Abstract
Consider the Lotka–Volterra competition model
for two species, say A and B, where u=u(t) denotes the total population of A, and v=v(t) the total population of B, respectively, at time t. Let the conditions
be satisfied. We note that these conditions imply γ 11 γ 22<γ 12 γ 21, that is, interspecies competition is stronger than intraspecies competition. Then the equations
have four nonnegative solutions
where \(\overline{u}=(c\gamma_{22}-d\gamma_{12})/D\) and \(\overline{v}=(d\gamma_{11}-c\gamma_{21})/D\) with D=γ 11 γ 22− γ 12 γ 21<0 (due to (15.2)). The solutions \(O,\,\overline{U}_{u}\) , and \(\overline{U}_{v}\) are extinct solutions; meanwhile, \(\overline{U}\) is a coexistent solution. By the theory ordinary equations, it is easy under (15.2) to see that the coexistent solution \(\overline{U}\) is always unstable. This means that in the case where interspecies competition dominates intraspecies one, it is impossible to explain the coexistence of the species A and B by the ordinary system (15.1).
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Yagi, A. (2010). Lotka–Volterra Competition Model with Cross-Diffusion. In: Abstract Parabolic Evolution Equations and their Applications. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04631-5_15
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DOI: https://doi.org/10.1007/978-3-642-04631-5_15
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