TY - JOUR
T1 - On the effects of carrying capacity and intrinsic growth rate on single and multiple species in spatially heterogeneous environments
AU - Guo, Qian
AU - He, Xiaoqing
AU - Ni, Wei Ming
N1 - Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020/8/1
Y1 - 2020/8/1
N2 - 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.
AB - 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.
KW - Asymptotic stability
KW - Carrying capacity
KW - Coexistence
KW - Intrinsic growth rate
KW - Reaction–diffusion equations
KW - Spatial heterogeneity
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U2 - 10.1007/s00285-020-01507-9
DO - 10.1007/s00285-020-01507-9
M3 - Article
C2 - 32621114
AN - SCOPUS:85087526326
SN - 0303-6812
VL - 81
SP - 403
EP - 433
JO - Journal of Mathematical Biology
JF - Journal of Mathematical Biology
IS - 2
ER -