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Resumenes 24 X µ ⎧ ⎪ ⎪ dX ⎪ ⎪ dt ⎪ : ⎨ ⎪ ⎪ ⎪ dY ⎪ ⎪ dt ⎪ ⎩ ⎛ = r⎜1 − ⎝ X K ⎛ αX ⎞ q ⎜ X − ⎟ ⎞ ⎝ X + β − ⎠ ⎟ X Y ⎠ αX X − + a X + β ⎛ ⎛ αX ⎞ ⎞ ⎜ p⎜ X − ⎟ ⎟ ⎜ ⎝ X + β ⎠ ⎟ = ⎜ − c ⎟ Y αX ⎜ X − + a ⎟ ⎜ ⎟ ⎝ X + β ⎠ 8 con µ = ( r , K, q, a, p, c, α, β ) ∈ ℜ+ , donde Y = Y() t representa el tamaño poblacional de los depredadores para t ≥ 0 y los parámetros, todos positivos, tienen diferentes significados biológicos. El sistema obtenido es del tipo Kolmogorov [Freedman 1980], y se determina una región de invarianza, los puntos de equilibrios, estableciendo la naturaleza de cada uno de ellos y se obtienen condiciones en los parámetros para la existencia o no de ciclos límites. La importancia del estudio del uso de refugio por parte de las presas es relevante en los proceso de conservación de especies en peligro de extinción con la creación de reservas (refugios) para preservarlas [Srinivasu and Gayitri 2005] Dinamics in the Rosenzweig-McArthur predation model considering saturated refuge for prey. Commonly in Population Dynamics it is affirmed that prey refuge use has a stabilizing effect in system when deterministic continuous-time models for predator-prey interactions are employed [Collings 1995, Maynard-Smith 1974, Ruxton 1995, Sih 1987]. If X = X(t) represents the prey population size, this affirmation it is possible to verify, describing the interaction with the Lotka-Volterra model, and assuming that the quantity of prey in cover Xr is proportional to population size , that is X r = α X , or else, the quantity of population at refugia is constant., i. e. X r = β [Harrison 1979, Taylor 1984]. When the predator-prey interaction is described with by the wellknown Rosenzweig-McArthur model [Murdoch 2003, Turchin, 2003], in González-Olivares and Ramos-Jiliberto, (2003) was demonstrated that for certain parameter constraints, the effect of refuge by a fraction of prey population implies oscillatory behavior due the apparition of limit cycles on system. At once, Ruxton [1995] constructs a model assuming that the rate in which prey move to refuge , is proportional to predator density, that is, = λ y , and it is prove that the refuge has a stabilizing effect. X r
Resumenes 25 In this work, we propose a new form for modeling the fraction of prey on refuge, using a saturated function which is monotonic increasing described by: α x X r = x + β whereα represents the maximun physical capacity of refuge and β is the half saturation constant, i. e., the prey population size necessary for attain the half of α . Moreover, we have that the per capita fraction of prey X r population on refuge is a decreasing function. x We introduce this new form for describe the refuge use in the Rosenzweig-McArthur model obtaining the vector field X µ describes by the following autonomous differential equations system: ⎧ ⎪ ⎛ αX ⎞ q⎜ X − ⎟ ⎪dX ⎛ X ⎞ ⎝ X + β ⎪ = r ⎠ ⎜1− ⎟X − Y ⎪ dt ⎝ K ⎠ αX X − + a ⎪ X + β ⎪ X µ : ⎨ ⎪ ⎪ ⎛ ⎛ αX ⎞ ⎞ ⎜ p⎜ X − ⎟ ⎟ ⎪dY ⎜ ⎝ X + β ⎠ ⎟ ⎪ = ⎜ − c⎟Y ⎪ dt αX ⎜ X − + a ⎟ ⎪ ⎜ ⎟ ⎩ ⎝ X + β ⎠ 8 with µ = ( r , K, q, a, p, c, α, β ) ∈ ℜ+ , where Y = Y() t represents the predator population size fort ≥ 0 and the parameters all positives, have different biological meanings. System obtained is of Kolmogorov type [Freedman 1980], and we determine a invariance region, the equilibrium points an its nature, obtaining conditions for the parameter about the existence or non existence of limit cycles. The importance of to study prey refuge use is essential for conservation of species in extinction damage creating protected areas (reserves) for preserve them [Srinivasu and Gayitri 2005]. Referencias [1] J. B. Collins, Bifurcations and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge, Bulletin of Mathematical Biology 57, 63-76, 1995. [2] H. I. Freedman, Deterministic Mathematical Model in Population Ecology. Marcel Dekker, 254 pp.1980. [3] E. González-Olivares and R. Ramos-Jiliberto, Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecological Modelling, Vol. 166, 135-146, 2003. [4] G. W. Harrison, Global stability of predator-prey interactions. J. Mathematical Biology 8, 139-171, 1979. [5] J. Maynard Smith, Models in Ecology. Cambridge, At the University Press, 1974.
Page 3 and 4: ORGANIZADO POR LA SOCIEDAD LATINOAM
Page 5 and 6: (en orden alfabético) COMITÉ CIEN
Page 7 and 8: CURSILLOS Cursillo Nº1: Dinámica
Page 9: CONFERENCIAS INVITADAS
Page 12 and 13: Conferencias 10 Estabilidad y persi
Page 14 and 15: Conferencias 12 (K, the ‘carrying
Page 16 and 17: Conferencias 14 Víctimas Inmigrant
Page 18 and 19: Conferencias 16 Influencia de la di
Page 21 and 22: Resumenes 19 Bifurcaciones de Hopf
Page 23 and 24: Resumenes 21 where y = y(t) represe
Page 25: Resumenes 23 Dinámica del modelo d
Page 29 and 30: Resumenes 27 Simulação computacio
Page 31 and 32: Resumenes 29 of the minisymposium:
Page 33 and 34: 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
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Resumenes 75 Un modelo matemático
Page 79 and 80:
Resumenes 77 Estudio comparativo de
Page 81 and 82:
Resumenes 79 Introduction The proje
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Resumenes 81 formulation to obtain
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Resumenes 83 tiene un único punto
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Resumenes 85 [4] Courchamp, F., Clu
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Resumenes 87 An important part of a
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Resumenes 89 A(H3N2). Starting in V
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Resumenes 91 i) Panmixia, donde se
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Resumenes 93 punto de equilibrio y
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Resumenes 95 Un modelo epidemiológ
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Resumenes 97 Aproximación saddlepo
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Resumenes 99 [3] Daniels, H. E. 196
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Resumenes 101 bloqueados. El nudo q
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Resumenes 103 Referencias • Stauf
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Resumenes 105 Un modelo dinámico d
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Resumenes 107 Un modelo en ecuacion
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Resumenes 109 The mite show a prefe
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Resumenes 111 Ondas viajantes en un
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Resumenes 113 [5] Pregnolatto S. de
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Resumenes 115 All this will enable
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Resumenes 117 Extensión de un mode
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Resumenes 119 ∂C ∂C C = 0. = 0.
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Resumenes 121 Comparación de Enfoq
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Resumenes 123 Refúgios Espaciais e
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Resumenes 125 La dinámica del sest
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Resumenes 127 Dinámica poblacional
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Resumenes 129 Aproximación y simul
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Resumenes 131 An age-structured fin
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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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