Competitive–cooperative models with various diffusion strategies

Abstract

The paper is concerned with different types of dispersal chosen by competing species. We introduce a model with the diffusion-type term ∇·[a∇ (u/P )] which includes some previously studied systems as special cases, where a positive space-dependent function P can be interpreted as a chosen dispersal strategy. The well-known result that if the first species chooses P proportional to the carrying capacity while the second does not then the first species will bring the second one to extinction, is also valid for this type of dispersal. However, we focus on the case when the ideal free distribution is attained as a combination of the two strategies adopted by the two species. Then there is a globally stable coexistence equilibrium, its uniqueness is justified. If both species choose the same dispersal strategy, non-proportional to the carrying capacity, then the influence of higher diffusion rates is negative, while of higher intrinsic growth rates is positive for survival in a competition.

Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment

“…Thus, the consideration of diffusion and cross-diffusion effect is very reasonable and more close to reality, see for example, [29] for mixed-culture biofilm model, [17] for the tumor-growth model and [14,16,31,40] for the competition model. There are some valuable results about the roles of diffusion and cross-diffusion in the modeling of the dynamics of strongly coupled reaction-diffusion systems [5,11,13,16,19,22,25,31,30,38,40]. For instance, Shigesada et al [31] proposed the strongly coupled elliptic system describing two species Lotka-Volterra competition model.…”

“…In 2014, Jia et al [16] discussed a Lotka-Volterra competition reaction-diffusion system with nonlinear diffusion effects. In 2016, Braverman and Kamrujjaman [5] introduced a competitive-cooperative models with various diffusion strategies. More recently, Li et al studied an effect of cross-diffusion on the stationary problem of a Leslie prey-predator model with a protection zone [22].…”

“…It follows from the definition, we know that L is a continuous and positive operator which maps the initial infection distribution Ψ to the distribution of the total members produced during the infection period. Consequently, we define the spectral radius of L as the basic reproduction number of system (5), that is,…”

Dynamics of Spruce budworms and single species competition models with bifurcation analysis

“…5 A very recent study of diffusion reaction in competition model is. 6 There are also other studies [7][8][9] related to diffusion where the effect of dispersion procedures on competitive populations are explored. In another study, the relation between growth rates, carrying capacity, and diffusion strategies is discussed, it is analyzed in 9 whether coexistence is promoted when diffusion strategies are diverse.…”

On the interplay of harvesting and various diffusion strategies for spatially heterogeneous populations

“…An interesting investigation in this area in the discrete case was recently reported in [13]. In the continuous case for two (or more) species, cooperation in resources consumption is possible when the densities of the two populations complement each other forming an ideal free pair [2,6]. For such a pair, the total density exactly matches the carrying capacity, and the diffusion strategies are aligned with these distributions.…”