Morse decompositions for a population model with harvesting. Case He-Se: Equal harvesting and equal survival rates of juveniles and adults - Open Research Data - MOST Wiedzy


Morse decompositions for a population model with harvesting. Case He-Se: Equal harvesting and equal survival rates of juveniles and adults


This dataset contains selected results of rigorous numerical computations conducted in the framework of the research described in the paper "Global dynamics in a stage-structured discrete population model with harvesting" by E. Liz and P. Pilarczyk: Journal of Theoretical Biology, Vol. 297 (2012), pp. 148–165, doi: 10.1016/j.jtbi.2011.12.012.

The purpose of the research was to analyze the effect of constant effort harvesting upon global dynamics of the discrete-time population model with juvenile and adult stages described in the paper, with a Ricker-type nonlinearity. The following parameters of the dynamical system were considered:

ha, hj – harvesting rates for adults and juveniles, respectively
sa, sj – survival rates of adults and juveniles, respectively

A few different scenarios were considered, each called a specific case. The current dataset contains data for Case He-Se: Equal harvesting and equal survival rates of juveniles and adults, that is:

hj = ha
sj = sa
(ha,sa) ∈ [0,1]✕[0,1]

The computations followed the general scheme explained in the paper "A database schema for the analysis of global dynamics of multiparameter systems" by Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk, published in SIAM J. Appl. Dyn. Syst., Vol. 8, No. 3 (2009), 757–789, doi: 10.1137/080734935.

The parameter space [0,1]✕[0,1] was sampled at the resolution of 500✕500. The phase space [0,1.35]✕[0,1.35] was sampled at the resolution of 1024✕1024. A collection of isolating neighborhoods that enclose Morse sets in a Morse decomposition was computed for each box of parameters, and a Conley-Morse graph was determined, with the Conley indices of the Morse sets computed where feasible. Clutching graphs between Morse decompositions found for adjacent boxes were also computed, and the parameter space was subdivided into classes of equivalent Morse decompositions. The complete computation on a Quad-Core AMD Opteron™ Processor 2376 2.3 GHz with Debian GNU/Linux took 1,169 hours of CPU time, and the memory usage was up to 405 MB. The parallelization framework introduced in the paper "Parallelization method for a continuous property" by P. Pilarczyk was used, as published in Foundations of Computational Mathematics, Vol. 10, No. 1 (2010), 93–114, doi: 10.1007/s10208-009-9050-8.

The dataset contains the Morse decompositions found for all the parameter boxes. A collection of combinatorial Morse sets, that is, isolating neighborhoods of the actual Morse sets in each Morse decomposition is encoded in terms of a PNG image. In this image, a single pixel corresponds to a square in the phase space. All the images are compressed together in the zipped archive file. The name of each file is in the format "pn_m.png", where n,m are the integer coordinates of the box, both in the range [0,499]. Each image is cropped in order to avoid the vast white area around the sets. Each combinatorial Morse set comprises of a collection of pixels drawn in a specific color. The colors used for the consecutive sets are listed at the bottom of the image as squares: black, blue, red, green, etc. The order is the same as in the corresponding Conley-Morse graphs available in a separate dataset. Note that the first set (black) is often a very small neighborhood of the origin that can be spotted as a single pixel at the bottom left corner of the image.

An interactive browser of all the Conley-Morse graphs and phase space portraits of the Morse decompositions provided in the current series of datasets is available at the website

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Year of publication:
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Fields of science:
  • Mathematics (Natural sciences)
  • Biological sciences (Natural sciences)
DOI ID 10.34808/e0km-ct36 open in new tab
A database of global dynamics for a population model with harvesting
Verified by:
Gdańsk University of Technology



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