We first consider a diffusive logistic model of a single species in a heterogeneous environment, with two parameters, r(x) for intrinsic growth rate and K(x) for carrying capacity. When r(x) and K(x) are proportional, i.e., r= cK, it is proved by Lou (J Differ Equ 223(2):400–426, 2006) that a population diffusing at any rate will reach a higher total equilibrium biomass than the population in an environment in which the same total resources are distributed homogeneously. This paper studies another case when r(x) is a constant, i.e., independent of K(x). In such case, a striking result is that for any dispersal rate, the logistic equation with spatially heterogeneous resources will always support a total population strictly smaller than the total carrying capacity at equilibrium, which is just opposite to the case r= cK. These two cases of single species models also lead to two different forms of Lotka–Volterra competition-diffusion systems. We then examine the consequences of the aforementioned difference on the two forms of competition systems. We find that the outcome of the competition in terms of the dispersal rates and spatial distributions of resources for the two forms of competition systems are again quite different. Our results indicate that in heterogeneous environments, the correlation between r(x) and K(x) has more profound impacts in population ecology than we had previously expected, at least from a mathematical point of view.
Bibliographical noteFunding Information:
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 stability
- Carrying capacity
- Intrinsic growth rate
- Reaction–diffusion equations
- Spatial heterogeneity