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 ofNile perch population.Thefishery controlmanagement should be exercised to avoid overharvesting of cichlid and tilapia fishes.


Introduction
In today's life, the relationship between predator and prey became an important aspect to discuss in ecology.The preypredator system has attracted many researchers to study the interaction between the species [1].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 [2], FAO [3], LVFO [4], Barack [5], and Barilwa [6].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 [2].
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 [7].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 [8], Chakraborty [9], and Yunfei and Yongzhen [10], while studies by Tapas et al. [11], Ganguli at el [7], Kar [8], Gian [12], Chaudhuri and Kar [13], and Kar [8] analysed the bioeconomic aspect of preypredator 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.

Materials and Methods
. .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,  1 (),  2 (), and  3 () 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  푖 .
(v) The predation functional response of the Nile perch towards both cichlid fishes  1 and tilapia fishes  2 is assumed to follow Michaelis-Menten kinetics and is modeled using a Holling type II functional form with predation coefficients  13 and  23 and the half saturation constants  and .
All parameters in the model are assumed to be positive and  1 and  2 are per capita intrinsic growth rates of cichlid and tilapia fishes, respectively, while  1 and  2 are environmental carrying capacities of cichlid and tilapia fishes, respectively,  12 and  21 are coefficients for interspecific competition,  13 and  23 are predation coefficients for cichlid fishes and tilapia fishes, respectively,  1 ,  2 , and  3 are effort harvesting rates,  is natural mortality rate of Nile perch, and  31 and  32 are conversion parameters for cichlid fishes and tilapia fishes by Nile perch, while  1 ,  2 , and  3 are catchability coefficients.
Theorem 1. e positive equilibrium point  6 ( * ,  * ,  * ) will exist if the following conditions are satisfied: where  푓 and  푔 are as defined in ( ) and ( ), respectively.
In terms of original parameter,  1 >  1 implies that  1 >  1  1 ; i.e., the growth rate of cichlid fishes must be greater than the harvesting effort imparted.Condition  3 >  2 gives  3  3 >  2  2 .That is, the harvesting rate of Nile perch should be greater than the harvesting rate of tilapia fishes.For economic purpose the condition  3 >  needs to be satisfied which implies that the rate at which the Nile perch is harvested must be greater than its death rate.
The Jacobian matrix ( 23) is evaluated at  1 with the following eigenvalues: The eigenvalues are negative if Hence, the equilibrium point  1 (, 0, 0) is locally asymptotically stable if the following conditions hold: For the local stability of  1 (, 0, 0),  1 , 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  1 (, 0, 0).
The Jacobian matrix ( 23) is evaluated at  2 and the following eigenvalues obtained: The eigenvalues above are negative if  2 >  2 .Hence, the equilibrium point  1 (, 0, 0) is locally asymptotically stable if conditions For the local stability of  1 (, 0, 0),  2 , the harvesting rate of tilapia fishes must be less than their intrinsic growth rate.(iv The eigenvalues of ( 3 ) are obtained by solving the characteristic equation; where This can be expressed in form of By Routh-Hurwitz criteria (Murray, 1989), the The eigenvalues ( 4 ) are obtained by solving the characteristics equation; where The characteristic equation is in the form where . By Rouths stability criterion, the equilibrium point  4 is stable if (i)  1 > 0,  2 > 0, and The eigenvalues evaluated at ( 5 ) are obtained by solving the characteristic equation where which simplifies to . .Bioeconomic Equilibrium.The term bionomic equilibrium is an amalgamation of the concepts of biological equilibrium as well as economic equilibrium Kar [8].From system (2), a biological equilibrium is given by / = / = / = 0.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  1 be the fishing cost per unit effort for cichlid fishes,  2 the fishing cost per unit effort for tilapia fishes,  3 the fishing cost per unit effort for Nile perch,  1 the price per unit biomass of cichlid fishes,  2 the price per unit biomass of tilapia fishes, and  3 the price per unit biomass of Nile perch;  1 ,  2 , and  3 are catchability coefficients of cichlid fishes, tilapia fishes, and Nile perch, respectively.
Then we have 1 ,  2 , and  3 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  =  1 +  2 +  3 , which is Then ( ∞ ,  ∞ ,  ∞ ,  1∞ ,  2∞ ,  3∞ ) is the bioeconomic equilibrium where  ∞ ,  ∞ ,  ∞ ,  1∞ ,  2∞ , and  3∞ 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 III.If  3 >  3  3 , 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.,  1 <  1  1  and  2 <  2  2 ).
Case IV.If  1 >  1  1 ,  2 >  2  2  and  3 >  3  3 , then the fishing cost is greater than revenues for all three species and the whole fishery will be closed.

Results and Discussion
. .Numerical Results.The model system is simulated using the inbuilt ODE solvers coded in Matlab programming language and figures are plotted using parameter values presented in Table 1.
Figure 1 presents the dynamics of the population interaction with respect to time.The figure indicates that cichlid fishes population grows faster compared to other species.The dynamics was also visualized in 3 as presented in Figure 2. Figure 3 shows the impact in harvesting cichlid fish population; it is observed that increasing harvesting effort leads to decrease Nile perch population as well.In Figure 4 the same scenario happens when tilapia fish population harvested tends also to decrease Nile perch population while increasing harvesting effort in Nile perch population tends to increase cichlid and tilapia fishes as presented in Figure 5. Figure 6 presents the exponential growth rate of cichlid and tilapia fishes in absence of Nile perch population.
. .Discussion.A mathematical model was proposed and analysed to study the dynamics of a two-prey-one predator system with harvesting aspects.The model was used to study the ecological dynamics of the Nile perch-cichlid-tilapia fishes prey-predator system of the lake Victoria fishery.The harvesting rate was found to play a crucial role in stabilizing the system.Figures 3 and 4 show that cichlid and tilapia fishes tend to extinction when the harvesting rates  1 and  2 exceed their intrinsic growth rates  1 and  2 .For sustainability harvesting of cichlid and tilapia fishes their intrinsic growth rates should be kept smaller.However, harvesting of the three species at a rate much lower than their intrinsic growth rate  would not lead to collapse of the system.This result is similar to one obtained by Chaudhuri and Kar [13] on the existence conditions for the system they studied.In absence of the predator, the two prey species can coexist and are stable if the interspecific competition among them is maintained at minimum level or negligible also if both prey species are not harvested beyond its intrinsic growth rate.Numerical analysis results indeed confirmed this as shown in Figure 6.
Theorem 1 showed that the three species would coexist if cichlid and tilapia fishes were not harvested beyond their intrinsic growth rates; the Nile perch converted the biomass of cichlid and tilapia fishes into fertility at a rate greater than the Nile perch's natural mortality rate and the time it took to handle the cichlid or tilapia fish.The findings of Dubey and Upadhyay [14] indicated that the predator's mortality rate and food conversion coefficients played a crucial role in determining the stability behaviour of the equilibrium points.
In order for the bioeconomic equilibrium to exist, the fishing cost per unity effort for all species, price per unity biomass, catchability coefficient, harvesting effort, intrinsic growth rate of prey species, and the mortality rate of predator play a vital role.Hence for the bioeconomic equilibrium to exist, the fishing cost per unity effort for all three species should be less than the revenue in their fishery.Similar results were also obtained by Ganguli and Kar [7] when they studied    the optimal harvesting of a prey-predator model with variable carrying capacity.One of the major observations from numerical simulation results is that the predator population density increased significantly when the harvesting rate of both prey species  decreased.This implies that a gradual increase in the number of cichlid and tilapia fishes would result in significant increase in the number of Nile perch and vice versa, which would in the long term lead the population density of the cichlid and tilapia fishes to fall to a level lower than the original one.Kar and Chaudhuri [16] also discovered that predator population density can increase significantly when the harvesting rate of prey species is decreased.

Conclusion
We deduce from both analytical and numerical results that if the harvesting rate of the cichlid and tilapia fishes exceeds their intrinsic growth rate, the population of the cichlid, tilapia fishes, and Nile perch would become extinct with time.However, analytical and numerical results also show that harvesting of the three species at a rate much lower than their intrinsic growth rate would not lead to collapse of the system.Thus, in order to use fish as a resource and produce maximum economic benefit while maintaining sustainable fishery species, the harvesting rate of species should never be allowed to exceed their growth rate.

Figure 1 :
Figure 1: Graph of prey  1 and  2 and predator  3 against time (in years).It shows the variation of population density with time.

Figure 2 :
Figure2: Graph of prey  1 and  2 and predator  3 against time (in years).The limit cycle shows that population density will slightly change but not oscillate the boundary.

Figure 3 :
Figure 3: Variation of the fishes population with different harvesting efforts in cichlid 1 with fixed values of 2 and 3.

Figure 4 :
Figure 4: The effect of Tilapia harvest with different values of 2 with fixed values of 1 and 3.

Figure 5 :
Figure 5: The effect of Nile perch harvest with different values of 3 with fixed values of 1 and 2.

Figure 6 :
Figure 6: The interaction between cichlid and tilapia fishes in absence of Nile perch.

Table 1 :
Parameter Values of the model.