# American Institute of Mathematical Sciences

doi: 10.3934/era.2020084

## Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting

 Laboratory of mathematics and their interactions, University Abdelhafid Boussouf, Mila, 43000, Algeria

* Corresponding author: Abdelouahab Mohammed Salah

Received  February 2020 Revised  July 2020 Published  August 2020

The objective of the current paper is to investigate the dynamics of a new bioeconomic predator prey system with only predator's harvesting and Holling type Ⅲ response function. The system is equipped with an algebraic equation because of the economic revenue. We offer a detailed mathematical analysis of the proposed model to illustrate some of the significant results. The boundedness and positivity of solutions for the model are examined. Coexistence equilibria of the bioeconomic system have been thoroughly investigated and the behaviours of the model around them are described by means of qualitative theory of dynamical systems (such as local stability and Hopf bifurcation). The obtained results provide a useful platform to understand the role of the economic revenue $v$. We show that a positive equilibrium point is locally asymptotically stable when the profit $v$ is less than a certain critical value $v^{*}_1$, while a loss of stability by Hopf bifurcation can occur as the profit increases. It is evident from our study that the economic revenue has the capability of making the system stable (survival of all species). Finally, some numerical simulations have been carried out to substantiate the analytical findings.

Citation: Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, doi: 10.3934/era.2020084
##### References:
 [1] M.-S. Abdelouahab, N.-E. Hamri and J. Wang, Hopf bifurcation and chaos in fractional-order modified hybrid optical system, Nonlinear Dynamics, 69 (2012), 275-284.  doi: 10.1007/s11071-011-0263-4.  Google Scholar [2] M.-S. Abdelouahab and R. Lozi, Hopf-like bifurcation and mixed mode oscillation in a fractional-order FitzHugh-Nagumo model, AIP Conference Proceedings 2183, 100003, (2019). doi: 10.1063/1.5136214.  Google Scholar [3] G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, $4^th$ edition, John Wiley and Sons, Inc., New York, 1989.  Google Scholar [4] B. S. Chen, X. X. Liao and Y. Q. Liu, Normal forms and bifurcations for the differential-algebraic systems, Acta Math. Appl. Sinica, 23 (2000), 429-443.   Google Scholar [5] C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, $2^nd$ edition, John Wiley and Sons, New York, 1990.  Google Scholar [6] H. S. Gordon, The economic theory of a common property resource: The fishery, Journal of Political Economy, 62 (1954), 124-142.  doi: 10.1086/257497.  Google Scholar [7] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar [8] J. Hale, Theory of Functional Differential Equations, $2^nd$ edition, Applied Mathematical Sciences, Vol. 3. Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar [9] P. Henrici, Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Location of Zeros, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.  Google Scholar [10] C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Canadian Entomology, 91 (1959a), 293-320.  doi: 10.4039/Ent91293-5.  Google Scholar [11] C. S. Holling, Some characteristics of simple types of predation and parasitism, Canadian Entomology, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.  Google Scholar [12] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Memoirs of the Entomological Society of Canada, 97 (1965), 5-60.  doi: 10.4039/entm9745fv.  Google Scholar [13] T. K. Kar and K. Chakraborty, Effort dynamics in a prey-predator model with harvesting, Int. J. Inf. Syst. Sci., 6 (2010), 318-332.   Google Scholar [14] W. Liu, L. Biwen, C. Fu and B. Chen, Dynamics of a predator-prey ecological system with nonlinear harvesting rate, Wuhan Univ. J. Nat. Sci., 20 (2015), 25-33.  doi: 10.1007/s11859-015-1054-4.  Google Scholar [15] W. Liu, C. J. Fu and B. Chen, Hopf bifurcation for a predator-prey biological economic system with Holling type Ⅱ functional response, J. Franklin Inst., 348 (2011), 1114-1127.  doi: 10.1016/j.jfranklin.2011.04.019.  Google Scholar [16] C. Liu, Q. Zhang, Y. Zhang and X. D. Duan, Bifurcation and control in a differential-algebraic harvested prey-predator model with stage structure for predator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 3159-3168.  doi: 10.1142/S0218127408022329.  Google Scholar [17] W. Zhu, J. Huang and W. Liu, The stability and Hopf bifurcation of the differential-algebraic biological economic system with single harvesting, 2015 Sixth International Conference on Intelligent Control and information Processing (ICICIP), Wuhan, (2015), 92–97. doi: 10.1109/ICICIP.2015.7388150.  Google Scholar

show all references

##### References:
 [1] M.-S. Abdelouahab, N.-E. Hamri and J. Wang, Hopf bifurcation and chaos in fractional-order modified hybrid optical system, Nonlinear Dynamics, 69 (2012), 275-284.  doi: 10.1007/s11071-011-0263-4.  Google Scholar [2] M.-S. Abdelouahab and R. Lozi, Hopf-like bifurcation and mixed mode oscillation in a fractional-order FitzHugh-Nagumo model, AIP Conference Proceedings 2183, 100003, (2019). doi: 10.1063/1.5136214.  Google Scholar [3] G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, $4^th$ edition, John Wiley and Sons, Inc., New York, 1989.  Google Scholar [4] B. S. Chen, X. X. Liao and Y. Q. Liu, Normal forms and bifurcations for the differential-algebraic systems, Acta Math. Appl. Sinica, 23 (2000), 429-443.   Google Scholar [5] C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, $2^nd$ edition, John Wiley and Sons, New York, 1990.  Google Scholar [6] H. S. Gordon, The economic theory of a common property resource: The fishery, Journal of Political Economy, 62 (1954), 124-142.  doi: 10.1086/257497.  Google Scholar [7] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar [8] J. Hale, Theory of Functional Differential Equations, $2^nd$ edition, Applied Mathematical Sciences, Vol. 3. Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar [9] P. Henrici, Applied and Computational Complex Analysis, Vol. 1: Power Series-Integration-Conformal Mapping-Location of Zeros, Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974.  Google Scholar [10] C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Canadian Entomology, 91 (1959a), 293-320.  doi: 10.4039/Ent91293-5.  Google Scholar [11] C. S. Holling, Some characteristics of simple types of predation and parasitism, Canadian Entomology, 91 (1959), 385-398.  doi: 10.4039/Ent91385-7.  Google Scholar [12] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Memoirs of the Entomological Society of Canada, 97 (1965), 5-60.  doi: 10.4039/entm9745fv.  Google Scholar [13] T. K. Kar and K. Chakraborty, Effort dynamics in a prey-predator model with harvesting, Int. J. Inf. Syst. Sci., 6 (2010), 318-332.   Google Scholar [14] W. Liu, L. Biwen, C. Fu and B. Chen, Dynamics of a predator-prey ecological system with nonlinear harvesting rate, Wuhan Univ. J. Nat. Sci., 20 (2015), 25-33.  doi: 10.1007/s11859-015-1054-4.  Google Scholar [15] W. Liu, C. J. Fu and B. Chen, Hopf bifurcation for a predator-prey biological economic system with Holling type Ⅱ functional response, J. Franklin Inst., 348 (2011), 1114-1127.  doi: 10.1016/j.jfranklin.2011.04.019.  Google Scholar [16] C. Liu, Q. Zhang, Y. Zhang and X. D. Duan, Bifurcation and control in a differential-algebraic harvested prey-predator model with stage structure for predator, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 3159-3168.  doi: 10.1142/S0218127408022329.  Google Scholar [17] W. Zhu, J. Huang and W. Liu, The stability and Hopf bifurcation of the differential-algebraic biological economic system with single harvesting, 2015 Sixth International Conference on Intelligent Control and information Processing (ICICIP), Wuhan, (2015), 92–97. doi: 10.1109/ICICIP.2015.7388150.  Google Scholar
Number of the interior equilibria of system (4) versus the economic profit $v$, for $0< v \leq 5.$
The biological coordinates $x_e$, $y_e$ of the two interior equilibria $X_{e1}$ and $X_{e2}$ versus the economic profit $v$
Representation of the trace $Tr$ and the determinant $Det$ of the Jacobian matrix $A$ at the two interior equilibria $X_{e1}$ and $X_{e2}$ for $v\in I_v$
Representation of the discriminants of $X_{e1}$ and $X_{e2}$ and the trace $Tr(A(X_{e1}))$ versus the economic profit $v$
Time evolution of prey $x$, and the phase trajectory of the system (4) for $v = 0.955<v^{*}_1,$ showing stable behaviour of the first positive equilibrium point $X_{e1}(v)$ with the initial conditions $x_{0} = x_e+0.22,\; y_{0} = y_e,\; E_{0} = E_e,$ surrounded by the bifurcating unstable limit cycle $\gamma$ and an unstable behaviour in the exterior of $\gamma.$
Time evolution of species $x,\;y$, the harvest effort $E$ and phase portrait of the system (4), for $v\approx v^{*}_1,$ indicating that $X_{e1}(v^*_1)$ is a center surrounded by a band of continues cycles
Time evolution of species $x,\;y$, the harvest effort $E$ and the phase trajectory of the system (4) depicting unstable behaviour of the positive equilibrium point $X_{e1}(v)$ for $v = 0.961>v^{*}_1$ with initial conditions $x_{0} = x_e+0.02,\; y_{0} = y_e,\; E_{0} = E_e$
Evaluation of the coefficients $p_i$ and $q_i$ of $P(x)$ and $Q(x)$ respectively
 Coefficients $p_i$ Coefficients $q_i$ $p_0$ $0.51518$ $q_0$ $2.62089$ $p_1$ $-2.36479$ $q_1$ $-1.77522$ $p_2$ $6.5172 + 3.84 v$ $q_2$ $8.7363$ $p_3$ $-22.6624$ $q_3$ $-2.18408$ $p_4$ $27.7848 + 12.8 v$ $p_5$ $-52.1281$ $p_6$ $31.0395$ $p_7$ $-9.54038$ $p_8$ $1.19255$
 Coefficients $p_i$ Coefficients $q_i$ $p_0$ $0.51518$ $q_0$ $2.62089$ $p_1$ $-2.36479$ $q_1$ $-1.77522$ $p_2$ $6.5172 + 3.84 v$ $q_2$ $8.7363$ $p_3$ $-22.6624$ $q_3$ $-2.18408$ $p_4$ $27.7848 + 12.8 v$ $p_5$ $-52.1281$ $p_6$ $31.0395$ $p_7$ $-9.54038$ $p_8$ $1.19255$
 [1] Susmita Sadhu. Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020342 [2] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [3] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [4] Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074 [5] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [6] Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050 [7] Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047 [8] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [9] Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261 [10] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [11] A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441 [12] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468 [13] Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275 [14] Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 [15] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [16] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [17] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [18] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [19] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [20] Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

Impact Factor: 0.263