ISSN:
1432-1416
Keywords:
Discrete dynamical systems
;
Lotka-Volterra competition
;
symmetric Hopf bifurcation
Source:
Springer Online Journal Archives 1860-2000
Topics:
Biology
,
Mathematics
Notes:
Abstract We consider the generalized Lotka-Volterra two-species system $$\begin{gathered} x_{n + 1} = x_n \exp \left( {r_1 \left( {1 - x_n } \right) - s_1 y_n } \right) \hfill \\ y_{n + 1} = y_n \exp \left( {r_2 \left( {1 - y_n } \right) - s_2 x_n } \right) \hfill \\ \end{gathered} $$ originally proposed by R. M. May as a model for competitive interaction. In the symmetric case that r 1 = r 2 and s 1 = s 2, a region of ultimate confinement is found and the dynamics therein are described in some detail. The bifurcations of periodic points of low period are studied, and a cascade of period-doubling bifurcations is indicated. Within the confinement region, a parameter region is determined for the stable Hopf bifurcation of a pair of symmetrically placed period-two points, which imposes a second component of oscillation near the stable cycles. It is suggested that the symmetric competitive model contains much of the dynamical complexity to be expected in any discrete two-dimensional competitive model.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00275495
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