Previously in , we considered a diffusive logistic equation with two parameters, r(x) - intrinsic growth rate and K(x) - carrying capacity. We investigated and compared two special cases of the way in which r(x) and K(x) are related for both the logistic equations and the corresponding Lotka-Volterra competition-diffusion systems. In this paper, we continue to study the Lotka-Volterra competition-diffusion system with general intrinsic growth rates and carrying capacities for two competing species in heterogeneous environments. We establish the main result that determines the global dynamics of the system under a general criterion. Furthermore, when the ratios of the intrinsic growth rate to the carrying capacity for each species are proportional - such ratios can also be interpreted as the competition coefficients - this criterion reduces to what we obtained in . We also study the detailed dynamics in terms of dispersal rates for such general case. On the other hand, when the two ratios are not proportional, our results in  show that the criterion in  cannot be fully recovered as counterexamples exist. This indicates the importance and subtleties of the roles of heterogeneous competition coefficients in the dynamics of the Lotka-Volterra competition-diffusion systems. Our results apply to competition-diffusion-advection systems as well.
|Original language||English (US)|
|Number of pages||27|
|Journal||Discrete and Continuous Dynamical Systems- Series A|
|State||Published - Nov 2020|
Bibliographical noteFunding Information:
Acknowledgments. The research of X. He is supported in part by NSFC(11601155) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000); the research of W.-M. Ni is partially supported by NSF grants DMS-1210400 and DMS-1714487, and NSFC grant No. 11431005. The authors are also grateful to the anonymous referees for the careful reading and helpful suggestions which greatly improves the original manuscript.
- Asymptotic analysis
- Carrying capacity
- Global dynamics
- Intrinsic growth rate
- Spatial heterogeneity