Abstract
A prey–predator discrete-time model with a Holling type I functional response is investigated by incorporating a prey refuge. It is shown that a refuge does not always stabilize prey–predator interactions. A prey refuge in some cases produces even more chaotic, random-like dynamics than without a refuge and prey population outbreaks appear. Stability analysis was performed in order to investigate the local stability of fixed points as well as the several local bifurcations they undergo. Numerical simulations such as parametric basins of attraction, bifurcation diagrams, phase plots and largest Lyapunov exponent diagrams are executed in order to illustrate the complex dynamical behavior of the system.
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Notes
The shape of the Holling type I functional response is linear. So if the predators are spiders and the preys are biting flies, the number of flies killed by one spider is proportional to the flies’ density [4].
The results of the stability analysis have been exhibited by using the software package Maxima 5.27.0 (http://maxima.sourceforge.net/).
All numerical simulations were executed using the software package E&F Chaos [21].
This low-resting metabolic rate may be due to the fact that they use hydrostatic pressure for extending their appendages [26].
In addition, spiders can further reduce their metabolic rate below their already low levels when they experience periods of food limitation [27].
Many spiders, including both web-building and wandering spiders, are sit-and-wait predators that spend very little time in active locomotion [28]. For example, for the wolf spider Pardosa amentata, the daily energy loss attributed to locomotion was estimated to be only 1% of the daily energy usage of spiders [29].
The energetic costs of web production are relatively small because web building is often a short process and some spiders are able to recycle web proteins, which can substantially reduce the metabolic cost of silk production [29].
For example for the wolf spider Pardosa lugubris, the females invest 26% in reproduction and males invest only 16% [30].
Positive Lyapunov exponents correspond to diverging neighboring orbits, negative Lyapunov exponents correspond to converging orbits and zero Lyapunov exponents correspond to bifurcations that occur in the system [34].
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Gkana, A., Zachilas, L. Incorporating prey refuge in a prey–predator model with a Holling type I functional response: random dynamics and population outbreaks. J Biol Phys 39, 587–606 (2013). https://doi.org/10.1007/s10867-013-9319-7
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DOI: https://doi.org/10.1007/s10867-013-9319-7