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Application of aggregation of variables methods to a class of two-time reaction-diffusion-chemotaxis models of spatially structured populations with constant diffusion

  • * Corresponding author: anouarelharrak1@gmail.com

    * Corresponding author: anouarelharrak1@gmail.com 
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  • The main goal of this paper is to adapt a class of complexity reduction methods called aggregation of variables methods to the construction of reduced models of two-time reaction-diffusion-chemotaxis models of spatially structured populations and to provide an error bound of the approximate dynamics. Aggregation of variables methods are general techniques that allow reducing the dimension of a mathematical dynamical system. Here we reduce a system of Partial Differential Equations to a simpler Ordinary Differential Equation system, provided that the evolution processes occur at two different time scales: a slow one for the demography and a fast one for migrations and chemotaxis, with a ratio $ \varepsilon>0 $ small enough. We give an approximation of the error between solutions of both original and reduced model for a generic function representing the demography. Finally, we provide an optimization of the error bound and validate numerically this result for a spatial inter-specific model with constant diffusion and population growth given by a logistic law in population dynamics.

    Mathematics Subject Classification: Primary: 37M99, 37N25; Secondary: 35K55.


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  • Figure 1.  Plot of the numerical solution of the perturbed problem (11), (A) and (C), and its approximate solution using aggregation of variables methods, (B) and (D), at two times $ t = 0.2 $ and $ t = 25 $ for $ \varepsilon = 1e-1 $

    Figure 2.  Plot of total population of the global model and its approximate solution using aggregation of variables methods for for $ \varepsilon = 1e-1 $. $ K_T: = \int_{\Omega}K(x)dx $ stands for the total carrying capacity of the environment and $ K^\ast $ for the new homogeneous one

    Figure 3.  Plot of errors, $ |N_{\varepsilon}(t)-N(t)| $, (A), and $ {\Vert n_{\varepsilon}(.,t)-\lambda(.)N(t) \Vert_\infty} $ with $ t\geq 1 $, (B), with respect to time $ t $ for different values of $ \varepsilon $; $ \varepsilon = 1e-1 $, $ \varepsilon = 1e-2 $, $ \varepsilon = 1e-3 $, and $ \varepsilon = 1e-4 $

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