Search

In this paper, we consider a nonlocal advection model for two populations on a bounded domain. The first part of the paper is devoted to the existence and uniqueness of solutions and the associated semi-flow properties. By employing the notion of solution integrated along the characteristics, we rigorously prove the segregation property of solutions. Furthermore, we construct an energy functional to investigate the asymptotic behavior of solutions. To resolve the lack of compactness of the positive orbits, we obtain a description of the asymptotic behavior of solutions by using the narrow convergence in the space of Young measures. The last section of the paper is devoted to numerical simulations, which confirm and complement our theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we discuss a cell–cell repulsion model based on a hyperbolic Keller–Segel equation with two populations, which aims at describing the cell growth and dispersion in the co-culture experiment from the work of Pasquier et al. (Biol Direct 6(1):5, 2011). We introduce the notion of solution integrated along the characteristics, which allows us to prove the existence and uniqueness of solutions and the segregation property for the two species. From a numerical perspective, we also observe that our model admits a competitive exclusion principle which is different from the classical competitive exclusion principle for the corresponding ODE model. More importantly, our model shows the complexity of the short term (6 days) co-cultured cell distribution depending on the initial distribution of each species. Through numerical simulations, we show that the impact of the initial distribution on the proportion of each species in the final population lies in the initial number of cell clusters and that the final proportion of each species is not influenced by the precise distribution of the initial distribution. We also find that a fast dispersion rate gives a short-term advantage while the vital dynamics contributes to a long-term population advantage. When the initial condition for the two species is not segregated, the numerical simulations suggest that asymptotic segregation occurs when the dispersion coefficients are not equal for two populations. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability and the bifurcation properties of the positive interior equilibrium for a reaction-diffusion equation with nonlocal advection. Under rather general assumption on the nonlocal kernel, we first study the local well posedness of the problem in suitable fractional spaces and we obtain stability results for the homogeneous steady state. As a special case, we obtain that “standard” kernels such as Gaussian, Cauchy, Laplace and triangle, will lead to stability. Next we specify the model with a given step function kernel and investigate two types of bifurcations, namely Turing bifurcation and Turing-Hopf bifurcation. In fact, we prove that a single scalar equation may display these two types of bifurcations with the dominant wave number as large as we want. Moreover, similar instabilities can also be observed by using a bimodal kernel. The resulting complex spatiotemporal dynamics are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]