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Global Harvesting and Stocking Dynamics in a Modified Rosenzweig–MacArthur Model

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Abstract

Effective management of predator–prey systems is crucial for sustaining ecological balance and preserving biodiversity, which requires full understanding the dynamics of such systems with harvesting and stocking. This paper aims to investigate the global dynamics of a Rosenzweig–MacArthur model considering the interplay of these intervention practices. We reveal that this model undergoes a sequence of bifurcations, including cusp of codimensions 2 and 3, saddle-node bifurcation, Bogdanov–Takens (BT) bifurcation of codimensions 2 and 3, and degenerate Hopf bifurcation of codimension 2. In particular, a codimension-2 cusp of limit cycles is found, which indicates the coexistence of three limit cycles. An interesting and novel scenario is discovered: two distinct homoclinic cycle curves connect their respective BT bifurcation points. This differs from most models where a single homoclinic cycle curve may connect both BT bifurcation points. Moreover, we find that two families of limit cycles converge toward a heteroclinic cycle, signaling the risk of overexploitation. From a biological perspective, the prey population may undergo extinction for all initial states under large constant harvesting rate. Further, the simultaneous stocking of both populations is not conducive to the coexistence of both species; the stocking of one population and the harvesting of the other will promote the coexistence of two populations; while the simultaneous harvesting of two populations may result in multiple limit cycles, which effectively underscore the positive effect of harvesting and stocking. Identifying the optimal timing to harvest or stock predators and prey is crucial to prevent system collapse. This work promotes to a deeper understanding of the dynamics of ecosystems when harvesting and stocking occurs simultaneously. Further, it reveals the important roles of harvesting and stocking, contributing to the effective management of predator–prey systems.

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Acknowledgements

The authors are very grateful to professor Jiang Yu for his helpful and insightful suggestions. Y. C. Xu’s research is partially supported by the National NSF of China (No. 11671114) and L. B. Rong’s research is partially supported by the National NSF of USA (DMS-1950254).

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Authors and Affiliations

  1. Department of Mathematics, Qufu Normal University, Qufu, 273165, Shandong, China

    Yue Yang & Fanwei Meng

  2. Department of Mathematics, China Jiliang University, Hangzhou, 310018, Zhejiang, China

    Yue Yang & Yancong Xu

  3. Department of Mathematics, University of Florida, Gainesville, FL, 32611, USA

    Libin Rong

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Yue Yang and Yancong Xu wrote the main manuscript text, Fanwei Meng and Libin Rong prepared figures. All authors reviewed the manuscript.

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Correspondence to Yancong Xu or Fanwei Meng.

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Appendices

Coefficients in the Proof of Theorem 1

$$\begin{aligned}{} & {} \hat{a}_{10}=\frac{\beta x^{*}(1-2x^{*})}{\delta (\alpha +x^{*})},\ \hat{a}_{01}=-\frac{x^{*}}{\alpha +x^{*}},\ \hat{a}_{20}=\frac{(2 x^{*}-1) (\beta x^{*}-\delta (\alpha +x^{*}))}{\delta (\alpha +x^{*})^2}-1,\\{} & {} \hat{a}_{11}=-\frac{\alpha }{(\alpha +x^{*})^{2}},\ \hat{a}_{30}=\frac{(2 x^{*}-1)(\delta (\alpha +x^{*})-\beta x^{*})}{\delta (\alpha +x^{*})^{3}},\ \hat{a}_{21}=\frac{\alpha }{(\alpha +x^{*})^{3}};\\{} & {} \hat{b}_{10}=\frac{\beta (2 x^{*}-1)(\beta x^{*}-\delta (\alpha +x^{*}))}{\delta (\alpha +x^{*})},\ \hat{b}_{01}=\frac{\beta x^{*}}{\alpha +x^{*}}-\delta ,\\{} & {} \hat{b}_{20}=-\frac{\beta (2 x^{*}-1) (\beta x^{*}-\delta (\alpha +x^{*}))}{\delta (\alpha +x^{*})^2},\ \hat{b}_{11}=\frac{\alpha \beta }{(\alpha +x^{*})^{2}},\\{} & {} \hat{b}_{30}=\frac{\beta (2x^{*}-1)(\beta x^{*}-\delta (\alpha +x^{*}))}{\delta (\alpha +x^{*})^{3}},\ \hat{b}_{21}=-\frac{\alpha \beta }{(\alpha +x^{*})^{3}};\\{} & {} \hat{c}_{11}=\frac{-2\hat{a}_{01}^{2}\hat{b}_{20}+\hat{a}_{01}\hat{b}_{01}(2\hat{a}_{20}-\hat{b}_{11}) +\hat{a}_{11}\hat{b}_{01}^{2}+\hat{a}_{10}(\hat{a}_{01}\hat{b}_{11}-\hat{a}_{11}\hat{b}_{01})}{\hat{a}_{01} (\hat{a}_{10}+\hat{b}_{01})},\\{} & {} \hat{c}_{02}=\frac{-\hat{a}_{01}^2 \hat{b}_{20}+\hat{a}_{01} \hat{b}_{01} (\hat{a}_{20}-\hat{b}_{11})+\hat{a}_{11} \hat{b}_{01}^2}{\hat{a}_{01} (\hat{a}_{10}+\hat{b}_{01})},\\{} & {} \hat{c}_{30}=\frac{\hat{a}_{10}(\hat{a}_{01}\hat{b}_{21}-\hat{a}_{21}\hat{b}_{01})+\hat{a}_{01} (\hat{a}_{30}\hat{b}_{01}-\hat{a}_{01}\hat{b}_{30})}{\hat{a}_{01}(\hat{a}_{10}+\hat{b}_{01})},\\ \end{aligned}$$
$$\begin{aligned}{} & {} \hat{c}_{21}=\frac{-3\hat{a}_{01}^{2}\hat{b}_{30}+\hat{a}_{01}\hat{b}_{01}(3\hat{a}_{30}-\hat{b}_{21}) +\hat{a}_{21}\hat{b}_{01}^{2}+\hat{a}_{10}(2\hat{a}_{01}\hat{b}_{21}-2\hat{a}_{21}\hat{b}_{01})}{\hat{a}_{01}(\hat{a}_{10}+\hat{b}_{01})},\\{} & {} \hat{c}_{12}=\frac{-3\hat{a}_{01}^{2}\hat{b}_{30}+\hat{a}_{01}\hat{b}_{01}(3\hat{a}_{30}-2\hat{b}_{21}) +2\hat{a}_{21}\hat{b}_{01}^{2}+\hat{a}_{10}(\hat{a}_{01}\hat{b}_{21}-\hat{a}_{21}\hat{b}_{01})}{\hat{a}_{01}(\hat{a}_{10}+\hat{b}_{01})},\\{} & {} \hat{c}_{03}=\frac{-\hat{a}_{01}^2 \hat{b}_{30}+\hat{a}_{01} \hat{b}_{01} (\hat{a}_{30}-\hat{b}_{21})+\hat{a}_{21} \hat{b}_{01}^2}{\hat{a}_{01} (\hat{a}_{10}+\hat{b}_{01})};\\{} & {} \hat{d}_{20}=\frac{\hat{a}_{01}^2 \hat{b}_{20}+\hat{a}_{10} \hat{a}_{01} (\hat{a}_{20}-\hat{b}_{11})-\hat{a}_{10}^2 \hat{a}_{11}}{\hat{a}_{01} (\hat{a}_{10}+\hat{b}_{01})},\\{} & {} \hat{d}_{11}=\frac{\hat{a}_{10}(\hat{a}_{11}\hat{b}_{01}+\hat{a}_{01}(2\hat{a}_{20}-\hat{b}_{11})) +\hat{a}_{01}(2\hat{a}_{01}\hat{b}_{20}+\hat{b}_{01}\hat{b}_{11})-\hat{a}_{10}^{2}\hat{a}_{11}}{\hat{a}_{01}(\hat{a}_{10}+\hat{b}_{01})},\\{} & {} \hat{d}_{02}=\frac{\hat{a}_{01}^{2}\hat{b}_{20}+\hat{a}_{01}(\hat{a}_{10}\hat{a}_{20}+\hat{b}_{01} \hat{b}_{11})+\hat{a}_{10}\hat{a}_{11}\hat{b}_{01}}{\hat{a}_{01}(\hat{a}_{10}+\hat{b}_{01})},\\{} & {} \hat{d}_{30}=\frac{\hat{a}_{01}^2 \hat{b}_{30}+\hat{a}_{10} \hat{a}_{01} (\hat{a}_{30}-\hat{b}_{21})-\hat{a}_{10}^2 \hat{a}_{21}}{\hat{a}_{01} (\hat{a}_{10}+\hat{b}_{01})},\\{} & {} \hat{d}_{21}=\frac{\hat{a}_{10}(\hat{a}_{21}\hat{b}_{01}+\hat{a}_{01}(3 \hat{a}_{30}-2\hat{b}_{21})) +\hat{a}_{01}(3\hat{a}_{01}\hat{b}_{30}+\hat{b}_{01}\hat{b}_{21})-2\hat{a}_{21}\hat{a}_{10}^{2}}{\hat{a}_{01}(\hat{a}_{10}+\hat{b}_{01})},\\{} & {} \hat{d}_{12}=\frac{\hat{a}_{10}(2\hat{a}_{21}\hat{b}_{01}+\hat{a}_{01}(3\hat{a}_{30}-\hat{b}_{21})) +\hat{a}_{01}(3\hat{a}_{01}\hat{b}_{30}+2\hat{b}_{01}\hat{b}_{21})-\hat{a}_{10}^{2}\hat{a}_{21}}{\hat{a}_{01}(\hat{a}_{10}+\hat{b}_{01})},\\{} & {} \hat{d}_{03}=\frac{\hat{a}_{01}^{2}\hat{b}_{30}+\hat{a}_{01}(\hat{a}_{10}\hat{a}_{30}+\hat{b}_{01} \hat{b}_{21})+\hat{a}_{10}\hat{a}_{21}\hat{b}_{01}}{\hat{a}_{01}(\hat{a}_{10}+\hat{b}_{01})}.\\ \end{aligned}$$
$$\begin{aligned}{} & {} a_{10}^{*}=-\frac{\delta (2x^{*}-1)}{\delta -2x^{*}+1},\ a_{01}^{*}=-\frac{(\delta -1)\delta +2x^{*}(2x^{*}-1)}{(\delta -2x^{*}+1) (\delta +4x^{*}-2)},\\{} & {} a_{20}^{*}=\frac{\delta (\delta +2 x^{*}-1) (6 x^{*2}-(\delta +5) x^{*}+1)}{x^{*} (\delta -2 x^{*}+1)^2 (\delta +4 x^{*}-2)},\\{} & {} a_{11}^{*}=\frac{2(x^{*}(6 x^{*}-\delta -5)+1)((\delta -1)\delta +2 x^{*} (2 x^{*}-1))}{x^{*}(\delta -2 x^{*}+1)^{2}(\delta +4 x^{*}-2)^{2}},\\{} & {} a_{30}^{*}=-\frac{(1-2 x^{*})^{2}((\delta -1)\delta +2 x^{*}(2 x^{*}-1))^{2}}{x^{*2}(\delta -2 x^{*}+1)^{3}(\delta +4 x^{*}-2)^{2}},\\{} & {} a_{21}^{*}=-\frac{2(x^{*}(6 x^{*}-\delta -5)+1)((\delta -1)\delta +2 x^{*} (2 x^{*}-1))^{2}}{x^{*2}(\delta -2 x^{*}+1)^{3}(\delta +4 x^{*}-2)^{3}},\\{} & {} a_{40}^{*}=\frac{(1-2 x^{*})^2 ((\delta -1) \delta +2 x^{*} (2x^{*}-1))^3}{x^{*3}(\delta -2x^{*}+1)^{4}(\delta +4x^{*}-2)^{3}},\\{} & {} a_{31}^{*}=\frac{2(x^{*}(6 x^{*}-\delta -5)+1)((\delta -1)\delta +2 x^{*} (2 x^{*}-1))^{3}}{x^{*3}(\delta -2 x^{*}+1)^{4}(\delta +4 x^{*}-2)^{4}};\\{} & {} b_{10}^{*}=\frac{(\delta +4 x^{*}-2)(\delta -2 \delta x^{*})^{2}}{(\delta -2 x^{*}+1) ((\delta -1)\delta +2 x^{*}(2 x^{*}-1))},\ b_{01}^{*}=\frac{\delta (2 x^{*}-1)}{\delta -2 x^{*}+1},\\{} & {} b_{20}^{*}=-\frac{(\delta -2 \delta x^{*})^{2}}{x^{*}(\delta -2 x^{*}+1)^{2}},\ b_{11}^{*}=\frac{2\delta ^{2}(x^{*}(\delta -6 x^{*}+5)-1)}{x^{*}(\delta -2 x^{*}+1)^{2} (\delta +4x^{*}-2)},\\ \end{aligned}$$
$$\begin{aligned}{} & {} b_{30}^{*}=\frac{(\delta -2 \delta x^{*})^{2}((\delta -1)\delta +2 x^{*}(2x^{*}-1))}{x^{*2}(\delta -2x^{*}+1)^{3}(\delta +4x^{*}-2)},\\{} & {} b_{21}=-\frac{2\delta ^{2}((\delta -1)\delta +2 x^{*}(2 x^{*}-1))(x^{*} (\delta -6 x^{*}+5)-1)}{x^{*2}(\delta -2 x^{*}+1)^{3}(\delta +4 x^{*}-2)^{2}},\\{} & {} b_{40}^{*}=-\frac{\delta ^{2}(1-2 x^{*})^{2}((\delta -1)\delta +2 x^{*}(2 x^{*}-1))^2}{x^{*3}(\delta -2 x^{*}+1)^{4}(\delta +4 x^{*}-2)^{2}},\\{} & {} b_{31}^{*}=\frac{2\delta ^{2}((\delta -1)\delta +2 x^{*}(2 x^{*}-1))^{2}(x^{*} (\delta -6 x^{*}+5)-1)}{x^{*3}(\delta -2 x^{*}+1)^{4}(\delta +4 x^{*}-2)^{3}};\\{} & {} c_{20}^{*}=-a_{20}^{*}b_{01}^{*}+a_{11}^{*}b_{10}^{*}-a_{10}^{*}b_{11}^{*}+a_{01}^{*}b_{20}^{*},\ c_{02}^{*}=\frac{a_{11}^{*}}{a_{01}^{*}},\\{} & {} c_{30}^{*}=-a_{30}^{*}b_{01}^{*}+a_{21}^{*}b_{10}^{*}-a_{20}^{*}b_{11}^{*}+a_{11}^{*}b_{20}^{*} -a_{10}^{*}b_{21}^{*}+a_{01}^{*}b_{30}^{*},\\{} & {} c_{21}^{*}=\frac{a_{10}^{*}(a_{11}^{*2}-2a_{01}^{*}a_{21}^{*})-a_{01}^{*}a_{11}^{*}a_{20}^{*}}{a_{01}^{*2}}+3a_{30}^{*}+b_{21}^{*},\ c_{12}^{*}=\frac{2a_{01}^{*}a_{21}^{*}-a_{11}^{*2}}{a_{01}^{*2}},\\{} & {} c_{40}^{*}=-a_{40}^{*}b_{01}^{*}+a_{31}^{*}b_{10}^{*}-a_{30}^{*}b_{11}^{*}+a_{21}^{*}b_{20}^{*} -a_{20}^{*} b_{21}^{*}+a_{11}^{*}b_{30}^{*}-a_{10}^{*}b_{31}^{*}+a_{01}^{*}b_{40}^{*}, \\{} & {} c_{31}^{*}=\frac{1}{a_{01}^{*3}}\{a_{01}^{*}(a_{01}^{*}(a_{01}^{*}(4a_{40}^{*} +b_{31}^{*})-2 a_{20}^{*}a_{21}^{*})+a_{20}^{*}a_{11}^{*2}-a_{01}^{*}a_{30}^{*}a_{11}^{*})\\{} & {} \qquad -a_{10}^{*}(a_{11}^{*3}-3 a_{01}^{*}a_{21}^{*}a_{11}^{*}\\{} & {} \qquad +3a_{01}^{*2}a_{31}^{*})\},\\{} & {} c_{22}^{*}=\frac{a_{11}^{*3}-3a_{01}^{*}a_{21}^{*}a_{11}^{*}+3a_{01}^{*2}a_{31}^{*}}{a_{01}^{*3}}. \end{aligned}$$

Coefficients in the Proof of Theorem 2 and Theorem 3

$$\begin{aligned}{} & {} \bar{a}_{00}=-\lambda _{1},\ \bar{a}_{10}=\frac{\delta (1-2x^{*})}{\delta -2 x^{*}+1},\ \bar{a}_{01}=-\frac{x^{*}}{\alpha +x^{*}},\ \bar{a}_{20}=\frac{(1-2x^{*})^{2}}{(\alpha +x^{*})(\delta -2x^{*}+1)}-1,\\{} & {} \bar{a}_{11}=-\frac{\alpha }{(\alpha +x^{*})^{2}};\ \bar{b}_{00}=-\lambda _{2},\ \bar{b}_{10}=\frac{(\alpha +x^{*})(\delta -2\delta x^{*})^{2}}{x^{*}(\delta -2x^{*}+1)^2},\ \bar{b}_{01}=\frac{\delta (2 x^{*}-1)}{\delta -2x^{*}+1},\\{} & {} \bar{b}_{20}=-\frac{(\delta -2\delta x^{*})^{2}}{x^{*}(\delta -2x^{*}+1)^{2}},\ \bar{b}_{11}=\frac{\alpha \delta ^{2}}{x^{*}(\alpha +x^{*})(\delta -2x^{*}+1)};\\{} & {} \bar{c}_{00}=\bar{a}_{01}\bar{b}_{00}-\bar{a}_{00}\bar{b}_{01},\ \bar{c}_{10}=\bar{a}_{11}\bar{b}_{00}-\bar{a}_{10}\bar{b}_{01}+\bar{a}_{01}\bar{b}_{10}-\bar{a}_{00} \bar{b}_{11},\\{} & {} \bar{c}_{01}=\bar{a}_{10}-\frac{\bar{a}_{00}\bar{a}_{11}}{\bar{a}_{01}}+\bar{b}_{01},\, \bar{c}_{20}=-\bar{a}_{20}\bar{b}_{01}+\bar{a}_{11}\bar{b}_{10}-\bar{a}_{10}\bar{b}_{11}+\bar{a}_{01} \bar{b}_{20},\\{} & {} \bar{c}_{11}=\frac{\bar{a}_{11}(\bar{a}_{00}\bar{a}_{11}-\bar{a}_{01}\bar{a}_{10})}{\bar{a}_{01}^{2}}+2 \bar{a}_{20}+\bar{b}_{11}, \ \bar{c}_{02}=\frac{\bar{a}_{11}}{\bar{a}_{01}};\ \bar{d}_{00}=\bar{c}_{00},\ \bar{d}_{10}=\bar{c}_{10}-2\bar{c}_{00}\bar{c}_{02},\\ \end{aligned}$$
$$\begin{aligned}{} & {} \bar{d}_{01}=\bar{c}_{01},\ \bar{d}_{20}=\bar{c}_{00}\bar{c}_{02}^{2}-2\bar{c}_{10}\bar{c}_{02}+\bar{c}_{20},\\{} & {} \bar{d}_{11}=\bar{c}_{11}-\bar{c}_{01}\bar{c}_{02};\ \bar{e}_{00}=\bar{d}_{00}-\frac{\bar{d}_{10}^{2}}{4\bar{d}_{20}},\ \bar{e}_{01}=\bar{d}_{01}-\frac{\bar{d}_{10}\bar{d}_{11}}{2\bar{d}_{20}},\ \bar{e}_{20}=\bar{d}_{20}, \bar{e}_{11}=\bar{d}_{11}.\\{} & {} \bar{a}_{00}^{*}=-\epsilon _{1},\ \bar{a}_{10}^{*}=\frac{\delta (1-2x^{*})}{\delta -2x^{*}+1},\ \bar{a}_{01}^{*}=-\frac{(\delta -1)\delta +2x^{*}(2x^{*}-1)}{(\delta -2x^{*}+1) (\delta +4x^{*}-2)},\\{} & {} \bar{a}_{20}^{*}=\frac{\delta (\delta +2 x^{*}-1) (6 x^{*2}-(\delta +5) x^{*}+1)}{x^{*} (\delta -2 x^{*}+1)^2 (\delta +4 x^{*}-2)},\\{} & {} \bar{a}_{11}^{*}=\frac{2(x^{*}(6 x^{*}-\delta -5)+1)((\delta -1)\delta +2 x^{*} (2 x^{*}-1))}{x^{*}(\delta -2 x^{*}+1)^{2}(\delta +4 x^{*}-2)^{2}},\\{} & {} \bar{a}_{30}^{*}=-\frac{(1-2 x^{*})^{2}((\delta -1)\delta +2 x^{*}(2 x^{*}-1))^2}{x^{*2}(\delta -2 x^{*}+1)^{3}(\delta +4 x^{*}-2)^{2}},\\{} & {} \bar{a}_{21}^{*}=-\frac{2(x^{*}(6 x^{*}-\delta -5)+1)((\delta -1)\delta +2 x^{*} (2 x^{*}-1))^{2}}{x^{*2}(\delta -2 x^{*}+1)^{3}(\delta +4 x^{*}-2)^{3}},\\{} & {} \bar{a}_{40}^{*}=\frac{(1-2 x^{*})^{2}((\delta -1)\delta +2 x^{*}(2 x^{*}-1))^3}{x^{*3}(\delta -2 x^{*}+1)^{4}(\delta +4 x^{*}-2)^{3}},\\{} & {} \bar{a}_{31}^{*}=\frac{2( x^{*}(6 x^{*}-\delta -5)+1)((\delta -1)\delta +2 x^{*} (2 x^{*}-1))^{3}}{x^{*3}(\delta -2 x^{*}+1)^{4}(\delta +4 x^{*}-2)^{4}};\\{} & {} \bar{b}_{00}^{*}=-\frac{ x^{*} (1-2 x^{*})^2 (\delta +4 x^{*}-2)}{12 x^{*2}-2 (\delta +5) x^{*}+2} \epsilon _2-\epsilon _3,\\{} & {} \bar{b}_{10}^{*}=-\frac{(1-2 x^{*})^2 (\epsilon _2 ((\delta -1) \delta +4 x^{*2}-2 x^{*})+\delta ^2 (\delta +4 x^{*}-2))}{(-\delta +2 x^{*}-1) ((\delta -1) \delta +4 x^{*2}-2 x^{*})},\\{} & {} \bar{b}_{01}^{*}=\frac{\epsilon _{2}((\delta -1)\delta +2 x^{*}(2 x^{*}-1))+\delta (2 x^{*}-1)(\delta +4 x^{*}-2)}{(\delta -2 x^{*}+1)(\delta +4 x^{*}-2)},\\{} & {} \bar{b}_{20}^{*}=-\frac{(1-2 x^{*})^2 (\epsilon _2 ((\delta -1) \delta +(2 x^{*}-1)2 x^{*})+\delta ^2 (\delta +4 x^{*}-2))}{x^{*} (\delta -2 x^{*}+1)^2 (\delta +4 x^{*}-2)},\\{} & {} \bar{b}_{11}^{*}=-\frac{2(x^{*}(6 x^{*}-\delta -5)+1) (\delta ^{2}(\delta +4 x^{*}-2)+\epsilon _{2}((\delta -1)\delta +2 x^{*}(2 x^{*}-1)))}{x^{*}(\delta -2 x^{*}+1)^{2} (\delta +4 x^{*}-2)^{2}},\\ \end{aligned}$$
$$\begin{aligned}{} & {} \bar{b}_{30}^{*}=\frac{(1{-}2 x^{*})^{2}((\delta {-}1)\delta +2 x^{*}(2 x^{*}{-}1)) (\delta ^{2}(\delta +4 x^{*}{-}2){+}\epsilon _{2}((\delta {-}1)\delta {+}2 x^{*}(2 x^{*}{-}1)))}{x^{*2}(\delta -2 x^{*}+1)^{3}(\delta +4 x^{*}-2)^{2}},\\{} & {} \bar{b}_{21}^{*}=\frac{1}{x^{*2}(\delta -2 x^{*}+1)^{3}(\delta +4 x^{*}-2)^{3}} (2(x^{*}(6 x^{*}-\delta -5)+1))((\delta -1)\delta \\{} & {} \quad \qquad +2 x^{*} (2 x^{*}-1)) (\delta ^{2} (\delta +4 x^{*}-2)+\epsilon _{2}((\delta -1)\delta +2 x^{*}(2 x^{*}-1))),\\{} & {} \bar{b}_{40}^{*}=\frac{(1{-}2 x^{*})^2 ((\delta {-}1) \delta {+}2 x^{*} (2 x^{*}-1))^2 (\epsilon _2 ((\delta {-}1) \delta {+}2 x^{*} (2 x^{*}{-}1)){+}\delta ^2 (\delta {+}4 x^{*}{-}2))}{x^{*3} (\delta -2 x^{*}+1)^4 (2-\delta -4 x^{*})^3},\\{} & {} \bar{b}_{31}^{*}=-\frac{1}{x^{*3} (\delta -2 x^{*}+1)^4 (\delta +4 x^{*}-2)^4} 2 ((\delta -1) \delta +2x^{*} (2x^{*}-1))^2 \\{} & {} \quad \qquad (6 x^{*2}-(\delta +5) x^{*}+1) (\epsilon _2 ((\delta -1) \delta +2x^{*} (2x^{*}-1))+\delta ^2 (\delta +4 x^{*}-2));\\{} & {} \bar{c}_{00}^{*}=\bar{a}^{*}_{01} \bar{b}^{*}_{00}-\bar{a}^{*}_{00} \bar{b}^{*}_{01},\ \bar{c}_{10}^{*}=\bar{a}^{*}_{11} \bar{b}^{*}_{00}-\bar{a}^{*}_{10} \bar{b}^{*}_{01} +\bar{a}^{*}_{01} \bar{b}^{*}_{10}-\bar{a}^{*}_{00} \bar{b}^{*}_{11},\\{} & {} \bar{c}_{01}^{*}=\frac{\bar{a}^{*}_{01} (\bar{a}^{*}_{10}+\bar{b}^{*}_{01})-\bar{a}^{*}_{00} \bar{a}^{*}_{11}}{\bar{a}^{*}_{01}},\\{} & {} \bar{c}_{20}^{*}=\bar{a}^{*}_{21} \bar{b}^{*}_{00}-\bar{a}^{*}_{20} \bar{b}^{*}_{01}+\bar{a}^{*}_{11} \bar{b}^{*}_{10}-\bar{a}^{*}_{10} \bar{b}^{*}_{11}+\bar{a}^{*}_{01} \bar{b}^{*}_{20}-\bar{a}^{*}_{00} \bar{b}^{*}_{21},\\{} & {} \bar{c}_{11}^{*}=2 \bar{a}^{*}_{20}+\frac{\bar{a}^{*}_{00} \bar{a}^{*2}_{11}-\bar{a}^{*}_{01} (\bar{a}^{*}_{10} \bar{a}^{*}_{11}+2 \bar{a}^{*}_{00} \bar{a}^{*}_{21})}{\bar{a}^{*2}_{01}}+\bar{b}^{*}_{11},\ \bar{c}_{02}^{*}=\frac{\bar{a}^{*}_{11}}{\bar{a}^{*}_{01}},\\{} & {} \bar{c}_{30}^{*}=\bar{a}^{*}_{31} \bar{b}^{*}_{00}-\bar{a}^{*}_{30} \bar{b}^{*}_{01}+\bar{a}^{*}_{21} \bar{b}^{*}_{10}-\bar{a}^{*}_{20} \bar{b}^{*}_{11}+\bar{a}^{*}_{11} \bar{b}^{*}_{20}-\bar{a}^{*}_{10} \bar{b}^{*}_{21}+\bar{a}^{*}_{01} \bar{b}^{*}_{30}-\bar{a}^{*}_{00} \bar{b}^{*}_{31},\\{} & {} \bar{c}_{21}^{*}=3 \bar{a}^{*}_{30}+\frac{-\bar{a}^{*}_{00} \bar{a}^{*3}_{11}+\bar{a}^{*}_{01} (\bar{a}^{*}_{10} \bar{a}^{*}_{11}+3 \bar{a}^{*}_{00} \bar{a}^{*}_{21}) \bar{a}^{*}_{11}-\bar{a}^{*2}_{01} (\bar{a}^{*}_{11} \bar{a}^{*}_{20}+2 \bar{a}^{*}_{10} \bar{a}^{*}_{21}+3 \bar{a}^{*}_{00}\bar{a}^{*}_{31})}{\bar{a}^{*3}_{01}}\\{} & {} \quad \qquad +\bar{b}^{*}_{21},\\ \end{aligned}$$
$$\begin{aligned}{} & {} \bar{c}_{12}^{*}=\frac{2 \bar{a}^{*}_{01} \bar{a}^{*}_{21}-\bar{a}^{*2}_{11}}{\bar{a}^{*2}_{01}},\\{} & {} \bar{c}_{40}^{*}=-\bar{a}^{*}_{40} \bar{b}^{*}_{01}+\bar{a}^{*}_{31} \bar{b}^{*}_{10}-\bar{a}^{*}_{30} \bar{b}^{*}_{11}+\bar{a}^{*}_{21} \bar{b}^{*}_{20}-\bar{a}^{*}_{20} \bar{b}^{*}_{21}+\bar{a}^{*}_{11} \bar{b}^{*}_{30}-\bar{a}^{*}_{10} \bar{b}^{*}_{31}+\bar{a}^{*}_{01} \bar{b}^{*}_{40},\\{} & {} \bar{c}_{31}^{*}=\frac{1}{\bar{a}^{*4}_{01}}(\bar{a}^{*}_{00} \bar{a}^{*4}_{11}-\bar{a}^{*}_{01} \bar{a}^{*2}_{11}(\bar{a}^{*}_{10} \bar{a}^{*}_{11}+4 \bar{a}^{*}_{00} \bar{a}^{*}_{21}) -\bar{a}^{*3}_{01} (2 \bar{a}^{*}_{20} \bar{a}^{*}_{21}+\bar{a}^{*}_{11} \bar{a}^{*}_{30}+3 \bar{a}^{*}_{10} \bar{a}^{*}_{31})\\{} & {} \quad \qquad +\bar{a}^{*2}_{01} (\bar{a}^{*}_{20} \bar{a}^{*2}_{11}+ (3 \bar{a}^{*}_{10} \bar{a}^{*}_{21}+4 \bar{a}^{*}_{00} \bar{a}^{*}_{31}) \bar{a}^{*}_{11}+2 \bar{a}^{*}_{00} \bar{a}^{*2}_{21}))+4 \bar{a}^{*}_{40}+\bar{b}^{*}_{31},\\{} & {} \bar{c}_{22}^{*}=\frac{\bar{a}^{*3}_{11}-3 \bar{a}^{*}_{01} \bar{a}^{*}_{21} \bar{a}^{*}_{11}+3 \bar{a}^{*2}_{01} \bar{a}^{*}_{31}}{\bar{a}^{*3}_{01}};\ \bar{d}_{10}^{*}=\bar{c}_{10}^{*}-\bar{c}_{00}^{*} \bar{c}_{02}^{*},\ \bar{d}_{01}^{*}=\bar{c}_{01}^{*},\\{} & {} \bar{d}_{20}^{*}=\bar{c}_{20}^{*}+\bar{c}_{00}^{*} \bar{c}_{02}^{*2}-\frac{\bar{c}_{10}^{*} \bar{c}_{02}^{*}}{2},\ \bar{d}_{11}^{*}=\bar{c}_{11}^{*},\ \bar{d}_{30}^{*}=\bar{c}_{30}^{*}+\frac{1}{2} (\bar{c}_{10}^{*}-2 \bar{c}_{00}^{*} \bar{c}_{02}^{*}) \bar{c}_{02}^{*2},\\{} & {} \bar{d}_{21}^{*}=\bar{c}_{21}^{*}+\frac{\bar{c}_{02}^{*} \bar{c}_{11}^{*}}{2},\ \bar{d}_{12}^{*}=\bar{c}_{12}^{*}+2 \bar{c}_{02}^{*2},\ \bar{d}_{40}^{*}=\bar{c}_{40}^{*}+\bar{c}_{00}^{*} \bar{c}_{02}^{*4}+\frac{1}{4} (\bar{c}_{02}^{*} (\bar{c}_{20}^{*}\\{} & {} \quad \qquad -2 \bar{c}_{02}^{*} \bar{c}_{10}^{*})+2 \bar{c}_{30}^{*}) \bar{c}_{02}^{*},\\{} & {} \bar{d}_{31}^{*}=\bar{c}_{31}^{*}+\bar{c}_{02}^{*} \bar{c}_{21}^{*},\ \bar{d}_{22}^{*}=\bar{c}_{22}^{*}-\bar{c}_{02}^{*3}+\frac{3 \bar{c}_{12}^{*} \bar{c}_{02}^{*}}{2};\ \bar{e}_{00}^{*}=\bar{d}_{00}^{*},\ \bar{e}_{10}^{*}=\bar{d}_{10}^{*},\ \bar{e}_{01}^{*}=\bar{d}_{01}^{*},\\{} & {} \bar{e}_{20}^{*}=\bar{d}_{20}^{*}-\frac{\bar{d}_{00}^{*} \bar{d}_{12}^{*}}{2},\ \bar{e}_{11}^{*}=\bar{d}_{11}^{*},\ \bar{e}_{30}^{*}=\bar{d}_{30}^{*}-\frac{\bar{d}_{10}^{*} \bar{d}_{12}^{*}}{3},\ \bar{e}_{21}^{*}=\bar{d}_{21}^{*},\\{} & {} \bar{e}_{40}^{*}=\bar{d}_{40}^{*}+\frac{\bar{d}_{00}^{*} \bar{d}_{12}^{*2}}{4} -\frac{\bar{d}_{12}^{*} \bar{d}_{20}^{*}}{6},\ \bar{e}_{31}^{*}=\bar{d}_{31}^{*}+\frac{\bar{d}_{11}^{*} \bar{d}_{12}^{*}}{6},\ \bar{e}_{22}^{*}=\bar{d}_{22}^{*}; \bar{f}_{00}^{*}=\bar{e}_{00}^{*},\ \bar{f}_{10}^{*}=\bar{e}_{10}^{*},\\{} & {} \bar{f}_{01}^{*}=\bar{e}_{01}^{*},\ \bar{f}_{20}^{*}=\bar{e}_{20}^{*},\ \bar{f}_{11}^{*}=\bar{e}_{11}^{*}, \ \bar{f}_{30}^{*}=\bar{e}_{30}^{*}-\frac{\bar{e}_{00}^{*} \bar{e}_{22}^{*}}{3},\ \bar{f}_{21}^{*}=\bar{e}_{21}^{*},\ \bar{f}_{40}^{*}=\bar{e}_{40}^{*}-\frac{\bar{e}_{10}^{*} \bar{e}_{22}^{*}}{4},\\{} & {} \bar{f}_{31}^{*}=\bar{e}_{31}^{*};\ \bar{g}_{00}^{*}=\bar{f}_{00}^{*},\ \bar{g}_{10}^{*}=\bar{f}_{10}^{*}-\frac{\bar{f}_{00}^{*} \bar{f}_{30}^{*}}{2\bar{f}_{20}^{*}},\ \bar{g}_{01}^{*}=\bar{f}_{01}^{*},\\ \end{aligned}$$
$$\begin{aligned}{} & {} \bar{g}_{20}^{*}=\bar{f}_{20}^{*}+\frac{9 \bar{f}_{00}^{*} \bar{f}_{30}^{*2}}{16 \bar{f}_{20}^{*2}}-\frac{3 (5 \bar{f}_{10}^{*} \bar{f}_{30}^{*}+4 \bar{f}_{00}^{*} \bar{f}_{40}^{*})}{20 \bar{f}_{20}^{*}},\ \bar{g}_{11}^{*}=\bar{f}_{11}^{*}-\frac{\bar{f}_{01}^{*}\bar{f}_{30}^{*}}{2\bar{f}_{20}^{*}},\\{} & {} \bar{g}_{30}^{*}=\frac{\bar{f}_{10}^{*}(35\bar{f}_{30}^{*2} -32\bar{f}_{20}^{*}\bar{f}_{40}^{*})}{40\bar{f}_{20}^{*2}},\ \bar{g}_{21}^{*}=\bar{f}_{21}^{*}-\frac{3 (20 \bar{f}_{11}^{*} \bar{f}_{20}^{*} \bar{f}_{30}^{*}+\bar{f}_{01}^{*} (16 \bar{f}_{20}^{*} \bar{f}_{40}^{*}-15 \bar{f}_{30}^{*2}))}{80\bar{f}_{20}^{*2}},\\{} & {} \bar{g}_{40}^{*}=\frac{\bar{f}_{10}^{*} \bar{f}_{30}^{*} (16 \bar{f}_{20}^{*} \bar{f}_{40}^{*}-15 \bar{f}_{30}^{*2})}{64 \bar{f}_{20}^{*3}},\ \bar{g}_{31}^{*}=\bar{f}_{31}^{*}+\frac{7\bar{f}_{11}^{*}\bar{f}_{30}^{*2}}{8\bar{f}_{20}^{*2}} -\frac{5\bar{f}_{21}^{*}\bar{f}_{30}^{*}+4\bar{f}_{11}^{*}\bar{f}_{40}^{*}}{5\bar{f}_{20}^{*}};\\{} & {} \bar{h}_{00}^{*}=\bar{g}_{00}^{*},\ \bar{h}_{10}^{*}=\bar{g}_{10}^{*},\ \bar{h}_{01}^{*}=\bar{g}_{01}^{*}-\frac{\bar{g}_{00}^{*}\bar{g}_{21}^{*}}{\bar{g}_{20}^{*}},\ \bar{h}_{20}^{*}=\bar{g}_{20}^{*},\ \bar{h}_{11}^{*}=\bar{g}_{11}^{*}-\frac{\bar{g}_{10}^{*}\bar{g}_{21}^{*}}{\bar{g}_{20}^{*}},\\{} & {} \bar{h}_{31}^{*}=\bar{g}_{31}^{*}-\frac{\bar{g}_{21}^{*}\bar{g}_{30}^{*}}{\bar{g}_{20}^{*}};\bar{j}_{00}^{*}=\bar{h}_{00}^{*} \bar{h}_{31}^{*\frac{4}{5}} \bar{h}_{20}^{*-\frac{7}{5}},\ \bar{j}_{10}^{*}=\bar{h}_{10}^{*} \bar{h}_{31}^{*\frac{2}{5}} \bar{h}_{20}^{*-\frac{6}{5}}, \ \bar{j}_{01}^{*}=\bar{h}_{01}^{*} \bar{h}_{31}^{*\frac{1}{5}} \bar{h}_{20}^{*-\frac{3}{5}},\\{} & {} \bar{j}_{11}^{*}=\bar{h}_{11}^{*} \bar{h}_{31}^{*-\frac{1}{5}} \bar{h}_{20}^{*-\frac{2}{5}}. \end{aligned}$$

Coefficients in the Proof of Theorem 5

$$\begin{aligned} \sigma _{22}= & {} 20 (\alpha +\tilde{x})^4 (\alpha h-\tilde{x}^2 (\alpha +2 \tilde{x}-1))^7-20 \tilde{x} (\alpha +\tilde{x})^3 (\alpha h-\tilde{x}^2 (\alpha +2 \tilde{x}-1))^6 \\{} & {} \times (2 \alpha h-19 \tilde{x}^3 +(7-22 \alpha ) \tilde{x}^2+\alpha \tilde{x} (-5 \alpha -2 \beta +5))\\{} & {} +(\alpha +\tilde{x})^4 (-5 \alpha h-2 \tilde{x}^3) (\alpha \beta \tilde{x}^2 (h+(\tilde{x}-1) \tilde{x})+ (\alpha h-\tilde{x}^2 (\alpha +2 \tilde{x}-1))^2)^3\\{} & {} -180 \tilde{x}^3 (\alpha +\tilde{x})^3 (\alpha +3 \tilde{x}-1) (-\alpha h+2 \tilde{x}^3+(\alpha -1) \tilde{x}^2)^5\\{} & {} \times (-\alpha h+5 \tilde{x}^3+(5 \alpha -2) \tilde{x}^2+\alpha \tilde{x} (\alpha +\beta -1))\\{} & {} +20 \tilde{x}^3 (\alpha +\tilde{x}) \left( -\alpha h+2 \tilde{x}^3+(\alpha -1) \tilde{x}^2\right) ^4 (2 \alpha h+17 \tilde{x}^3+(26 \alpha -5) \tilde{x}^2\\{} & {} +\alpha \tilde{x} (7 \alpha -2 \beta -7)) (-\alpha h+5 \tilde{x}^3+(5 \alpha -2) \tilde{x}^2+\alpha \tilde{x} (\alpha +\beta -1))^2\\{} & {} -20 \tilde{x}^4 (-\alpha h+2 \tilde{x}^3+(\alpha -1) \tilde{x}^2)^3 (\alpha h+4 \tilde{x}^3+(7 \alpha -1) \tilde{x}^2\\{} & {} +\alpha \tilde{x} (2 \alpha -\beta -2)) (-\alpha h+5 \tilde{x}^3+(5 \alpha -2) \tilde{x}^2+\alpha \tilde{x} (\alpha +\beta -1))^3\\{} & {} +(\alpha +\tilde{x})^2 (\alpha \beta \tilde{x}^2 (h+(\tilde{x}-1) \tilde{x})+(\alpha h- \tilde{x}^2 (\alpha +2 \tilde{x}-1))^2)^2\\{} & {} \times (30 (\alpha +\tilde{x})^2 (\alpha h-\tilde{x}^2 (\alpha +2 \tilde{x}-1))^3+10 \tilde{x} (\alpha +\tilde{x})\\{} & {} \times (-\alpha h+2 \tilde{x}^3+(\alpha -1) \tilde{x}^2)^2 (-\alpha h+23 \tilde{x}^3+(30 \alpha -9) \tilde{x}^2\\{} & {} +\alpha \tilde{x} (8 \alpha +\beta -8))+\tilde{x}^3 (\alpha +\tilde{x}) (\alpha h-\tilde{x}^2 (\alpha +2 \tilde{x} -1)) (-10 (\alpha -1) \alpha h\\{} & {} +2 \alpha \tilde{x} (5 (\alpha -1) (7 \alpha +\beta -7)-24 h)+510 \tilde{x}^4+(821 \alpha -407) \tilde{x}^3\\{} & {} + \tilde{x}^2 (419 \alpha ^2+\alpha (48 \beta -499)+80))+(1-\alpha ) \tilde{x}^4\\{} & {} \times (\alpha h+13 \tilde{x}^3+(19 \alpha -4) \tilde{x}^2+\alpha \tilde{x} (5 \alpha -\beta -5))\\{} & {} \times (\alpha h-14 \tilde{x}^3+(5-17 \alpha ) \tilde{x}^2-\alpha \tilde{x} (4 \alpha +\beta -4)))\\{} & {} -(\alpha h-\tilde{x}^2 (\alpha +2 \tilde{x}-1)) (\alpha ^2 h+3\tilde{x}^4+(2 \alpha -1) \tilde{x}^3\\{} & {} +\alpha \beta \tilde{x}^2) (\alpha \beta \tilde{x}^2 (h+(\tilde{x}-1) \tilde{x})+(\alpha h-\tilde{x}^2 (\alpha +2 \tilde{x}-1))^2) (\tilde{x}^2 (\alpha +\tilde{x})\\{} & {} \quad \times (\alpha h-\tilde{x}^2 (\alpha +2 \tilde{x}-1)) (3 \tilde{x}^2 (\alpha +\tilde{x}) (7 \alpha h-7 \tilde{x} (\alpha \beta +\tilde{x} (\alpha +2 \tilde{x}-1))\\{} & {} -46 \tilde{x} (\alpha +\tilde{x}) (\alpha +3 \tilde{x}-1))+19 \tilde{x} (-\alpha -3 \tilde{x}+1) (\alpha +\tilde{x})\\{} & {} \times (\alpha h-\tilde{x} (\alpha \beta +\tilde{x} (\alpha +2 \tilde{x}-1)))-5 (\alpha h-\tilde{x} (\alpha \beta +\tilde{x} (\alpha +2 \tilde{x}-1)))^2\\{} & {} +120 \tilde{x}^2 (\alpha +\tilde{x})^2 (\alpha +3 \tilde{x}-1)^2)+45 (\alpha +\tilde{x})^3 (\alpha h-\tilde{x}^2 (\alpha +2 \tilde{x}-1))^3\\{} & {} +\tilde{x}^3 (\alpha h+4 \tilde{x}^3+(7 \alpha -1) \tilde{x}^2+\alpha \tilde{x} (2 \alpha -\beta -2)) \\{} & {} \times (5 (\alpha h-\tilde{x} (\alpha \beta +\tilde{x} (\alpha +2 \tilde{x}-1)))^2+19 \tilde{x} (-\alpha -3 \tilde{x}+1) \\{} & {} \times (\alpha +\tilde{x}) (\alpha h-\tilde{x} (\alpha \beta +\tilde{x} (\alpha +2 \tilde{x}-1)))+20 \tilde{x}^2 (\alpha +\tilde{x})^2 (\alpha +3 \tilde{x}-1)^2)\\{} & {} - \tilde{x} (\alpha +\tilde{x})^2 (\alpha h-\tilde{x}^2 (\alpha +2 \tilde{x}-1))^2 (5 \alpha h-301 \tilde{x}^3+(135-426 \alpha ) \tilde{x}^2\\{} & {} -5 \alpha \tilde{x} (26 \alpha +\beta -26))). \end{aligned}$$

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Yang, Y., Xu, Y., Meng, F. et al. Global Harvesting and Stocking Dynamics in a Modified Rosenzweig–MacArthur Model. Qual. Theory Dyn. Syst. 23, 196 (2024). https://doi.org/10.1007/s12346-024-01056-2

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