zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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: $$\dot{x} = \rho x(1-x) - y p(x) - \lambda,\quad \dot{y} = y (-\delta + p(x)), \tag{1}$$ where $x\geq 0$, $y\geq 0$, and $$p(x) = {x^2 \over \alpha x^2 + \beta x + 1}. \tag{2}$$ 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 $\rho > 4 \lambda$ and no equilibrium for $\rho < 4 \lambda$. The two points merge in a saddle-node for $\rho = 4 \lambda$. 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:
 34C60 Qualitative investigation and simulation of models (ODE) 34C23 Bifurcation (ODE) 92D25 Population dynamics (general)
Full Text:
References:
 [1] Arnold, V. I.: Geometrical methods in the theory of ordinary differential equations, (1983) · Zbl 0507.34003 [2] Annik Martin, Predator-prey models with delays and prey harvesting, Master of Science thesis, Dalhousie University Halifax, Nova Scotia, 1999. · Zbl 1008.34066 [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) · Zbl 0967.92015 [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) · Zbl 0487.47039 [9] Chow, S. N.; Li, C.; Wang, D.: Normal forms and bifurcation of planar vector fields, (1994) · Zbl 0804.34041 [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) · Zbl 0448.92023 [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) · Zbl 1082.37002 [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