Abstract:

The datasets associated can be used for the long-term dynamics of the models presented in Ackleh et al., 2019. The study here deals with a more in-depth analysis of these models, including global stability of equilibria, the existence of cycles, and chaos. The main focus of this work is to investigate how the speed of evolution may impact the population dynamics. For the non-evolutionary model developed in Ackleh et al., 2019, we first show that the system has an unstable interior equilibrium and the solution converges to a 6-cycle. This situation is described in figure-1 in the associated manuscript Ackleh et al., 2020. Figure-2 and figure-3 in the text illustrate a unique globally asymptotically stable interior equilibrium for a specific set of parameter values for the non-evolutionary model. For the pure trait model (with no predator-population) of the evolutionary case, it is shown in figure-4 and figure-5 that by increasing the value of the speed of the evolution, it is possible for this model to exhibit a classical period-doubling route to chaos. In figure-5(b), these scenarios are supported by the Lyapunov exponents for the pure-trait model. Figure-7 in the example-3 provided in the text gives the numerical illustration of a specific parameter value that may be finite or infinite for which the dynamics may alter, resulting in the destabilization of the interior equilibrium. Finally, figure-7 shows that the increasing the size of the speed of the evolution, it is possible to have stable cycles and chaos. This example also demonstrates that when the speed of evolution is fast, evolution may have a destabilizing effect on the population dynamics. This dataset supports the publication: Ackleh, A. S., Hossain, M. I., Veprauskas, A., & Zhang, A. (2020). Long-term dynamics of discrete-time predator-prey models: stability of equilibria, cycles and chaos. Journal of Difference Equations and Applications, 26(5), 693–726. doi:10.1080/10236198.2020.1786818.

Suggested Citation:

Ackleh, Azmy S., Md Istiaq Hossain, Amy Veprauskas, and Aijun Zhang. 2021. Dataset for: Long-term dynamics of discrete-time predator–prey models: stability of equilibria, cycles and chaos. Distributed by: GRIIDC, Harte Research Institute, Texas A&M University–Corpus Christi. doi:10.7266/QZVXBJYJ

Publications:

Ackleh, A. S., Hossain, M. I., Veprauskas, A., & Zhang, A. (2020). Long-term dynamics of discrete-time predator–prey models: stability of equilibria, cycles and chaos. Journal of Difference Equations and Applications, 26(5), 693–726. doi:10.1080/10236198.2020.1786818

Purpose:

Investigate how the speed of evolution may impact the population dynamics of discrete-time predator-prey models through more in-depth analysis of these models, including global stability of equilibria, the existence of cycles and chaos.

Data Parameters and Units:

T:time (representing hours, days, years, etc.), n(t): prey density (numbers of prey per unit area), p(t): predator density (number of predators per unit area), u(t): the prey population mean trait describing the amount of toxicant resistance possessed by prey when prey is evolving to toxicant resistance, ϵ_0: the level of the initial toxicant value. The parameter ν measures the speed of the evolution. V(n,p) used in the datasets provides z-coordinate for the surface plot of the V function for n and p grid values given on [0,3]×[0,16], V_1 (n,p) and V_2 (n,p) values given in the dataset are used for Figure-3 and are the critical curves of V(n,p). The dataset also includes the codes used to reproduce figures in the associated publication Ackleh et al., 2020.

Methods:

Dataset contains numerical simulation. For the detailed methods, please refer to the associated publication Ackleh et al., 2020.

Provenance and Historical References:

Ackleh, A. S., Hossain, M. I., Veprauskas, A., & Zhang, A. (2019). Persistence and stability analysis of discrete-time predator-prey models: a study of population and evolutionary dynamics. Journal of Difference Equations and Applications, 25(11), 1568–1603. doi:10.1080/10236198.2019.1669579.

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