Stability Analysis of a Population Dynamics Model with Allee Effect

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Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA Stability Analysis of a Population Dynamics Model with Allee Effect Canan Celik Abstract— In this study, we focus on the stability analysis of equilibrium points of population dynamics with delay when the Allee effect occurs at low population density is considered. Mainly, mathematical results and numerical simulations illustrate the stabilizing effect of the Allee effects on population dynamics with delay. Keywords: Delay difference equations, Allee effect, population dynamics, stability analysis, bifurcation 1 Introduction Dynamic population models are generally described by the differential and difference equations with or without delay. These models have been considered by many authors, for these related results we refer to [1], [4], [6]-[10]. In 1931, Allee [1] demonstrated that a negative density dependence, the so called Allee effect, occurs when population growth rate is reduced at low population size. The Allee effect refers to a population that has a maximal per capita growth rate at low density. This occurs when the per capita growth rate increases as density increases, and decreases after the density passes a certain value which is often called threshold. This effect can be caused by difficulties in, for example, mate finding, social dysfunction at small population sizes, inbreeding depression, food exploitation, and predator avoidance of defence. The Allee effects have been observed on different organisms, such as vertebrates, invertebrates and plants. (see, for instance, [2], [3]). The purpose of this paper is to study the following general non-linear delay difference equation with or without the Allee effect Nt+1 = F (λ, Nt , Nt−T ), (1) where λ is per capita growth rate which is always positive, Nt represents the population density at time t, T is the time sexual maturity. Here, F has the following form ∗ increases. Mainly, we work on the stability analysis of this model and compare the stability of this model with or without Allee effects. Eq. (1) is an appropriate model for single species without an Allee effect. Therefore, a natural question arising here is that ”How the stability of equilibrium points are effected when an Allee effect is incorporated in Eq. (1)” . In this work, we answer this question, especially, for the case when T = 1. 2 Stability analysis of Eq. (1) for T = 1 Before we give the main results of this paper, we shall remind the following well-known linearized stability theorem (see, for instance, [5] and [9]) for the following nonlinear delay difference equation Nt+1 = F (Nt , Nt−1 ). (2) Theorem A (Linearized Stability). Let N ∗ be an equilibrium point of Eq. (2). Then N ∗ is locally stable if and only if |p| < 1 − q < 2, where p := ∂F ∂F (N ∗ , N ∗ ) and q := (N ∗ , N ∗ ). ∂Nt ∂Nt−1 We now consider the following non-linear difference equation with delay Nt+1 = λNt f (Nt−1 ) =: F (λ; Nt , Nt−1 ), λ > 0, (3) where λ is per capita growth rate, Nt is the density at time t, and f (Nt−1 ) is the function describing interactions (competitions) among individuals. Firstly, we assume that f satisfies the following conditions: F (λ; Nt , Nt−T ) := λNt f (Nt−T ) where f (Nt−T ) is the function describing interactions (competitions) among the mature individuals. It is generally assumed that f continuously decreases as density Economics and Technology University, Faculty of Arts and Sciences, Department of Mathematics, Ankara, Turkey, 06530, E-mail address: 1 f ′ (N ) < 0 for N ∈ [0, ∞); that is, f continuously decreases as density increases. 2 f (0) is a positive finite number. ∗ TOBB ISBN:978-988-98671-6-4 Then we have the following result. WCECS 2007 Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA Theorem 1 Let N ∗ be a (positive) equilibrium point of Eq. (3) with respect to λ. Then N ∗ is locally stable if and only if f ′ (N ∗ ) > −1. (4) N∗ f (N ∗ ) Proof. By hypothesis, we have 1 = λf (N ∗ ). (5) Let p := FNt (λ; N ∗ , N ∗ ). Then, the equality (5) implies that p = 1. Also, observe that q := FNt−1 (λ; N ∗ , N ∗ ) = N ∗ f ′ (N ∗ ) . f (N ∗ ) (6) Theorem A says that N ∗ is locally stable if and only |p| < 1 − q < 2. Since p = 1, we conclude that N ∗ is locally stable if and only if −1 < q < 0. However, since f is a decreasing function for all N , we get from (6) that the inequality q < 0 is always valid. So the proof is completed. Now we find a sufficient condition that increasing λ in Eq. (5) decreases the stability of the corresponding equilibrium points. Theorem 2 Let λ1 and λ2 be positive numbers such that λ1 < λ2 , and let N (1) and N (2) be corresponding positive equilibrium points of Eq. (3) with respect to λ1 and λ2 , respectively. Then the local stability of N (2) is weaker than N (1) provided that (2) N Z ′ [N (log f (N ))′ ] dN < 0 (7) N (1) holds; that is, increasing λ decreases the local stability of the equilibrium point in Eq. (3) if (7) holds. Proof. By the definitions of N (1) and N (2) , it is easy to see that 1 = λ1 f (N (1) ) and 1 = λ2 f (N (2) ). (8) Since λ1 < λ2 , it follows from (8) that f (N (1) ) > f (N (2) ). Also, since f is decreasing function, we have N (1) < N (2) . Now, for each i = 1, 2, qi := FNt−1 (λi ; N (i) , N (i) ). Then, we can easily get that qi = N (i) f ′ (N (i) ) for i = 1, 2 f (N (i) ) holds. So, by Theorem 2.1, each N (i) is locally stable if and only if N (i) f ′ (N (i) ) > −1 for i = 1, 2. f (N (i) ) (9) If the condition (7) holds, then we have N f ′ (N (2) ) (N (1) ) > N (2) . (1) f (N ) f (N (2) ) (1) f ′ ISBN:978-988-98671-6-4 (10) Now by considering (9) and (10) we can say that the local stability in N (2) is weaker than N (1) , which completes the proof. The following condition is weaker than (7), but it enables us to control easily for increasing λ being a destabilizing parameter. Corollary 3 Let λ1 , λ2 , N (1) and N (2) be the same as in Theorem 2.2. Then the local stability of N (2) is weaker than that of N (1) provided that £ ′ ¤′ (11) N (log f (N )) < 0 for all N ∈ [N (1) , N (2) ]. Remark If there is no time delay in model (3), i.e., Nt+1 = λNt f (Nt ) =: F (λ; Nt ), λ > 0, (12) then we have the same result as in theorem 2 where the stability condititon reduces to N (i) f ′ (N (i) ) > −2 for i = 1, 2. f (N (i) ) (13) See Example 2 below. Example 1 Consider the difference equation ¶ µ Nt−1 Nt+1 = λNt 1 − , λ > 0 and K > 0, K (14) with the initial values N−1 and N0 . An obvious drawback of this specific model is that if Nt−1 > K and Nt+1 < 0. However, if we assume 0 < N−1 < K and 0 < N0 < K, then, by choosing appropriate λ > 0, we can guarantee that Nt > 0 for any time t. According to this example f (N ) = 1−N/K. In this case, we have, for all 0 < N < K, £ ′ ¤′ N (log f (N )) = − K 2 (K − N ) < 0. So, by Corollary 2, we easily see that if the λ increases, then the local stability in the corresponding equilibrium point of Eq. (14) decreases. Example 2 Consider the difference equation with no delay term Nt+1 = λNt exp[r(1 − Nt /K)] (15) λ > 0, r > 0 and K > 0,with the initial value N0 . According to this model f (N ) = exp[r(1 − Nt /K)]. In this case, we have, for all N > 0, £ r ′ ¤′ N (log f (N )) = −N < 0. K So, Corollary 2 immediately implies that if λ increases, then the local stability of the corresponding equilibrium point of Eq. (15) decreases. WCECS 2007 Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA 3 Allee effects on the discrete delay Since N1∗ ∈ (0, Nc ), it is easy to see that q1∗ > 0, which implies that N1∗ must be unstable. On the other hand, model (3) for N2∗ ∈ (Nc , +∞) the inequality q2∗ < 0 always holds so that N2∗ is locally stable if and only if (17) holds. Allee effect at time t − 1 3.1 To incorporate an Allee effect into the discrete delay model (3) we first consider the following non-linear difference equation with delay − Nt+1 = λ∗ Nt a(Nt−1 )f (Nt−1 ) =: F (a ) (λ; Nt , Nt−1 ), (16) where λ∗ > 0 and the function f satisfies the conditions 1 and 2 . As stated in the first chapter, the biological facts lead us to the following assumptions on a : 3 if N = 0, then a(N ) = 0; that is, there is no reproduction without partners, 4 a′ (N ) > 0 for N ∈ (0, ∞); that is, Allee effect decreases as density increases, 5 limN →∞ a(N ) = 1; that is, Allee effect vanishes at high densities. By the conditions 1 − 5 , Eq. (16) has at most two positive equilibrium points which satisfy the equation of 1 = λ∗ a(N )f (N ) := h(N ). Assume now that two equilibrium points N1∗ and N2∗ (N1∗ < N2∗ ) exist so that we have h(N1∗ ) = h(N2∗ ) = 1. In this case, by the mean value theorem, there exists a critical point Nc such that h′ (Nc ) = 0 and N1∗ < Nc < N2∗ . Now, choose λ = λ∗ a(N2∗ ) and consider the following Nt+1 = λNt f (Nt−1 ) =: F (λ; Nt , Nt−1 ) (19) N2∗ Then observe that is also positive equilibrium point of Eq. (19). Furthermore, since a(N2∗ ) < 1 and λ = λ∗ a(N2∗ ), we get λ < λ∗ . We have the following result. Theorem 5 The Allee effect increases the stability of the equilibrium point, that is, the local stability of the equilibrium point in Eq. (16) is stronger than that of in Eq. (19). Proof. By the definition of a, for each i = 1, 2, we get µ ′ ∗ ¶ a (N2 ) ∗ N2 > 0, a(N2∗ ) which yields N2∗ f ′ (N2∗ ) < N2∗ f (N2∗ ) µ a′ (N2∗ ) f ′ (N2∗ ) + a(N2∗ ) f (N2∗ ) ¶ . (20) Now considering (20), Theorem 2 and Theorem 3.1 we can say that the stability in Eq. (16) is stronger than the stability in Eq. (19). Observe that Theorem 5 implies the following result. Corollary 6 There is a locally stable equilibrium point for Eq. (16) such that it is unstable for Eq. (3) Then we have the following theorem. 3.2 Theorem 4 The equilibrium point N1∗ of Eq. (16) is unstable. On the other hand, N2∗ is locally stable if and only if the inequality ¶ µ ′ ∗ a (N2 ) f ′ (N2∗ ) ∗ + > −1 (17) N2 a(N2∗ ) f (N2∗ ) In this part we incorporate an Allee effect into the discrete delay model (3) as follows: holds. Proof. Notice that since for each i = 1, 2, Ni∗ is positive equilibrium point of Eq. (16), we have (a− ) p∗i := FNt (λ∗ , Ni∗ , Ni∗ ) = λ∗ a(Ni∗ )f (Ni∗ ) = 1 Allee effect at time t + Nt+1 = λNt a(Nt )f (Nt−1 ) =: F (a ) (λ; Nt , Nt−1 ). (21) Assume now that N1∗ and N2∗ (N1∗ < N2∗ ) are positive equilibrium points of Eq. (21). Then we have the following theorem. Theorem 7 We get that N1∗ is an unstable equilibrium point of Eq. (21). On the other hand, N2∗ is locally stable if and only if the inequality and N2∗ (a− ) qi∗ := FNt−1 (λ∗ , Ni∗ , Ni∗ ) = Ni∗ µ a (Ni∗ ) a(Ni∗ ) ′ ′ + f (Ni∗ ) f (Ni∗ ) Then, by Theorem A, we get Ni∗ is stable ⇔ |p∗i | < 1 − qi∗ < 2 ⇔ −1 < qi∗ < 0 for i = 1, 2. ISBN:978-988-98671-6-4 ¶ . (18) f ′ (N2∗ ) > −1. f (N2∗ ) (22) holds. Proof. Since, for each i = 1, 2, Ni∗ is positive equilibrium point of Eq. (16), we have (a+ ) p∗i := FNt (λ∗ , Ni∗ , Ni∗ ) = 1 + Ni∗ a′ (Ni∗ ) a(Ni∗ ) WCECS 2007 Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA and (a− ) qi∗ := FNt−1 (λ∗ , Ni∗ , Ni∗ ) = Ni∗ f ′ (Ni∗ ) . f (Ni∗ ) is stable ⇔ ⇔ ⇔ |p∗i | < 1 − qi∗ < 2 f ′ (N ∗ ) a′ (N ∗ ) 1 + Ni∗ a(N ∗i ) < 1 − Ni∗ f (N ∗i ) i i a′ (N ∗ ) f ′ (N ∗ ) Ni∗ a(N ∗i ) < −Ni∗ f (N ∗i ) < 1 i i < 2. 0.3 0.6 0.25 0.2 a′ (Ni∗ ) f ′ (Ni∗ ) + a(Ni∗ ) f (Ni∗ ) < 0 < 1 + Ni∗ f ′ (Ni∗ ) . f (Ni∗ ) Since N1∗ ∈ (0, Nc ), it is easy to see that ¶ µ ′ ∗ a (N1 ) f ′ (N1∗ ) + > 0, N1∗ a(N1∗ ) f (N1∗ ) which implies that N1∗ is an unstable equilibrium point. Furthermore, since N2∗ ∈ (Nc , +∞), the inequality µ ′ ∗ ¶ a (N2 ) f ′ (N2∗ ) ∗ N2 <0 + a(N2∗ ) f (N2∗ ) Population density Ni∗ ¶ (c) 0.65 i.e., µ 0.7 (a) 0.55 0 10 20 30 Time 40 50 (b) 0.5 Population density Ni∗ Population density Then, by Theorem A, we get for each i = 1, 2 0.5 0.45 0.4 0.35 0.4 0.3 0.3 0.2 0.25 0 10 20 30 Time 40 50 0.2 0 50 Time 100 Figure 1: Density-time graphs of the model (14) with K = 1 and the initial conditions N−1 = 0.2 and N0 = 0.3. (a) λ = 1.4. (b) λ = 1.7. (c) λ = 1.95. always holds. So N2∗ is locally stable if and only if (22) holds. Let N ∗ be a positive equilibrium point of Eq. (3). So, by the choice of λ = λ∗ a(N ∗ ), N ∗ is also an equilibrium point of Eq. (21). In this case, combining Theorem 1 with Theorem 7, we get the following result at once. 0.65 Corollary 8 Equilibrium point N ∗ is locally stable for Eq. (21) if and only if it is locally stable for Eq. (3). 0.6 4 Numerical simulations In this section, we numerically verify our analytical results (or theorems) obtained in previous sections by using MATLAB programming. In Matlab programs, by taking the initial conditions of the population density as N−1 and N0 in the difference equations we compute Ni ’s for the model that we discussed in previous sections. After computing Ni ’s, mainly we illustrate the graph of the trajectory of models in 2D graph in which we can see the stability of equilibrium points. Note that in each graph, we connect the discrete points of population by straight lines. As we prove analytically in Theorem 2, if we graph the population density function Nt with respect to t (time) in the model (14) of Example 1, we can easily see in Figure 1- (a)-(b)-(c) that as the parameter λ increases, the local stability of the equilibrium points decreases. Population density 0.55 0.5 0.45 0.4 0.35 0.3 without Allee effect with Allee effect 0.25 0.2 0 10 20 30 Time 40 50 60 Figure 2: Density-time graphs of the models Nt+1 = λNt (1−Nt−1 /K) and Nt+1 = λ∗ Nt a(Nt−1 )(1−Nt−1 /K) with K = 1, λ = 1.9, a(Nt−1 ) = Nt−1 /(α + Nt−1 ), α = 0.03, λ = λ∗ a(N ∗ ) and the initial conditions N−1 = 0.2 and N0 = 0.3. In Figure 2 we graph the 2D trajectory of the population dynamics model (14) respectively with and without Allee ISBN:978-988-98671-6-4 WCECS 2007 Proceedings of the World Congress on Engineering and Computer Science 2007 WCECS 2007, October 24-26, 2007, San Francisco, USA effect at time t−1, which verifies our Theorem 4. In these figures we take the Allee effect function as a(Nt−1 ) = Nt−1 /(α + Nt−1 ), where α is a positive constant. As we can see from the graph that population density function obviously verify that when we impose the Allee effect at time t−1 into our model in Example 1, the local stability of the equilibrium point increases and trajectory of the solution approximates to the corresponding equilibrium point much faster. [1] Allee WC. Animal Aggretions: A Study in General Sociology, University of Chicago Press, Chicago, 1931. 2 1.5 Nt+1/Nt [2] Beddington JR. ”Age distribution and the stability of simple discrete time population models”, J. Theor. Biol. V47, pp.65-74,74. 1 [3] Birkhead TR. ”The effect of habitat and density on breeding success in the common guillemot (uria aalge)”, J. Anim. Ecol. V46, pp.751-64, 77. 0.5 2 2.1 2.2 2.3 Growth rate 2.4 2.5 2.6 Figure 3: Bifurcation diagram of the models in Figure 2 with K = 1, α = 0.06, λ = 1.9 : 0.001 : 2.3, λ = λ∗ a(N ∗ ) and the initial conditions N−1 = 0.2 and N0 = 0.3. Finally, Figure-3 shows bifurcation diagram of the model in Example 1 as a function of intrinsic growth rate λ without the Allee effect (on the left) and with the Allee effect (on the right). In this numeric simulations, N−1 = 0.2 and N0 = 0.3 are taken as the initial conditions as in the previous calculations. Here we again take the Allee function as a(Nt−1 ) = Nt−1 /(α + Nt−1 ), where α is a positive constant. It is obvious from the graph that the comparison of bifurcation diagrams still verifies the stabilizing effect of Allee effect. Besides this result, we also observed that the Allee effect diminishes the fluctuations in the chaotic dynamic which is different from the results obtained in the former studies. 5 Acknowledgement: This study includes some results of ”Allee effects on population dynamics with delay”, (C. Çelik, H. Merdan, O. Duman, Ö. Akın), Chaos, Solitons and Fractals. References 2.5 0 1.9 effect into the system, the stability of equilibrium points were also studied. Keeping the normalized growth rate the same, we compared the stability of the same equilibrium point corresponding to the model with and without Allee effect. Conclusions and remarks Previous studies demonstrate that Allee effects play an important role on the stability analysis of equilibrium points of a population dynamics model (see, for example, [10] and [6]). Generally, an Allee effect has a stabilizing effect on population dynamics. In this paper, we consider a second order discrete models (i.e. with delay), where increasing per capita growth rate decreases the stability of the fixed point. First, we characterized the stability of equilibrium point(s) of this model. Imposing an Allee ISBN:978-988-98671-6-4 [4] Chen J and Blackmore D. ”On the exponentially self-regulating population model”, Chaos, Solitons & Fractals, V14, pp.1433-50, 02. [5] Cunningham K, Kulenović MRS, Ladas G and Valicenti SV. ”On the recursive sequences xn+1 = (α + βxn )/(Bxn + Cxn−1 )”, Nonlinear Anal. , V47, pp.4603-14, 01. [6] Fowler MS and Ruxton GD. ”Population dynamic consequences of Allee effects”, J. Theor. Biol., V215, pp.39-46, 02. [7] Gao S and Chen L. ”The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses”, Chaos, Solitons & Fractals, V24, pp.1013-23, 05. [8] Hadjiavgousti D and Ichtiaroglou S. ”Existence of stable localized structures in population dynamics through the Allee effect”, Chaos, Solitons & Fractals, V21, pp.119-31, 04. [9] Kulenović MRS, Ladas G and Prokup NR. ”A rational difference enfocar la importancia de la producción mediática de los niños en su descubrimiento del mundo, sobre todo utilizando el periódico escolar y la imprenta. Asimismo las asociaciones de profesores trabajaron en esta línea e incluso la enseñanza católica se comprometió desde los años sesenta realizando trabajos originales en el marco de la corriente del Lenguaje Total. Páginas 43-48 45 Comunicar, 28, 2007 En el ámbito de los medios, también desde el principio del siglo XX hay ciertas corrientes de conexión. Pero es a lo largo de los años sesenta cuando se constituyeron asociaciones de periodistas apasionados por sus funciones de mediadores, que fomentaron la importancia ciudadana de los medios como algo cercano a los jóvenes, a los profesores y a las familias. Así se crearon la APIJ (Asociación de Prensa Información para la Juventud), la ARPEJ (Asociación Regional de Prensa y Enseñanza para la Juventud), el CIPE (Comité Interprofesional para la Prensa en la Escuela) o la APE (Asociación de Prensa y Enseñanza), todas ellas para la prensa escrita Estas asociaciones fueron precedidas por movimientos surgidos en mayo de 1968, como el CREPAC que, utilizando películas realizadas por periodistas conocidos, aclaraba temas que habían sido manipulados por una televisión demasiado próxima al poder político y realizaba encuentros con grupos de telespectadores. cipio del siglo XX, y nos han legado textos fundadores muy preciados, importantes trabajos de campo y muchos logros educativos y pedagógicos. La educación en medios ha tenido carácter de oficialidad de múltiples maneras, aunque nunca como una enseñanza global. Así la campaña «Operación Joven Telespectador Activo» (JTA), lanzada al final de los años setenta y financiada de manera interministerial para hacer reflexionar sobre las prácticas televisuales de los jóvenes, la creación del CLEMI (Centro de Educación y Medios de Comunicación) en el seno del Ministerio de Educación Nacional en 1983, la creación de la optativa «Cine-audiovisual» en los bachilleratos de humanidades de los institutos en 1984 (primer bachillerato en 1989) y múltiples referencias a la educación de la imagen, de la prensa, de Internet. La forma más visible y rápida de evaluar el lugar de la educación en medios es valorar el lugar que se le ha reservado en los libros de texto del sistema educa- 2. Construir la educación en los medios sin nombrarla El lugar que ocupa la edu- La denominación «educación en medios», que debería cación en los medios es muy ambiguo, aunque las cosas están cambiando recientemente. entenderse como un concepto integrador que reagrupase todos los medios presentes y futuros, es a menudo percibida En principio, en Francia, co- por los «tradicionalistas de la cultura» como una tendencia mo en muchos otros países, la educación en los medios no es hacia la masificación y la pérdida de la calidad. una disciplina escolar a tiempo completo, sino que se ha ido conformado progresivamente a través de experiencias y reflexiones teóricas que han tivo en Francia. Una inmersión sistemática nos permi- permitido implantar interesantes actividades de carác- te constatar que los textos oficiales acogen numerosos ter puntual. Se ha ganado poco a poco el reconoci- ejemplos, citas, sin delimitarla con precisión. miento de la institución educativa y la comunidad es- colar. Podemos decir que ha conquistado un «lugar», 3. ¿Por qué la escuela ha necesitado casi un siglo en el ámbito de la enseñanza transversal entre las dis- para oficilializar lo que cotidianamente se hacía en ciplinas existentes. ella? Sin embargo, la escuela no está sola en esta aspi- Primero, porque las prácticas de educación en me- ración, porque el trabajo en medios es valorado igual- dios han existido antes de ser nombradas así. Recor- mente por el Ministerio de Cultura (campañas de foto- demos que no fue hasta 1973 cuando aparece este grafía, la llamada «Operación Escuelas», presencia de término y que su definición se debe a los expertos del colegios e institutos en el cine ), así como el Minis- Consejo Internacional del Cine y de la Televisión, que terio de la Juventud y Deportes que ha emprendido en el seno de la UNESCO, definen de esta forma: numerosas iniciativas. «Por educación en medios conviene entender el estu- Así, esta presencia de la educación en los medios dio, la enseñanza, el aprendizaje de los medios moder- no ha sido oficial. ¡La educación de los medios no apa- nos de comunicación y de expresión que forman parte rece oficialmente como tal en los textos de la escuela de un dominio específico y autónomo de conocimien- francesa hasta 2006! tos en la teoría y la práctica pedagógicas, a diferencia Este hecho no nos puede dejar de sorprender ya de su utilización como auxiliar para la enseñanza y el que las experiencias se han multiplicado desde el prin- aprendizaje en otros dominios de conocimientos tales Páginas 43-48 46 Comunicar, 28, 2007 como los de matemáticas, ciencias y geografía». A pe- mente en todas las asignaturas. Incluso los nuevos cu- sar de que esta definición ha servido para otorgarle un rrículos de materias científicas en 2006 para los alum- reconocimiento real, los debates sobre lo que abarca y nos de 11 a 18 años hacen referencia a la necesidad no, no están totalmente extinguidos. de trabajar sobre la información científica y técnica y En segundo lugar, porque si bien a la escuela fran- el uso de las imágenes que nacen de ella. cesa le gusta la innovación, después duda mucho en Desde junio de 2006, aparece oficialmente el tér- reflejar y sancionar estas prácticas innovadoras en sus mino «educación en medios» al publicar el Ministerio textos oficiales. Nos encontramos con una tradición de Educación los nuevos contenidos mínimos y las sólidamente fundada sobre una transmisión de conoci- competencias que deben adquirir los jóvenes al salir mientos muy estructurados, organizados en disciplinas del sistema educativo. escolares que se dedican la mayor parte a transmitir Este documento pretende averiguar cuáles son los conocimientos teóricos. La pedagogía es a menudo se- conocimientos y las competencias indispensables que cundaria, aunque los profesores disfrutan de una ver- deben dominar para terminar con éxito su escolaridad, dadera libertad pedagógica en sus clases. El trabajo seguir su formación y construir su futuro personal y crítico sobre los medios que estaba aún en elaboración profesional. Siete competencias diferentes han sido te- necesitaba este empuje para hacerse oficial. nidas en cuenta y en cada una de ellas, el trabajo con Aunque el trabajo de educación en los medios no los medios es reconocido frecuentemente. Para citar esté reconocido como disciplina, no está ausente de un ejemplo, la competencia sobre el dominio de la len- gua francesa definen las capa- cidades para expresarse oral- La metodología elaborada en el marco de la educación en mente que pueden adquirirse con la utilización de la radio e, medios parece incluso permitir la inclinación de la sociedad incluso, se propone fomentar de la información hacia una sociedad del conocimiento, como defiende la UNESCO. En Francia, se necesitaría unir el interés por la lectura a través de la lectura de la prensa. La educación en los medios las fuerzas dispersas en función de los soportes mediáticos y orientarse más hacia la educación en medios que al dominio adquiere pleno derecho y entidad en la sección sexta titulada «competencias sociales y cívi- técnico de los aparatos. cas» que indica que «los alum- nos deberán ser capaces de juz- gar y tendrán espíritu crítico, lo que supone ser educados en los las programaciones oficiales, ya que, a lo largo de un medios y tener conciencia de su lugar y de su influencia estudio de los textos, los documentalistas del CLEMI en la sociedad». han podido señalar más de una centena de referencias a la educación de los medios en el seno de disciplinas 4. Un entorno positivo como el francés, la historia, la geografía, las lenguas, Si nos atenemos a las cifras, el panorama de la las artes plásticas : trabajos sobre las portadas de educación en medios es muy positivo. Una gran ope- prensa, reflexiones sobre temas mediáticos, análisis de ración de visibilidad como la «Semana de la prensa y publicidad, análisis de imágenes desde todos los ángu- de los medios en la escuela», coordinada por el CLE- los, reflexión sobre las noticias en los países europeos, MI, confirma año tras año, después de 17 convocato- información y opinión rias, el atractivo que ejerce sobre los profesores y los Esta presencia se constata desde la escuela mater- alumnos. Concebida como una gran operación de nal (2 a 6 años) donde, por ejemplo, se le pregunta a complementariedad source of circulating FGF-21. The lack of association between circulating and muscle-expressed FGF-21 also suggests that muscle FGF-21 primarily works in a local manner regulating glucose metabolism in the muscle and/or signals to the adipose tissue in close contact to the muscle. Our study has some limitations. The number of subjects is small and some correlations could have been significant with greater statistical power. Another aspect is that protein levels of FGF-21 were not determined in the muscles extracts, consequently we cannot be sure the increase in FGF-21 mRNA is followed by increased protein expression. In conclusion, we show that FGF-21 mRNA is increased in skeletal muscle in HIV patients and that FGF-21 mRNA in muscle correlates to whole-body (primarily reflecting muscle) insulin resistance. These findings add to the evidence that FGF-21 is a myokine and that muscle FGF-21 might primarily work in an autocrine manner. Acknowledgments We thank the subjects for their participation in this study. Ruth Rousing, Hanne Willumsen, Carsten Nielsen and Flemming Jessen are thanked for excellent technical help. The Danish HIV-Cohort is thanked for providing us HIV-related data. 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