#### Abstract

In this paper, a fishery model for African catfish and Nile tilapia is formulated. This model is used to compare financial profit and biomass outtakes in a two-species system versus single species systems. We consider a stage-structured fish population model consisting of the aforementioned fish species together with two food resources. The model dynamics include cannibalism, predator-prey, feeding, reproduction, maturation, development, mortality, and harvesting. We prove consistency of the model in the sense that the solutions will stay bounded and nonnegative over time. Conditions for local stability of fish-free equilibrium point are established. The simulation results reveal asymptotically stable solutions with coexistence of African catfish, Nile tilapia, and two food resources. The major conclusion from our findings is that fisheries should culture both species to maximize the biomass outtake and financial profit.

#### 1. Introduction

Fish is a source of animal protein, providing many key nutrients and calories that are needed for physical and mental development [1, 2]. Fish contribute percent of animal protein and percent of all proteins and are crucial for over billion people in developing countries [3].

In East Africa, the major species of fish caught are Nile perch (*Lates niloticus*), Nile tilapia (*Oreochromis niloticus*), African catfish (*Clarias gariepinus*), and silver fish (*Rastrineobola argentea*) to mention but a few. The fishery industry contributes to the gross domestic product (GDP) and has continued to be an important source of foreign exchange earnings through fish exports to the East African region and international markets [4]. Due to the importance of fish to so many people, the demand is very high and has resulted in overfishing and hence a decrease in fish population stock [5, 6]. Therefore, there is a need to develop sustainable fisheries management practices in East Africa.

Mathematical modeling plays a significant role in the sustainable management of renewable resources. It describes phenomena leading to better prediction, management, and control techniques. Modeling and analysis of multiple species ecological problems were first done by Volterra [7]. Since then, many unstructured ecological models have been formulated and analyzed to study different phenomena [8–12]. Unstructured models, however, treat all individuals as identical and account for reproduction and death as life history processes that are important for population dynamics.

Over the last four decades, there has been a trend towards increased realism of ecological theory. This trend has been supported by introducing physiologically structured population models that distinguish individuals based on physiological characteristics, e.g., size, age, weight, and length. These models include feeding, development, reproduction, and mortality life history processes [13–19].

Stage-structured population models, which still carry a lot of interesting information, are simpler versions of physiologically structured population models and have increased in popularity [9, 19–24]. Field and laboratory studies have attributed several interesting features of population dynamics of cannibalistic populations, specifically to the occurrence of cannibalism [13, 19, 25–30]. Nevertheless, we have not encountered any mathematical models that consider several species including cannibalism.

In this study, we formulate a stage-structured fishery model that resembles the feeding habits of African catfish and Nile tilapia to compare financial profit and biomass outtakes in a two-species system versus single-species systems. The paper is organized as follows: In Section 2, we present the food and feeding habits of African catfish and Nile tilapia. In Section 3, we formulate a stage-structured fishery model with harvesting. In Section 4, we prove the basic properties of the model system (18). Section (5) gives the mathematical analysis of the model. In Section 6, we present the numerical simulations and results of the model equations. We end with a discussion of the results in Section 7.

#### 2. Food and Feeding Habits of African Catfish and Nile Tilapia

African catfish (*Clarias gariepinus*) is a group of domesticated African catfishes of the family of Clariidae, air breathing catfishes. It is widely distributed in African and middle East fresh waters [31–33]. In Uganda, it is found in most natural water bodies, including swamps, streams, rivers, and lakes [34].

Nile tilapia (*Oreochromis niloticus*) is a species of tilapia, a Cichlid fish native to Africa and Israel. As a food resource, the Nile tilapia is the most important fish species in the tropical and subtropical freshwater, often forming the basis of commercially fisheries in many African countries [35]. In Uganda, Nile tilapia is the second most important farmed aquaculture after African catfish [36]. It is native of lakes Albert, Edward, and George and introduced in the 1950s into Lakes Kyoga and Victoria [37].

Food and feeding habits of fresh water fish species make up the foundation for the development of a management program of fish capture and culture. According to Otieno et al. [38], fish feed mainly on food items that can fit their mouth and what their stomach can digest. As fish grows, the stomach becomes larger, and its digestive system becomes more developed. African catfish is an opportunistic omnivorous feeder ingesting a wide variety of food items such as algae, macrophytes, zooplankton, insects, fish, detritus, amphibians, and sand grains [39, 40].

Tesfahun [41] showed that African catfish preferred to feed on animals while plant items were ingested during pursuing other prey organisms. On top of some common resources, such as zoo plankton, smaller specimens tend to ingest more insects while larger specimens mainly prefer fish prey.

Nile tilapia is omnivorous in its feeding habit with plant origin foods as the most dominant food item [41].

#### 3. Model Formulation

In this section, we will develop our model system (18). The model we are considering is devoted to the interactions and dynamics of African catfish, Nile tilapia, and two food resources with harvesting. Two food resources are used to capture the difference in feeding habits. African catfish are classified into four stages: small juvenile , large juvenile , small adult , and large adult . Nile tilapia are composed of two stages: juvenile and adult . The stages divide up the the population in body mass intervals. The two food resources are an animal-based food resource (e.g., zooplankton) and a plant-based food resource (e.g., phytoplankton). African catfish feed on both food resources. In addition, small adult African catfish feed on juvenile Nile tilapia and small juvenile African catfish; large adult African catfish feed on adult Nile tilapia and large juvenile African catfish. Nile tilapia only feed on the plant-based food resources.

The variables used in the model are summarized in Table 1.

Basing on the food and feeding pattern of African catfish and Nile tilapia in Section 2, the assumed feeding habit pattern is summarized in Table 2.

The biomass of small juvenile African catfish increases due to somatic growth at a rate , reproduction at a rate from large adult African catfish, and reproduction at a rate from small adult African catfish, where is the reproduction fraction and is the maturation rate from small adult African catfish to large adult African catfish. Its biomass decreases due to maturation into the large juvenile stage at a rate , harvesting at a rate , cannibalism at a rate , and death due to natural causes at a rate . The reproduction fraction, , is a parameter determining the fraction of how much of the energy production should be dedicated to reproduction and how much should be dedicated to maturation.

The biomass of large juvenile African catfish increases due to the maturation of small juvenile African catfish into the large juvenile stage at a rate and the somatic growth at a rate . Its biomass decreases due to maturation of the large juvenile at a rate , cannibalism at a rate , harvesting at a rate , and death due to natural causes at a rate .

The biomass of small adult African catfish biomass increases due to maturation of large juvenile African catfish into the small adult stage at a rate and somatic growth at a rate . Its biomass decreases due to maturation of small adult African catfish at a rate to large adult African catfish, reproduction at a rate , harvesting at a rate , and natural death at a rate .

The biomass of large adult African catfish increases due to the maturation of small adult African catfish into the large adult stage at a rate . Its biomass decreases due to harvesting at a rate and natural death at a rate . Biomass increase due to somatic growth is lost in reproduction.

The biomass of juvenile Nile tilapia increases due to reproduction of adult Nile tilapia at a rate and somatic growth at a rate . Its biomass decreases due to maturation into the adult stage at a rate , predation at a rate , harvesting at a rate , and death due to natural causes at a rate .

The biomass of adult Nile tilapia increases due to maturation into the adult stage at a rate , and it decreases due to predation at a rate harvesting at a rate and natural death at a rate . Biomass increase due to somatic growth is lost in reproduction.

According to De Roos et al. [21], in the absence of African catfish and Nile tilapia, the food resource is assumed to have semichemostat growth rate , where is the intrinsic growth rate and is the carrying capacity of food resource, .

While the food resource follows the logistic growth rate , where is the intrinsic growth rate and is the carrying capacity of food resource .

Due to external influx and spatial conditions such as coverage, not all food resources are available to the fish population, see, e.g., [18] for more details and discussions. That is, the fish cannot reach all food ; thus, we introduce a threshold constant, ; the available animalistic food is defined by

This threshold forces the food resource away from zero, thus eliminating the zero growth rate at the origin. A zero growth rate at the origin creates instability in the solution, as well as the possibility of extinction of that food resource. Since the plant food resource, , follows a semichemostat growth, it has a nonhorizontal tangent line at the origin; therefore, no analytical nor numerical instability occurs, and hence, we are using a zero threshold constant on the plant food resource. We note here that if we instead had used logistic growth rate models without threshold for the food resources, then cyclic solutions would most probably appear, for a deeper discussion of this and related concepts, see, e.g., [42].

##### 3.1. Biological Assumptions

In our model, we make the following assumptions: (i)African catfish are assumed to have birth size (in fact, the birth size in our model corresponds to the size at which the fish transform from larval into fry, since this is the stage when the fish starts utilizing the food resource) . The size of small juveniles is in the interval , the size of large juveniles is in the interval , the size of the small adults is in the interval , and the size of large adult African catfish is (ii)Nile tilapia are assumed to have birth size . The size of the juveniles is in the interval , and the size of adults is (iii)Large adult African catfish of size and adult Nile tilapia of size do not grow but invest all energy in reproduction(iv)Small adult African catfish uses the available energy for growth, maturation, and reproduction(v)Small juvenile African catfish, large juvenile African catfish, and juvenile Nile tilapia use all available energy for growth and maturation(vi)The growth rate and reproduction rate depend on food availability(vii)Maturation rates depend on the net biomass production rates(viii)Consumption of food follows the Holling type II functional response. The net biomass production rates are assumed to equal the balance between ingestion and mass-specific maintenance rates(ix)At low food densities, the energy intake of juveniles and adults is insufficient to cover their maintenance requirements. As a result, they do not produce biomass. They only produce biomass when their food intake is sufficient to cover their maintenance demands(x)The net biomass production rates for juveniles and adults are assumed to equal the balance between ingestion and maintenance rates for African catfish and Nile tilapia, respectively

##### 3.2. Vital Rates

We adopt De Roos et al.’s [21] derivation of net biomass production rates of individuals in different stages and the maturation rate from one stage to another.

The net biomass production rates of juvenile and adult African catfish are given by where

Here, is the conversion efficiency of food ingested, and are the maximum ingestion rates per unit biomass of juvenile and adult African catfish, respectively. is the mass specific maintenance rate of African catfish, and is the half saturation food level of African catfish.

The feeding rate is defined in such a way that, for large values of biomass, say , we have and for it follows that . Note also that .

The net biomass production rates of juvenile and adult Nile tilapia are given by where is the conversion efficiency of food ingested by Nile tilapia and and are the maximum ingestion rates per unit biomass of juvenile and adult Nile tilapia, respectively. is the half saturation constant of Nile tilapia, and is the mass-specific maintenance rate of Nile tilapia.

Natural mortality rates of African catfish and Nile tilapia are given by and , respectively.

Cannibalistic mortality rates (the cannibalistic and predation mortality rates are in fact proportional to the biomass of the consumed population stage; in some literature, the biomass of the consumed population is included in these expressions) of small adult African catfish on small juvenile African catfish, , and large adult African catfish on large juvenile African catfish, , are given by

Predation rates of small adult African catfish on juvenile Nile tilapia, , and adult African catfish on adult Nile tilapia, , are given by

We use the maturation function to define the following maturation rates for African catfish and for Nile tilapia

##### 3.3. The Schematic Diagram of the Model

Based on the model description and assumptions made in Section 3, the dynamics of the system is represented in Figure 1.

##### 3.4. Model Equations

From the assumptions and interrelation between the variables in Table 1 and the parameters in Tables 3–5, as shown in Figure 1, the rates at which the biomass of small juvenile African catfish, large juvenile African catfish, small adult African catfish, large adult African catfish, juvenile Nile tilapia, adult Nile tilapia, and food resources changes are given by the following ordinary differential equations, using the expressions given in Equations (4)–(13) where subject to the initial conditions:

and .

#### 4. Basic Properties of the Model

We need to show that the model is well-posed in a biologically feasible domain. Since the variables represent living species, nonnegativity ensures that they never become negative. Boundedness explains that there is a natural restriction to the growth of the population as a consequence of limited resources.

##### 4.1. Positivity

We state a theorem which proves that all solutions of the model system (18) are nonnegative for . This proof could be made by considering cases, in the spirit of [43], but since our model would require 15 such cases, we instead prove the nonnegativity using decoupled ODEs.

Let we say if all elements in are nonnegative.

To prove the above statement, we need some preliminary theory. Let .

Proposition 1. *Let and be integrable. All solutions to the ordinary differential equations
satisfies either case (1) for all if or case (2) for all if .*

*Proof. *Rewriting the equation we get after multiplication with the integrating factor integrating both sides gives
or equivalently
From this inequality, the two statements above follows instantly.

Theorem 2. *The solutions, , to system (18) satisfy for all if the initial data is nonnegative, i.e., . If for some , the initial data then for all .*

The ecological interpretation of this theorem is that, on the one hand, if one species is present in the model at the start of simulations, then its biomass is always positive. On the other hand, if we have high harvest pressure or high cannibalism, the species biomass might decline exponentially towards extinction.

*Proof (of Theorem 2). *That all solutions are nonnegative to system (18) follows as a direct consequence of case (2) in Proposition 1 since all coupled ODEs in system (18), studied as single de-coupled ODEs can be written in the form given in Proposition 1 including the fulfillment of the hypotheses. In the same way, applying case (1) in Proposition 1 proves the second part of the statement.

##### 4.2. Boundedness

Theorem 3. *All solutions of the model system (18) with initial conditions are uniformly bounded within the region .*

*Proof. *Let be the total biomass of system (18) at time such that
Taking the derivatives of gives
which along the solution path of the model system (18) yields
Substituting Equations (24) and (25) into Equation (34), we get (for convenience, we now use and )
where
For any , we add to both sides of Equation (35) to get
Since are nonnegative continuous functions, each function bounded by a positive constant and , for example, . Thus, we define . This now implies that
We study the right hand side
which is a second degree polynomial in , and hence, attains its maximum value at
Substituting from Equation (41) into inequality (38) yields
Using the product rule of derivatives on the left hand side and integrating from to on both sides give the bound
where is the total biomass at time and
By Theorem 2, we get the bound
for all .

Hence, the solutions of the model in are confined in the region
. Therefore, we conclude that the model is biologically and mathematically well-posed in the region .

#### 5. Analysis of the Model

##### 5.1. Existence of Equilibrium Points

The points at which the derivatives of the system (18) are zero will be referred to as equilibrium points or steady state solutions. The system has six possible feasible equilibrium points. The vector below indicates the values of .

*E*^{1} The fish-free with plant based food resource equilibrium point .

*E*^{2} The fish-free with two food resources equilibrium point .

*E*^{3} The African catfish free equilibrium point .

*E*^{4} The African catfish free equilibrium point .

*E*^{5} The Nile tilapia free equilibrium point .

*E*^{6} The coexistence equilibrium point .

##### 5.2. Local Stability Analysis

In this subsection, we investigate the local stability of each of the equilibrium solutions of the model system (18) obtained by calculating the Jacobian matrix corresponding to each of the equilibrium points.

Theorem 4. *The equilibrium points with zero animalistic food resource and are all unstable.*

*Proof. *The Jacobian matrix evaluated at equilibrium point is given by
where

, , , , , , and . We observe that the seventh and eighth columns contain only diagonal terms which form two eigenvalues and out of eight eigenvalues. Since is positive, we thus have a nonstable equilibrium point.

Similarly, using the method above, the equilibrium point is unstable.

Theorem 5. *The fish-free equilibrium point is locally asymptotically stable if and and unstable otherwise.*

*Proof. *The local stability of the equilibrium point is investigated using the eigenvalues or trace determinant criteria of the Jacobian matrix. An equilibrium point is locally asymptotically stable if the Jacobian matrix evaluated at that point has negative eigenvalues or has a negative trace and a positive determinant.

The Jacobian matrix evaluated at equilibrium point is given by
where

, , , , , , , , , , , , , , and .

Observe that if the harvesting is high enough, then for . We need to show that all eigenvalues of are negative. As the seventh and eighth columns contain only diagonal terms which form the two eigenvalues and , the other six eigenvalues can be obtained from the submatrix formed by excluding the seventh and eighth rows and columns of . Hence, we have
The eigenvalues of the matrix are the roots of the characteristic equation . That is
=0, which is equal to solving (since this is a block diagonal matrix) where
From
We get a second degree characteristic equation given by
that is
Using the Routh-Hurwitz criterion, the solutions have negative real parts if
Now
is equivalent to
That is, this inequality is satisfied if
is equivalent to
That is, if , we get
This inequality is satisfied if and only if
Here, is the recovery potential of Nile tilapia based on growth in biomass.

Considering
We get a fourth degree characteristic equation given by
where
Using Routh-Hurwitz criteria [44, 45], the roots of the characteristic Equation (63) are negative or have negative real parts if
Criteria 1. That is, is true, provided , which is true if the harvesting rates are high enough.

Criteria 2. We have to show that ; that is, we need to check if the left hand side is greater than zero; the left hand side is
All terms in the right hand side of Equation (67) are greater than zero (since ) with the possible exception of . From the definitions, is equivalent to which is also greater than zero, hence showing that .

Criteria 3. The analytical proof of
is very hard. Instead, we observe the graphs in Figure 2 where they show that this criterion is satisfied, since at the harvesting rate zero, the graph of against harvesting rate starts at a positive value and is increasing. Figure 2(a) shows the values of this expression for a larger set of harvesting rates. Figure 2(b) shows that this expression is positive even for small values of the harvesting rate.

Finally, Criteria 4: is the same as
This is equivalent to
where
The function can be viewed as the recovery potential of African catfish based on growth in biomass. The recovery potential represents the net generational biomass production (per unit body mass) in a virgin environment [24]. Thus, by the Routh-Hurwitz criterion, the fish-free equilibrium is asymptotically locally stable iff and , defined by Equations (61) and (71) which completes the proof.

**(a)**

**(b)**

The Jacobian matrices evaluated at the equilibrium points , , and lead to characteristic polynomials of degree eight whose eigenvalues are too complex to be obtained analytically. Therefore, we revert to numerical simulations, see Section 6.

#### 6. Numerical Simulations

In this section, we carry out numerical simulations using MATLAB to investigate coexistence and stability of African catfish, Nile tilapia, and the two types of food resources at steady state. We further investigate proportional harvesting in a system that has both African catfish and Nile tilapia, only African catfish, and only Nile tilapia, respectively, in terms of biomass yield and financial profits at steady state.

##### 6.1. Model Parameterization

The parameter values are obtained from the literature review when possible. The maturation size of African catfish is 28-30 cm in total body length [46], and the maturation size of Nile tilapia is 20.8 cm [47].

We used the standard quarter-power scaling law [48–50] to estimate: the maximum ingestion rate with proportionality constant for the maximum ingestion rate 0.01 [49, 51], maintenance rate with proportionality constant 0.01 [48, 50], and mortality rate with proportionality constant 0.001 [52]. Following Lessmark [53] and Byström and García-Berthou [54], we used uniform half saturation constants for Nile tilapia and African catfish, evaluated by the mean weight of, respectively, species.

In the model, the birth size of African catfish is 0.02 grams [55]. The maturation size of large juveniles and large adults is assumed to be 30 cm and 100 cm, respectively. Since our model is based on weight, we use the following relation between length, (cm), and weight, (grams) where and for the African catfish [10] and and for the Nile tilapia [56]. Conversion efficiency rate is assumed to be independent of food type and fish species and is set to 0.5 [49, 50]. The maximum ingestion rates of adult individuals are assumed to be 0.85 times the ingestion rates of juveniles for both species [23]. Finally, we assume an equal natural mortality rate at all stages.

The parameter values used in the model are summarized in Tables 3–5.

##### 6.2. Numerical Stability Analysis of , , and

The remaining equilibrium points to investigate the stability are , , and .

The stability of equilibrium point occurs when , . We tested the stability by running the model with different harvesting rates , using one hundred different initial values of the form , taken randomly from the parameter space. In all these simulations, we ended up with the equilibrium solution ; we therefore conclude that is a stable equilibrium point provided that the harvesting rate of the catfish population is high enough and and are smaller than 0.005.

In an analogous manner, we established the stability of when both and exceed 0.01 and for .

Finally, when all harvesting rates are low, the simulations always converge to the equilibrium point .

##### 6.3. Simulation Results

Our simulations are stable in the sense that when positive initial values are used, the solutions approach the positive steady state (all simulations shown in other figures in this paper have reached steady state) as indicated by Figure 3 which shows the dynamics of the solutions to system (18). This signifies coexistence and stability of African catfish, Nile tilapia, and the two different food resources at steady state.

Recently, Lundström et al. [23] adopted the model introduced by De Roos et al. [21], in which Lundström et al. compared the properties between a stage-structured fish population model with an age-structured population model. They investigated how harvesting strategies that qualify for *pretty good yield* can account for conservation and concluded that equal harvesting rate of juveniles and adults is always a good strategy. We present some of these properties, using equal harvesting in all stages in our model: Figure 4 for African catfish and Nile tilapia and Figure 5 Nile tilapia only.

One result by Lundström et al. [23] is that when studying the impact of harvesting on population size structure, they found a positive change in the fraction of juveniles compared to adults in the population. In contrast to this, we find that the impact of population size structure may have a negative dependence on the harvest rate, see Figure 4. This negative impact occurs because African catfish is cannibalistic and predates on Nile tilapia, in that adults feed on juveniles, and this reduces the fraction of juveniles in the system. When we simulate the single fish species Nile tilapia (see Figure 5), our model shows similar results as in [23]; in particular, the impact on size structure is positively dependent on the harvesting rate, which signifies that increasing the uniform harvesting rate there is an increase in the fraction of juveniles.

In fisheries, one can use different mesh sizes in the nets; we therefore have modeled system (18) under a specific type of harvesting rate. In Figures 6 and 7, we consider unbalanced harvesting rates on juveniles and adults for both African catfish and Nile tilapia, i.e., harvesting pressure , where the proportion . The figures display the biomass yield (Figure 6) and financial profit (Figure 7) with respect to different harvesting rates, , and proportions, . The maximum sustainable biomass yield and maximum sustainable financial profits are attained when is large; that is, we mainly harvest adults.

In fisheries, one can harvest the fish stock by catch and release, for example, with a bottle trap or a fyke net; we therefore model system (18) with variable harvesting rates on all fish compartments. Table 6 shows the results after using pattern search to find optimal harvest rates that maximize biomass yield and financial profit for both species in the system, only Nile tilapia in the system, and only African catfish in the system. Our finding is that harvesting in a system that has both Nile tilapia and African catfish gives the highest biomass yield and financial profit (the optimal values for biomass yield and financial profit in Table 6 are found at different harvesting rates), compared to harvesting in a system having Nile tilapia only or African catfish only.

#### 7. Discussion

The major aim of this study was to compare financial profit and biomass outtakes in a two-species system versus single-species systems. To facilitate this comparison, a nonlinear deterministic stage-structured fishery model for the dynamics of African catfish, Nile tilapia, and two food resources has been developed. African catfish is classified into four stages to capture the cannibalistic behavior, while Nile tilapia is categorized into two stages; in addition, the fish feed on two separate food resources in a preferred order. We established the region where the model system (18) is biologically feasible and mathematically well posed. The analytical results revealed that the fish-free two food resource equilibrium point is asymptotically stable if the recovery potentials and . Numerical simulations indicated one positive equilibrium solution for each set of parameters regardless of positive initial values.

We established the two-species system to be better in both financial profit and biomass outtakes, compared to a single-species fish stock. Considering equal harvesting rates on both African catfish and Nile tilapia, results show that the impact of harvesting on population size structure is negative. The negative impact signifies a decrease in the fraction of juveniles in the system. This is because African Catfish is cannibalistic and predates on Nile tilapia.

Applying optimal harvest rates for harvesting both Nile tilapia and African catfish, African catfish only, and Nile tilapia only to maximize biomass yield and financial profits, our findings to some extent agree with [63, 64], in that, small fisheries using ponds or small lakes should implement both African catfish and Nile tilapia to harvest more biomass yield and gain more financial profits.

We emphasize that our findings are based on the model system (18); to prove these results, one should compare two species versus one species in fisheries, alternatively corroborating our findings, using different models. This study can be extended by analyzing a fishery model where there are more than two fish species and more food resources to make the model more realistic. Another direction for future research would be to investigate under what conditions the hydra effect occurs in cannibalistic predator prey models, cf. [65, 66].

#### Data Availability

The parameter input values used in the simulations were obtained from literature. They are summarized in Tables 3–5.

#### Conflicts of Interest

The authors declare that they have no conflict of interest.

#### Acknowledgments

We are very grateful to the financial supported extended by Sida Phase IV bilateral program with Makerere University (2015-2020 project 316) “Capacity building in Mathematics and its application.”