Abstract
We consider a system of two differential equations modeling chemotaxis. The system consists of a parabolic equation describing the behavior of a biological species “u” coupled to an ODE patterning the concentration of a chemical substance “v”. The growth of the biological species is limited by a logistic-like term where the carrying capacity presents a time-periodic asymptotic behavior. The production of the chemical species is described in terms of a regular function h, which increases as “u” increases. Under suitable assumptions we prove that the solution is globally bounded in time by using an Alikakos-Moser iteration, and it fulfills a certain periodic asymptotic behavior. Besides, numerical simulations are performed to illustrate the behavior of the solutions of the system showing that the model considered here can provide very interesting and complex dynamics.
Highlights
The mathematical model that we study in this article describes the behavior of a biological species “u” in terms of a PDE of parabolic type
Under suitable assumptions, that the solution of a chemotaxis system is globally bounded in time by using an Alikakos-Moser iteration, and it fulfills a periodic asymptotic behavior
A possible future work is the consideration of the nonconstant chemosensitivity, χ(u, v), as in [23,24,25,26,27,28,29,30,31,32,33,34,35,36], and to consider biological systems with two species, two chemotactic terms and one chemical substance verifying a similar equation as in (1)
Summary
The mathematical model that we study in this article describes the behavior of a biological species “u” in terms of a PDE of parabolic type. The equation includes the linear diffusion of “u” which moves following the direction of the chemical gradient of a non-diffusive substance “v”. The evolution of “v” is given in terms of a general function “h” satisfying some technical assumptions presented . The proof is based on the properties of the functional c ln cdx + Notice that, in this case, three of the variables leading the movement satisfy ordinary differential equations (see Stinner, Surulescu and Uatay [27], Tao and Winkler [28], Zhigun, Surulescu and Hunt [29] and Zhigun, Surulescu and Uatay [30] for similar models).
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