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Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III. (English) Zbl 1217.34080

This paper studies a generalized Gause model with prey harvesting and a generalized Holling response function of type III:

x ˙=ρx(1-x)-yp(x)-λ,y ˙=y(-δ+p(x)),(1)

where x0, y0, and

p(x)=x 2 αx 2 +βx+1·(2)

This basic result is a bifurcation diagram to equation (1).

The authors show that the x-axis of system (1) is invariant. The system has 2 singular points, C and D, on the positive x-axis for ρ>4λ and no equilibrium for ρ<4λ. The two points merge in a saddle-node for ρ=4λ. In the first quadrant, there is at most one singular point E which is always of anti-saddle type (i.e., a node, focus, weak focus or center). The singular point E disappears from the first quadrant by a saddle-node bifurcation by merging, with either C, or D. The point E can undergo a Hopf bifurcation of order at most two. When the order is two, the second Lyapunov coefficient is positive (the weak focus is repelling).


MSC:
34C60Qualitative investigation and simulation of models (ODE)
34C23Bifurcation (ODE)
92D25Population dynamics (general)
References:
[1]Arnold, V. I.: Geometrical methods in the theory of ordinary differential equations, (1983)
[2]Annik Martin, Predator-prey models with delays and prey harvesting, Master of Science thesis, Dalhousie University Halifax, Nova Scotia, 1999.
[3]Bazykin, A. D.: Nonlinear dynamics of interacting populations, World sci. Ser. nonlinear sci. Ser. A monogr. Treatises 11 (1998)
[4]Brauer, F.; Castillo-Chavez, C.: Mathematical models in population biology and epidemiology, (2001)
[5]Broer, H. W.; Naudot, V.; Roussarie, R.; Saleh, K.: Dynamics of a predator-prey model with non-monotonic response function, Discrete contin. Dyn. syst. 18, 221-251 (2007) · Zbl 1129.92061 · doi:10.3934/dcds.2007.18.221
[6]Clark, C. W.: Mathematical bioeconomics: the optimal management of renewable resources, (1990) · Zbl 0712.90018
[7]Caubergh, M.; Dumortier, F.: Hopf-Takens bifurcations and centres, J. differential equations 202, 1-31 (2004) · Zbl 1059.34026 · doi:10.1016/j.jde.2004.03.018
[8]Chow, S. N.; Hale, J. K.: Methods of bifurcation theory, (1982)
[9]Chow, S. N.; Li, C.; Wang, D.: Normal forms and bifurcation of planar vector fields, (1994)
[10]Dumortier, F.; Roussarie, R.; Rousseau, C.: Elementary graphics of cyclicity one or two, Nonlinearity 7, 1001-1043 (1994) · Zbl 0855.58043 · doi:10.1088/0951-7715/7/3/013
[11]Dumortier, F.; Roussarie, R.; Sotomayor, J.: Generic 3-parameter families of planar vector fiels: unfolding of saddle, focus and elliptic singularities with nilpotent linear parts, Lecture notes in math. 1480, 1-164 (1991)
[12]Dai, G. R.; Tang, M.: Coexistence region and global dynamics of a harvested predator-prey system, SIAM J. Appl. math. 58, No. 1, 193-210 (1998) · Zbl 0916.34034 · doi:10.1137/S0036139994275799
[13]R.M.D. Etoua, Étude d’un modèle de Gause généralisé avec récolte de proies et fonction de Holling type III généralisée, Thèse de PhD, Université de Montréal, Novembre 2008.
[14]R.M.D. Etoua, Étude des familles standard des déploiements du col nilpotent dont l’axe des abscisses est invariant, preprint, 2010.
[15]Freedman, H. I.: Deterministic mathematical models in population ecology, (1980)
[16]Freedman, H. I.; Wolkowicz, G. S. K.: Predator-prey systems with group defence: the paradox of enrichment revisited, Bull. math. Biol. 48, No. 5/6, 493-508 (1986) · Zbl 0612.92017
[17]Gause, G. F.: The struggle for existence, (1935)
[18]Godeau, R.: Algèbre supérieure, (1962)
[19]Kuznetsov, Y. A.: Elements of applied bifurcation theory, Appl. math. Sci. 112 (2004)
[20]Lotka, A.: Elements of physical biology, (1925) · Zbl 51.0416.06
[21]Lamontagne, Y.; Coutu, C.; Rousseau, C.: Bifurcation analysis of a predator-prey system with generalised Holling type III function response, J. dynam. Differential equations 20, No. 3, 535-571 (2008) · Zbl 1160.34047 · doi:10.1007/s10884-008-9102-9
[22]Jr., Manuel C. Molles: Ecology: concepts and applications, (2002)
[23]Perko, L.: Differential equations and dynamical systems, Texts appl. Math. 7 (2002)
[24]Shi, Shongling: A method of constructing cycles without contact around a weak focus, J. differential equations 41, 301-312 (1981) · Zbl 0442.34029 · doi:10.1016/0022-0396(81)90039-5
[25]Volterra, V.: Fluctuations in the abundance of species considered mathematically, Nature 118, 558-560 (1926) · Zbl 52.0453.03 · doi:10.1038/118558a0
[26]Wolkowicz, G. S. K.: Bifurcation analysis of a predator-prey system involving group defence, SIAM J. Appl. math. 48, No. 3, 592-606 (1988) · Zbl 0657.92015 · doi:10.1137/0148033
[27]Xiao, D.; Jennings, L. S.: Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. math. 65, No. 3, 737-753 (2005) · Zbl 1094.34024 · doi:10.1137/S0036139903428719
[28]Xiao, D.; Ruan, S.: Global analysis in predator-prey system with nonmonotonic functional response, SIAM J. Appl. math. 61, No. 4, 1445-1472 (2001) · Zbl 0986.34045 · doi:10.1137/S0036139999361896
[29]Hsu, Sze-Bi; Huang, Tzy-Wei: Global stability for a class of predator-prey systems, SIAM J. Appl. math. 55, No. 3, 763-783 (1995) · Zbl 0832.34035 · doi:10.1137/S0036139993253201
[30]Zhu, H.; Campbell, S. A.; Wolkowicz, G. S. K.: Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. math. 63, No. 2, 636-682 (2002) · Zbl 1036.34049 · doi:10.1137/S0036139901397285
[31]Rousseau, C.; Zhu, H.: PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert’s 16th problem, J. differential equations 196, 169-208 (2004) · Zbl 1046.34055 · doi:10.1016/S0022-0396(03)00119-0