Abstract
Ecological systems have all the properties to produce chaotic dynamics. To predict the chaotic behavior in an ecological system and its possible control mechanism is interesting. Aziz-Alaoui [1] considered a tri-trophic food-chain model with modified Leslie-Gower type growth rate for top-predator population and established the chaotic dynamics exhibited by the model system for a certain choice of parameter values. We have modified the said model by incorporating density dependent death rate for predator population. Our mathematical findings reveal the fact that there are two coexisting equilibrium points one of which is a source and the other one is a sink. The positive equilibrium point which is sink is actually globally asymptotically stable under certain parametric conditions. Numerical experiment analysis shows that the model system are capable to produce chaotic dynamics when the rate of intra specific completion is very low and chaotic dynamics disappears for a certain value of the rate of intra specific completion for predator species. Our results suggest that the consideration of density dependent death rate for predator species have the ability to control the chaotic dynamics.
Highlights
The dynamical problems associated with mathematical modeling of various ecological systems may appear to be simple at first sight but the detailed and systematic analysis often leads us to very interesting and fascinating dynamics
Numerical experiment analysis shows that the model system are capable to produce chaotic dynamics when the rate of intra specific completion is very low and chaotic dynamics disappears for a certain value of the rate of intra specific completion for predator species
Our results suggest that the consideration of density dependent death rate for predator species have the ability to control the chaotic dynamics
Summary
The dynamical problems associated with mathematical modeling of various ecological systems may appear to be simple at first sight but the detailed and systematic analysis often leads us to very interesting and fascinating dynamics. A similar chaotic behavior is observed by Hastings and Powell [6] for a simple three dimensional food chain model within deterministic environment. To the best of our knowledge, there are few literatures which have considered the inhibitory effect of “intra specific density dependence”, but it has the ability to regulate the population dynamics significantly [19] Based upon this idea we are intended to revisit the dynamical model for prey-specialist predator-generalist predator model system studied by Aziz-Alaoui [1] incorporating intra-specific density dependent death rate for specialist predator and show that the intra-specific density dependence has ability to control the chaotic oscillation observed in the original model. Density Dependent Predator Death Prevalence Chaos in a Tri-Trophic Food Chain Model
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