that harvesting of the prey at the MSY level in a predatorprey system might result in the extinction of the predator, due to the reduced source of food supply. As a result, any management policy must consider the interactions between species when deciding on harvesting quotes, as the population dynamics between species is inevitably linked. Hence, it is not clear whether a direct application of MSY theory toward one species, such as the invasive blue catfish, would be sustainable for the whole ecosystem. Further research on the intricate ecological and economic dynamics between the predator and prey is needed. Mathematical modeling via differential equations provides a method to simulate the dynamics of growth, death, competition, interaction, harvesting, with some economic considerations.
Fisheries management is often done as one-species management and is focused on maximizing the yield of this targeted species ignoring the inter-species interactions or any other ecosystem effects. Such one-dimensional policy often leads to deterioration of the stocks and in the case of Atlantic menhaden has already resulted in critical decrease of the menhaden's population in the Bay area [2]. Only recently, the Atlantic States Marine Fisheries Commission (ASMFC) has taken the first step to formally consider the importance of one species to others, particularly the importance of menhaden to other predators, including the striped bass. In their new management policy, they committed to including ecological reference points on menhaden's population and reduced the harvesting quota on menhaden by 10 percent for 2021 and 2022 fishing seasons [3].
This article builds upon the previous work [11] of the authors on three important species in the Chesapeake Bay, involving the Atlantic menhaden as the prey to two competing predators, the striped bass and the blue catfish ( Figure 1).  In [11], a mathematical model was built from a Lotka-Volterra model with the following assumptions: exponential growth on the menhaden, dependence of bass on the menhaden, competition between the bass and catfish, and exponential growth on the catfish. The reasoning behind the exponential growth assumption for the catfish is the fact that this fish is an invasive fish in the Bay and is a generalist predator, meaning that it does not depend on menhaden only as a prey. In the absence of menhaden, it can prey on many other species and as a result the catfish population is increasing very fast and currently, they make up about 75 percent of the biomass in the Chesapeake Bay area [8,10].
Here, we continue the study of the three-species model we introduced in [11] with the aim of balancing ecological and economic harvesting goals. Let = ( ) be the population of the Atlantic menhaden, = ( ) be the population of striped bass, and = ( ) be the population of blue catfish. We assume the following: 1. logistic growth on prey (menhaden), 2. the two predators (bass) and (catfish) are in competition, both preying on , 3. the predator declines at a rate proportional to its size in the absence of , and 4. the predator is an invasive species and grows exponentially at a rate proportional to its size.
The system of differential equations describing the three-species biological interactions under the above assumptions is given by: with initial data values (0) ≥ 0, (0) ≥ 0, (0) ≥ 0. The parameters are in Table 1.
growth parameter of preying effect of on preying effect of on death parameter of the predator growth parameter of environmental carrying capacity of competition effect of on competition effect of on Table 1: The ecological parameters in model (A). All parameters are positive.
The paper is organized as follows. In Section 2, we investigate the bio-economic equilibria (also known as bionomic equilibria) of the system and derive conditions for their existence. A bionomic equilibrium is a point that considers not only the biological or ecological equilibrium, but also the economic equilibrium, with the latter being investigated using a profit analysis technique from Economics. In Section 3, we compute conditions for MSY policy in order to guarantee that none of our three species become extinct. Numerically-simulated graphs are presented to aid analysis and discussion. In Section 4, we look at the issue of balancing economic and ecological harvesting goals.

Bionomic Equilibrium
Bionomic equilibrium comes from biological equilibrium and economic equilibrium. Biological equilibrium is attained when ′ = ′ = ′ = 0 while economic equilibrium is attained when total revenue earned by selling the harvested biomass is equal to the total cost for the effort in harvesting the biomass. The idea comes from [12] and [14].
The goal of this section is to find conditions under which bionomic equilibria exist. Toward this goal, we look at a way to measure the overall net revenue by adding the net revenue for each of our three fish species. The overall net revenue, also known as economic rent (or even profit), of the dynamical system involves parameters on cost per unit effort, price per unit, and catchability coefficient for each of the three species (see Table 2): at any given time where 1 , 2 , 3 are the harvesting effort coefficients of the Atlantic menhaden, the striped bass, and the catfish, respectively.
Thus, we need to consider the dynamical system 1 fishing cost per unit effort for Atlantic menhaden 2 fishing cost per unit effort for striped bass 3 fishing cost per unit effort for catfish 1 price per unit biomass of Atlantic menhaden 2 price per unit biomass of striped bass 3 price per unit biomass of catfish 1 catchability coefficient of Atlantic menhaden 2 catchability coefficient of striped bass 3 catchability coefficient of catfish with initial data values (0) ≥ 0, (0) ≥ 0, (0) ≥ 0. In other words, conditions on the existence of bionomic equilibria can be obtained by solving the system Observe that we did not consider the biological equilibrium 0 ( , , ) = (0, 0, 0) because in this case, the overall net revenue becomes negative. Also, we focus on the biologically relevant interior equilibria with , , ≠ 0. Observe that the bionomic equilibrium is characterized not only by , , but also by 1 , 2 , 3 Suppose the bionomic equilibria is ( , , , 1 , 2 , 3 ). In order for all equations to be satisfied at , we need to consider eight cases. Full details of the proofs for Cases 1 and 2 are given; proofs for the other cases are similar.

Case 1. No harvesting efforts on the prey.
In this case, we have that 1 = 0 and both 2 and 3 are positive, that is, the Atlantic menhaden is not harvested but the striped bass and the catfish are both harvested. We assume that the decision to harvest both predators rely on the fact that net revenue for each of the predators is non-negative. In other words, it does not make sense, economically speaking, to harvest the striped bass if the cost 2 to harvest them is higher than the price 2 2 earned by selling them; similarly, we want 3 ≤ 3 3 . Thus, we take = 2 2 2 and = 3 3 3 . These conditions on 1 , , and guarantee that the fourth equation in (2.1) is equal to zero.
Computing the harvesting effort coefficients 2 and 3 of the striped bass and the catfish, respectively, from the second and third equations in (2.1) we get: . (2.2) Since we want the harvesting efforts 2 and 3 to be positive, we solve for from both equations in (2.2) to obtain: Now, using the first equation in (2.1), we obtain an explicit expression for : The biologically relevant case is when > 0, hence, 1 − . that is, These conditions on the carrying capacity of the prey determine the existence of the bionomic equilibrium for Case 1: When there are no harvesting efforts on the prey, the bionomic equilibrium ( , , , 0, 2 , 3 ) exists, where is given by (2.4), = 2 When there are no harvesting efforts on the first predator, the bionomic equilibrium ( , , , 1 , 0, 3 ) exists, where = 1 1 1 , = 3 3 3 , and the harvesting efforts on the prey and the second predator are given in (2.5).
When there are no harvesting efforts on the second predator, the bionomic equilibrium ( , , , 1 , 2 , 0) exists where = 1 1 1 , = 2 2 2 and the harvesting efforts on the prey and the first predator are given by There are five more cases to consider. Unlike Cases 2 and 3, these remaining cases have exact expressions for all components of the bionomic equilibrium ( , , , 1 , 2 , 3 ). We need to require that each component of is positive in order to insure that exists.
Student Exploration Activity. There are eight cases to consider when analyzing the bionomic equilibria in different harvesting scenarios. Three of the cases were explicitly computed in the discussion. Look at the other five cases: verify that that the components of the bionomic equilibria are as listed in Table 3.

Maximum Sustainable Yield
Analyzing the maximum sustainable yield (MSY) is one of the scientific approaches in managing harvesting and fishing efforts. It balances over-exploitation and under-exploitation, where over-exploitation involves harvesting a resource at a level that threatens its extinction while under-exploitation is the harvesting of a resource that is below the level that the population can withstand.
Computing the bionomic equilibria( , , , 1 , 2 3 )in different harvesting scenarios. Case 4. No harvesting efforts on the prey and the first predator.
No harvesting efforts on the prey and the second predator.
No harvesting efforts on both predators.
No harvesting efforts on all species. The ecological issue that we need to check is extinction of any of our three species when harvesting is implemented. Thus, from the original system (A), we consider the dynamical system To compute the MSY, we look at the harvesting yield as a function of the harvesting effort. We consider several relevant cases: harvesting of the Atlantic menhaden only (that is, no harvesting efforts on the two predators so that 2 = 3 = 0), harvesting of the striped bass and catfish only ( 1 = 0), and the special case of harvesting the catfish only ( 1 = 2 = 0). Analysis of the other cases follows analogous arguments. We start by looking at the equilibrium of (C) and considering the interior cases only. Observe that, ecologically, we interpret any of the cases = 0, = 0, = 0 to mean extinction of the corresponding species. Hence, consider the system: Harvesting Prey Only.
In this case, take 2 = 3 = 0 and solve for , , as functions of = 1 in (3.1). Suppose = + ( + ). We have: Let us denote the slope of ( ) by , that is, write Note that the biologically relevant scenario is the case when (0) > 0. Also, we observe that = < 0. Since and are also negative, we see that an increase in harvesting efforts of the prey will result to a decrease in the biomass of all three fish species.
As an example of how to interpret the results so far from the ecological perspective, suppose, for instance, that the competition between the two predators are unequal with the second predator more aggressive than the first (that is, > ) and suppose that the preying effect of either predators on the prey are the same (that is, = ). Then the biomass of the first predator decreases faster than the biomass of the second predator (that is, / < / ) due to the harvesting of the prey and the competition between the predators. Now, the yield function ( ) in this case is defined as ( ) = ( ). Since ( ) is a linear function, ( ) is a quadratic function of that achieves a maximum yield at the effort given by * = − (0)  A graphical illustration of the ideas will provide context to Theorem 3.1. In Figure 2, the yield function ( ) = ( ) is a parabola that has a highest point because ( ) is a linear function with negative slope. Observe that ( ), ( ) are also linear functions with negative slopes. The maximum sustainable yield * is the -coordinate of the highest point of the yield function. Maximum sustainability in harvesting all three species is attained when harvesting effort * on the prey only is to the left of the points * and * . In other words, one has to insure that harvesting the prey only should occur before and become extinct. • if * > * then harvesting prey only at effort * causes the extinction of the second predator, and • if * > { * , * } then harvesting prey only at effort * causes the extinction of both predators.
Choosing particular values for the model's parameters = 1, = 1, = 0.2, = 0.01, = 0.01, = 0.05 and assuming that the catfish is more competitive than the bass, i.e. = 0.8, and = 0.5, we can plot the yield function and the biomass of the three fish with respect to the harvesting effort as shown in Figure 3. As we can see from the graph, in this particular case harvesting the prey only at MSY level is sustainable and does not lead any one of the three species to extinction.

Harvesting Predators Only.
In this case, take 1 = 0 and solve for , , . For simplicity, assume 2 = 3 = . Suppose = + ( + ). We obtain which are linear functions on . Let us denote the slope of ( ) by so that ( ) = + (0). The biologically relevant scenario is the case when (0) > 0. Observe that > 0 so that increasing the harvesting efforts on the predators does not result to a decrease in the biomass of the prey. Note that The theorem implies that when conditions on are as stated, then increasing the harvesting efforts on the predators will not result to a decrease in the biomass of either predator.
To analyze the situation where either or is negative, let us now consider the yield function when harvesting the predators (quadratic function): or, upon re-arranging, we have  Figure 4: When harvesting both predators, the yield function is represented by a parabola that has a maximum. When harvesting the second predator only, the yield function is represented by a parabola that does not have a maximum because ′ ( ) > 0.
In order to have a maximum yield, we must require that the parabola opens downward, that is The maximum yield occurs when the harvesting effort is * = −1 2 where, in order for * to be positive, we must also require that ( + ) (0) + − > 0. Denote Observe that * > 0 provided − 1 < 0. This implies that is always negative. In order for * to be positive, we must consider two cases: − 1 < 0 and − (0) < 0, or, − 1 > 0 and − (0) > 0. In the previous case, < 1/ so that is negative or that the biomass of decreases with increasing harvesting efforts on . In the latter, is positive, which tells us that even though we are harvesting , it is still possible that its biomass continues to increase. Illustrations of the different scenarios are given in Figure 4 and a numerically simulated graph is given in Figure 5. When both predators are harvested at MSY level, the system is sustainable, even though the biomass of the striped bass gets critically low, while the biomass of the prey (menhaden) is increasing. • suppose / < 0 and / < 0. When harvesting the predators, maximum sustainable yield is achieved, provided * < min{ * , * }, where * , * , * are given in (3.4) and (3.5).

Harvesting the Second Predator Only.
In the special case that only the second predator is harvested, we continue the discussion from above but this time we have 1 = 2 = 0 and 3 = . Then the corresponding linear functions for the biomasses , , as functions of the harvesting effort are: We observe that / is positive: even with increasing harvesting efforts on the predator , its biomass increase. Also, / is positive implies increasing prey biomass. What seems to be counter-intuitive is that / < 0: the biomass of the first predator decreases when we harvest the second predator only. To see that the mathematics does not defy the ecology, consider that an increase in the biomass of the second predator implies an increase in the competition effect of the second predator on the first, i.e. >> , as can be observed in the magnitude of = −( + ). As it is well know from the ecological principle of competitive extinction [4] when two competing species occupy the same niche, one of the two species, which is less competitive, will either get extinct or adapt to a new habitat. Hence, the behavior we obtained in this case is in accordance with this ecological principle and as long as the catfish biomass is increasing, we can not expect increase in the biomass of the striped bass. Such an adverse effect of the generalist predator invasive non-native catfish on the in the Chesapeake Bay could also justify why "although the striped bass stock is not overfished and overfishing is not occurring according model projections [spawning stock biomass] could fall below the threshold in the future" [6].
The second counter intuitive result is the increase in the biomass of the second predator, the catfish, even though it is harvested. This result however does not contradict our previous findings in [11], where we have shown that only with a very aggressive harvesting of the catfish such that the harvesting effort (ℎ) is comparable or greater that the intrinsic growth rate of the catfish (ℎ ≥ ) its growth could be controlled. Such aggressive harvesting resulted in a phase-shifted behavior of the catfish and the prey, and on a long run the striped bass still got extinct supporting the principle of competitive extinction. Furthermore, since / > 0, it follows that the quadratic yield function ( ) = ( ) will not have a maximum value and hence the harvesting of the catfish only could be done at any level but will still not be sufficient to halt its invasive effect.
When harvesting the catfish only, as shown in Figure 6, we see that biomass of both catfish and menhaden is increasing as the effort increases, while the biomass of the bass is decreasing. Furthermore, MSY can not be attained due to the fact that yield is constantly increasing. Hence, in order to avoid the danger of driving the striped bass to extinction, harvesting the catfish only must be halted at some point.  In this case, let us assume that 1 = 2 = 3 = ≠ 0. Solving for , , as a function of , we obtain where = + ( + ). Observe that among the three, only (0) is guaranteed to be positive. We consider the situation where , , are all negative; in this case, the biomass of the species decrease when combined harvesting effort increases. The goal is to find conditions so that none of the species become extinct. Also, the biologically relevant scenario is when the (0), (0), (0) > 0.
Using the same values for the model's parameters as before, we graph the yield and the biomass of the three fish species when the harvesting effort is applied to all three species as shown in Figure 7. As expected from the theoretical results, the biomass of all three fish is decreasing as the effort increases. We can also see that in this particular case, even though * < * , * , * , and hence the system is sustainable, the biomass of the both predators is getting critically low which will be too detrimental to their long term sustainability.
Student Exploration Activity. In the analyses presented when harvesting predators only, it is assumed that the harvesting efforts 1 and 2 are equal. Analogously, when harvesting all three species, it is assumed that 1 = 2 = 3 (which is why the case is called combined harvesting). Consider the case when harvesting efforts are not equal. In these cases, what are the corresponding yield functions? Geometrically, what kind of graphs will aid analyses of these cases?

Balancing the Economic and Ecological Harvesting Goals
At this point, it is natural to ask how are the results on the bionomic equilibrium related to the results on the maximum sustainable yield? Observe that the computations in Section 2 involve specific points ( , , , 1 , 2 , 3 ) while computations in Section 3 involve functions ( ), ( ), ( ) on the effort . While Section 2 tackles the analysis of the economic structure via the total net revenue ( ) as a function of harvesting, Section 3 considers the investigation of a very important ecological issue, extinction, as a function of harvesting. In this section, we address the problem of relating these economic and ecological considerations, in particular, at what harvesting efforts are we able to maintain the ecological and economic structures of our system?
Let us look at the case when we harvest the prey only, that is, 1 = ≠ 0 with 2 = 3 = 0. Analysis of the other cases follows analogous arguments. Consider the overall net revenue when 2 = 3 = 0 from Case 6 (B) in Section 2 and express it in terms of the yield ( ): (4.1) Oftentimes, this overall net revenue function in Economics is also called the profit because 1 ( ) gives the revenue (money coming in due to harvesting ) while 1 gives the cost (money coming out due to harvesting ). In the table in Table 3 (for simplicity we set the harvesting coefficient 1 = 1), we see that ( 1 ) = 0 where, with ( , , ) = ( 1 1 , 1 ( + ), 1 (− + )). Let us seek the maximum of the profit function ( ); computing the derivative, we have From Section 3, we know that the biomass ( ) of the prey is a linear function of , that is, ( ) = + (0) so that ′ ( ) = 0 if and only if 1 ( + (0) + ) − 1 = 0, i.e., Since ′′ ( ) < 0 for all because < 0 in the case when we harvest prey only, we see that achieves a maximum at the effort¯. In this case, we say that¯gives us the maximum economic yield (MEY).
We then ask, how does the MEY relate to the MSY when we harvest the prey only? Recall that the harvesting effort to achieve MSY when harvesting prey only is given by so that using (4.3), it follows that¯= 1 2 1 + * < * , because < 0. To complete our analysis, we next investigate the bionomic equilibrium effort 1 (4.2) in relation to the MEY (4.3) and the MSY (4.4). Since ( 1 ) = 0, it follows from (4.1) that In other words, the -coordinate of the bionomic equilibrium in case we harvest the prey only is just one point on the linear function ( ), that is, the point ( 1 , ) lies on the graph of the line ( ) and the vertical line where = 1 . Moreover, we have that We have to consider two cases: In the first case, we have that In the second case, we have that 1 < * . Let us summarize our results into a theorem: 3. the maximum sustainable yield is achieved at the effort * where * is given in (4.4).
In particular, the maximum economic yield can always be achieved at harvesting efforts that are less than the efforts to reach maximum sustainable yield. Moreover, if > (0)/2, that is, the ratio of the cost per unit effort to the price per unit biomass is more than half of the biomass when no harvesting of prey is implemented, then harvesting efforts that exceed¯may lead to a zero profit even before maximum sustainable yield is achieved. In other words, from both the economic and ecological sense, it does not make sense to harvest more fish when doing so becomes more costly.
Student Exploration Activity. Consider Figures 2 and 8. Create a graph that combines the two graphs in one coordinate system.

Conclusion
Mathematical modeling is a powerful tool that can be used for understanding the behavior of real-world systems. Here we used mathematical modeling to study the dynamics of two predators and one prey system in the Chesapeake Bay and considering the seemingly contradicting goals of the fishermen (gain profit) and the ecosystem (avoid extinction). This work could be used as an individual or group project in a mathematical modeling class. As mathematics is sometimes seen as a solitary activity, assigning group mathematical modeling projects encourage students to work together and will emphasize that modeling is an inherently collaborative process. Moreover, this project highlights an interdisciplinary approach towards solving a real-life issue by weaving mathematical concepts such as differential equations, equilibrium points and graphs of parabolas and lines with economical concepts such as maximum yield, maximum profit, and economic equilibria.
As most fisheries management focus on yield maximization (higher profit for fishermen, more jobs, increased food security), it is very important to understand the long-term effects of such a policy on sustainability of the species population. Here we use one prey (Atlantic menhaden) and two competing predators (striped bass and the invasive catfish) model to investigate the impact of maximum sustainable yield policy. We have shown that harvesting the prey only at MSY level under certain conditions on the model's parameters may result in extinction of one or even both predators; while harvesting the predators at MSY levels is beneficial for the prey as their biomass is increasing as the harvesting effort increases. We have also shown that harvesting both predators, the native striped bass and the invasive catfish, at MSY level is sustainable and while the biomass of the striped bass is decreasing with the increased effort, the biomass of the catfish could decrease or increase depending on the model parameters. Another interesting, but unexpected result of this study was obtained in the case when the harvesting effort was applied only on the catfish, and even though the other predator was not harvested, its biomass was still decreasing with increasing the effort on the catfish. Even though these results may seem puzzling, they are in compliance with the well known principle of competitive exclusion [4] that if two competing species occupy the same niche in an ecosystem, they cannot co-exist indefinitely, one will either get extinct or will adapt to fill a different niche of the ecosystem. In conclusion, our study showed that even a very aggressive harvesting will not be sufficient to halt the invasive effects of the non-native catfish in the Chesapeake Bay area and some other approaches may be needed in order to reduce their intrinsic growth rate.
Finally, the mathematics and economics computations in the final section provide evidence that when harvesting prey only, efforts to achieve maximum economic yield (MEY) is less than the harvesting efforts to achieve MSY. In other words, fishing beyond efforts that yield MEY will not guarantee more economic returns but will require more costs (jobs, fishing boats, fuel). Moreover, it is even possible that fishing beyond efforts that yield MEY may lead to an effort 1 that yields zero overall net revenue when harvesting prey only.