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  • br Numerical simulations br In this section


    4. Numerical simulations
    In this section, we complete the theoretical analysis with nu-merical simulations of system (3) showing how the patterning be-havior of solution changes depending only on few parameters, as commented in the previous sections. All simulations have been performed with the software Freefem++, which is based on fi-nite Thonzonium Bromide methods (Hecht, 2012). They have been carried out on a circular domain of radius R = 20, discretized with at least 500 points mesh.
    4.1. Typical behavior
    When the parameters are such that they are near the insta-bility region (6), the typical behavior of the solutions is the one showed in Fig. 4, where we set A = 70, ε = 0.01, r0 = 0.1 and used the logistic chemotactical sensitivity function with nmax = 1, i.e.,
    ϕ (n) = n(1 − n)+ . Starting from a small perturbation of the con-stant initial distribution n0 = 0.1, in few time steps the density grows uniformly in the domain until IC(t) in (6) is satisfied, in
    this case around t ≈ 2.5. Pattern formation then occurs very quickly, compared to the growth dynamics. When patterns are formed, the solution evolves very slowly and the most visible phenomenon is the merging of the spheroidal aggregates. Eventually, they become less regular, round structures, as at t = 17.5.
    4.2. Dependence on the initial mass
    If the initial mass is too small, the initial phase of growth takes much longer and the patterns arise later. For example in Fig. 5, where the solution of the model with the initial distribu-tion value n0 = 0.05 (leaving all the other parameters unchanged) is displayed, the first patterns appear only around t ≈ 7.5.
    4.3. Dependence on the parameter A
    In Fig. 6, we show how the kind of patterns observed strongly depends on the value of the quantity χ /D1. In these simulations, we chose Thonzonium Bromide A = 200 and the other parameters as in Fig. 4. In this case the diffusivity of cells is not as strong as the chemotactical attraction, which dominates the dynamics leading to smaller but more numerous spheroids. Moreover, in the first phases of dynam-ics, slightly more non-spheroidal and of variable geometry patterns arise.
    Finally, Fig. 7 shows solutions of the system (3) with ϕ(n) = ne−n. For this chemotactical sensitivity function, we find again spheroidal aggregates, but with less variable structures: in this case, variability lies in the maximums of the solution (higher than the ones in the logistic case), but only very round aggregates can be observed. 
    While the exponential function is usually advocated for because it does not impose an a priori maximal density, we here highlight a drawback of this choice: since the packing can continue even at high density, this fixes the size of patterns as new cells do not go at the periphery but instead concentrate at higher and higher den-sity at the middle. Thus, this choice of function offers less variabil-ity for the size of spheroids in two dimensions.
    Note that the numerical schemes do not preserve desirable properties, such as positivity of the solution. In fact, negative val-ues in the densities can be seen mostly at the interfaces between zones of high densities (the spheroids), and zones of low densi-ties (the rest). We were able to take fine enough grids to limit these numerical artifacts but we point out that some positivity-preserving numerical schemes have been designed to solve sys-tems of the type of (1), see for example (Almeida et al., 2019) and (Liu et al., 2018).
    5. Discussion and conclusions
    In order to understand the experimental observation of spheroidal aggregates in cultures of breast cancer cells, we have proposed a mathematical model which includes a chemotactic ef-