Modeling Dynamics of Prey-Predator Fishery Model with Harvesting: A Bioeconomic Model
A mathematical model is proposed and analysed to study the dynamics of two-prey one predator system of fishery model with Holling type II function response. The effect of harvesting was incorporated to both populations and thoroughly analysed. We study the ecological dynamics of the Nile perch, cichlid, and tilapia fishes as prey-predator system of lake Victoria fishery in Tanzania. In both cases, by nondimensionalization of the system, the equilibrium points are computed and conditions for local and global stability of the system are obtained. Condition for local stability was obtained by eigenvalue approach and Routh-Hurwitz Criterion. Moreover, the global stability of the coexistence equilibrium point is proved by defining appropriate Lyapunov function. Bioeconomic equilibrium is analysed and numerical simulations are also carried out to verify the analytical results. The numerical results indicate that the three species would coexist if cichlid and tilapia fishes will not be overharvested as these populations contribute to the growth rates of Nile perch population. The fishery control management should be exercised to avoid overharvesting of cichlid and tilapia fishes.
In today's life, the relationship between predator and prey became an important aspect to discuss in ecology. The prey-predator system has attracted many researchers to study the interaction between the species . Thus, we use mathematical ecology aspect to study the interacting species. The current study considers lake Victoria fishery found in Tanzania as a case study. However we have not gone to the field, but the current trends on the dynamics of species in the lake have been obtained from reading different literature such as NPFMP , FAO , LVFO , Barack , and Barilwa . This literature explains the current trend of the lake and the fishery management of the lake Victoria. The lake is comprised of a lot of species such as stocked Nile perch, Lates niloticus, tilapia fish, Oreochromis niloticus, the cyprinid, Rastrineobola argentea, catfishes, insects, cichlids, crocodiles, and many zooplanktons and phytoplanktons NPFMP .
The particular study focuses on Nile perch as the predator while cichlid fishes and tilapia fishes are considered as prey populations. All three species are encountering the harvesting aspect. However harvesting without limitations may have detrimental effects on fish population because it decreases the population and sometimes leads a certain species to extinction, Ganguli . The lake Victoria fishery today is either overexploited or in a state of full exploitation because of greater fishing effort and increased competition between fishers, vessels, or nations over the resource. The particular study intends to apply mathematical techniques to ensure the sustainability of the species in lake Victoria without compromising the biological, economic, and social objectives for the benefit of present and future generations. Prey-predator model in fishery was also studied by Kar , Chakraborty , and Yunfei and Yongzhen , while studies by Tapas et al. , Ganguli at el , Kar , Gian , Chaudhuri and Kar , and Kar  analysed the bioeconomic aspect of prey- predator system and observed that increasing harvesting efforts result in population decreases. The particular study intends to analyse the bioeconomic impact for the lake Victoria fishery activities in Tanzania.
2. Materials and Methods
2.1. Model Description, Formulation, and Analysis
It will be assumed that the Nile perch depends completely on cichlid and tilapia fishes as their favorite food (because of easy to capture and their taste) where cichlid and tilapia fishes have unlimited sources of food. The dynamics therefore follow the Holling type II function response. In this case, , , and represent the population of cichlid fish, tilapia fish, and Nile perch, respectively, at any time , where all species involved in modeling are also encountering harvesting aspect. The growth rate of cichlid fishes and tilapia fishes follows the logistic law and the birth rate should always be positive. Terms representing interspecific competition among the prey species are included in the model and the model is then divided into three nonlinear autonomous ordinary differential equations describing how the population densities of the three species would vary with time. The following assumptions are made in order to construct the model:(i)Cichlid and tilapia fishes have an unlimited food supply in the lake Victoria.(ii)The Nile perch is completely dependent on the cichlid and tilapia fishes as the only favorite food source.(iii)Interspecific competition among tilapia and cichlid fishes is exploitative.(iv)In absence of the predator, prey species grow logistically. That is, the population of the cichlid and tilapia fishes would increase exponentially until it reaches the maximum density of the Lake, which is its environmental carrying capacity .(v)The predation functional response of the Nile perch towards both cichlid fishes and tilapia fishes is assumed to follow Michaelis-Menten kinetics and is modeled using a Holling type II functional form with predation coefficients and and the half saturation constants and .
By considering the underlying assumptions of the incorporated populations, we formulate the system of model equations as: with initial data values .
All parameters in the model are assumed to be positive and and are per capita intrinsic growth rates of cichlid and tilapia fishes, respectively, while and are environmental carrying capacities of cichlid and tilapia fishes, respectively, and are coefficients for interspecific competition, and are predation coefficients for cichlid fishes and tilapia fishes, respectively, , , and are effort harvesting rates, is natural mortality rate of Nile perch, and and are conversion parameters for cichlid fishes and tilapia fishes by Nile perch, while , , and are catchability coefficients.
For ease of computation, we are rescaling model (1) to reduce the number of parameters as follows: take , then the system of model (1) becomeswith initial values , and
2.2. Equilibrium Points of System (2)
The equilibrium states of the model are obtained by setting and we assume that the predator has positive mortality rate . The following are the possible equilibrium points of the system , , , , , and . Therefore,(i) The equilibrium point with From system (2), in the absence of tilapia fish and Nile perch we have and this exists when (ii) The equilibrium point with From system (2), in the absence of cichlid fishes and Nile perch ( and ) we have and this exists if (iii) The equilibrium point with and From system (2), in the absence of Nile perch () we have This exists if, for and , Also exists if, for and , (iv) The equilibrium point with and From system (2), in the absence of tilapia fishes () we have This exists if which is possible when and In terms of original parameter it implies and (v) The equilibrium point with and . From system (2), in the absence of cichlid fishes () we have This exists if and this is possible when and In terms of original parameter it means and . The condition implies that and (vi) Coexistence equilibrium point
Following the procedure by Dubey , the endemic equilibrium point is obtained as follows:From (7) we haveFrom (8) we haveFrom (8) and (9) we haveFrom (10) and (11) we haveand from (10) and (12) we get Equations (13) and (14) are two functions of and . To prove the existence of , the conditions under which and meet in the interior of the positive plane at the point are found. Now the values of , and can be obtained from (7), then from (9) we observe that, as , tends to . is the value of at which the function would cut the axis in the plane. So is given byWe notice that is the same as of . From (8), as , y tends to given bywhere and are the points at which the functions and would cut the y-axis in the plane, respectively. Also from (13), where and We note that if and and this requires and . Similarly from (14) where and We also note that if and and this requires and . Since for , we have and for , we have , then and will meet if . We therefore state the existence of the positive equilibrium point in the following theorem.
Theorem 1. The positive equilibrium point will exist if the following conditions are satisfied:where and are as defined in (15) and (16), respectively.
In terms of original parameter, implies that ; i.e., the growth rate of cichlid fishes must be greater than the harvesting effort imparted. Condition gives . That is, the harvesting rate of Nile perch should be greater than the harvesting rate of tilapia fishes. For economic purpose the condition needs to be satisfied which implies that the rate at which the Nile perch is harvested must be greater than its death rate.
2.3. Local Stability of Equilibrium Points
To analyse the local stability of the equilibrium point we consider the Jacobian matrix;where (i), the Jacobian matrix evaluated at gives the eigenvalues, , , and . We see that and are always positive and so is unstable.(ii). The Jacobian matrix (23) is evaluated at with the following eigenvalues: The eigenvalues are negative if and . Hence, the equilibrium point is locally asymptotically stable if the following conditions hold: , , and . The condition implies . For the local stability of , , the harvesting rate of cichlid fishes must be less than their intrinsic growth rate. Other inequalities show parameters that are vital for the local stability of .(iii). The Jacobian matrix (23) is evaluated at and the following eigenvalues obtained: The eigenvalues above are negative if . Hence, the equilibrium point is locally asymptotically stable if conditions hold. The condition implies . For the local stability of , , the harvesting rate of tilapia fishes must be less than their intrinsic growth rate.(iv) = . The eigenvalues of are obtained by solving the characteristic equation; where , , , , , , and . This can be expressed in form ofBy Routh-Hurwitz criteria (Murray, 1989), the are negative if .(v) = . The eigenvalues are obtained by solving the characteristics equation; where The characteristic equation is in the form where , , and . By Rouths stability criterion, the equilibrium point is stable if (i) , , and ; (ii) . Otherwise it is unstable.(vi). The eigenvalues evaluated at are obtained by solving the characteristic equation where , , and . The characteristic equation is in the form ; where , , and . By Rouths stability criterion, the equilibrium point is stable if (i) , , and and (ii) . Otherwise it is unstable.
2.4. Global Stability of the Coexistence Equilibrium Point
To analyse global stability of coexistence equilibrium point, a suitable Lyapunov function is chosen, from which conditions for the global asymptotic stability of the coexistence point are derived. The approach is based on work by Chaudhuri  and Dubey .
Theorem 2. The coexistence equilibrium point is globally asymptotically stable if
(i) , (ii) , and (ii) .
Proof. Consider the following Lyapunov function: Simplifying the above Lyapunov function, where , ,and .
Differentiating with respect to time, i.e., which simplifies to Thus, is a quadratic form which can be expressed as , where and A is symmetric matrix given bywith , , , , , and . We note that the point is globally asymptotically stable if ; that is, the matrix A is positive definite Chaudhuri . Now the matrix A is positive if , , , , , and . gives , gives , gives , gives , and gives . This completes the proof.
2.5. Bioeconomic Equilibrium
The term bionomic equilibrium is an amalgamation of the concepts of biological equilibrium as well as economic equilibrium Kar . From system (2), a biological equilibrium is given by . The economic equilibrium is said to be achieved when the total revenue obtained by selling the harvested biomass (TR) equals the total cost for the effort devoted to harvesting (TC).
Let be the fishing cost per unit effort for cichlid fishes, the fishing cost per unit effort for tilapia fishes, the fishing cost per unit effort for Nile perch, the price per unit biomass of cichlid fishes, the price per unit biomass of tilapia fishes, and the price per unit biomass of Nile perch; , , and are catchability coefficients of cichlid fishes, tilapia fishes, and Nile perch, respectively.
Then we have , , and .
, , and are the economic rent (net revenue) of cichlid fishes, tilapia fishes, and Nile perch, respectively. Therefore, the economic rent (net revenue) at any time is given by , which isThen is the bioeconomic equilibrium where , , , , , and are the bioeconomic values of cichlid fishes, tilapia fishes, Nile perch, harvesting effort of cichlid fishes, harvesting effort of tilapia fishes, and harvesting effort of Nile perch, respectively, and it is given by the simultaneous equation:In order to determine the bioeconomic equilibrium, we now consider the following cases.
Case I. If , that is, the fishing cost per unit effort for cichlid fishes is greater than the revenue in the cichlid fish fishery, then fishermen will be in loss and naturally they would withdraw their participation from cichlid fish and the fishery will be closed . Only tilapia fish and Nile perch fishery remain operational (i.e., and ). Now we have and . From (39) when , we haveSince and , then (44) can be written in quadratic form as Thereforewhere , , .
From (46),Therefore, ifAlso, Thus, ifTherefore, the bioeconomic equilibrium exists if conditions (48) and (50) hold.
Case II. If , that is, the fishing cost per unit effort for tilapia fishes is greater than the revenue in the tilapia fish fishery, then fishermen will be in loss and naturally, they would withdraw their participation from tilapia fish fishery and the fishery will be closed . Only cichlid fishes and Nile perch fishery remain operational (i.e., and ). Now we have and . From (46) when , we haveSince and , then (52) can be written in quadratic form asThereforewhereFrom (39), ifAlso,Thus, ifTherefore, the bioeconomic equilibrium exists if conditions (57) and (59) hold.
Case III. If , that is, the fishing cost per unit effort for Nile perch is greater than the revenue in the Nile perch fishery, then fishermen will be in loss and naturally, they would withdraw their participation from Nile perch fishery and the fishery will be closed. Only cichlid fish and tilapia fish fishery remain operational (i.e., and ). Now we have and . Substituting and in (44), we getThus, ifAlso,