In this manuscript, a novel predator-prey system combining prey refuge with fuzzy parameters is formulated. Sufficient conditions for the existence and stability of biological equilibria are derived. The existence of bionomic equilibria is discussed under fuzzy biological parameters. The optimal harvesting policy, by Pontryagin's maximal principle, is also investigated under imprecise inflation and discount in fuzzy environment. Meticulous numerical simulations are performed to validate our theoretical analysis in detail.
Citation: Qinglong Wang, Shuqi Zhai, Qi Liu, Zhijun Liu. Stability and optimal harvesting of a predator-prey system combining prey refuge with fuzzy biological parameters[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 9094-9120. doi: 10.3934/mbe.2021448
In this manuscript, a novel predator-prey system combining prey refuge with fuzzy parameters is formulated. Sufficient conditions for the existence and stability of biological equilibria are derived. The existence of bionomic equilibria is discussed under fuzzy biological parameters. The optimal harvesting policy, by Pontryagin's maximal principle, is also investigated under imprecise inflation and discount in fuzzy environment. Meticulous numerical simulations are performed to validate our theoretical analysis in detail.
| [1] |
G. F. Gause, N. P. Smaragdova, A. A. Witt, Further studies of interaction between predators and prey, J. Anim. Ecol., 5 (1936), 1–18. doi: 10.2307/1087
|
| [2] | J. M. Smith, Models in Ecology, Cambridge University Press, Cambridge, 1974. |
| [3] | C. Bianca, Modeling complex systems with particles refuge by thermostatted kinetic theory methods, Abstr. Appl. Anal., 2013 (2013), 717–718. |
| [4] |
C. Bianca, C. Dogba, L. Guerrini, A thermostatted kinetic framework with particle refuge for the modeling of tumors hiding, Appl. Math. Inf. Sci., 8 (2014), 469–473. doi: 10.12785/amis/080203
|
| [5] |
Q. L. Wang, Z. J. Liu, X. A. Zhang, R. A. Cheke, Incorporating prey refuge into a predator-prey system with imprecise parameter estimates, Comp. Appl. Math., 36 (2017), 1067–1084. doi: 10.1007/s40314-015-0282-8
|
| [6] |
H. L. Li, L. Zhang, C. Hu, Y. L. Jiang, Z. D. Teng, Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, J. Appl. Math. Comput., 54 (2017), 435–449. doi: 10.1007/s12190-016-1017-8
|
| [7] |
E. G. Olivares, R. R. Jiliberto, Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecol. Model., 166 (2003), 135–146. doi: 10.1016/S0304-3800(03)00131-5
|
| [8] |
T. K. Kar, Stability analysis of a prey-predator model incorporating a prey refuge, Commun. Nonlinear Sci. Numer. Simul., 10 (2005), 681–691. doi: 10.1016/j.cnsns.2003.08.006
|
| [9] | Y. J. Huang, F. D. Chen, L. Zhong, Stability analysis of a prey-predator model with Holling type Ⅲ response function incorporating a prey refuge, Appl. Math. Comput., 182 (2006), 672–683. |
| [10] |
J. P. Tripathi, S. Abbas, M. Thakur, Dynamical analysis of a prey-predator model with Beddington-DeAngelis type function response incorporating a prey refuge, Nonlinear Dyn., 80 (2015), 177–196. doi: 10.1007/s11071-014-1859-2
|
| [11] |
R. J. Han, L. N. Guin, B. X. Dai, Consequences of refuge and diffusion in a spatiotemporal predator-prey model, Nonlinear Anal. Real World Appl., 60 (2021), 103311. doi: 10.1016/j.nonrwa.2021.103311
|
| [12] |
H. K. Qi, X. Z. Meng, Threshold behavior of a stochastic predator-prey system with prey refuge and fear effect, Appl. Math. Lett., 113 (2021), 106846. doi: 10.1016/j.aml.2020.106846
|
| [13] | Y. Zhang, J. Zhang, Optimal harvesting for a stochastic competition system with stage structure and distributed delay, Electron. J. Qual. Theory Differ. Equations, 2021 (2021), 1–22. |
| [14] |
G. S. Mahapatra, P. Santra, Prey-predator model for optimal harvesting with functional response incorporating prey refuge, Int. J. Biomath., 9 (2016), 1650014. doi: 10.1142/S1793524516500145
|
| [15] |
W. X. Li, K. Wang, Optimal harvesting policy for general stochastic logistic population model, J. Math. Anal. Appl., 368 (2010), 420–428. doi: 10.1016/j.jmaa.2010.04.002
|
| [16] | W. X. Li, K. Wang, H. Su, Optimal harvesting policy for stochastic logistic population model, Appl. Math. Comput., 218 (2011), 157–162. |
| [17] | A. R. Palma, E. G. Olivares, Optimal harvesting in a predator-prey model with Allee effect and Sigmoid functional response, Appl. Math. Model., 5 (2012), 1864–1874. |
| [18] | C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Wiley, New York, 1976. |
| [19] | C. W. Clark, Bioeconomic Modelling and Fisheries Management, John Wiley and Sons, New York, 1985. |
| [20] |
T. K. Kar, K. S. Chaudhuri, Harvesting in a two-prey one predator fishery: a bioeconomic model, ANZIAM J., 45 (2004), 443–456. doi: 10.1017/S144618110001347X
|
| [21] | Z. R. He, N. Zhou, Optimal harvesting for a nonlinear hierarchical age-structured population model, (Chin.) J. Syst. Sci. Math. Sci., 40 (2020), 2248–2263. |
| [22] |
D. Pal, G. S. Mahaptra, G. P. Samanta, Optimal harvesting of prey-predator system with interval biological parameters: A bioeconomic model, Math. Biosci., 241 (2013), 181–187. doi: 10.1016/j.mbs.2012.11.007
|
| [23] | D. Pal, G. S. Mahapatra, A bioeconomic modeling of two-prey and one-predator fishery model with optimal harvesting policy through hybridization approach, Appl. Math. Comput., 242 (2014), 748–763. |
| [24] |
D. Pal, G. S. Mahapatra, Dynamic behavior of a predator-prey system of combined harvesting with interval-valued rate parameters, Nonlinear Dynam., 83 (2016), 2113–2123. doi: 10.1007/s11071-015-2469-3
|
| [25] |
Q. Z. Xiao, B. X. Dai, L. Wang, Analysis of a competition fishery model with interval-valued parameters: extinction, coexistence, bionomic equilibria and optimal harvesting policy, Nonlinear Dynam., 80 (2015), 1631–1642. doi: 10.1007/s11071-015-1967-7
|
| [26] |
S. Y. Chen, Z. J. Liu, L. W. Wang, J. Hu, Stability of a delayed competitive model with saturation effect and interval biological parameters, J. Appl. Math. Comput., 64 (2020), 1–15. doi: 10.1007/s12190-020-01341-8
|
| [27] |
K. Qi, Z. J. Liu, L. W. Wang, Q. L. Wang, Survival and stationary distribution of a stochastic facultative mutualism model with distributed delays and strong kernels, Math. Biosci. Eng., 18 (2021), 3160–3179. doi: 10.3934/mbe.2021157
|
| [28] |
W. X. Ning, Z. J. Liu, L. W. Wang, R. H. Tan, A stochastic mutualism model with saturation effect and impulsive toxicant input in a polluted environment, J. Appl. Math. Comput., 65 (2021), 177–197. doi: 10.1007/s12190-020-01387-8
|
| [29] |
M. Liu, K. Wang, Persistence and extinction of a stochastic single-specie model under regime switching in a polluted environment, J. Theoret. Biol., 264 (2010), 934–944. doi: 10.1016/j.jtbi.2010.03.008
|
| [30] |
M. Liu, K. Wang, Dynamics of a Leslie-Gower Holling-type Ⅱ predator-prey system with Lévy jumps, Nonlinear Anal., 85 (2013), 204–213. doi: 10.1016/j.na.2013.02.018
|
| [31] |
M. Liu, K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations, Discrete Contin. Dyn. Syst., 33 (2013), 2495–2522. doi: 10.3934/dcds.2013.33.2495
|
| [32] | L. A. Zadeh, Fuzzy sets, Inf. Control, 8 (1965), 338–353. |
| [33] |
L. A. Zadeh, Toward a generalized theory of uncertainty (GTU)-An outline, Inf. Sci., 172 (2005), 1–40. doi: 10.1016/j.ins.2005.01.017
|
| [34] | S. L. Chang, L. A. Zadeh, On fuzzy mapping and control, IEEE Trans. Syst. Man Cybern., 2 (1972), 30–34. |
| [35] |
O. Kaleva, Fuzzy differential equations, Fuzzy Sets Syst., 24 (1987), 301–317. doi: 10.1016/0165-0114(87)90029-7
|
| [36] |
B. Bede, I. J. Rudas, A. L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Inf. Sci., 177 (2007), 1648–1662. doi: 10.1016/j.ins.2006.08.021
|
| [37] |
A. Khastan, J. J. Nieto, A boundary value problem for second order fuzzy differential equations, Nonlinear Anal., 72 (2010), 3583–3593. doi: 10.1016/j.na.2009.12.038
|
| [38] |
L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal., 71 (2009), 1311–1328. doi: 10.1016/j.na.2008.12.005
|
| [39] |
M. Puri, D. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl., 91 (1983), 552–558. doi: 10.1016/0022-247X(83)90169-5
|
| [40] | B. Bede, S.G. Gal, Solutions of fuzzy differential equations based on generalized differentiability, Commun. Math. Anal., 9 (2010), 22–41. |
| [41] |
E. Hullermeier, An approach to modelling and simulation of uncertain dynamical systems, Int. J. Uncertain. Fuzziness Knowl. Based Syst., 5 (1997), 117–137. doi: 10.1142/S0218488597000117
|
| [42] |
T. Allahviranloo, M. B. Ahmadi, Fuzzy Laplace transforms, Soft Comput., 14 (2010), 235–243. doi: 10.1007/s00500-008-0397-6
|
| [43] |
E. J. Villamizar-Roa, V. Angulo-Castillo, Y. Chalco-Cano, Existence of solutions to fuzzy differential equations with generalized Hukuhara derivative via contractive-like mapping principles, Fuzzy Sets Syst., 265 (2015), 24–38. doi: 10.1016/j.fss.2014.07.015
|
| [44] |
D. Pal, G. S. Mahapatra, G. P. Samanta, Stability and bionomic analysis of fuzzy prey-predator harvesting model in presence of toxicity: a dynamic approach, Bull. Math. Biol., 78 (2016), 1493–1519. doi: 10.1007/s11538-016-0192-y
|
| [45] | X. W. Yu, S. L. Yuan, T. H. Zhang, About the optimal harvesting of a fuzzy predator-prey system: a bioeconomic model incorporating prey refuge and predator mutual interference, Comput. Math. Appl., 94 (2018), 2143–2160. |
| [46] |
X. Y. Meng, Y. Q. Wu, Dynamical analysis of a fuzzy phytoplankton-zooplankton model with refuge, fishery protection and harvesting, J. Appl. Math. Comput., 63 (2020), 361–389. doi: 10.1007/s12190-020-01321-y
|
| [47] |
D. Pal, G. S. Mahapatra, G. P. Samanta, A study of bifurcation of prey-predator model with time delay and harvesting using fuzzy parameters, J. Biol. Syst., 26 (2018), 339–372. doi: 10.1142/S021833901850016X
|
| [48] |
R. C. Bassanezi, L. C. Barros, A. Tonelli, Attractors and asymptotic stability for fuzzy dynamical systems, Fuzzy Sets Syst., 113 (2000), 473–483. doi: 10.1016/S0165-0114(98)00142-0
|
| [49] |
M. T. Mizukoshi, L. C. Barros, R. C. Bassanezi, Stability of fuzzy dynamic systems, Int. J. Uncertain, Fuzziness Knowl. -Based Syst., 17 (2009), 69–83. doi: 10.1142/S0218488509005747
|
| [50] |
M. S. Guo, X. P. Xue, R. L. Li, Impulsive functional differential inclusions and fuzzy population models, Fuzzy Sets Syst., 138 (2003), 601–615. doi: 10.1016/S0165-0114(02)00522-5
|
| [51] |
V. Lupulescu, On a class of fuzzy functional differential equations, Fuzzy Sets Syst., 160 (2009), 1547–1562. doi: 10.1016/j.fss.2008.07.005
|
| [52] |
D. Sadhukhan, L. N. Sahoo, B. Mondal, M. Maiti, Food chain model with optimal harvesting in fuzzy environment, J. Appl. Math. Comput., 34 (2010), 1–18. doi: 10.1007/s12190-009-0301-2
|
| [53] | D. Pal, G. S. Mahapatra, S. K. Mahato, G. P. Samanta, A mathematical model of a prey-predator type fishery in the presence of toxicity with fuzzy optimal harvesting, J. Appl. Math. Inf., 38 (2020), 13–36. |
| [54] |
D. Pal, G. S. Mahapatra, G. P. Samanta, Stability and bionomic analysis of fuzzy parameter based prey-predator harvesting model using UFM, Nonlinear Dyn., 79 (2015), 1939–1955. doi: 10.1007/s11071-014-1784-4
|
| [55] |
J. Dijkman, H. Haeringen, S. DeLange, Fuzzy numbers, J. Math. Anal. Appl., 92 (1983), 301–341. doi: 10.1016/0022-247X(83)90253-6
|
| [56] | R. Jafari, W. Yu, Uncertainty nonlinear systems modeling with fuzzy equations, in 2015 IEEE 16th International Conference on Information Reuse and Integration, San Francisco, (2015), 182–188. |
| [57] | M. Mizumoto, K. Tanaka, The four operations of arithmetic on fuzzy numbers, Syst. Comput. Controls, 7 (1976), 73–81. |
| [58] | D. Ralescu, A survey of the representation of fuzzy concepts and its applications, in Advances in Fuzzy Set Theory and Applications, North-Holland, Amsterdam-New York, (1979), 77–91. |
| [59] | K. M. Miettinen, Non-Linear Multi-Objective, Optimization, Kluwer's International Series, 1999. |
| [60] | L. S. Pontryagin, V. G. Boltyonsku, R. V. Gamkrelidre, E. F. Mishchenko, Math. Theory Optim. Processes, Wiley, New York, 1962. |
| [61] |
K. Maity, M. Maiti, A numerical approach to a multi-objective optimal inventory control problem for deteriorating multi-items under fuzzy inflation and discounting, Comput. Math. Appl., 55 (2008), 1794–1807. doi: 10.1016/j.camwa.2007.07.011
|
Figures(8) / Tables(9)
Qinglong Wang, Shuqi Zhai, Qi Liu, Zhijun Liu. Stability and optimal harvesting of a predator-prey system combining prey refuge with fuzzy biological parameters[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 9094-9120. doi: 10.3934/mbe.2021448
DownLoad: