0 0.5 1 1.5 2 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Prey
Predator
Figure 5.1: Phase-portrait for the system (2.1.2) with parameter valuesc= 1.1, m= 1 and b= 3.
5.3 Stability of the Modified Ratio-Dependent Model with Prey
0 0.5 1 1.5 2 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Prey
Predator
Figure 5.2: Phase-portrait for the system (5.2.1) with parameter valuesc= 1.1, m= 1, b= 3, q = 1 and E= 0.2.
5.3.1 Existence of Equilibrium Points
The system (5.3.1) admits the trivial equilibrium point (0,0), two axial equilibria, (1−qE,0) and (0,(b−α)ξ) and two interior equilibria (xi, yi), i= 1,2 given by the solution of following equations,
1−x− cy
x+y+αξ −qE= 0 b[x+ξ]
x+y+αξ −1 = 0.
(5.3.2)
Solving (5.3.2) we obtain (xi, yi), i= 1,2 where each xi ∈(0,1−qE) is the root of quadratic equation x2+ (c−1−cb +ξ+qE)x+ξ(qE+c−1−αcb ) = 0 (see Appendix C), which is given by
xi = 1−qE−c+cb −ξ
2 +(−1)i 2
r
qE+c−1−c b+ξ2
−4ξ
qE+c−1−αc b
and yi = (b−1)xi+ (b−α)ξ.
The following relation can be derived from (5.3.2) (see Appendix C), xi= 1−c+c
b −qE+c(α−1)ξ b(xi+ξ).
This relation is satisfied by the interior equilibria (xi, yi) when they exist. From this relation we can conclude that whenever α >1, the supply of additional food to predators, reduces the pressure of
0 0.5 1 1.5 2 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Prey
Predator
Figure 5.3: Phase-portrait for the system (5.2.1) with parameter valuesc= 1.1, m= 1, b= 3, q = 1 and E= 0.5.
predation (on prey) as well as the pressure (on both prey and predator) resulting from harvesting.
Defining ∆ as the expression under the square root term ofxi, we obtain the conditions (see Table 5.3) on the model parameters, required for the existence of the above equilibrium points.
Under the condition 0< xi <1−qE, we haveyi >0. To see this, consideryi = (1−xqE−xi)(xi+αξ)
i+c−1+qE . Observe that ifc≥1−qE, thenyi>0. For the case whenc <1−qEand noting thatxi = 1−c−qE is an asymptote to the prey isocline yi = (1−xqE−xi)(xi+αξ)
i+c−1+qE , we obtain xi >1−c−qE. Thusyi >0 forc <1−qE also.
5.3.2 Local Stability Analysis The Jacobian matrix for (5.3.1) is given by
J =
(1−x)−x+y+αξcy −qE+xh
−1 + (x+y+αξ)cy 2i
−(x+y+αξ)cx(x+αξ)2
bmy(y+αξ−ξ)
(x+y+αξ)2 m b[x+ξ]
x+y+αξ−1
−(x+y+αξ)bm(x+ξ)y2
.
Equilibrium point Existential conditions
(0,0) -
(1−qE,0) qE <1
(0,(b−α)ξ) b > α
(x1, y1) qE+c−1−cb +ξ <0,qE+c−1−αcb >0, ∆>0,x1 <1−qE or
qE+c−1−cb +ξ <0, ∆ = 0,x1 <1−qE
(x2, y2) qE+c−1−cb +ξ <0,qE+c−1−αcb >0, ∆>0,x2 <1−qE or
qE+c−1−cb +ξ <0,qE+c−1−αcb = 0,x2<1−qE or
qE+c−1−cb +ξ <0, ∆ = 0,x2 <1−qE or
qE+c−1−αcb <0,x2 <1−qE
Table 5.3: Conditions on parameters for existence of equilibrium points of the modified ratio- dependent system with prey harvesting
Hence the Jacobian matrix at the equilibria of the system (5.3.1) is of the following form, J(0,0) =
1−qE 0 0 mα(b−α)
, J(1−qE,0) =
"
−(1−qE) −1c(1−−qEqE)+αξ
0 m((b−1)(1−1−qE+αξqE)+(b−α)ξ)
# , J(0,(b−α)ξ) =
1−qE−c+cαb 0
m(b−α)(b−1)
b −m(bb−α)
,
J(xi,yi) =
xih
−1 + (x cyi
i+yi+αξ)2
i −(xcxi+yi(xii+αξ)+αξ)2
bmyi(yi+αξ−ξ)
(xi+yi+αξ)2 −(xbm(xi+yii+ξ)y+αξ)i2
.
For population conservation (while harvesting) to be ensured, we need the equilibria (0,0),(1− qE,0) and (0,(b−α)ξ) to be unstable in nature. The equilibrium (0,0) is always unstable in nature provided the axial equilibrium point (1−qE,0) exists. However, (1−qE,0) always exists in order to ensure the existence of the interior equilibrium point. Thus (0,0) is always unstable.
The conditions (b−1)(1−qE) + (b−α)ξ >0 and 1−qE−c+ cαb >0 ensures the saddle nature of (1−qE,0) and (0,(b−α)ξ) respectively. Note that, the condition for the saddle nature of (1−qE,0) is satisfied whenever any one of the interior equilibrium point exists. This is because 0< yi = (b−1)xi+ (b−α)ξ <(b−1)(1−qE) + (b−α)ξ.
The stability of interior equilibria (xi, yi) can be analyzed by using the determinant and trace of the Jacobian matrixJ(xi,yi) given by
detJ(xi,yi) = bcmxiyi
(xi+yi+αξ)3
ξ(α−1) +b
c(xi+ξ)2
,
tr J(xi,yi) = −(b+ 1)x2i −[(bξ−1) +m(b−1)]xi−m(b−α)ξ
b(xi+ξ) .
We study the stability of interior equilibrium points (for all the three cases defined in Table 3.2) when (0,(b−α)ξ) either exists with saddle nature or does not exist at all. Here, only interior equilibrium point (x2, y2) can exists, ifx2 <1−qE holds. We summarizes the conditions as follows,
Cases Interior equilibrium point exists Stable
Case 1 (x2, y2) if tr J(x2,y2)<0 Case 2 (x2, y2) if tr J(x2,y2)<0
Case 3 (x2, y2) always
5.3.3 Global Stability Analysis
Theorem 5.3.1. Suppose that the interior equilibrium point(x2, y2)is locally asymptotically stable and the conditions αξ≥1−qE and α >
1−1−cqE
b hold, then(x2, y2) is also globally stable.
Proof. We first define the following function, L2(x, y) = ∂
∂x(B2(x, y)f2(x, y)) + ∂
∂y(B2(x, y)g2(x, y)), wheref2(x, y) =x(1−x)−x+y+αξcxy −qEx , g2(x, y) =m b(x+ξ)
x+y+αξ −1
y and B2(x, y) = x+y+αξxy . After simplification, we obtain,
L2(x, y) = (1−qE−αξ)
y − (2x+y)
y −m
x.
Now, L2(x, y) < 0 whenever x > 0 and y > 0, since αξ ≥ 1−qE. Thus, by the Dulac’s crite- rion [49] the system (5.3.1) will not have any non-trivial periodic orbit in R2+. Note that, when
1−1−cqE
b < α < b, then the trivial equilibrium point is a repeller and the two axial equilib- ria, (1−qE,0) and (0,(b−α)ξ), are both saddle and have x−axis and y−axis as their respective stable manifolds. On the other hand, when α > b, then both (0,0) and (1 −qE,0) are sad- dle and have y−axis and x−axis as their respective manifolds. Using these, in conjunction with the Poincare-Bendixson Theorem [49] gives us that the interior equilibrium point (x2, y2) will be globally stable.
Finally, we summarize in Table 5.4 the conditions on effort E in order to achieve ecologically sustainable harvesting for the system (5.3.1).
Conditions Ensures Applicable Cases
E < 1q (0,0) is saddle/repeller and (1−qE,0) exists Case 1, Case 2, Case 3 E < 1q
1 + (bb−−α)ξ1
(1−qE,0) is saddle Case 1, Case 2, Case 3
E < 1q(1−c+αcb) (x2, y2) exists providedx2<1−qE Case 1, Case 2, Case 3 and (0,(b−α)ξ) is saddle (if it is exists)
1
q(1−αξ)≤E <1q(1−c+αcb ) (x2, y2) is globally stable (if it exists) Case 1, Case 2 and tr J(x2,y2)<0
1
q(1−αξ)≤E <1q(1−c+αcb ) (x2, y2) is globally stable (if it exists) Case 3
Table 5.4: Conditions on harvesting effort for the system (5.3.1).
Clearly, here, one can see that ecologically sustainable harvesting is possible under all three cases whereas without additional food it possible only under Case 3. Also, one can observe that bounds on harvesting effort E depends on the additional food parameters, namely, α and ξ. A choice of α > 1, results in an increase in the upper bound for ecologically sustainable harvesting effort (corresponding to the global stability of interior equilibrium point) as compared to the case without additional food. On the other hand, a choice of ξ > 1/b ensures that the condition tr J(x2,y2) <0 (required for global stability of (x2, y2) under Case 1 and Case 2) is satisfied. This illustrates the advantage of providing additional food to the predators over the scenario of no additional food being provided. A consequence of this is that a higher effort level can be achieved with the choice of an appropriate level of additional food supply. For example, we choose the same parameter values, c= 1.1, m= 1, b= 3 and q= 1 as in Section 4. Recall that in this example, the ecological conservation required that the effort levelE be kept below 0.2667. As an illustration, we choose E = 0.5, which results in the extinction of both the populations, when no additional food is provided. However an appropriate level of additional food (for example, α = 2 andξ = 0.5 for which coexistence is achieved for 0 < E < 0.633) ensures the stability of the interior equilibrium point and hence conservation of both the species. The phase portrait for this example is given in Figure 5.4.
0 0.5 1 1.5 2 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Prey
Predator
Figure 5.4: Phase-portrait for the system (5.3.1) with parameter values c= 1.1, m= 1, b= 3, q = 1, E = 0.5, α= 2 andξ = 0.5.