Global dynamics and spatio-temporal patterns in a two-species chemotaxis system with two chemicals

Abstract

In this paper, we consider the signal-dependent diffusion and sensitivity in a chemotaxis-competition population system with two different signals in a two-dimensional bounded domain. We consider more general signal production functions and assume that the signal-dependent diffusion is a decreasing function which may be degenerate with respect to the density of the corresponding signal. We first obtain the global existence and uniform-in-time bound of classical solutions and show that the blow-up effect can be precluded for signal-dependent diffusion and sensitivity with certain properties. Then, by constructing Lyapunov functionals, we study the global attractivity of nonzero (boundary/positive) homogeneous steady states under three different strengths of competition. In particular, we obtain that the nonzero boundary constant steady states are globally asymptotically stable when they are globally attractive, which means no pattern formation occurs, while for interior constant steady state, its global attractivity can imply the global stability for some special signal production functions. Finally, numerical simulations show that for large signal sensitivity, different signal production functions can lead to various complex spatial-temporal patterns around the positive homogeneous steady state. In particular, for a given signal production mechanism, various patterns are observed for different population growth rates.

Stability analysis of a stage‐structure model with spatial heterogeneity

“…The presence of time delay makes it difficult to study the stability of the coexisting steady‐state solution of system (). Moreover, the spatially non‐homogeneity of the parameters in system () makes it impossible to discuss () as in the previous literature 22–29 . In Lou 30 and Cantrell and Cosner, 31 the existence of a positive steady‐state solution, as well as the effects of dispersal and spatial heterogeneity, is obtained for a logistic population model with a function m ( x ) representing the intrinsic growth rate of the species.…”

“…Moreover, the spatially non-homogeneity of the parameters in system (2) makes it impossible to discuss (2) as in the previous literature. [22][23][24][25][26][27][28][29] In Lou 30 and Cantrell and Cosner, 31 the existence of a positive steady-state solution, as well as the effects of dispersal and spatial heterogeneity, is obtained for a logistic population model with a function m(x) representing the intrinsic growth rate of the species. The main reason for the introduction of the spatial variation in m(x) is to investigate how the favorable (i.e., m(x) > 0) and unfavorable (i.e., m(x) < 0) habitats affect the dynamics of population.…”

Dynamics of a delayed Lotka-Volterra model with two predators competing for one prey

“…In this model the populations of predator and prey permanently oscillate for almost all positive initial conditions. Recently, there have been some excellent works on global dynamics of resource competitive models (see [4,5,11,16,17,19,24,22,25,30,31,32,33,34]), which are important in understanding of the mechanism of natural selection: the principle of competitive exclusion (see [1,3,10,26,27,28,35,36]) or the coexistence of competing species. For example, Volterra [29] observed that the coexistence of two or more predators competing for fewer prey resources is impossible.…”