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Register a new userEarly and late stage profiles for a new chemotaxis model with density-dependent jump probability and quorum-sensing mechanisms
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In this paper, we derive a new chemotaxis model with degenerate diffusion and density-dependent chemotactic sensitivity, and we provide a more realistic description of cell migration process for its early and late stages. Different from the existing studies focusing on the case of non-degenerate diffusion, the new model with degenerate diffusion causes us some essential difficulty on the boundedness estimates and the propagation behavior of its compact support. In the presence of logistic damping, for the early stage before tumour cells spread to the whole body, we first estimate the expanding speed of tumour region as $O(t^{beta})$ for $0<beta<frac{1}{2}$. Then, for the late stage of cell migration, we further prove that the asymptotic profile of the original system is just its corresponding steady state. The global convergence of the original weak solution to the steady state with exponential rate $O(e^{-ct})$ for some $c>0$ is also obtained.
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We investigate the propagating profiles of a degenerate chemotaxis model describing the bacteria chemotaxis and consumption of oxygen by aerobic bacteria, in particular, the effect of the initial attractant distribution on bacterial clustering. We prove that the compact support of solutions may shrink if the signal concentration satisfies a special structure, and show the finite speed propagating property without assuming the special structure on attractant concentration, and obtain an explicit formula of the population spreading speed in terms of model parameters. The presented results suggest that bacterial cluster formation can be affected by chemotactic attractants and density-dependent dispersal.
The aim of this paper is to analyze a model for chemotaxis based on a local sensing mechanism instead of the gradient sensing mechanism used in the celebrated minimal Keller-Segel model. The model we study has the same entropy as the minimal Keller-Segel model, but a different dynamics to minimize this entropy. Consequently, the conditions on the mass for the existence of stationary solutions or blow-up are the same, however we make the interesting observation that with the local sensing mechanism the blow-up in the case of supercritical mass is delayed to infinite time. Our observation is made rigorous from a mathematical point via a proof of global existence of weak solutions for arbitrary large masses and space dimension. The key difference of our model to the minimal Keller-Segel model is that the structure of the equation allows for a duality estimate that implies a bound on the $(H^1)$-norm of the solutions, which can only grow with a square-root law in time. This additional $(H^1)$-bound implies a lower bound on the entropy, which contrasts markedly with the minimal Keller-Segel model for which it is unbounded from below in the supercritical case. Besides, regularity and uniqueness of solutions are also studied.
Well-posedness and uniform-in-time boundedness of classical solutions are investigated for a three-component parabolic system which describes the dynamics of a population of cells interacting with a chemoattractant and a nutrient. The former induces a chemotactic bias in the diffusive motion of the cells and is accounted for by a density-suppressed motility. Well-posedness is first established for generic positive and non-increasing motility functions vanishing at infinity. Growth conditions on the motility function guaranteeing the uniform-in-time boundedness of solutions are next identified. Finally, for sublinearly decaying motility functions, convergence to a spatially homogeneous steady state is shown, with an exponential rate for consumption rates behaving linearly near zero.
The current series of three papers is concerned with the asymptotic dynamics in the following chemotaxis model $$partial_tu=Delta u-chi abla(u abla v)+u(a(x,t)-ub(x,t)) , 0=Delta v-lambda v+mu u (1)$$where $chi, lambda, mu$ are positive constants, $a(x,t)$ and $b(x,t)$ are positive and bounded. In the first of the series, we investigate the persistence and asymptotic spreading. Under some explicit condition on the parameters, we show that (1) has a unique nonnegative time global classical solution $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with $u(x,t_0;t_0,u_0)=u_0(x)$ for every $t_0in R$ and every $u_0in C^{b}_{rm unif}(R^N)$, $u_0geq 0$. Next we show the pointwise persistence phenomena in the sense that, for any solution $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ of (1) with strictly positive initial function $u_0$, then$$0<inf_{t_0in R, tgeq 0}u(x,t+t_0;t_0,u_0)lesup_{t_0in R, tgeq 0} u(x,t+t_0;t_0,u_0)<infty$$and show the uniform persistence phenomena in the sense that there are $0<m<M$ such that for any strictly positive initial function $u_0$, there is $T(u_0)>0$ such that$$mle u(x,t+t_0;t_0,u_0)le M forall,tge T(u_0), xin R^N.$$We then discuss the spreading properties of solutions to (1) with compactly supported initial and prove that there are positive constants $0<c_{-}^{*}le c_{+}^{*}<infty$ such that for every $t_0in R$ and every $u_0in C^b_{rm unif}(R^N), u_0ge 0$ with nonempty compact support, we have that$$lim_{ttoinfty}sup_{|x|ge ct}u(x,t+t_0;t_0,u_0)=0, forall c>c_+^*,$$and$$liminf_{ttoinfty}sup_{|x|le ct}u(x,t+t_0;t_0,u_0)>0, forall 0<c<c_-^*.$$We also discuss the spreading properties of solutions to (1) with front-like initial functions. In the second and third of the series, we will study the existence, uniqueness, and stability of strictly positive entire solutions and the existence of transition fronts, respectively.
We study a system of PDEs modeling the population dynamics of two competitive species whose spatial movements are governed by both diffusion and mutually repulsive chemotaxis effects. We prove that solutions to this system are globally well-posed, without any smallness assumptions on the chemotactic coefficients. Moreover, in the weak competition regime, we prove that neither species can be driven to extinction as the time goes to infinity, regardless of how strong the chemotaxis coefficients are. Finally, long-time behaviors of the system are studied both analytically in the weakly nonlinear regime, and numerically in the fully nonlinear regime.
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