March  2021, 29(1): 1641-1660. 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, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084
References:
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M.-S. AbdelouahabN.-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

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B. S. ChenX. 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

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C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, $2^nd$ edition, John Wiley and Sons, New York, 1990.  Google Scholar

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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

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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

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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. LiuL. BiwenC. 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. LiuC. 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. LiuQ. ZhangY. 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. AbdelouahabN.-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. ChenX. 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. LiuL. BiwenC. 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. LiuC. 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. LiuQ. ZhangY. 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

Figure 1.  Number of the interior equilibria of system (4) versus the economic profit $ v $, for $ 0< v \leq 5. $
Figure 2.  The biological coordinates $ x_e $, $ y_e $ of the two interior equilibria $ X_{e1} $ and $ X_{e2} $ versus the economic profit $ v $
Figure 3.  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 $
Figure 4.  Representation of the discriminants of $ X_{e1} $ and $ X_{e2} $ and the trace $ Tr(A(X_{e1})) $ versus the economic profit $ v $
Figure 5.  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. $
Figure 6.  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
Figure 7.  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 $
Table 1.  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 $
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