# Modeling Dynamics of Prey-Predator Fishery Model with Harvesting: A Bioeconomic Model

Authors: Charles Raymond; Alfred Hugo; Monica Kung’aro
Journal: Journal of Applied Mathematics
 (2019)

---

## Abstract

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.


---

## Body

## 1. Introduction

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 [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 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,x1(t), x2(t), and x3(t) represent the population of cichlid fish, tilapia fish, and Nile perch, respectively, at any time t, 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 capacityKi.(v)
The predation functional response of the Nile perch towards both cichlid fishesx1 and tilapia fishes x2 is assumed to follow Michaelis-Menten kinetics and is modeled using a Holling type II functional form with predation coefficients α13 and α23 and the half saturation constants β and γ.By considering the underlying assumptions of the incorporated populations, we formulate the system of model equations as:(1)dx1dt=λ1x11-x1K1-α12x1x2-α13x1x31+βx1-q1E1x1dx2dt=λ2x21-x2K2-α23x1x2-α23x2x31+γx2-q2E2x2dx3dt=-wx3+α31α13x1x31+βx1+α32α23x2x31+γx2-q3E3x3with initial data values x1(0)⩾0,x2(0)⩾0,x3(0)⩾0.All parameters in the model are assumed to be positive andλ1 and λ2 are per capita intrinsic growth rates of cichlid and tilapia fishes, respectively, while K1 and K2 are environmental carrying capacities of cichlid and tilapia fishes, respectively, α12 and α21 are coefficients for interspecific competition, α13 and α23 are predation coefficients for cichlid fishes and tilapia fishes, respectively, E1, E2, and E3 are effort harvesting rates, w is natural mortality rate of Nile perch, and α31 and α32 are conversion parameters for cichlid fishes and tilapia fishes by Nile perch, while q1, q2, and q3 are catchability coefficients.For ease of computation, we are rescaling model (1) to reduce the number of parameters as follows: take x=x1/K1,y=x2/K2,z=wx3, then the system of model (1) becomes(2)dxdt=λ1x1-x-σ1xy-P1xz1+Q1x-r1xdydt=λ2y1-y-σ2xy-P2yz1+Q2y-r2ydzdt=-wz+e1xz1+Q1x+e2yz1+Q2y-r3zwith initial values x(0)⩾0,y(0)⩾0, and z(0)⩾0.


### 2.2. Equilibrium Points of System (2)

The equilibrium states of the model are obtained by settingdx/dt=dy/dt=dz/dt=0 and we assume that the predator has positive mortality rate w. The following are the possible equilibrium points of the system P1(x∗,0,0), P2(0,y∗,0), P3(x∗,y∗,0), P4(x∗,0,z∗), P5(0,y∗,z∗), and P6(x∗,y∗,z∗). Therefore,(i)
The equilibrium pointP1(x∗,0,0)withx∗>0From system (2), in the absence of tilapia fish and Nile perch we have P1(x∗,0,0)=P1(λ1-r1/λ1,0,0) and this exists when r1<λ1(ii)
The equilibrium pointP2(0,y∗,0)withy∗>0From system (2), in the absence of cichlid fishes and Nile perch (x=0 and z=0) we have P2(0,y∗,0)=P2(λ2-r2/λ2,0,0) and this exists if r2<λ2(iii)
The equilibrium pointP3(x∗,y∗,0)withx∗>0andy∗>0From system (2), in the absence of Nile perch (z=0) we have (3)P3x∗,y∗,0=P3λ2λ1-r1-σ1λ2-r2λ1λ2-σ1σ2,λ1λ2-r2+σ2r1-λ1λ1λ2-σ1σ2,0This exists if, forr1<λ1 and r2<λ2, λ2(λ1-r1)>σ1(λ2-r2)⇒λ2(λ1-q1E1)>α12K2(λ2-q2E2)AlsoP3(x∗,y∗,0) exists if, for λ1<r1 and r2<λ2, σ2(r1-λ1)>-λ1(λ2-r2)(iv)
The equilibrium pointP4(x∗,0,z∗)withx∗>0andz∗>0From system (2), in the absence of tilapia fishes (y=0) we have(4)P4x∗,0,z∗=P4w+r3e1-Q1w-r3,0,-λ1Q1w+r32P1e1-Q1w-r32+λ1Q1-r1Q1-λ1w+r3P1e1-Q1w-r3+λ1-r1P1This exists if(5)w+r3-λ1Q1w+r3-λ1Q1-r1Q1-λ1e1-wQ1-r3Q1+e1-wQ1-r3Q12λ1-r1>0which is possible whenr3>-w and λ1>r1In terms of original parameter it impliesE3q3>-w and λ1>E1q1(v)
The equilibrium pointP5(0,y∗,z∗)withy∗>0andz∗>0. From system (2), in the absence of cichlid fishes (x=0) we have(6)P50,y∗,z∗=P50,w+r3e2-Q2w-r3,-λ2Q2w+r32P2e2-Q2w-r32+λ2Q2-r2Q2-λ2w+r3P2e2-Q2w-r3+λ2-r2P2This exists if(w+r3)-λ2Q2(w+r3)-(λ2Q2-r2Q2-λ2)(e2-wQ2-r3Q2)+(e2-wQ2-r3Q2)2(λ2-r2)>0 and this is possible when r3>-w and λ2>r2 In terms of original parameter it means E3q3>-w and λ2>E2q2. The condition implies that E3>-w and λ2>E2(vi)
Coexistence equilibrium pointP6(x∗,y∗,z∗)Following the procedure by Dubey [14], the endemic equilibrium point is obtained as follows:(7)λ1x1-x-σ1xy-P1xz1+Q1x-r1x=0(8)λ2y1-x-σ2xy-P2xz1+Q1x-r2y=0(9)-wz+e1xz1+Q1x+e2yz1+Q2y-r3z=0From (7) we have(10)z=λ11-x-σ1y-r11+Q1xP1From (8) we have(11)z=λ21-y-σ2x-r21+Q2yP2From (8) and (9) we have(12)z=λ21-y-σ2x-r2+w+r31+Q2y-e2yP2-e1x1+Q1yP21+Q1xFrom (10) and (11) we have(13)fx,y=w+r31+Q2y-e2y1+Q1x-e1x1+Q2yP21+Q1xand from (10) and (12) we get (14)gx,y=λ21-y-σ2x-r2+w+r31+Q2y-e2yP2-e1x1+Q1yP21+Q1x-λ11-x-σ1y-r11+Q1xP1Equations (13) and (14) are two functions of x and y. To prove the existence of P6(x∗,y∗,z∗), the conditions under which f(x,y) and g(x,y) meet in the interior of the positive (x,y) plane at the point (x∗,y∗) are found. Now the values of x∗, y∗ and z∗ can be obtained from (7), then from (9) we observe that, as x→0, y tends to yf. yf is the value of y at which the function f(x,y) would cut the y axis in the (x,y) plane. So yf is given by(15)yf=w+r3e2-Q2w-Q2r3We notice that yf is the same as y of P5(0,y∗,z∗). From (8), as x→0, y tends to yg given by(16)yg=-D2+D22-4D1D32D1where (17)D1=-λ2Q2P2D2=-λ2+λ2Q2-σ2r2+wQ2+r3Q2-e2P2+σ1P1D3=P1w+r3-r2+λ2+P2r1-λ2P1P2yf and yg are the points at which the functions f(x,y) and g(x,y) would cut the y-axis in the (x,y) plane, respectively. Also from (13), dy/dx=-∂f/∂x/∂f/∂y where (18)∂f∂x=P21+Q1w+r31+Q2y-e2yQ1P221+Q1x2-P2Q1w+r31+Q2y-e2y1+Q1x-e1x1+Q2yP221+Q1x2and (19)∂f∂y=wQ2+r3Q2-e21+Q1x-e2Q2xP21+Q1xWe note that dy/dx>0 if ∂f/∂x>0 and ∂f/∂y<0 and this requires r3>-w and λ1>r1. Similarly dy/dx=-∂g/∂x/∂g/∂y from (14) where (20)∂g∂x=-σ21+Q2yP2+e11+Q2yP21+Q1x2-Q1λ1-σ1y-r1-λ1-2λ1Q1xP1and (21)∂g∂y=w+r3-r2-σ2xQ2P2-e1Q2xP21+Q1xWe also note that dy/dx<0 if ∂g/∂x<0 and ∂g/∂y<0 and this requires r3>-w and r3>r2. Since for f(x,y), we have dy/dx>0 and for g(x.y), we have dy/dx<0, then f(x,y) and g(x,y) will meet if yf<yg. We therefore state the existence of the positive equilibrium point P6(x∗,y∗,z∗) in the following theorem.Theorem 1.
The positive equilibrium pointP6(x∗,y∗,z∗) will exist if the following conditions are satisfied:(22)r3>w,λ1>r1,r3>r2,yf<ygwhere yf and yg are as defined in (15) and (16), respectively.In terms of original parameter,λ1>r1 implies that λ1>E1q1; i.e., the growth rate of cichlid fishes must be greater than the harvesting effort imparted. Condition r3>r2 gives q3E3>q2E2. That is, the harvesting rate of Nile perch should be greater than the harvesting rate of tilapia fishes. For economic purpose the condition r3>w 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;(23)JPi=A∗∗σ1x-P1x1+Q1x-σ1yB∗∗-P2y1+Q2ye1z1+Q1x2e2z1+Q2y2C∗∗where (24)A∗∗=λ1-2λ1x-σ1y-P1z1+Q1x2-r1,B∗∗=λ2-2λ2y-σ2x-P2z1+Q2y2-r2C∗∗=-w+e1x1+Q1x+e2y1+Q2y-r3(i)
P0(0,0,0), the Jacobian matrix evaluated at J(P0) gives the eigenvalues, λ1-r1, λ2-r2, and -(w+r3). We see that (λ2-r2)>0 and (λ1-r1)>0 are always positive and so P0(0,0,0) is unstable.(ii)
P1(x∗,0,0)=λ1-r1/λ1,0,0.The Jacobian matrix (23) is evaluated at P1 with the following eigenvalues:(25)L1=-λ1-r1,L2=λ2-σ2λ1-r1λ1-P2λ1-r1λ1-r2,L3=-w-P1λ1-r1λ1+Q1λ1-r1-r3.The eigenvalues are negative ifλ1>r1 and λ2>(σ2-P2)(λ1-r1)/λ1.Hence, the equilibrium pointP1(x,0,0) is locally asymptotically stable if the following conditions hold: λ1>r1, Q2>P2, and λ2>(Q2-P2)(λ1-r1)/λ1. The condition λ1>r1 implies E1<λ1. For the local stability of P1(x,0,0), E1, 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 P1(x,0,0).(iii)
P2(0,y∗,0)=(0,λ2-r2/λ2,0).The Jacobian matrix (23) is evaluated at P2 and the following eigenvalues obtained:(26)L1=λ1-σ1λ2-r2λ2-r1,L2=r2-λ2,L3=-w-P2λ2-r2λ2+Q2λ2-r2-r3.The eigenvalues above are negative ifλ2>r2. Hence, the equilibrium point P1(x,0,0) is locally asymptotically stable if conditions λ2>r2 hold. The condition λ2>r2 implies E2<λ2. For the local stability of P1(x,0,0), E2, the harvesting rate of tilapia fishes must be less than their intrinsic growth rate.(iv)
P3(x∗,y∗,0) = λ2(λ1-r1)-σ1(λ2-r2)/λ1λ2-σ1σ2,λ1(λ2-r2)+σ2(r1-λ1)/λ1λ2-σ1σ2,0.The eigenvalues ofJ(P3) are obtained by solving the characteristic equation; (27)λ3-A∗+E∗+G∗λ2+A∗G∗+E∗G∗+A∗E∗-D∗B∗λ+G∗D∗B∗-G∗A∗E∗=0whereA∗=λ1-2λ1x∗-σ1y∗-r1, B∗=-σ1y∗, C∗=-P1x∗/1+Q1x∗, D∗=-σ1y∗, E∗=λ2-2λ2y∗-σ2x∗-r2, F∗=-P2y∗/1+Q2y∗, and G∗=-w+e1x∗/1+Q1x∗+e2y∗/1+Q2y∗-r3.This can be expressed in form of(28)λ3+a1λ2+a2λ+a3=0By Routh-Hurwitz criteria (Murray, 1989), theλ′s are negative if a1>0,a3>0,a1a2-a3>0. (v)
P 4 ( x ∗ , 0 , z ∗ ) = P4w+r3/e1-wQ1-r3Q1,0,-λ1Q1(w+r3)2/P1(e1-wQ1-r3Q1)2+(λ1Q1-r1Q1-λ1)(w+r3)/P1(e1-wQ1-r3Q1)+λ1-r1/P1. The eigenvaluesJ(P4) are obtained by solving the characteristics equation; (29)L3-A2∗+B2∗+C2∗L2+A2∗B2∗+C2∗+B2∗C2∗+P1e1x∗z∗1+Q1x∗3L-A2∗B2∗C2∗-B2∗P1e1x∗z∗1+Q1x∗∗=0 where(30)A2∗=λ1-r1-2λ1x∗-P1z∗1+Q1x∗2,B2∗=λ2-σ2x∗-P2z∗-r2,C2∗=-w+e1x∗1+Q1x∗-r3. The characteristic equation is in the form(31)L3+a1L2+a2L+a3=0 wherea1=-(A2∗+B2∗+C2∗), a2=A2∗(B2∗+C2∗)+B2∗C2∗+P1e1x∗z∗/(1+Q1x∗)3, and a3=-A2∗B2∗C2∗-B2∗P1e1x∗z∗/(1+Q1x∗)∗. By Rouths stability criterion, the equilibrium point P4 is stable if (i) a1>0, a2>0, and a3>0; (ii) a1a2>a3. Otherwise it is unstable. (vi)
P 5 ( 0 , y ∗ , z ∗ ) = P 5 w + r 3 / e 2 - w Q 2 - r 3 Q 2 , 0 , - λ 2 Q 2 ( w + r 3 ) 2 / P 2 ( e 2 - w Q 2 - r 3 Q 2 ) 2 + ( λ 2 Q 2 - r 2 Q 2 - λ 2 ) ( w + r 3 ) / P 2 ( e 2 - w Q 2 - r 3 Q 2 ) + λ 2 - r 2 / P 2. The eigenvalues evaluated atJ(P5) are obtained by solving the characteristic equation (32)L3-A3∗+B3∗+C3∗L2+B3∗C3∗+A3∗B3∗+C3∗+P2e2z∗y∗1+Q2y∗3L-A3∗B3∗C3∗-P2e2z∗y∗1+Q2y∗3=0 whereA3∗=λ1-σ1y∗-P1z∗-r1, B3∗=λ2-r2-2λ2y∗-P2z∗/(1+Q2x∗)2, and C3∗=-w+e2y∗/1+Q2y∗-r3. The characteristic equation is in the form L3+a1L2+a2L+a3=0; wherea1=-(A3∗+B3∗+C3∗), a2=B3∗C3∗+A3∗(B3∗+C3∗)+P2e2z∗y∗/(1+Q2y∗)3, and a3=-A3∗B3∗C3∗-P2e2z∗y∗/(1+Q2y∗)3. By Rouths stability criterion, the equilibrium point P5 is stable if (i) a1>0, a2>0, and a3>0 and (ii) a1a2>a3. Otherwise it is unstable.


### 2.4. Global Stability of the Coexistence Equilibrium PointP6(x∗,y∗,z∗)

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 pointP6(x∗,y∗,z∗) are derived. The approach is based on work by Chaudhuri [13] and Dubey [14].Theorem 2.
The coexistence equilibrium pointP6(x∗,y∗,z∗) is globally asymptotically stable if
(i)λ1>0,λ2>0, (ii) P1>e1,P2>e2, and (ii) λ1λ2>(σ1+σ2)2.Proof.
Consider the following Lyapunov function:(33)Vx,y,z=x-x∗-x∗log⁡xx∗+y-y∗-y∗log⁡yy∗+z-z∗-z∗log⁡zz∗Simplifying the above Lyapunov function, (34)Vx,y,z=x-x∗-x∗log⁡x-x∗log⁡x∗+y-y∗-y∗log⁡y-y∗log⁡y∗+z-z∗-z∗log⁡z-z∗log⁡z∗Vx,y,z=x-x∗-x∗log⁡x+x∗log⁡x∗+y-y∗-y∗log⁡y+y∗log⁡y∗+z-z∗-z∗log⁡z+z∗log⁡z∗where ∂V/∂x=1-x∗/x=x-x∗/x, ∂V/∂y=1-y∗/y=y-y∗/y,and ∂V/∂z=1-z∗/z=z-z∗/z.
DifferentiatingV with respect to time, i.e., (35)dVdt=∂V∂xdxdt+∂V∂ydydt+∂V∂zdzdtdVdt=x-x∗xλ1x1-x-σ1xy-P1xz1+Q1x-r1x+y-y∗yλ2y1-x-σ2xy-P2xz1+Q1x-r2y+z-z∗z-wz+e1xz1+Q1x+e2yz1+Q2y-r3zdVdt=x-x∗λ11-x-σ1y-P1z1+Q1x-r1+y-y∗λ21-x-σ2x-P2z1+Q1x-r2+z-z∗-w+e1x1+Q1x+e2y1+Q2y-r3dVdt=x-x∗λ1-λ1x-σ1y-P1z1+Q1x-r1-λ1-+λ1x∗+σ1y∗+P1z∗1+Q1x∗+r1+y-y∗λ2-λ2y-σ2x-P2z1+Q2y-r2-λ2+λ2y∗+σ2x∗+P2z∗1+Q2y∗+r2+z-z∗-w+e1x1+Q1x+e2y1+Q2y-r3+w-e1x∗1+Q1x∗-e2y∗1+Q2y∗+r3dVdt=x-x∗λ1x∗-x+σ1Z∗-z+y-y∗λ2y2-y+σ2x∗-x+P2z∗-z1+Q2y+z∗-ze1x-x∗1+Q1x+e2y-y∗1+Q2ywhich simplifies to (36)dVdt=-x-x∗2λ1-x-x∗y-y∗σ1-σ2-x-x∗z-z∗P1-e11+Q1x-y-y∗2λ2-y-y∗z-z∗P2-e21+Q2yThus, dV/dt is a quadratic form which can be expressed as dV/dt=-XTAX, where XT=(x-x∗,y-y∗,z-z∗) and A is symmetric matrix given by(37)A=a11a12a13a12a22a23a13a23a33with a11=λ1, a12=σ1-σ2, a13=(P1-e1)/(1+Q1x), a22=λ2, a23=P2-e2/1+Q2y, and a33=0. We note that the point P6(x∗,y∗,z∗) is globally asymptotically stable if dV/dt<0; that is, the matrix A is positive definite Chaudhuri [13]. Now the matrix A is positive if a11>0, a13=0, a12>0, a22>0, a23=0, and a11a22-a122>0. a11>0 gives λ1>0, a13=0 gives P1=e1, a12>0 gives σ1>σ2, a22>0 gives λ2>0, and a11a22-a122>0 gives λ1λ2>(σ1+σ2)2. 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 [8]. From system (2), a biological equilibrium is given by dx/dt=dy/dt=dz/dt=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).Letc1 be the fishing cost per unit effort for cichlid fishes, c2 the fishing cost per unit effort for tilapia fishes, c3 the fishing cost per unit effort for Nile perch, p1 the price per unit biomass of cichlid fishes, p2 the price per unit biomass of tilapia fishes, and p3 the price per unit biomass of Nile perch; q1, q2, and q3 are catchability coefficients of cichlid fishes, tilapia fishes, and Nile perch, respectively.Then we haveπ1=(p1q1x-c1)E1, π2=(p2q2y-c2)E2, and π3=(p3q3x-c3)E3.π 1, π2, and π3 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 π=π1+π2+π3, which is(38)π=p1q1x-c1E1+p2q2y-c2E2+p3q3z-c3E3Then P(x∞,y∞,z∞,E1∞,E2∞,E3∞) is the bioeconomic equilibrium where x∞, y∞, z∞, E1∞, E2∞, and E3∞ 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:(39)λ11-x-σ1y-P1z1+Q1x-q1E1=0(40)λ21-y-σ2x-P2z1+Q2y-q2E2=0(41)-w+e1x1+Q1x+e2y1+Q2y-q3E3=0(42)π=p1q1x-c1E1+p2q2y-c2E2+p3q3z-c3E3=0In order to determine the bioeconomic equilibrium, we now consider the following cases.Case I.
Ifc1>p1q1, 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 (E1=0). Only tilapia fish and Nile perch fishery remain operational (i.e., c2<p2q2y and c3<p3q3z). Now we have y∞=c2/p2q2 and z∞=c3/p3q3. From (39) when E1=0, we have(43)λ1-λ1x∞-σ1y∞-P1z∞1+Q1x∞=0λ11+Q1x∞-λ1x∞1+Q1x∞-σ1y∞1+Q1x∞-P1z∞=0λ1+λ1Q1x∞-λ1x∞+λ1Q1x∞2-σ1y∞+σ1Q1x∞y∞-P1z∞=0(44)-λ1Q1x∞2+λ1Q1-λ1-σ1Q1y∞x∞+λ1-σ1y∞-P1z∞=0Since y∞=c2/p2q2 and z∞=c3/p3q3, then (44) can be written in quadratic form as (45)λ1Q1x∞2-λ1Q1-λ1-σ1Q1c2p2q2x∞+λ1-σ1c2p2q2-P1c3p3q3=0Therefore(46)x∞1,2=-R2±R22-4R1R22R3where R1=λ1Q1, R2=λ1Q1p2q2-λ1p2q2-σ1Q1c2/p2q2, R3=λ1-σ1c2/p2q2-P1c3/p3q3.
From (46),(47)E1∞=λ2q21-y∞-σ2q2x∞-P2z∞q21+Q2y∞Therefore, E1∞>0 if(48)λ2q2p2-λ2c2-σ2p2q2x∞p2q22>P2p2q2c3p3q3p2q2-Q2p3q3c2Also, (49)E2∞=-wzq3+e1xzq3+Q1q3x+e2yzq3+Q2q3y=-wc3p3q32+e1c3x∞p3q32+Q1p3q32x∞+e2c2c3p2q2p22q22p3q32+Q2p2q2p3q32c2Thus, E2∞>0 if(50)e1c3x∞p22q22p3q32+Q2p2q2p3q32c2+e2c2c3p2q2p3q32+Q1p3q32x∞p3q32-Q1p3q32x∞p22q22q32p3+Q2p2q2p3q32c2>wc3p3q32Therefore, the bioeconomic equilibrium exists if conditions (48) and (50) hold.Case II.
Ifc2>p2q2, 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 (E2=0). Only cichlid fishes and Nile perch fishery remain operational (i.e., c1<p1q1x and c3<p3q3z). Now we have x∞=c1/p1q1 and z∞=c3/p3q3. From (46) when E2=0, we have(51)λ2-λ1y∞-σ2x∞-P2z∞1+Q1y∞=0λ21+Q2y∞-λ2y∞1+Q2y∞-σ2x∞1+Q2y∞-P2z∞=0λ2+λ2Q2y∞-λ2y∞+λ2Q2y∞2-σ2x∞+σ2Q2x∞y∞-P2z∞=0(52)-λ2Q2y∞2+λ2Q2-λ2-σ2Q2x∞y∞+λ2-σ2x∞-P2z∞=0Since x∞=c1/p1q1 and z∞=c3/p3q3, then (52) can be written in quadratic form as(53)λ2Q2y∞2-λ2Q2-λ2-σ2Q2c1p1q1y∞+λ1-σ2c1p1q1-P2c3p3q3=0Therefore(54)y∞=-S2±S22-4S1S22S3where(55)S1=λ2Q2S2=λ2Q2p1q1-λ2p1q1-σ2Q2c1p1q1S3=λ2-σ2c1p1q1-P2c3p3q3From (39),(56)λ11-x∞-σ1y∞-P1z∞1+Q1x∞-q1E1∞=0E1∞=λ1q1-λ1c1p1q12-σ1y∞q1-P1p1q1c3p3q3p1q1-Q1p3q3c1
E 1 ∞ > 0 if(57)λ1q1p1-λ1c1-σ1p1q1y∞p1q12>P1p1q1c3p3q3p1q1-Q1p3q3c1Also,(58)E3∞=-wzq3+e1xzq3+Q1q3x+e2yzq3+Q2q3y=-wc3p3q32+e1c3y∞p3q32+Q2p3q32y∞+e2c1c3p1q1p12q12p3q32+Q1p1q1p3q32c1Thus, E2∞>0 if(59)e2c3y∞p12q12p3q32+Q1p1q1p3q32c1+e2c1c3p1q1p3q32+Q2p3q32y∞p3q32-Q2p3q32y∞p12q12q32p3+Q1p1q1p3q32c1>wc3p3q32Therefore, the bioeconomic equilibrium exists if conditions (57) and (59) hold.Case III.
Ifc3>p3q3, 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., c1<p1q1x and c2<p2q2z). Now we have x∞=c1/p1q1 and y∞=c2/p2q2. Substituting x∞ and y∞in (44), we get(60)E1∞=λ11q1-c1p1q12-σ1c2p1q12-Q1q1c1+P1p1q12p2q2z∞p2q2q1p1q12-Q1q1c1Thus, E1∞>0 if(61)λ11q1-c1p1q12>σ1c2p1q12-Q1q1c1+P1p1q12p2q2z∞p2q2q1p1q12-Q1q1c1Also,(62)E2∞=λ11q1-c2p2q22-σ2c1p2q22-Q2q2c2+P2p2q22p1q1z∞p1q1q2p2q22-Q2q2c2Thus, E2∞>0 if(63)λ21q2-c2p2q22>σ2c1p2q22-Q2q2c2+P2p2q22p1q1z∞p1q1q2p2q22-Q2q2c2Therefore, the bioeconomic equilibrium exists if conditions (61) and (63) hold.Case IV.
Ifc1>p1q1x,c2>p2q2y and c3>p3q3z, then the fishing cost is greater than revenues for all three species and the whole fishery will be closed.Case V.
Ifc1<p1q1x,c2<p2q2y and c3<p3q3z, then the fishing cost is less than revenues for all three species; that is, the fishery is more profitable and hence it would attract more fishermen and the whole fishery will be in operation. Now we have x∞=c1/p1q1,y∞=c2/p2q2, and z∞=c3/p3q3, then(64)E1∞=λ11q1-c1p1q12-σ1c2p1q12p3q3-Q1p3q3q1c1+P1p1q1c3p2q2q1p3q3P1q12-Q1p3q3q1c1Thus, E1∞>0, if(65)λ11q1-c1p1q12>σ1c2p1q12p3q3-Q1p3q3q1c1+P1p1q1c3p2q2q1P3q3P1q12-Q1p3q3q1c1And,(66)E2∞=λ21q2-c2p2q22-σ2c1p2q22p3q3-Q2p3q3q2c2+P2p2q2c3p1q1q2p3q3p2q22-Q2p3q3q2c2Thus, E2∞>0,(67)λ21q2-c2p2q22>σ2c1p2q22p3q3-Q2p3q3q2c2+P2p2q2c3p1q1q2P3q3p2q22-Q2p3q3q2c2Also, (68)E3∞=-wc3p3q3+e1c1c3p1q1p12q12p3q3+p3q3p1q1c1+e2c2c3p2q2p3q3p22q22-p3q3p2q2c2Thus, E3∞>0 if (69)e1c1c3p1q1p3q3p22q22-p3q3p2q2c2+e2c2c3p2q2p12q12p3q3+p3q3p1q1c1p12q12p3q3+p3q3p1q1c1p3q3p22q22-p3q3p2q2c2>0Thus, the nontrivial bioeconomic equilibrium point P∞(x∞,y∞,z∞,E1∞,E2∞,E3∞) exists if and only if conditions (65), (67), and (69) hold together.


## 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,x1(t), x2(t), and x3(t) represent the population of cichlid fish, tilapia fish, and Nile perch, respectively, at any time t, 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 capacityKi.(v)
The predation functional response of the Nile perch towards both cichlid fishesx1 and tilapia fishes x2 is assumed to follow Michaelis-Menten kinetics and is modeled using a Holling type II functional form with predation coefficients α13 and α23 and the half saturation constants β and γ.By considering the underlying assumptions of the incorporated populations, we formulate the system of model equations as:(1)dx1dt=λ1x11-x1K1-α12x1x2-α13x1x31+βx1-q1E1x1dx2dt=λ2x21-x2K2-α23x1x2-α23x2x31+γx2-q2E2x2dx3dt=-wx3+α31α13x1x31+βx1+α32α23x2x31+γx2-q3E3x3with initial data values x1(0)⩾0,x2(0)⩾0,x3(0)⩾0.All parameters in the model are assumed to be positive andλ1 and λ2 are per capita intrinsic growth rates of cichlid and tilapia fishes, respectively, while K1 and K2 are environmental carrying capacities of cichlid and tilapia fishes, respectively, α12 and α21 are coefficients for interspecific competition, α13 and α23 are predation coefficients for cichlid fishes and tilapia fishes, respectively, E1, E2, and E3 are effort harvesting rates, w is natural mortality rate of Nile perch, and α31 and α32 are conversion parameters for cichlid fishes and tilapia fishes by Nile perch, while q1, q2, and q3 are catchability coefficients.For ease of computation, we are rescaling model (1) to reduce the number of parameters as follows: take x=x1/K1,y=x2/K2,z=wx3, then the system of model (1) becomes(2)dxdt=λ1x1-x-σ1xy-P1xz1+Q1x-r1xdydt=λ2y1-y-σ2xy-P2yz1+Q2y-r2ydzdt=-wz+e1xz1+Q1x+e2yz1+Q2y-r3zwith initial values x(0)⩾0,y(0)⩾0, and z(0)⩾0.


## 2.2. Equilibrium Points of System (2)

The equilibrium states of the model are obtained by settingdx/dt=dy/dt=dz/dt=0 and we assume that the predator has positive mortality rate w. The following are the possible equilibrium points of the system P1(x∗,0,0), P2(0,y∗,0), P3(x∗,y∗,0), P4(x∗,0,z∗), P5(0,y∗,z∗), and P6(x∗,y∗,z∗). Therefore,(i)
The equilibrium pointP1(x∗,0,0)withx∗>0From system (2), in the absence of tilapia fish and Nile perch we have P1(x∗,0,0)=P1(λ1-r1/λ1,0,0) and this exists when r1<λ1(ii)
The equilibrium pointP2(0,y∗,0)withy∗>0From system (2), in the absence of cichlid fishes and Nile perch (x=0 and z=0) we have P2(0,y∗,0)=P2(λ2-r2/λ2,0,0) and this exists if r2<λ2(iii)
The equilibrium pointP3(x∗,y∗,0)withx∗>0andy∗>0From system (2), in the absence of Nile perch (z=0) we have (3)P3x∗,y∗,0=P3λ2λ1-r1-σ1λ2-r2λ1λ2-σ1σ2,λ1λ2-r2+σ2r1-λ1λ1λ2-σ1σ2,0This exists if, forr1<λ1 and r2<λ2, λ2(λ1-r1)>σ1(λ2-r2)⇒λ2(λ1-q1E1)>α12K2(λ2-q2E2)AlsoP3(x∗,y∗,0) exists if, for λ1<r1 and r2<λ2, σ2(r1-λ1)>-λ1(λ2-r2)(iv)
The equilibrium pointP4(x∗,0,z∗)withx∗>0andz∗>0From system (2), in the absence of tilapia fishes (y=0) we have(4)P4x∗,0,z∗=P4w+r3e1-Q1w-r3,0,-λ1Q1w+r32P1e1-Q1w-r32+λ1Q1-r1Q1-λ1w+r3P1e1-Q1w-r3+λ1-r1P1This exists if(5)w+r3-λ1Q1w+r3-λ1Q1-r1Q1-λ1e1-wQ1-r3Q1+e1-wQ1-r3Q12λ1-r1>0which is possible whenr3>-w and λ1>r1In terms of original parameter it impliesE3q3>-w and λ1>E1q1(v)
The equilibrium pointP5(0,y∗,z∗)withy∗>0andz∗>0. From system (2), in the absence of cichlid fishes (x=0) we have(6)P50,y∗,z∗=P50,w+r3e2-Q2w-r3,-λ2Q2w+r32P2e2-Q2w-r32+λ2Q2-r2Q2-λ2w+r3P2e2-Q2w-r3+λ2-r2P2This exists if(w+r3)-λ2Q2(w+r3)-(λ2Q2-r2Q2-λ2)(e2-wQ2-r3Q2)+(e2-wQ2-r3Q2)2(λ2-r2)>0 and this is possible when r3>-w and λ2>r2 In terms of original parameter it means E3q3>-w and λ2>E2q2. The condition implies that E3>-w and λ2>E2(vi)
Coexistence equilibrium pointP6(x∗,y∗,z∗)Following the procedure by Dubey [14], the endemic equilibrium point is obtained as follows:(7)λ1x1-x-σ1xy-P1xz1+Q1x-r1x=0(8)λ2y1-x-σ2xy-P2xz1+Q1x-r2y=0(9)-wz+e1xz1+Q1x+e2yz1+Q2y-r3z=0From (7) we have(10)z=λ11-x-σ1y-r11+Q1xP1From (8) we have(11)z=λ21-y-σ2x-r21+Q2yP2From (8) and (9) we have(12)z=λ21-y-σ2x-r2+w+r31+Q2y-e2yP2-e1x1+Q1yP21+Q1xFrom (10) and (11) we have(13)fx,y=w+r31+Q2y-e2y1+Q1x-e1x1+Q2yP21+Q1xand from (10) and (12) we get (14)gx,y=λ21-y-σ2x-r2+w+r31+Q2y-e2yP2-e1x1+Q1yP21+Q1x-λ11-x-σ1y-r11+Q1xP1Equations (13) and (14) are two functions of x and y. To prove the existence of P6(x∗,y∗,z∗), the conditions under which f(x,y) and g(x,y) meet in the interior of the positive (x,y) plane at the point (x∗,y∗) are found. Now the values of x∗, y∗ and z∗ can be obtained from (7), then from (9) we observe that, as x→0, y tends to yf. yf is the value of y at which the function f(x,y) would cut the y axis in the (x,y) plane. So yf is given by(15)yf=w+r3e2-Q2w-Q2r3We notice that yf is the same as y of P5(0,y∗,z∗). From (8), as x→0, y tends to yg given by(16)yg=-D2+D22-4D1D32D1where (17)D1=-λ2Q2P2D2=-λ2+λ2Q2-σ2r2+wQ2+r3Q2-e2P2+σ1P1D3=P1w+r3-r2+λ2+P2r1-λ2P1P2yf and yg are the points at which the functions f(x,y) and g(x,y) would cut the y-axis in the (x,y) plane, respectively. Also from (13), dy/dx=-∂f/∂x/∂f/∂y where (18)∂f∂x=P21+Q1w+r31+Q2y-e2yQ1P221+Q1x2-P2Q1w+r31+Q2y-e2y1+Q1x-e1x1+Q2yP221+Q1x2and (19)∂f∂y=wQ2+r3Q2-e21+Q1x-e2Q2xP21+Q1xWe note that dy/dx>0 if ∂f/∂x>0 and ∂f/∂y<0 and this requires r3>-w and λ1>r1. Similarly dy/dx=-∂g/∂x/∂g/∂y from (14) where (20)∂g∂x=-σ21+Q2yP2+e11+Q2yP21+Q1x2-Q1λ1-σ1y-r1-λ1-2λ1Q1xP1and (21)∂g∂y=w+r3-r2-σ2xQ2P2-e1Q2xP21+Q1xWe also note that dy/dx<0 if ∂g/∂x<0 and ∂g/∂y<0 and this requires r3>-w and r3>r2. Since for f(x,y), we have dy/dx>0 and for g(x.y), we have dy/dx<0, then f(x,y) and g(x,y) will meet if yf<yg. We therefore state the existence of the positive equilibrium point P6(x∗,y∗,z∗) in the following theorem.Theorem 1.
The positive equilibrium pointP6(x∗,y∗,z∗) will exist if the following conditions are satisfied:(22)r3>w,λ1>r1,r3>r2,yf<ygwhere yf and yg are as defined in (15) and (16), respectively.In terms of original parameter,λ1>r1 implies that λ1>E1q1; i.e., the growth rate of cichlid fishes must be greater than the harvesting effort imparted. Condition r3>r2 gives q3E3>q2E2. That is, the harvesting rate of Nile perch should be greater than the harvesting rate of tilapia fishes. For economic purpose the condition r3>w 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;(23)JPi=A∗∗σ1x-P1x1+Q1x-σ1yB∗∗-P2y1+Q2ye1z1+Q1x2e2z1+Q2y2C∗∗where (24)A∗∗=λ1-2λ1x-σ1y-P1z1+Q1x2-r1,B∗∗=λ2-2λ2y-σ2x-P2z1+Q2y2-r2C∗∗=-w+e1x1+Q1x+e2y1+Q2y-r3(i)
P0(0,0,0), the Jacobian matrix evaluated at J(P0) gives the eigenvalues, λ1-r1, λ2-r2, and -(w+r3). We see that (λ2-r2)>0 and (λ1-r1)>0 are always positive and so P0(0,0,0) is unstable.(ii)
P1(x∗,0,0)=λ1-r1/λ1,0,0.The Jacobian matrix (23) is evaluated at P1 with the following eigenvalues:(25)L1=-λ1-r1,L2=λ2-σ2λ1-r1λ1-P2λ1-r1λ1-r2,L3=-w-P1λ1-r1λ1+Q1λ1-r1-r3.The eigenvalues are negative ifλ1>r1 and λ2>(σ2-P2)(λ1-r1)/λ1.Hence, the equilibrium pointP1(x,0,0) is locally asymptotically stable if the following conditions hold: λ1>r1, Q2>P2, and λ2>(Q2-P2)(λ1-r1)/λ1. The condition λ1>r1 implies E1<λ1. For the local stability of P1(x,0,0), E1, 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 P1(x,0,0).(iii)
P2(0,y∗,0)=(0,λ2-r2/λ2,0).The Jacobian matrix (23) is evaluated at P2 and the following eigenvalues obtained:(26)L1=λ1-σ1λ2-r2λ2-r1,L2=r2-λ2,L3=-w-P2λ2-r2λ2+Q2λ2-r2-r3.The eigenvalues above are negative ifλ2>r2. Hence, the equilibrium point P1(x,0,0) is locally asymptotically stable if conditions λ2>r2 hold. The condition λ2>r2 implies E2<λ2. For the local stability of P1(x,0,0), E2, the harvesting rate of tilapia fishes must be less than their intrinsic growth rate.(iv)
P3(x∗,y∗,0) = λ2(λ1-r1)-σ1(λ2-r2)/λ1λ2-σ1σ2,λ1(λ2-r2)+σ2(r1-λ1)/λ1λ2-σ1σ2,0.The eigenvalues ofJ(P3) are obtained by solving the characteristic equation; (27)λ3-A∗+E∗+G∗λ2+A∗G∗+E∗G∗+A∗E∗-D∗B∗λ+G∗D∗B∗-G∗A∗E∗=0whereA∗=λ1-2λ1x∗-σ1y∗-r1, B∗=-σ1y∗, C∗=-P1x∗/1+Q1x∗, D∗=-σ1y∗, E∗=λ2-2λ2y∗-σ2x∗-r2, F∗=-P2y∗/1+Q2y∗, and G∗=-w+e1x∗/1+Q1x∗+e2y∗/1+Q2y∗-r3.This can be expressed in form of(28)λ3+a1λ2+a2λ+a3=0By Routh-Hurwitz criteria (Murray, 1989), theλ′s are negative if a1>0,a3>0,a1a2-a3>0. (v)
P 4 ( x ∗ , 0 , z ∗ ) = P4w+r3/e1-wQ1-r3Q1,0,-λ1Q1(w+r3)2/P1(e1-wQ1-r3Q1)2+(λ1Q1-r1Q1-λ1)(w+r3)/P1(e1-wQ1-r3Q1)+λ1-r1/P1. The eigenvaluesJ(P4) are obtained by solving the characteristics equation; (29)L3-A2∗+B2∗+C2∗L2+A2∗B2∗+C2∗+B2∗C2∗+P1e1x∗z∗1+Q1x∗3L-A2∗B2∗C2∗-B2∗P1e1x∗z∗1+Q1x∗∗=0 where(30)A2∗=λ1-r1-2λ1x∗-P1z∗1+Q1x∗2,B2∗=λ2-σ2x∗-P2z∗-r2,C2∗=-w+e1x∗1+Q1x∗-r3. The characteristic equation is in the form(31)L3+a1L2+a2L+a3=0 wherea1=-(A2∗+B2∗+C2∗), a2=A2∗(B2∗+C2∗)+B2∗C2∗+P1e1x∗z∗/(1+Q1x∗)3, and a3=-A2∗B2∗C2∗-B2∗P1e1x∗z∗/(1+Q1x∗)∗. By Rouths stability criterion, the equilibrium point P4 is stable if (i) a1>0, a2>0, and a3>0; (ii) a1a2>a3. Otherwise it is unstable. (vi)
P 5 ( 0 , y ∗ , z ∗ ) = P 5 w + r 3 / e 2 - w Q 2 - r 3 Q 2 , 0 , - λ 2 Q 2 ( w + r 3 ) 2 / P 2 ( e 2 - w Q 2 - r 3 Q 2 ) 2 + ( λ 2 Q 2 - r 2 Q 2 - λ 2 ) ( w + r 3 ) / P 2 ( e 2 - w Q 2 - r 3 Q 2 ) + λ 2 - r 2 / P 2. The eigenvalues evaluated atJ(P5) are obtained by solving the characteristic equation (32)L3-A3∗+B3∗+C3∗L2+B3∗C3∗+A3∗B3∗+C3∗+P2e2z∗y∗1+Q2y∗3L-A3∗B3∗C3∗-P2e2z∗y∗1+Q2y∗3=0 whereA3∗=λ1-σ1y∗-P1z∗-r1, B3∗=λ2-r2-2λ2y∗-P2z∗/(1+Q2x∗)2, and C3∗=-w+e2y∗/1+Q2y∗-r3. The characteristic equation is in the form L3+a1L2+a2L+a3=0; wherea1=-(A3∗+B3∗+C3∗), a2=B3∗C3∗+A3∗(B3∗+C3∗)+P2e2z∗y∗/(1+Q2y∗)3, and a3=-A3∗B3∗C3∗-P2e2z∗y∗/(1+Q2y∗)3. By Rouths stability criterion, the equilibrium point P5 is stable if (i) a1>0, a2>0, and a3>0 and (ii) a1a2>a3. Otherwise it is unstable.


## 2.4. Global Stability of the Coexistence Equilibrium PointP6(x∗,y∗,z∗)

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 pointP6(x∗,y∗,z∗) are derived. The approach is based on work by Chaudhuri [13] and Dubey [14].Theorem 2.
The coexistence equilibrium pointP6(x∗,y∗,z∗) is globally asymptotically stable if
(i)λ1>0,λ2>0, (ii) P1>e1,P2>e2, and (ii) λ1λ2>(σ1+σ2)2.Proof.
Consider the following Lyapunov function:(33)Vx,y,z=x-x∗-x∗log⁡xx∗+y-y∗-y∗log⁡yy∗+z-z∗-z∗log⁡zz∗Simplifying the above Lyapunov function, (34)Vx,y,z=x-x∗-x∗log⁡x-x∗log⁡x∗+y-y∗-y∗log⁡y-y∗log⁡y∗+z-z∗-z∗log⁡z-z∗log⁡z∗Vx,y,z=x-x∗-x∗log⁡x+x∗log⁡x∗+y-y∗-y∗log⁡y+y∗log⁡y∗+z-z∗-z∗log⁡z+z∗log⁡z∗where ∂V/∂x=1-x∗/x=x-x∗/x, ∂V/∂y=1-y∗/y=y-y∗/y,and ∂V/∂z=1-z∗/z=z-z∗/z.
DifferentiatingV with respect to time, i.e., (35)dVdt=∂V∂xdxdt+∂V∂ydydt+∂V∂zdzdtdVdt=x-x∗xλ1x1-x-σ1xy-P1xz1+Q1x-r1x+y-y∗yλ2y1-x-σ2xy-P2xz1+Q1x-r2y+z-z∗z-wz+e1xz1+Q1x+e2yz1+Q2y-r3zdVdt=x-x∗λ11-x-σ1y-P1z1+Q1x-r1+y-y∗λ21-x-σ2x-P2z1+Q1x-r2+z-z∗-w+e1x1+Q1x+e2y1+Q2y-r3dVdt=x-x∗λ1-λ1x-σ1y-P1z1+Q1x-r1-λ1-+λ1x∗+σ1y∗+P1z∗1+Q1x∗+r1+y-y∗λ2-λ2y-σ2x-P2z1+Q2y-r2-λ2+λ2y∗+σ2x∗+P2z∗1+Q2y∗+r2+z-z∗-w+e1x1+Q1x+e2y1+Q2y-r3+w-e1x∗1+Q1x∗-e2y∗1+Q2y∗+r3dVdt=x-x∗λ1x∗-x+σ1Z∗-z+y-y∗λ2y2-y+σ2x∗-x+P2z∗-z1+Q2y+z∗-ze1x-x∗1+Q1x+e2y-y∗1+Q2ywhich simplifies to (36)dVdt=-x-x∗2λ1-x-x∗y-y∗σ1-σ2-x-x∗z-z∗P1-e11+Q1x-y-y∗2λ2-y-y∗z-z∗P2-e21+Q2yThus, dV/dt is a quadratic form which can be expressed as dV/dt=-XTAX, where XT=(x-x∗,y-y∗,z-z∗) and A is symmetric matrix given by(37)A=a11a12a13a12a22a23a13a23a33with a11=λ1, a12=σ1-σ2, a13=(P1-e1)/(1+Q1x), a22=λ2, a23=P2-e2/1+Q2y, and a33=0. We note that the point P6(x∗,y∗,z∗) is globally asymptotically stable if dV/dt<0; that is, the matrix A is positive definite Chaudhuri [13]. Now the matrix A is positive if a11>0, a13=0, a12>0, a22>0, a23=0, and a11a22-a122>0. a11>0 gives λ1>0, a13=0 gives P1=e1, a12>0 gives σ1>σ2, a22>0 gives λ2>0, and a11a22-a122>0 gives λ1λ2>(σ1+σ2)2. 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 [8]. From system (2), a biological equilibrium is given by dx/dt=dy/dt=dz/dt=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).Letc1 be the fishing cost per unit effort for cichlid fishes, c2 the fishing cost per unit effort for tilapia fishes, c3 the fishing cost per unit effort for Nile perch, p1 the price per unit biomass of cichlid fishes, p2 the price per unit biomass of tilapia fishes, and p3 the price per unit biomass of Nile perch; q1, q2, and q3 are catchability coefficients of cichlid fishes, tilapia fishes, and Nile perch, respectively.Then we haveπ1=(p1q1x-c1)E1, π2=(p2q2y-c2)E2, and π3=(p3q3x-c3)E3.π 1, π2, and π3 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 π=π1+π2+π3, which is(38)π=p1q1x-c1E1+p2q2y-c2E2+p3q3z-c3E3Then P(x∞,y∞,z∞,E1∞,E2∞,E3∞) is the bioeconomic equilibrium where x∞, y∞, z∞, E1∞, E2∞, and E3∞ 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:(39)λ11-x-σ1y-P1z1+Q1x-q1E1=0(40)λ21-y-σ2x-P2z1+Q2y-q2E2=0(41)-w+e1x1+Q1x+e2y1+Q2y-q3E3=0(42)π=p1q1x-c1E1+p2q2y-c2E2+p3q3z-c3E3=0In order to determine the bioeconomic equilibrium, we now consider the following cases.Case I.
Ifc1>p1q1, 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 (E1=0). Only tilapia fish and Nile perch fishery remain operational (i.e., c2<p2q2y and c3<p3q3z). Now we have y∞=c2/p2q2 and z∞=c3/p3q3. From (39) when E1=0, we have(43)λ1-λ1x∞-σ1y∞-P1z∞1+Q1x∞=0λ11+Q1x∞-λ1x∞1+Q1x∞-σ1y∞1+Q1x∞-P1z∞=0λ1+λ1Q1x∞-λ1x∞+λ1Q1x∞2-σ1y∞+σ1Q1x∞y∞-P1z∞=0(44)-λ1Q1x∞2+λ1Q1-λ1-σ1Q1y∞x∞+λ1-σ1y∞-P1z∞=0Since y∞=c2/p2q2 and z∞=c3/p3q3, then (44) can be written in quadratic form as (45)λ1Q1x∞2-λ1Q1-λ1-σ1Q1c2p2q2x∞+λ1-σ1c2p2q2-P1c3p3q3=0Therefore(46)x∞1,2=-R2±R22-4R1R22R3where R1=λ1Q1, R2=λ1Q1p2q2-λ1p2q2-σ1Q1c2/p2q2, R3=λ1-σ1c2/p2q2-P1c3/p3q3.
From (46),(47)E1∞=λ2q21-y∞-σ2q2x∞-P2z∞q21+Q2y∞Therefore, E1∞>0 if(48)λ2q2p2-λ2c2-σ2p2q2x∞p2q22>P2p2q2c3p3q3p2q2-Q2p3q3c2Also, (49)E2∞=-wzq3+e1xzq3+Q1q3x+e2yzq3+Q2q3y=-wc3p3q32+e1c3x∞p3q32+Q1p3q32x∞+e2c2c3p2q2p22q22p3q32+Q2p2q2p3q32c2Thus, E2∞>0 if(50)e1c3x∞p22q22p3q32+Q2p2q2p3q32c2+e2c2c3p2q2p3q32+Q1p3q32x∞p3q32-Q1p3q32x∞p22q22q32p3+Q2p2q2p3q32c2>wc3p3q32Therefore, the bioeconomic equilibrium exists if conditions (48) and (50) hold.Case II.
Ifc2>p2q2, 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 (E2=0). Only cichlid fishes and Nile perch fishery remain operational (i.e., c1<p1q1x and c3<p3q3z). Now we have x∞=c1/p1q1 and z∞=c3/p3q3. From (46) when E2=0, we have(51)λ2-λ1y∞-σ2x∞-P2z∞1+Q1y∞=0λ21+Q2y∞-λ2y∞1+Q2y∞-σ2x∞1+Q2y∞-P2z∞=0λ2+λ2Q2y∞-λ2y∞+λ2Q2y∞2-σ2x∞+σ2Q2x∞y∞-P2z∞=0(52)-λ2Q2y∞2+λ2Q2-λ2-σ2Q2x∞y∞+λ2-σ2x∞-P2z∞=0Since x∞=c1/p1q1 and z∞=c3/p3q3, then (52) can be written in quadratic form as(53)λ2Q2y∞2-λ2Q2-λ2-σ2Q2c1p1q1y∞+λ1-σ2c1p1q1-P2c3p3q3=0Therefore(54)y∞=-S2±S22-4S1S22S3where(55)S1=λ2Q2S2=λ2Q2p1q1-λ2p1q1-σ2Q2c1p1q1S3=λ2-σ2c1p1q1-P2c3p3q3From (39),(56)λ11-x∞-σ1y∞-P1z∞1+Q1x∞-q1E1∞=0E1∞=λ1q1-λ1c1p1q12-σ1y∞q1-P1p1q1c3p3q3p1q1-Q1p3q3c1
E 1 ∞ > 0 if(57)λ1q1p1-λ1c1-σ1p1q1y∞p1q12>P1p1q1c3p3q3p1q1-Q1p3q3c1Also,(58)E3∞=-wzq3+e1xzq3+Q1q3x+e2yzq3+Q2q3y=-wc3p3q32+e1c3y∞p3q32+Q2p3q32y∞+e2c1c3p1q1p12q12p3q32+Q1p1q1p3q32c1Thus, E2∞>0 if(59)e2c3y∞p12q12p3q32+Q1p1q1p3q32c1+e2c1c3p1q1p3q32+Q2p3q32y∞p3q32-Q2p3q32y∞p12q12q32p3+Q1p1q1p3q32c1>wc3p3q32Therefore, the bioeconomic equilibrium exists if conditions (57) and (59) hold.Case III.
Ifc3>p3q3, 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., c1<p1q1x and c2<p2q2z). Now we have x∞=c1/p1q1 and y∞=c2/p2q2. Substituting x∞ and y∞in (44), we get(60)E1∞=λ11q1-c1p1q12-σ1c2p1q12-Q1q1c1+P1p1q12p2q2z∞p2q2q1p1q12-Q1q1c1Thus, E1∞>0 if(61)λ11q1-c1p1q12>σ1c2p1q12-Q1q1c1+P1p1q12p2q2z∞p2q2q1p1q12-Q1q1c1Also,(62)E2∞=λ11q1-c2p2q22-σ2c1p2q22-Q2q2c2+P2p2q22p1q1z∞p1q1q2p2q22-Q2q2c2Thus, E2∞>0 if(63)λ21q2-c2p2q22>σ2c1p2q22-Q2q2c2+P2p2q22p1q1z∞p1q1q2p2q22-Q2q2c2Therefore, the bioeconomic equilibrium exists if conditions (61) and (63) hold.Case IV.
Ifc1>p1q1x,c2>p2q2y and c3>p3q3z, then the fishing cost is greater than revenues for all three species and the whole fishery will be closed.Case V.
Ifc1<p1q1x,c2<p2q2y and c3<p3q3z, then the fishing cost is less than revenues for all three species; that is, the fishery is more profitable and hence it would attract more fishermen and the whole fishery will be in operation. Now we have x∞=c1/p1q1,y∞=c2/p2q2, and z∞=c3/p3q3, then(64)E1∞=λ11q1-c1p1q12-σ1c2p1q12p3q3-Q1p3q3q1c1+P1p1q1c3p2q2q1p3q3P1q12-Q1p3q3q1c1Thus, E1∞>0, if(65)λ11q1-c1p1q12>σ1c2p1q12p3q3-Q1p3q3q1c1+P1p1q1c3p2q2q1P3q3P1q12-Q1p3q3q1c1And,(66)E2∞=λ21q2-c2p2q22-σ2c1p2q22p3q3-Q2p3q3q2c2+P2p2q2c3p1q1q2p3q3p2q22-Q2p3q3q2c2Thus, E2∞>0,(67)λ21q2-c2p2q22>σ2c1p2q22p3q3-Q2p3q3q2c2+P2p2q2c3p1q1q2P3q3p2q22-Q2p3q3q2c2Also, (68)E3∞=-wc3p3q3+e1c1c3p1q1p12q12p3q3+p3q3p1q1c1+e2c2c3p2q2p3q3p22q22-p3q3p2q2c2Thus, E3∞>0 if (69)e1c1c3p1q1p3q3p22q22-p3q3p2q2c2+e2c2c3p2q2p12q12p3q3+p3q3p1q1c1p12q12p3q3+p3q3p1q1c1p3q3p22q22-p3q3p2q2c2>0Thus, the nontrivial bioeconomic equilibrium point P∞(x∞,y∞,z∞,E1∞,E2∞,E3∞) exists if and only if conditions (65), (67), and (69) hold together.


## 3. Results and Discussion

### 3.1. 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 Table1.Table 1
Parameter Values of the model.
Parameter Value source λ 1 2.07 [13] λ 2 2.09 [13] K 1 200 [13] K 2 100 [13] α 12 0.001 [15] α 13 0.02 [15] α 21 0.002 [16] α 23 0.03 [16] α 31 1.5 [17] β 0.1 [17] γ 0.2 [17] q 1 0.14 [17] q 2 0.13 [18] q 3 0.125 [18] w 1 [17]Figure1 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 3D 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.Figure 1
Graph of preyx1 and x2 and predator x3 against time (in years). It shows the variation of population density with time.Figure 2
Graph of preyx1 and x2 and predator x3 against time (in years). The limit cycle shows that population density will slightly change but not oscillate the boundary.Figure 3
Variation of the fishes population with different harvesting efforts in cichlidE1 with fixed values of E2 and E3.Figure 4
The effect of Tilapia harvest with different values ofE2 with fixed values of E1 and E3.Figure 5
The effect of Nile perch harvest with different values ofE3 with fixed values of E1 and E2.Figure 6
The interaction between cichlid and tilapia fishes in absence of Nile perch.


### 3.2. 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. Figures3 and 4 show that cichlid and tilapia fishes tend to extinction when the harvesting rates E1 and E2 exceed their intrinsic growth rates λ1 and λ2. 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.Theorem1 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.


## 3.1. 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 Table1.Table 1
Parameter Values of the model.
Parameter Value source λ 1 2.07 [13] λ 2 2.09 [13] K 1 200 [13] K 2 100 [13] α 12 0.001 [15] α 13 0.02 [15] α 21 0.002 [16] α 23 0.03 [16] α 31 1.5 [17] β 0.1 [17] γ 0.2 [17] q 1 0.14 [17] q 2 0.13 [18] q 3 0.125 [18] w 1 [17]Figure1 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 3D 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.Figure 1
Graph of preyx1 and x2 and predator x3 against time (in years). It shows the variation of population density with time.Figure 2
Graph of preyx1 and x2 and predator x3 against time (in years). The limit cycle shows that population density will slightly change but not oscillate the boundary.Figure 3
Variation of the fishes population with different harvesting efforts in cichlidE1 with fixed values of E2 and E3.Figure 4
The effect of Tilapia harvest with different values ofE2 with fixed values of E1 and E3.Figure 5
The effect of Nile perch harvest with different values ofE3 with fixed values of E1 and E2.Figure 6
The interaction between cichlid and tilapia fishes in absence of Nile perch.


## 3.2. 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. Figures3 and 4 show that cichlid and tilapia fishes tend to extinction when the harvesting rates E1 and E2 exceed their intrinsic growth rates λ1 and λ2. 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.Theorem1 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.


## 4. 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.



---

*Source: 2601648-2019-06-02.xml*