SPATIAL DEGENERACY VS FUNCTIONAL RESPONSE
2016
In this paper, we are concerned with a predator-prey model with
Beddington-DeAngelis functional response in heterogeneous
environment. By the bifurcation theory and some estimates, the
global bifurcation of positive stationary solution is shown. Our
result shows that new stationary patterns are produced by the
spatial degeneracy and the Beddington-DeAngelis functional response.
Essentially different from the known results, the two factors
generate two critical values for the prey growth rate $\lambda.$ As
$\lambda$ crosses each critical value, the positive stationary
solution set undergoes a drastic change. In particular, when
$\lambda$ is suitably large, the interaction between the two
factors yields nonexistence of positive stationary solutions for
any $\mu,$ which is in strong contrast to the existence for suitable
ranges of $\mu$ corresponding to the Lotka-Volterra or Holling-II
functional response. Moreover, which one of the two factors plays a
dominating role in the stationary patterns is shown. In addition, we
give the asymptotic behavior of the positive stationary solutions as
$\mu\rightarrow\infty$. Finally, both uniqueness and multiplicity of
the positive stationary solutions are shown as well as their
stability.
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