5.5 Optimal Harvesting Policy for the Modified Ratio-Dependent
To determine the singular optimal equilibrium solution (x∗, y∗) we take δ = 0 and substitute the interior equilibrium points of the systems (5.3.1) and (5.5.2) into ∂∂EH = 0 which results in the following cubic equation (see Appendix C),
(−2bpq)x3+ (bc1+ (−4bξ+b+c−bc)pq)x2+ 2(bc1+pq(b+c−bc−bξ))ξ)x
+ ξ(−c(1−α)c1+bc1ξ+pq(b−c(b−α))ξ) = 0. (5.5.3) When sign of the coefficient of x3 and constant term in a cubic equation are opposite, then the equation has atleast one positive root. Thus, in the above case, if (cαb −cb+ξ)c1+pqξ(1−c+cαb )>0 then above cubic equation has at least one positive root. Let x∗ be the positive root of this cubic equation. Thenx∗ is called the singular optimal equilibrium prey-population level. The necessary conditions for the singular control to be optimal is that the generalized Legendre condition [40]
(2λ1−p)
1− cy
(x+y+αξ)2
+ (λ1cx+λ2bm(y+αξ−ξ)) y
(x+y+αξ)3 ≥0
is satisfied along singular solution (see Appendix C). Hence the optimal harvesting policy will be,
E MRD(t) =
Emax if x > x∗ E∗ if x=x∗ 0 if x < x∗.
Note that, in order to ensure an economically better harvesting policy with the additional food as compared to harvesting without additional food being supplied to the predators in the system, the integrand in (5.5.1) must be greater than the integrand in (5.4.1). This can be ensured as long as the cost per unit quantity biomass of additional food satisfies,
c2 < 1
ξ (pqx−c1) E MRD−E RD . It can be shown that equation (5.5.3) reduces to (see Appendix C)
2bpq(x∗+ξ)2(x∗−x) =e cξ(c1+pqξ)(α−1).
Consequently, x∗ <(>)xe ⇐⇒ α <(>)1. This means that in order to undertake prey harvesting with additional food at a level,x∗below (above)ex(the level without additional food), the predators need to be supplied with good (poor) quality of additional food. There is no benefit in terms of harvesting stock level, by providing same quality of additional food relative to the prey,i.e.,α= 1, since this results in x∗ =xeand E∗ =E. Apart from this, there is added financial burden arisinge from supply of additional food in this case.
One can observe from the cubic equation (5.5.3) that the singular optimal prey-equilibrium levelx∗ is dependent on the biological control parameters α andξ. We will now illustrate through
numerical examples, that for the modified model with additional food, the optimal harvesting policy can be more effectively decided by an appropriate choice of α and ξ. For this purpose, we refer to the example on prey harvesting without additional food case, as given in the previous section. Recall that for the parameter values b = 3, c = 1.1, m = 1, p = 1, q = 1 and c1 = 0.005, no prey harvesting can be done until the prey size reaches a level of xe = 0.1358. However, for this set of parameter values, the system supported by additional food to the predators results in bioeconomically sustainable harvesting policy for somex(t)<ex. In fact in this case, the harvesting effort can be more than the corresponding harvesting effort for the system without additional food.
To begin with, we set x∗ = 0.0001. Then, in this case α and ξ has to satisfy the relation (−0.2856 + 1.1α)ξ2 + (−0.00533712 + 0.0055α)ξ + 0.8144 ×10−8 = 0. By choosing α = 0.94 and ξ = 0.0001515 (which satisfies this relation), one can harvest at the maximum effort level Emax = 0.2447, for all time, provided x(0) > x∗. Clearly, the economic returns here is better than the returns without the additional food. Now, choosing α = 0.9,0.8,0.7 with corresponding ξ = 0.0005277,0.001569,0.003065 respectively, one can prey harvest at the maximum harvest rate of 0.23,0.1933,0.1567 respectively, providedx(0)> x∗. These numerical values ofαsuggest that, as we decreases the αvalue (improvement in the quality of additional food) maximum harvest rate is keeps decreasing. This phenomenon is possibly due to the increase in the predator population level (resulting from better quality of additional food) but the quantity being fixed, thereby resulting in more predation of prey by the predators. This can also be mathematically justified, by recalling that y2 = (b−1)x2+ (b−α)ξ. As we decrease the value ofα(for fixedξ),y2 (predator equilibrium level) will increase, while prey equilibrium level x2 will decrease. For the parameter values α = 0.94 and ξ = 0.0001515, the optimal trajectory for system (5.3.1) with initial population densities (x0, y0) = (0.9,0.4) is shown in Figure 5.6.
We now consider a second example, when x < xe ∗, with same parameters values chosen earlier, namely, b = 3, c = 1.1, m = 1, p = 1, q = 1 and c1 = 0.005. In this case, for some x(t) > x,e harvesting cannot be undertaken till the prey population reaches a level of atleastx∗. But, whenever harvesting is possible,Emax andE∗ can be increased by increasing the value ofα andξ. This may be possible from an ecological point of view, since supply of poor quality additional food could result in the predators being distracted, thereby easing the predation pressure on the prey, whose size consequently increases. Thus, the harvest effort can be increased in such a scenario. We setx∗= 0.2, in which caseαandξhas to satisfy the relation (−1.485+1.1α)ξ2+(−0.1595+0.0055α)ξ−0.0154 = 0. By choosingα = 2 andξ = 0.2836 and ifx(0)> x∗ one can start harvesting at the maximum effort level Emax = 0.6333, till x reduces to x∗ = 0.2. After reaching x∗ = 0.2, we switch to the singular effort levelE∗ = 0.2817.
For the parameter values α= 2 andξ = 0.2836, the optimal trajectory for system (5.3.1) with initial population densities (x0, y0) = (0.9,0.4) is shown in Figure 5.7. The optimal trajectory is
a combination of two trajectories, one (dashed) corresponding to the maximum harvesting effort Emax = 0.6333 while the second one corresponds to the singular effort E∗ = 0.2817. The switch from maximum harvesting effortEmax to singular effort Ee happens at time t= 2.9.
E MRD(t) =
(Emax= 0.6333 if t≤2.9, E∗ = 0.2817 if t >2.9.
Note that, in this case also, we get an improved harvesting policy, even with the low quality
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Prey
Predator
(0.9, 0.4)
Figure 5.6: Optimal path emanating from point (0.9,0.4) subject to the system (5.3.1) with pa- rameter values c= 1.1, m= 1, b= 3, q = 1, p= 1, c1 = 0.005, α= 0.94 and ξ = 0.0001515.
additional food as compared to prey nutrients, sinceα= 2>1 and while keeping prey population level at a level higher than the level without the additional food case. This is illustrative of how the supply of additional food can play a critical role in formulating an efficient harvesting policy along with population conservation.