Abstract
The interaction between prey and predator is well-known within natural ecosystems. Due to their multifariousness and strong link population dynamics, predators contain distinct features of ecological communities. Keeping in view the Nicholson-Bailey framework for host-parasitoid interaction, a discrete-time predator–prey system is formulated and studied with implementation of type-II functional response and logistic prey growth in form of the Beverton-Holt map. Persistence of solutions and existence of equilibria are discussed. Moreover, stability analysis of equilibria is carried out for predator–prey model. With implementation of bifurcation theory of normal forms and center manifold theorem, it is proved that system undergoes transcritical bifurcation around its boundary equilibrium. On the other hand, if growth rate of consumers is taken as bifurcation parameter, then system undergoes Neimark-Sacker bifurcation around its positive equilibrium point. Methods of chaos control are introduced to avoid the populations from unpredictable behavior. Numerical simulation is provided to strengthen our theoretical discussion.
Highlights
Many natural ecosystems in our universe are related to host-parasitoid interaction and most of the hosts and their parasitoids have life cycles of synchronized form
This paper is concerned to investigation of some qualitative aspects of a discrete-time prey–predator model
The model is a modification of classical Nicholson-Bailey framework related to host-parasitoid interaction with implementation of Beverton-Holt growth function for prey population and type-II functional response
Summary
Many natural ecosystems in our universe are related to host-parasitoid interaction and most of the hosts and their parasitoids have life cycles of synchronized form. Piecewise constant arguments or exponential–discretization (see [18, 24]) and Euler approximations (see [33, 39, 55]) are more frequently used methods to obtain discrete–time counterparts of predator—prey models Both of these methods are lacking the dynamical consistency with their continuous counterparts. Weide et al [56] proposed some discrete prey–predator models by adopting NB framework and implementation of type-II functional response and logistic prey growth in form of the Beverton-Holt map. These models reveal behaviors, that is, paradox of enrichment, hydra effect and Neimark-Sacker bifurcation, similar to Rosenzweig–MacArthur model.
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