Article Contents
Article Contents

Dynamics of a Leslie-Gower predator-prey system with hunting cooperation and prey harvesting

• * Corresponding author: Lingling Liu

The first author is supported by NSFC #12101470, WIT #K2021077. The second author is supported by NSFC #11771308, #11871041, SWPU #2017CXTD02, #18TD0013, #2019CXTD08

• In this paper, we study the dynamics of a Leslie-Gower predator-prey system with hunting cooperation among predator population and constant-rate harvesting for prey population. It is shown that there are a weak focus of multiplicity up to three and a cusp of codimension at most two for various parameter values, and the system exhibits two saddle-node bifurcations, a Bogdanov-Takens bifurcation of codimension two and a Hopf bifurcation as the bifurcation parameters vary. The results developed in this article reveal far more complex dynamics compared to the Leslie-Gower system and show how the prey harvesting and the hunting cooperation affect the dynamics of the system. In particular, there exist some critical values of prey harvesting and hunting cooperation such that the predator and prey populations are at risk of extinction if the intensities of harvesting and hunting cooperation are greater than these critical values. Moreover, numerical simulations are presented to illustrate our theoretical results.

Mathematics Subject Classification: Primary: 34C23, 37G10; Secondary: 34C45.

 Citation:

• Figure 1.  Saddle-nodes of system (5) when $\alpha = \beta = \gamma = 1$. (A) Saddle-nodes $E_{1*}$ when $h = 0.25$. (B) Saddle-nodes $E_{2*}$ when $h = 0.1126$

Figure 2.  Codimension 2 cusp when $\beta = 1$, $\gamma = 0.3$, $\alpha = 0.82645$ and $h = 0.1144$

Figure 3.  The Bogdanov-Takens bifurcation diagram and corresponding phase portraits of system (5) when $\beta = 1$ and $\gamma = 0.3$. (A) Bifurcation diagram. (B) No equilibrium when $(\alpha, h) = (1.2, 0.1108)\in\mathcal{I}_1$. (C) An unstable focus $E_{22}$ and a saddle $E_{21}$ when $(\alpha, h) = (1.5, 0.108)\in\mathcal{I}_2$. (D) An unstable limit cycle around a stable focus $E_{22}$ and a saddle $E_{21}$ when $(\alpha, h) = (1.5, 0.1078)\in\mathcal{I}_3$. (E) A homoclinic orbit when $(\alpha, h) = (1.2, 0.11057)\in\mathcal{HL}$. (F) A stable focus $E_{22}$ and a saddle $E_{21}$ when $(\alpha, h) = (1.5, 0.107)\in\mathcal{I}_4$

Figure 4.  Limit cycles induced by Hopf bifurcation at $E_{22}$ in system (5). (A) An unstable limit cycle around $E_{22}$ with $\beta = 1.8$, $\gamma = 0.3$, $\alpha = 10.49382716$ and $h = 0.038$. (B) Two limit cycles with $\beta = 8$, $\gamma = 0.1$, $\alpha = 86.65647719836$ and $h = 0.0013277768$. (C) The orbit from $P_2$ spirals outward. (D) The orbit from $P_3$ spirals inward

Table 1.  Dynamical behaviors near $E_{2*}$

 Parameters Equilibria and properties Closed orbits and homoclinic orbits $(\beta, \gamma)$ $(h, \alpha)$ $(0,+\infty)$ $\times[\frac{1}{2}, \frac{2}{3})$ or $(0,\frac{2\gamma}{1-2\gamma})$ $\times(0, \frac{1}{2})$ $\mathcal{I}_{1}$ No equilibria No $\mathcal{SN}^+\cup\mathcal{SN}^-$ $E_{2*}$(saddle node) No $\mathcal{I}_{2}$ $E_{21}$(saddle) $E_{22}$(unstable focus or node) No $\mathcal{H}$ $E_{21}$(saddle) $E_{22}$(unstable weak focus) No $\mathcal{I}_3$ $E_{21}$(saddle) $E_{22}$(stable focus) A unstable limit cycle $\mathcal{HL}$ $E_{21}$(saddle) $E_{22}$(stable focus) A homoclinic orbit $\mathcal{I}_{4}$ $E_{21}$(saddle) $E_{22}$(stable focus or node) No $(h_*, \alpha_*)$ $E_{2*}$(cusp) No
•  [1] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theoret. Biol., 139 (1989), 311-326.  doi: 10.1016/S0022-5193(89)80211-5. [2] N. Bacaër, A Short History of Mathematical Population Dynamics, Springer Verlag, New York, 2011. doi: 10.1007/978-0-85729-115-8. [3] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.  doi: 10.2307/3866. [4] L. Berec, Impacts of foraging facilitation among predators on predator-prey dynamics, Bull. Math. Biol., 72 (2010), 94-121.  doi: 10.1007/s11538-009-9439-1. [5] J. Carr, Applications of Center Manifold Theory, Springer, New York, 1981. [6] X. Chen and W. Zhang, Decomposition of algebraic sets and applications to weak centers of cubic systems, J. Comput. Appl. Math., 232 (2009), 565-581.  doi: 10.1016/j.cam.2009.06.029. [7] C. Cosner, D. L. DeAngelis, J. S. Ault and D. B. Olson, Effects of spatial grouping on the functional response of predators, Theor. Popul. Biol., 56 (1999), 65-75.  doi: 10.1006/tpbi.1999.1414. [8] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.  doi: 10.2307/1936298. [9] Y.-J. Gong and J.-C. Huang, Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting, Acta Math. Appl. Sin. Engl. Ser., 30 (2014), 239-244.  doi: 10.1007/s10255-014-0279-x. [10] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983. doi: 10.1007/978-1-4612-1140-2. [11] R. P. Gupta, M. Banerjee and P. Chandra, Bifurcation analysis and control of Leslie-Gower predator-prey model with Michaelis-Menten type prey-harvesting, Differ. Equ. Dyn. Syst., 20 (2012), 339-366.  doi: 10.1007/s12591-012-0142-6. [12] R. P. Gupta, P. Chandra and M. Banerjee, Dynamical complexity of a prey-predator model with nonlinear predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 423-443.  doi: 10.3934/dcdsb.2015.20.423. [13] M. P. Hassell and G. C. Varley, New inductive population model for insect parasites and its bearing on biological control, Nature, 223 (1969), 1133-1137.  doi: 10.1038/2231133a0. [14] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 97 (1965), 5-60.  doi: 10.4039/entm9745fv. [15] D. Hu and H. Cao, Stability and bifurcation analysis in a predator-prey system with Michaelis-Menten type predator harvesting, Nonlinear Anal. Real World Appl., 33 (2017), 58-82.  doi: 10.1016/j.nonrwa.2016.05.010. [16] J. Huang, Y. Gong and J. Chen, Multiple bifurcations in a predator-prey system of Holling and Leslie type with constant-yield prey harvesting, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350164, 24 pp. doi: 10.1142/S0218127413501642. [17] J. Huang, Y. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2101-2121.  doi: 10.3934/dcdsb.2013.18.2101. [18] S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201. [19] R. Kimun, K. Wonlyul and H. Mainul, Bifurcation analysis in a predator-prey system with a functional response increasing in both predator and prey densities, Nonlinear Dynam., 94 (2018), 1639-1656. [20] L. Kong and C. Zhu, Bogdanov-Takens bifurcations of codimensions 2 and 3 in a Leslie-Gower predator-prey model with Michaelis-Menten-type prey harvesting, Math. Meth. Appl. Sci., 40 (2017), 6715-6731.  doi: 10.1002/mma.4484. [21] A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Appl. Math. Lett., 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X. [22] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 1995. doi: 10.1007/978-1-4757-2421-9. [23] K. Q. Lan and C. R. Zhu, Phase portraits, Hopf bifurcations and limit cycles of the Holling-Tanner models for predator-prey interactions, Nonlinear Anal. Real World Appl., 12 (2011), 1961-1973.  doi: 10.1016/j.nonrwa.2010.12.012. [24] P. H. Leslie and J. C. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234.  doi: 10.1093/biomet/47.3-4.219. [25] A. Lotka, Elements of Physical Biology, Williams and Williams, Baltimore, 1925. [26] R. M. May,  Stability and Complexity in Model Ecosystems, Princeton University Press, New Jersey, 1973. [27] R. M. May, J. R. Beddington, C. W. Clark, S. J. Holt and R. M. Laws, Management of multispecies fisheries, Science, 205 (1979), 267-277.  doi: 10.1126/science.205.4403.267. [28] S. Pal, N. Pal, S. Samanta and J. Chattopadhyay, Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model, Math. Biosci. Eng., 16 (2019), 5146-5179.  doi: 10.3934/mbe.2019258. [29] E. Sáez and E. González-Olivares, Dynamics of a predator-prey model, SIAM J. Appl. Math., 59 (1999), 1867-1878.  doi: 10.1137/S0036139997318457. [30] D. Scheel and C. Packer, Group hunting behavior of lions: A search for cooperation, Anim. Behav., 41 (1991), 697-709.  doi: 10.1016/S0003-3472(05)80907-8. [31] G. T. Skalski and J. F. Gilliam, Functional responses with predator interference: Viable alternatives to the Holling type II model, Ecology, 82 (2001), 3083-3092. [32] M. Teixeira Alves and F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theor. Biol., 419 (2017), 13-22.  doi: 10.1016/j.jtbi.2017.02.002. [33] P. Turchin,  Complex Population Dynamics: A Theoretical/Empirical Synthesis, Princeton University Press, New Jersey, 2003. [34] V. Volterra, Fluctuations in the abundance of a species considered mathematically, Nature, 119 (1927), 12-13.  doi: 10.1038/119012b0. [35] D. Xiao and L. S. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753.  doi: 10.1137/S0036139903428719. [36] D. Xiao, W. Li and M. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl., 324 (2006), 14-29.  doi: 10.1016/j.jmaa.2005.11.048. [37] D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Ins. Commun., 21 (1999), 493-506. [38] Y. Yao, Dynamics of a prey-predator system with foraging facilitation in predators, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050009, 24 pp. doi: 10.1142/S0218127420500091. [39] P. Ye and D. Wu, Dynamics of a prey-predator system with foraging facilitation in predators, Impacts of strong Allee effect and hunting cooperation for a Leslie-Gower predator-prey system, Chinese J. Phys., 68 (2020), 49-64.  doi: 10.1016/j.cjph.2020.07.021. [40] Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, American Mathematical Society, Providence, RI, 1992. [41] C. Zhu and K. Lan, Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 289-306.  doi: 10.3934/dcdsb.2010.14.289.

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