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Resumenes 142 ser modelado de otras formas (ver Thieme 2003, Wang et al .1999, Zhou et al. 2005). El modelo que estudiamos es descrito por el sistema de ecuaciones diferenciales del tipo Kolmogorov (Freedman 1980) ⎧ 2 dx ⎛ x ⎞ qx ⎪ = r⎜1 − ⎟( x − m) x − y 2 ⎪ dt ⎝ K ⎠ x + a ⎪ X µ : ⎨ ⎪ 2 ⎪dy ⎛ px ⎞ = ⎪ ⎜ − c ⎟ y 2 ⎩ dt ⎝ x + a ⎠ donde x = x(t) e y = y(t) indican los tamaños poblacionales (número de individuos, densidad o biomasa) de las presas y depredadores respectivamente, y todos los parámetros son positivos, teniendo diferentes significados y por razones biológicas a y m < K. Si m = 0, se dice que la población está afectada por el efecto Allee débil (González-Olivares et al. 2005). Obtenemos condiciones en los parámetros para a) Determinar cantidad de los puntos de equilibrio en el primer cuadrante. b) Determinar la naturaleza de los puntos críticos. c) Establecer la existencia de curvas separatrices, curvas homoclínicas o heteroclínicas. d) Determinar la cantidad de ciclos límites Implications of the Allee effect in a Gause type predator-prey models with sigmoid functional response In this work we analize a deterministic continuous-time predator-prey models of Gause type (Freedman, 1980) considering two important aspects: En este trabajo analizamos un modelo de depredación determinista tiempo continuo que se i) The functional response o consumption rate is Holling type III or sigmoid (Turchin, 2003) ii) In the prey growth function the Allee effect it is considered. A study for this type of models is maked in (Conway and Smoller 1986), but some of its results are not fulfilled in the model that we study. In Population Dynamic, any ecological mechanism that can lead to a positive relationship between a component of individual fitness and either the number or density of conspecifics can be termed a mechanism of the Allee effect (Kent et al. 2003, Stephens and Sutherland 1999) or depensation (Clark 1990, Dennis 1989), or negative competition effect (Wang et al. 1999). Populations can exhibit allee dynamics due to a wide range of biological phenomenon such as reduced antipredator vigilance, social thermoregulation, genetic drift (Stephens and Sutherland, 1999), mating difficulty (McCarthy 1997), and so on. The Allee effect is a important and interesting phenomenon in both biological and mathematical sense. From biological point wise, Allee effect it produces to low population densities when the individual growth rate is a increasing function with density (Dennis 1989), and for this its possibilities of extinction increases (Courchamp et al. 1999); the importance of this
Resumenes 143 phenomenon has been emphasized by recent developments in diverses areas of ecology and conservation, and can be regarding as the basis of animal sociability (Stephens and Sutherland 1999). Mathematically, more simple models can show enough about this dynamic (Stephens and Sutherland 1999), and it can implied the apparition of new equilibrium points that changes the structural stability of system and to occur changesin the stability of other equilibrium points. Some authors estimates that the dense-dependient population models for single specie that assume the Allee effect, must consider that the population growth rate of this populations decreases if the population size is below a threshold level m (Brauer and Castillo-Chávez 2001). However, the Allee effect can be modeled by different forms as is presented in Sin embargo, el efecto Allee, que da cuenta del fenómeno que aparece en ciertas poblaciones de animales a bajas densidades puede ser modelado de otras formas como es presentado in (Thieme 2003, Wang et al .1999, Zhou et al. 2005). Model that we study is described by the Kolmogorov type differential equation system (Freedman 1980). ⎧ 2 dx ⎛ x ⎞ qx ⎪ = r⎜1 − ⎟( x − m) x − y 2 ⎪ dt ⎝ K ⎠ x + a ⎪ X µ : ⎨ ⎪ 2 ⎪dy ⎛ px ⎞ = ⎪ ⎜ − c ⎟ y 2 ⎩ dt ⎝ x + a ⎠ where x = x(t) and y = y(t) indicate the predator and prey population size respectively for t > 0 (number of individual, density or biomass), and the parameters are all positives, having different biological meanings. By biological reason a and m < K. If m = 0, it say that the population is affect by a weak Allee effect (González-Olivares et al. 2005). We obtain restrictions dependent of parameter values for: a) To determine the quantity of equilibrium points at the first quadrant. b) To determine the nature of those equilibrium points. c) To stablish the existente of separatrix curves and homoclinic or heteroclinic curves . d) To stablish the quantity of limit cycles. Referencias [1] Brauer, F., and Castillo-Chávez, C., 2001 Mathematical models in Population Biology and Epidemiology, TAM 40, Springer.Verlag. [2] Clark, C. W., 1990. Mathematical Bioeconomic: The optimal management of renewable resources, (second edition). John Wiley and Sons. [3] Conway E. D. and Smoller, J. A., 1986, Global Analysis of a System of Predator-Prey Equations. SIAM J. Applied Mathematics, Vol. 46, No.4, 630- 642. [4] Courchamp, F., Clutton-Brock, T., and Grenfell, B., 1999. Inverse dependence and the Allee effect, Trends in Ecology and Evolution Vol 14, No. 10, 405-410.
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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
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Resumenes 33 The minimum ratio is t
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Resumenes 35 [1] On the role of end
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Resumenes 37 Modelo matemático de
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Resumenes 39 Figure 1: Plano de fas
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Resumenes 41 Dinámica de poblacion
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Resumenes 43 El método de las solu
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Resumenes 45 Dinámica de un tumor
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Resumenes 47 Several cancer models
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Resumenes 49 process is physically
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Resumenes 51 For the analysis of fr
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Resumenes 53 Sustainability in floo
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Resumenes 55 Sincronismo em redes d
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Resumenes 57 Um modelo de autômato
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Resumenes 59 Um modelo matemático
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Resumenes 61 Redes de mapas acoplad
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Resumenes 63 Sobre la dinámica de
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Resumenes 65 Un modelo de depredaci
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Resumenes 67 There exists wide evid
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Resumenes 69 Modelado del control n
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Resumenes 71 Modelo de red neuronal
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Resumenes 73 Efectos de la radiaci
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Resumenes 75 Un modelo matemático
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Resumenes 77 Estudio comparativo de
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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
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
Page 137 and 138: Resumenes 135 Modelando el efecto A
Page 139 and 140: Resumenes 137 the Allee effect are
Page 141 and 142: Resumenes 139 Tratamientos de endom
Page 143: Resumenes 141 Implicaciones del efe
Page 147 and 148: Resumenes 145 Dinámica individual
Page 149 and 150: Resumenes 147 • f mixing ⇔ f mi
Page 151 and 152: Resumenes 149 Introduction The inve
Page 153 and 154: Resumenes 151 DNA microarrays are t
Page 155 and 156: Resumenes 153 V. Cholerae: Primeros
Page 157 and 158: Resumenes 155 Análisis cuantitativ
Page 159 and 160: Resumenes 157 Dinámica de un model
Page 161 and 162: Resumenes 159 Obtención de paráme
Page 163 and 164: Resumenes 161 [4] Nagy K. A (1987)
Page 165 and 166: Resumenes 163 diferencial automáti
Page 167 and 168: Resumenes 165 The localized peaks S
Page 169 and 170: Resumenes 167 obteniéndose que sie
Page 171 and 172: Resumenes 169 [CLARK 90] Clark, C.
Page 173 and 174: Resumenes 171 Numerical algorithms
Page 175 and 176: Resumenes 173 The second method of
Page 177 and 178: Resumenes 175 los parámetros son t
Page 179 and 180: Resumenes 177 Bifurcación hacia at
Page 181 and 182: Resumenes 179 Malaria is the tropic
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