DOI: 10.34808/58vn-v492
title: Conley-Morse graphs for a population model with harvesting. Case He-S1: Equal harvesting of juveniles and adults, survival rates of juveniles and adults add up to 1

description: 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, respectivelysa, 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-S1: Equal harvesting of juveniles and adults, survival rates of juveniles and adults add up to 1, that is:
hj = hasj = 1 – 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 1,349 hours of CPU time, and the memory usage was up to 780 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 Conley-Morse graphs computed for all the parameter boxes. Each graph is encoded in the text format compatible with the "dot" program from the Graphviz Graph Visualization Software package (https://graphviz.org/). All the text files are compressed together in the zipped archive file. The name of each file is in the format "gn_m.txt", where n,m are the integer coordinates of the box, both in the range [0,499]. For example, in order to obtain a PDF file with the visualizaton of the graph contained in the file "g241_221.txt" one could run the command "dot g241_221.txt -T pdf -o g241_221.pdf". Each node of the graph contains the information on the corresponding isolating neighborhood of a Morse set: the consecutive number of the neighborhood (starting with 0), the number of boxes comprising the neighborhood, and if a Conley index pair was constructed successfully then additionally the homology of the index pair, followed by the index map at each homology level, and the eigenvalues of the map at each homology level. The nodes have different shapes and colors for easy visual identification of the set type; for example, a yellow box indicates an attractor, and a cyan-filled oval stands for an isolating neighborhood with a non-trivial exit set. The directed edges between the nodes show direct connections between the sets (the transitive reduction should be taken for the full set of possible connections).
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 http://www.pawelpilarczyk.com/harvesting/.