Bifurcation of a discrete predator-prey model with increasing functional response and constant-yield prey harvesting. (English) Zbl 1512.92066
Summary: Using the forward Euler method, we derive a discrete predator-prey system of Gause type with constant-yield prey harvesting and a monotonically increasing functional response in this paper. First of all, a detailed study for the existence and local stability of fixed points of the system is obtained by invoking an important lemma. Mainly, by utilizing the center manifold theorem and the bifurcation theory some sufficient conditions are obtained for the saddle-node bifurcation and the flip bifurcation of this system to occur. Finally, with the use of Matlab software, numerical simulations are carried out to illustrate the theoretical results obtained and reveal some new dynamics of the system-chaos occuring.
MSC:
| 92D25 | Population dynamics (general) |
| 34C23 | Bifurcation theory for ordinary differential equations |
Keywords:
discrete predator-prey system; forward Euler method; flip bifurcation; saddle-node bifurcation; center manifold theoremReferences:
| [1] | M, A detailed study of the Beddington-DeAngelis predator-prey model, Math. Biosci., 234, 1-16 (2011) · Zbl 1402.92345 · doi:10.1016/j.mbs.2011.07.003 |
| [2] | A, Singular perturbations of the Holling Ⅰ predator-prey system with a focus, J. Differ. Equation, 269, 5434-5462 (2020) · Zbl 1457.34085 · doi:10.1016/j.jde.2020.04.011 |
| [3] | S, Relaxation oscillations for Leslie-type predator-prey model wemith Holling Type Ⅰ response functional function, Appl. Math. Lett., 120, 1-6 (2021) · Zbl 1480.34064 · doi:10.1016/j.aml.2021.107328 |
| [4] | B, Dynamic complexities of a Holling Ⅰ predator-prey model concerning periodic biological and chemical control, Chaos Solitons Fractals, 22, 123-134 (2004) · Zbl 1058.92047 · doi:10.1016/j.chaos.2003.12.060 |
| [5] | M, Dynamics of a Leslie-Gower Holling-type ii predator-prey system with levy jumps, Nonlinear Anal., 85, 204-213 (2013) · Zbl 1285.34047 · doi:10.1016/j.na.2013.02.018 |
| [6] | Y, Analysis of a stochastic two-predators one prey system with modified Leslie-Gower and holling-type Ⅱ schemes, J. Appl. Anal. Comput., 7, 713-727 (2017) · Zbl 1488.92055 · doi:10.1016/j.physa.2019.122761 |
| [7] | X, Survivability and stochastic bifurcations for a stochastic Holling type Ⅱ predator-prey model, Commun. Nonlinear Sci. Numer. Simul., 83, 1-20 (2020) · Zbl 1453.92282 · doi:10.1016/j.cnsns.2019.105136 |
| [8] | M, Global analysis in Bazykins model with Holling Ⅱ functional response and predator competition, J. Differ. Equation, 280, 99-138 (2021) · Zbl 1464.34070 · doi:10.1016/j.jde.2021.01.025 |
| [9] | A, Modeling the depletion of dissolved oxygen due to algal bloom in a lake by taking Holling type-Ⅲ interaction, Appl. Math. Comput., 217, 8367-8376 (2011) · Zbl 1215.92066 · doi:10.1016/j.amc.2011.03.034 |
| [10] | R, Stability and permanence of a discrete-time two-prey one-predator system with Holling Type-Ⅲ functional response, Chaos Solitons Fractals, 117, 240-248 (2018) · Zbl 1442.92124 · doi:10.1016/j.chaos.2018.10.032 |
| [11] | C, Heteroclinic and homoclinic orbits for a slow-fast predator-prey model of generalized Holling type Ⅲ, J. Differ. Equation, 267, 3397-3441 (2019) · Zbl 1418.34103 · doi:10.1016/j.jde.2019.04.008 |
| [12] | J, Bifurcations in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, J. Differ. Equation, 257, 1721-1752 (2014) · Zbl 1326.34082 · doi:10.1016/J.JDE.2014.04.024 |
| [13] | D, Bifurcation and bio-economic analysis of a prey-generalist predator model with Holling type Ⅳ functional response and nonlinear age-selective prey harvesting, Chaos Solitons Fractals, 122, 229-235 (2019) · Zbl 1448.92185 · doi:10.1016/j.chaos.2019.02.010 |
| [14] | Y, Bifurcations of a predator-prey system of Holling and Leslie types, Chaos Solitons Fractals, 34, 606-620 (2007) · Zbl 1156.34029 · doi:10.1016/j.chaos.2006.03.068 |
| [15] | S, A food chain model with impulsive perturbations and Holling Ⅳ functional response, Chaos Solitons Fractals, 26, 855-866 (2005) · Zbl 1066.92061 · doi:10.1016/j.chaos.2005.01.053 |
| [16] | S, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61, 1445-1472 (2001) · Zbl 0986.34045 · doi:10.1137/S0036139999361896 |
| [17] | C, Bifurcation analysis of a predator-prey model with predator intraspecific interactions and ratio-dependent functional response, Appl. Math. Comput., 402, 1-20 (2021) · Zbl 1510.92151 · doi:10.1016/j.amc.2021.126152 |
| [18] | X, Stochastic bifurcations, a necessary and sufficient condition for a stochastic Beddington-DeAngelis predator-prey model, Appl. Math. Lett., 117, 1-7 (2021) · Zbl 1466.34051 · doi:10.1016/j.aml.2021.107069 |
| [19] | D, Global dynamics of a Beddington-DeAngelis amensalism system with weak Allee effect on the first species, Appl. Math. Comput., 408, 1-19 (2021) · Zbl 1489.34066 · doi:10.1007/s12190-021-01533-w |
| [20] | G, Periodic solutions for a neutral delay Hassell-Varley type predator-prey system, Appl. Math. Comput., 264, 443-452 (2015) · Zbl 1410.34204 · doi:10.1016/j.amc.2015.04.110 |
| [21] | D, On a non-selective harvesting prey-predator model with Hassell-Varley type functional response, Appl. Math. Comput., 246, 678-695 (2014) · Zbl 1338.92116 · doi:10.1016/j.amc.2014.08.081 |
| [22] | C, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91, 385-398 (1959) · doi:10.4039/Ent91385-7 |
| [23] | C, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol., 56, 65-75 (1999) · Zbl 0928.92031 · doi:10.1006/tpbi.1999.1414 |
| [24] | K, Bifurcation analysis in a predator-prey system with a functional response increasing in both predator and prey densities, Nonlinear Dyn., 94, 1639-1656 (2018) · Zbl 1422.92127 · doi:10.1007/s11071-018-4446-0 |
| [25] | T, Hunting cooperation and Allee effects in predators, J. Theoret. Biol., 419, 13-22 (2017) · Zbl 1370.92151 · doi:10.1016/j.jtbi.2017.02.002 |
| [26] | F, Turing patterns in a reaction-diffusion system modeling hunting cooperation, Math. Comput. Simul., 165, 172-180 (2019) · Zbl 1540.92098 · doi:10.1016/j.matcom.2019.03.010 |
| [27] | Y, Cooperative hunting in a discrete predator-prey system, J. Biol. Dyn., 13, 247-264 (2019) · Zbl 1441.92031 · doi:10.1080/17513758.2018.1555339 |
| [28] | J, Chaos and crises in a model for cooperative hunting: a symbolic dynamics approach, Chaos, 19, 1-12 (2009) · doi:10.1063/1.3243924 |
| [29] | S, Effect of hunting cooperation and fear in a predator-prey model, Ecol. Complex., 39, 1-18 (2019) · doi:10.1016/j.ecocom.2019.100770 |
| [30] | N, Bifurcations and organized structures in a predator-prey model with hunting cooperation, Chaos Solitons Fractals, 140, 1-11 (2020) · Zbl 1495.92060 · doi:10.1016/j.chaos.2020.110184 |
| [31] | Z, Bifurcation analysis in a predator-prey system with an increasing functional response and constant-yield prey harvesting, Math. Comput. Simul., 190, 976-1002 (2021) · Zbl 1540.92143 · doi:10.1016/j.matcom.2021.06.024 |
| [32] | W, Neimark-Sacker bifurcation of a semi-discrete hematopoiesis model, J. Appl. Anal. Comput., 8, 1679-1693 (2018) · Zbl 1457.37109 · doi:10.11948/2018.1679 |
| [33] | C, Stability and Neimark-Sacker bifurcation of a semi-discrete population model, J. Appl. Anal. Comput., 4, 419-435 (2014) · Zbl 1323.92193 · doi:10.11948/2014024 |
| [34] | S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Second edition, Springer-verlag, New York, 2003. · Zbl 1027.37002 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
