Yuanshi Wang, Hong Wu
Abstract:
This article considers mutualisms in a lattice gas system of two species.
The species are mutualistic since each one can provide resources to the other.
They are also competitive since they compete for empty sites on the same lattice.
The mutualisms are assumed to have a saturated response, and the intraspecific
competition is considered because of self-limitation. The mutualism system is
characterized by differential equations, which are derived from reactions on
lattice and are extension of a previous model. Global stability analysis
demonstrates that (i) When neither species can survive alone, they can coexist
if mutualisms between them are strong and population densities are large,
which exhibits the Allee effect in obligate mutualism; (ii) When one species
can survive alone but the other cannot, the latter one will survive if
the mutualistic effect from the former is strong. Even if the effect is
intermediate, the latter species can survive by strengthening its mutualistic
effect on the former and enhancing its population density; (iii) When either
species can survive alone, a weak mutualism will lead to extinction of one
species. When in coexistence, intermediate strength of mutualism is shown
to be beneficial under certain parameter range, while over- or under-
mutualism is not good. Furthermore, extremely strong/weak mutualism is
exhibited to result in extinction of one/both species.
While seven typical dynamics are displayed by numerical simulation in
a previous work, they are proved in this work and the eighth one is exhibited.
Numerical simulations validate and extend our conclusions.
Submitted November 19, 2014. Published January 5, 2015.
Math Subject Classifications: 34C37, 92D25, 37N25.
Key Words: Stability; persistence; cooperation; saddle-node bifurcation;
Holling Type II functional response.
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Yuanshi Wang School of Mathematics and Computational Science Sun Yat-sen University, Guangzhou 510275, China email: mcswys@mail.sysu.edu.cn |
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Hong Wu School of Mathematics and Computational Science Sun Yat-sen University, Guangzhou 510275, China email: wuhong@mail.sysu.edu.cn |
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