Summary.A class of functions describing the Allee effect and local catastrophes in population dynamics is introduced and the behaviour of the resulting one-dimensional discrete dynamical system is investigated in detail. The main topic of the paper is a treatment of the two-dimensional system arising when an Allee function is coupled with a function describing the population decay in a so-called sink. New types of bifurcation phenomena are discovered and explained. The relevance of the results for metapopulation dynamics is discussed.
In this paper we consider a family of system with 2 predators feeding on one prey. We show how to construct a positively invariant set in which it is possible to define a Poincaré map for examining the behaviour of the system, mainly in the case when both predators survive. We relate it to examples from earlier works.
We consider a Rosenzweig-MacArthur predator-prey system which incorporates logistic growth of the prey in the absence of predators and a Holling type II functional response for interaction between predators and preys. We assume that parameters take values in a range which guarantees that all solutions tend to a unique limit cycle and prove estimates for the maximal and minimal predator and prey population densities of this cycle. Our estimates are simple functions of the model parameters and hold for cases when the cycle exhibits small predator and prey abundances and large amplitudes. The proof consists of constructions of several Lyapunov-type functions and derivation of a large number of non-trivial estimates which are also of independent interest.
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