Ecological invasion in competition–diffusion systems when the exotic species is either very strong or very weak

Abstract

Reaction-diffusion systems with a Lotka-Volterra-type reaction term, also known as competition-diffusion systems, have been used to investigate the dynamics of the competition among m ecological species for a limited resource necessary to their survival and growth. Notwithstanding their rather simple mathematical structure, such systems may display quite interesting behaviours. In particular, while for [Formula: see text] no coexistence of the two species is usually possible, if [Formula: see text] we may observe coexistence of all or a subset of the species, sensitively depending on the parameter values. Such coexistence can take the form of very complex spatio-temporal patterns and oscillations. Unfortunately, at the moment there are no known tools for a complete analytical study of such systems for [Formula: see text]. This means that establishing general criteria for the occurrence of coexistence appears to be very hard. In this paper we will instead give some criteria for the non-coexistence of species, motivated by the ecological problem of the invasion of an ecosystem by an exotic species. We will show that when the environment is very favourable to the invading species the invasion will always be successful and the native species will be driven to extinction. On the other hand, if the environment is not favourable enough, the invasion will always fail.

Complex pattern formation driven by the interaction of stable fronts in a competition-diffusion system

“…In our case in particular, thanks to assumptions (A1) and (A2), the final state is (r 1 , 0, 0) for general initial conditions. We remark that in the limiting situations where r 3 is either very large or very small, the same results can also be proven analytically [11].…”

Structured environments foster competitor coexistence by manipulating interspecies interfaces

“…super-diffusive, (ii) chemotaxis in exogenous chemical gradients, (iii) flow/advection, or (iv) the production, potency, transport and decay of secreted toxins that diffuse faster than organismal diffusion. This system of PDEs (which is not new [72,73]) represents a baseline set of assumptions and corresponding phenomena from which to build more complex models [74] of structured and locally competitive population dynamics. In particular, it is part of a larger class of boundary-forming models that localize competitive interactions to zones between competitors, and thus it is a reasonable starting point for understanding the effects of steric structure in such contexts.…”

Stacked invasion waves in a competition-diffusion model with three species

“…For the three-species system, a number of works has been focused on the problem of persistence in bounded domains, where the existence of stable non-trivial equilibria is of particular interest [37]. We refer for example to [29] for a global stability result to a multiple species system with discrete diffusion, and to [16,48] for the influence of an exotic species on other native species. More results on diffusive Lotka-Volterra systems in bounded domains can be found in [10,45,59].…”