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Resumenes 168 phenomenon has been emphasized by recent developments in diverse areas of ecology and conservation, and can be regarding as the basis of animal sociability [STE 99]. Mathematically, more simple models can show enough about this dynamic [STE 99], and it can implied the apparition of new equilibrium points that changes the structural stability of system and to occur changes in the stability of other equilibrium points. Usually the Allee effect it is expresses modifying the logistic growth function by including a new factor x - m as in [BRAUER 01, CLARK 90], obtaining that. dx ⎛ x ⎞ = r⎜1 − ⎟( x − m)x dt ⎝ K ⎠ where x = x(t) indicates the prey population size that in this work we name multiplicative Allee effect. In [MEN 04] the consequences that it implies the Allee effect in the well kwown Rosenzweig-McArthur model [TURC 03], modifying the logistic growth function of prey for a function that includes a new factor (x - m) to represent the Allee effect or additional mortality, being x = x(t) the prey population size. If x < m , the growth of prey population is negative and tends to 0. Here, the alternative proposed in [STE 99, WANG 99] is used, modifying the natural growth function by adding an extra mortality term for nx represent the Allee effect, which is described by the function A( x) = , x + b which increase with density; but when x tends to 0, this function tends to zero. In the book by Thieme [THI 03] a mathematical deduction is made of this form that we call additive Allee effect. Now, the interaction is represented for the bidimensional Kolmogorov type differential equations system given by [FREED 1980]: dx ⎛ x n ⎞ qxy = rx⎜1− − ⎟− dt ⎝ K x + b ⎠ x + a dy ⎛ px ⎞ = ⎜ −c⎟y 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. It can demonstrate that two fom for Allee effect are topologically equivalent, but we study if the consequences of aditve form are similar to multiplicative form. We make a qualitative analysis of model using theory of dynamical systems and we obtain conditions in the parameter space for to stablish the bifurcation diagram, determinig the quantity of equilibrium point at the first cuadrant, its nature, the existence of sepratrix curves, hetroclinic or homoclinic curves, the number of limit cycles, and so on. Referencias [BRAUER 01] Brauer, F., and Castillo-Chávez, C., 2001 Mathematical models in Population Biology and Epidemiology, TAM 40, Springer.Verlag.
Resumenes 169 [CLARK 90] Clark, C. W., 1990. Mathematical Bioeconomic. The optimal management of renewable resources, John Wiley and Sons. [COURC 99] 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. [DENN 89] Dennis, B., 1989. Allee effects: population growth, critical density, and the chance of extinction, Natural Resource Modeling, Vol 3, No. 4, 481-538. [FREED 80] Freedman, H. I., 1990. Deterministic Mathematical Model in Population Ecology. Marcel Dekker. [KEN 03] Kent, A., Doncaster, C. P. and Sluckin, T. , 2003. Consequences for predators of rescue and Allee effects on prey, Ecological Modelling, Vol 162, 233-245. [LIER 01] Liermann, M. and R. Hilborn, 2001, Depensation: evidence models and implications, Fish and Fisheries, Vol. 2 33-58. [MEN 04] Meneses-Alcay, H. and González-Olivares, E., 2004. Consequences of the Allee effect on Rosenzweig-McArthur predator-prey model, In R. Mondaini (Ed.) Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology, E-Papers Serviços Editoriais Ltda, Rio de Janeiro, Vol. 2 264-277. [STE 99] Stephens, P. A. and Sutherland, W. J. , 1999. Consequences of the Allee effect for behaviour, ecology and conservation. Trends in Ecol. Evo., Vol.14 n. 10, 401-405. [THI 03] Thieme, H. R. Mathematics in Population Biology, Princeton University Press. [TURC 03] Turchin, P., 2003 Complex population dynamics: a theoretical/ empirical synthesis, Princeton University Press . [WANG 99] Wang, G., Liang, X-G and Wang, F-Z. , 1999. The competitive dynamics of populations subject to an Allee effect, Ecological Modelling, Vol 124 183- 192.
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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
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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
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 and 144: Resumenes 141 Implicaciones del efe
Page 145 and 146: Resumenes 143 phenomenon has been e
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: Resumenes 167 obteniéndose que sie
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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