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Resumenes 84 ⎧dx ⎛ x ⎞ qx ⎪ = r⎜1 − ⎟( x − m) x − y ⎪ dt ⎝ K ⎠ x + a ⎪ X µ : ⎨ ⎪ ⎪dy ⎛ y ⎞ = s ⎪ ⎜ ⎜1− ⎟ y ⎩ dt ⎝ n x ⎠ 7 where ( , , , , , , ) + ℜ ∈ = n s a b q K r µ and the parameters have different biological meanings. By means of a diffeomorphism (Chicone 1999, Guckenheimer and Holmes 1983), we make the study in the topologically equivalent system given by: ⎧du 2 ⎪ = ( ( 1− u)( u − M )( u + A) − v) u dτ ⎪ Yη : ⎨ ⎪ dv ⎪ = B( u − v)( u + A) v ⎩dτ We prove that the Allee effect produces a great modification in the behavior of May-Holling-Tanner model, analysed in (Sáez and González-Olivares 1999), where it was demonstrated the existence of two limit cycles surrounding the unique equilibrium point at the interior of the first quadrant, for a set of parameter values, showing that local stability does not imply global stability of the equilibrium point. In the model studied here varied dynamics appear that are not topologically equivalent to the original one, for some we give the respective ecological interpretation. Among other results we have that: 1. For any parameter values the singularity (0, 0) is a non-hyperbolic attractor (Chicone 1999, Guckenheimer and Holmes 1983), and (1,0) is an hyperbolic saddle point. 2. There exists a set of parameter for which two equilibrium point in the interior of the first quadrant exists; also an unique or one having multiplicity two can exist. 3. When we have an unique positive equilibrium point of multiplicity two, it occurs the codimension 2 saddle-node bifurcation (Chicone 1999). 4. When two positive equilibrium points exist one of them is always a hyperbolic saddle point and the nature of the other depends on sign of trace; in this case there exists an unique limit cycle obtained via Hopf bifurcation. System Yn is highly sensitive to initial conditions and as the origin is always an attractor the possibility of extinction of both populations is large depending mainly on the predator-prey ratio. Referencias [1] Arrowsmith, D. K., Place, C. M. 1992. Dynamical System. Differential equations, maps and chaotic behaviour. Chapman and Hall. [2] Chicone C. 1999. Ordinary differential equations with applications, Text in Applied Mathematics 34, Springer. [3] Clark, C. W. 1990. Mathematical Bioeconomic: The optimal management of renewable resources, (second edition). John Wiley and Sons.
Resumenes 85 [4] Courchamp, F., Clutton-Brock, T., Grenfell B., 1999. Inverse dependence and the Allee effect, Trends in Ecology and Evolution Vol 14, No. 10, 405- 410. [5] González-Olivares, E., Meneses-Alcay, H., and González-Yañez, B., 2005, Metastable dynamics by considering strong and weak Allee effect on prey in Rosenzweig-McArthur predator-prey model, submitted. [6] Guckenheimer, J., Holmes, P. 1983. Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer-Verlag. [7] Meneses-Alcay, H., González-Olivares E., 2004. Consequences of the Allee effect on Rosenzweig-McArthur predator-prey model, In R. Mondini (Ed.) Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology, E-Papers Servicos Editoriais Ltda, Río de Janeiro, Volumen 2, 264-277. [8] Sáez, E., González-Olivares, E., 1999. Dynamics on a Predator-prey Model. SIAM Journal of Applied Mathematics, Vol. 59, No. 5, 1867-1878. [9] Stephens, P. A., Sutherland, W. J. 1999. Consequences of the Allee effect for behavior, ecology and conservation, Trends in Ecology and Evolution, Vol. 14 No. 10, 401-405. [10] Wollkind, D. J., Collings, J.B. and Logan, J. 1988. Metastability in temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees, Bull. of Mathematical Biology. Vol. 50 No.4, 379- 409. [11] Turchin, P. 2003. Complex population dynamics: a theoretical/ empirical synthesis, Princeton University Press.
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ORGANIZADO POR LA SOCIEDAD LATINOAM
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(en orden alfabético) COMITÉ CIEN
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CURSILLOS Cursillo Nº1: Dinámica
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CONFERENCIAS INVITADAS
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Conferencias 10 Estabilidad y persi
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Conferencias 12 (K, the ‘carrying
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Conferencias 14 Víctimas Inmigrant
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Conferencias 16 Influencia de la di
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Resumenes 19 Bifurcaciones de Hopf
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Resumenes 21 where y = y(t) represe
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Resumenes 23 Dinámica del modelo d
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Resumenes 25 In this work, we propo
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Resumenes 27 Simulação computacio
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Resumenes 29 of the minisymposium:
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Resumenes 31 Honeybee, Apis mellife
Page 35 and 36: Resumenes 33 The minimum ratio is t
Page 37 and 38: Resumenes 35 [1] On the role of end
Page 39 and 40: Resumenes 37 Modelo matemático de
Page 41 and 42: Resumenes 39 Figure 1: Plano de fas
Page 43 and 44: Resumenes 41 Dinámica de poblacion
Page 45 and 46: Resumenes 43 El método de las solu
Page 47 and 48: Resumenes 45 Dinámica de un tumor
Page 49 and 50: Resumenes 47 Several cancer models
Page 51 and 52: Resumenes 49 process is physically
Page 53 and 54: Resumenes 51 For the analysis of fr
Page 55 and 56: Resumenes 53 Sustainability in floo
Page 57 and 58: Resumenes 55 Sincronismo em redes d
Page 59 and 60: Resumenes 57 Um modelo de autômato
Page 61 and 62: Resumenes 59 Um modelo matemático
Page 63 and 64: Resumenes 61 Redes de mapas acoplad
Page 65 and 66: Resumenes 63 Sobre la dinámica de
Page 67 and 68: Resumenes 65 Un modelo de depredaci
Page 69 and 70: Resumenes 67 There exists wide evid
Page 71 and 72: Resumenes 69 Modelado del control n
Page 73 and 74: Resumenes 71 Modelo de red neuronal
Page 75 and 76: Resumenes 73 Efectos de la radiaci
Page 77 and 78: Resumenes 75 Un modelo matemático
Page 79 and 80: Resumenes 77 Estudio comparativo de
Page 81 and 82: Resumenes 79 Introduction The proje
Page 83 and 84: Resumenes 81 formulation to obtain
Page 85: Resumenes 83 tiene un único punto
Page 89 and 90: Resumenes 87 An important part of a
Page 91 and 92: Resumenes 89 A(H3N2). Starting in V
Page 93 and 94: Resumenes 91 i) Panmixia, donde se
Page 95 and 96: Resumenes 93 punto de equilibrio y
Page 97 and 98: Resumenes 95 Un modelo epidemiológ
Page 99 and 100: Resumenes 97 Aproximación saddlepo
Page 101 and 102: Resumenes 99 [3] Daniels, H. E. 196
Page 103 and 104: Resumenes 101 bloqueados. El nudo q
Page 105 and 106: Resumenes 103 Referencias • Stauf
Page 107 and 108: Resumenes 105 Un modelo dinámico d
Page 109 and 110: Resumenes 107 Un modelo en ecuacion
Page 111 and 112: Resumenes 109 The mite show a prefe
Page 113 and 114: Resumenes 111 Ondas viajantes en un
Page 115 and 116: Resumenes 113 [5] Pregnolatto S. de
Page 117 and 118: Resumenes 115 All this will enable
Page 119 and 120: Resumenes 117 Extensión de un mode
Page 121 and 122: Resumenes 119 ∂C ∂C C = 0. = 0.
Page 123 and 124: Resumenes 121 Comparación de Enfoq
Page 125 and 126: Resumenes 123 Refúgios Espaciais e
Page 127 and 128: Resumenes 125 La dinámica del sest
Page 129 and 130: Resumenes 127 Dinámica poblacional
Page 131 and 132: Resumenes 129 Aproximación y simul
Page 133 and 134: Resumenes 131 An age-structured fin
Page 135 and 136: Resumenes 133 Respuestas demográfi
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Resumenes 135 Modelando el efecto A
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Resumenes 137 the Allee effect are
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Resumenes 139 Tratamientos de endom
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Resumenes 141 Implicaciones del efe
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Resumenes 143 phenomenon has been e
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Resumenes 145 Dinámica individual
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Resumenes 147 • f mixing ⇔ f mi
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Resumenes 149 Introduction The inve
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Resumenes 151 DNA microarrays are t
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Resumenes 153 V. Cholerae: Primeros
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Resumenes 155 Análisis cuantitativ
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Resumenes 157 Dinámica de un model
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Resumenes 159 Obtención de paráme
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Resumenes 161 [4] Nagy K. A (1987)
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Resumenes 163 diferencial automáti
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Resumenes 165 The localized peaks S
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Resumenes 167 obteniéndose que sie
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Resumenes 169 [CLARK 90] Clark, C.
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Resumenes 171 Numerical algorithms
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Resumenes 173 The second method of
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Resumenes 175 los parámetros son t
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Resumenes 177 Bifurcación hacia at
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Resumenes 179 Malaria is the tropic
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