Balanced harvesting in two predators one prey system

Abstract

In this research, we discuss balancing two essential ecosystem services, known as resilience and yield, in a harvested two predators one prey system. We study how these two ecosystem services respond to different types of harvesting plans applied to the system. In individual harvesting, we observe that prey harvesting is suitable for generating more yield while either predator harvesting gives more resilience. Several patterns are possible based on the intensity of harvesting efforts applied to prey or the predator in simultaneous harvesting. This study shows that a balanced harvest between prey and predators may give more than the selective harvesting of any species yielding and stabilizing the ecosystem. Even if this fair strategy is not a cooperative situation for both resilience and yield, it may be the most favorable approach to scale those services. Finally, we may state that it would help us correlate the safest position (corresponds to the maximizing resilience yield) and the good views (which corresponds to the maximum sustainable return) to safeguard our ecosystems.

Appendix

The Jacobian matrix at the coexisting steady state corresponding to the system (1) is

$$\begin{aligned} J_{(u^{*},v^{*},w^{*})} = \left( \begin{array}{ccc} a_{11} &{} a_{12} &{} a_{13} \\ a_{21} &{} 0 &{} 0 \\ a_{31} &{} 0 &{} a_{33} \\ \end{array} \right) \end{aligned}$$

where, $a_{11} = - \frac{r u^{*}}{K} + \dfrac{\alpha _1 u^{*} v^{*}}{(h+u^{*})^2}$, $a_{12} =- \frac{\alpha _1 u^{*}}{h+u^{*}}$, $a_{13} =-\alpha _2 u^{*}$, $a_{21} = \frac{\alpha _1 h v^{*}}{(h+u^{*})^2}$, $a_{31}=\alpha _{2} w^{*}$, and $a_{33}= - \gamma w^{*}$.

The characteristic equation of the Jacobian matrix is

$$\begin{aligned} \lambda ^3+Q_1 \lambda ^2 +Q_2 \lambda + Q_3=0, \end{aligned}$$

(4)

where

$$\begin{aligned} Q_1=-(a_{11}+a_{33}), Q_2=-a_{12}a_{21} -a_{13}a_{31}+a_{11}a_{33}, Q_3=a_{12}a_{21}a_{33}. \end{aligned}$$

If condition-1: $Q_1>0$,  condition-2: $Q_3>0$ and condition-3: $Q_1Q_2-Q_3>0$ are fulfilled, then by the Routh-Hurwitz criterion, the eigenvalues have negative real part.

Now, $Q_1=-(a_{11}+a_{33})= \frac{r u^{*}}{K} +\gamma w^{*}- \dfrac{\alpha _1 u^{*} v^{*}}{(h+u^{*})^2}$

$Q_2=-a_{12}a_{21} -a_{13}a_{31}+a_{11}a_{33}=u^{*} (\frac{\alpha _1^2 h v^{*}}{(h + u^{*})^3} + \alpha _2^2 w^{*} - \frac{\alpha _1 v^{*} w^{*} \gamma }{(h + u^{*})^2} + \frac{w^{*} \gamma r}{K})$,

$Q_3=a_{12}a_{21}a_{33}=\frac{\alpha _1^2 h u^{*} v^{*} w^{*} \gamma }{(h + u^{*})^3}>0$

Again, $Q_1Q_2-Q_3=-(a_{11}+a_{33})(-a_{12}a_{21} -a_{13}a_{31}+a_{11}a_{33})-(a_{12}a_{21}a_{33}) =a_{11} a_{12} a_{21} + a_{11} a_{13} a_{31} - a_{11}^2 a_{33} + 2 a_{12} a_{21} a_{33} + a_{13} a_{31} a_{33} - a_{11} a_{33}^2=-\frac{\alpha _1^3 h (u^{*}v^{*} )^2}{(h + u^{*} )^5} - \frac{\alpha _1 \alpha _2^2 (u^{*})^2 v^{*} w^{*} }{(h + u^{*} )^2} + \frac{\alpha _1^2 (u^{*} v^{*})^2 w^{*} \gamma }{(h + u^{*} )^4}+ \alpha _2^2 u^{*} (w^{*})^2 \gamma - \frac{\alpha _1 u^{*} v^{*} w^2 \gamma ^2}{(h + u^{*} )^2} + \frac{\alpha _1^2 h (u^{*})^2 v^{*} r}{(h + u^{*} )^3 K} + \frac{\alpha _2^2 (u^{*})^2 w^{*} r}{K} - \frac{2 \alpha _1 (u^{*})^2 v^{*} w^{*} \gamma r}{(h + u^{*} )^2 K} + \frac{u^{*} (w^{*})^2 \gamma ^2 r}{K} + \frac{(u^{*})^2 w^{*} \gamma r^2}{K^2}$

We see that the second condition is satisfied as $Q_3$ is positive. We observe that the first and the third condition is not fulfilled as the signs of $Q_1$ and $Q_1 Q_2-Q_3$ are not identified as positive by inspection. But using the last condition of each Proposition 3, we see that $a_{11}<0$ and hence $Q_1, Q_2, Q_3$ and $Q_1Q_2 - Q_3$ are all positive. Thus, all the conditions of the Routh-Hurwitz criterion are satisfied. Hence, the interior equilibrium is asymptotically stable.

Similarly, using the last condition of each Proposition 4 , 5 and 6, we can show that the interior equilibrium is asymptotically stable in each case.

Next, to determine the resilience, we have to evaluate the roots of the above characteristic equation and they are $-\dfrac{Q_1}{3} - \frac{2^{1/3}}{3} \frac{(-Q_1^2 + 3 Q_2)}{ee}+\frac{ee}{3 2^{1/3}}$, $-\dfrac{Q_1}{3} - \omega ^2 \frac{2^{1/3}}{3} \frac{(-Q_1^2 + 3 Q_2)}{ee}+\omega \frac{ee}{3 2^{1/3}}$, and $-\dfrac{Q_1}{3} - \omega \frac{ 2^{1/3}}{3} \frac{(-Q_1^2 + 3 Q_2)}{ee}+ \omega ^2\frac{ ee}{3 2^{1/3}}$, where, $ee=(-2 Q_1^3 + 9 Q_1 Q_2 \!-\! 27 Q_3 \!+\! 3 \sqrt{3} \sqrt{-Q_1^2 Q_2^2 + 4 Q_2^3 \!+\! 4 Q_1^3 Q_3 \!-\! 18 Q_1 Q_2 Q_3 \!+\! 27 Q_3^2} )^{1/3}.$ Following Pimm and Lawton [34] and Kar et al. [22], we can determine the largest real part of all the eigenvalues and so the resilience as its absolute value.